In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)leg...In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)legs define an orthogonal path of edges in the solid with perpendicular adjacent edges along the path. The length of the hypotenuse and the volume of the solid can be calculated without the Cayley-Menger determinant, by direct extension of the corresponding right triangle formulas. I give a proof of the existence of these shapes, describe the distribution of right angles in them, give an algebraic proof of the Coxeter trisection of a right tetrahedron into three smaller right tetrahedra, and generalize this construction to n-dimensional spaces. Finally, I investigate the connection between the Coxeter partition and the Hadwiger conjecture on the partition of the simplex into orthoschemes, which I call Pythagorean simplexes.展开更多
In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the...In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the property of the equidistant conjugate points of a triangle is generalized to high dimensional simplex.展开更多
In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q ...In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q 〈 ∞ and 1/P - 1/q = （α＋β）/n. Furthermore, the boundedness of Hβ,b on the homogenous Herz space Kq^α,p（R^n） is obtained.展开更多
Dynamic fault tree analysis is widely used for the reliability analysis of the complex system with dynamic failure characteristics. In many circumstances, the exact value of system reliability is difficult to obtain d...Dynamic fault tree analysis is widely used for the reliability analysis of the complex system with dynamic failure characteristics. In many circumstances, the exact value of system reliability is difficult to obtain due to absent or insufficient data for failure probabilities or failure rates of components. The traditional fuzzy operation arithmetic based on extension principle or interval theory may lead to fuzzy accumulations. Moreover, the existing fuzzy dynamic fault tree analysis methods are restricted to the case that all system components follow exponential time-to-failure distributions. To overcome these problems, a new fuzzy dynamic fault tree analysis approach based on the weakest n-dimensional t-norm