Starting from the so-called “blue clearing” phenomenon, this paper establishes a link between disturbances of the Martian gravitational potential, the Allais effect of syzygy, astral influences and the Raman Stokes ...Starting from the so-called “blue clearing” phenomenon, this paper establishes a link between disturbances of the Martian gravitational potential, the Allais effect of syzygy, astral influences and the Raman Stokes effect. This phenomenon is apparently peculiar to the Martian atmosphere. Photographs of Mars taken in blue light normally show only the atmosphere itself and clouds high above the surface. On occasion of oppositions, however, blue photographs will penetrate in varying degrees to the surface of Mars. Curiously, a burst of brightness and storms then occur on Mars. The atmosphere and clouds can be seen and photographed at short wavelengths by Earth-based telescopes equipped with a Wratten 47 filter. It happens that the blue screen of the filter suddenly begins to disappear and that the Martian surface becomes visible. The exact mechanism that produces blue clearing when Earth is between the Sun and Mars is highly speculative. We believe that the “Allais syzygy effect” may explain this phenomenon. The opposition would generate a “gravito-electromagnetic tension”, which would spawn fluctuations in the gravitational potential of Mars, accompanied and linked to an electromagnetic effect. The outcome would be to trigger dust storms and exacerbate a disorderly excitement of molecules in the atmosphere. The thermal agitation facilitates the absorption of energy and the formation of small condensations that cause light scattering. Assuming that the Martian gravity decreased slightly, a Stokes Raman scattering would manifest at intramolecular level of the Martian atmosphere: the emitted photon has a lower energy than the absorbed photon. Therefore, it is mainly the waves corresponding to the spectral regions yellow, orange or red that are diffused, what eliminates short wavelengths. We deduce that the size of the inhomogeneities resulting from thermal excitation turns out to be greater than the length of the light waves of blue or purple regions of the spectrum.展开更多
We investigate the behavior of the extension dimension of subcategories of abelian categories under recollements.LetΛ',Λ,Λ"be art in algebras such that(modΛ',mod A,modΛ")is a recollement,and let...We investigate the behavior of the extension dimension of subcategories of abelian categories under recollements.LetΛ',Λ,Λ"be art in algebras such that(modΛ',mod A,modΛ")is a recollement,and let D'and D"be subcategories of modΛand modΛ"respectively.For any n,m≥0,under some conditions,we get dimΩ^(k)(D)≤dimΩ^(n)(D')+dimΩ^(m)(D")+1,where k=max{m,n}and D is the subcategory of modΛglued by D'and D";moreover,we give a sufficient condition such that the converse inequality holds true.As applications,some results for Igusa-Todorov subcategories and syzygy finite sub categories are obtained.展开更多
Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless te...Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.展开更多
The main objective of this paper is to study the dimension trees and further the homological dimensions of the extension algebras -- dual and trivially twisted extensions -- with a unified combinatorial approach using...The main objective of this paper is to study the dimension trees and further the homological dimensions of the extension algebras -- dual and trivially twisted extensions -- with a unified combinatorial approach using the two combinatorial algorithms -- Topdown and Bottomup. We first present a more complete and clearer picture of a dimension tree, with which we are then able, on the one hand, to sharpen some results obtained before and furthermore reveal a few more hidden subtle homological phenomenons of or connections between the involved algebras; on the other hand, to provide two more efficient combinatorial algorithms for computing dimension trees, and consequently the homological dimensions as an application. We believe that the more refined complete structural information on dimension trees will be useful to study other homological properties of this class of extension algebras.展开更多
Let f0, f1, f2, f3 be linearly independent homogeneous quadratic forms in the standard Z-graded ring R ：= K[s, t, u], and gcd（f0, f1, f2, f3） = 1. This defines a rational map Ф ： P2 → P3. The Rees algebra Rees（...Let f0, f1, f2, f3 be linearly independent homogeneous quadratic forms in the standard Z-graded ring R ：= K[s, t, u], and gcd（f0, f1, f2, f3） = 1. This defines a rational map Ф ： P2 → P3. The Rees algebra Rees（I） = R I I2 … of the ideal I = （f0, fl, f2, fs） is the graded R-algebra which can be described as the image of an R-algebra homomorphism h ： R[x, y, z, w] → Rees（I）. This paper discusses the free resolutions of I, and the structure of ker（h）.展开更多
Let Ro be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal ra and residue class field K = Rim. For a graded ideal I in R we show that for k 〉〉 0： （1） the Artin-Rees number of the syzy...Let Ro be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal ra and residue class field K = Rim. For a graded ideal I in R we show that for k 〉〉 0： （1） the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and （2） R is a polynomial ring over the regular Ro, the ring R/Ik is Golod, its Poincar4-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of I^k for k 〉〉 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.展开更多
文摘Starting from the so-called “blue clearing” phenomenon, this paper establishes a link between disturbances of the Martian gravitational potential, the Allais effect of syzygy, astral influences and the Raman Stokes effect. This phenomenon is apparently peculiar to the Martian atmosphere. Photographs of Mars taken in blue light normally show only the atmosphere itself and clouds high above the surface. On occasion of oppositions, however, blue photographs will penetrate in varying degrees to the surface of Mars. Curiously, a burst of brightness and storms then occur on Mars. The atmosphere and clouds can be seen and photographed at short wavelengths by Earth-based telescopes equipped with a Wratten 47 filter. It happens that the blue screen of the filter suddenly begins to disappear and that the Martian surface becomes visible. The exact mechanism that produces blue clearing when Earth is between the Sun and Mars is highly speculative. We believe that the “Allais syzygy effect” may explain this phenomenon. The opposition would generate a “gravito-electromagnetic tension”, which would spawn fluctuations in the gravitational potential of Mars, accompanied and linked to an electromagnetic effect. The outcome would be to trigger dust storms and exacerbate a disorderly excitement of molecules in the atmosphere. The thermal agitation facilitates the absorption of energy and the formation of small condensations that cause light scattering. Assuming that the Martian gravity decreased slightly, a Stokes Raman scattering would manifest at intramolecular level of the Martian atmosphere: the emitted photon has a lower energy than the absorbed photon. Therefore, it is mainly the waves corresponding to the spectral regions yellow, orange or red that are diffused, what eliminates short wavelengths. We deduce that the size of the inhomogeneities resulting from thermal excitation turns out to be greater than the length of the light waves of blue or purple regions of the spectrum.
基金Supported by NSFC(Grant Nos.11971225,12171207,12001168)Henan University of Engineering(Grant Nos.DKJ2019010,XTYR-2021JZ001)the Key Research Project of Education Department of Henan Province(Grant No.21A110006)。
文摘We investigate the behavior of the extension dimension of subcategories of abelian categories under recollements.LetΛ',Λ,Λ"be art in algebras such that(modΛ',mod A,modΛ")is a recollement,and let D'and D"be subcategories of modΛand modΛ"respectively.For any n,m≥0,under some conditions,we get dimΩ^(k)(D)≤dimΩ^(n)(D')+dimΩ^(m)(D")+1,where k=max{m,n}and D is the subcategory of modΛglued by D'and D";moreover,we give a sufficient condition such that the converse inequality holds true.As applications,some results for Igusa-Todorov subcategories and syzygy finite sub categories are obtained.
基金National Natural Science Foundation of China (Grant Nos. 1177132& 11771405, 1157117& 11372124)Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716, 15300717).
文摘Third order three-dimensional symmetric and traceless tensors play an important role in physics and tensor representation theory. A minimal integrity basis of a third order three-dimensional symmetric and traceless tensor has four invariants with degrees two, four, six, and ten, respectively. In this paper, we show that any minimal integrity basis of a third order three-dimensional symmetric and traceless tensor is also an irreducible function basis of that tensor, and there is no syzygy relation among the four invariants of that basis, i.e., these four invariants are algebraically independent.
基金Supported by the Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD)
文摘The main objective of this paper is to study the dimension trees and further the homological dimensions of the extension algebras -- dual and trivially twisted extensions -- with a unified combinatorial approach using the two combinatorial algorithms -- Topdown and Bottomup. We first present a more complete and clearer picture of a dimension tree, with which we are then able, on the one hand, to sharpen some results obtained before and furthermore reveal a few more hidden subtle homological phenomenons of or connections between the involved algebras; on the other hand, to provide two more efficient combinatorial algorithms for computing dimension trees, and consequently the homological dimensions as an application. We believe that the more refined complete structural information on dimension trees will be useful to study other homological properties of this class of extension algebras.
文摘Let f0, f1, f2, f3 be linearly independent homogeneous quadratic forms in the standard Z-graded ring R ：= K[s, t, u], and gcd（f0, f1, f2, f3） = 1. This defines a rational map Ф ： P2 → P3. The Rees algebra Rees（I） = R I I2 … of the ideal I = （f0, fl, f2, fs） is the graded R-algebra which can be described as the image of an R-algebra homomorphism h ： R[x, y, z, w] → Rees（I）. This paper discusses the free resolutions of I, and the structure of ker（h）.
文摘Let Ro be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal ra and residue class field K = Rim. For a graded ideal I in R we show that for k 〉〉 0： （1） the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and （2） R is a polynomial ring over the regular Ro, the ring R/Ik is Golod, its Poincar4-Betti series is rational and the Betti numbers of the free resolution of K over R/I^k are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of I^k for k 〉〉 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.
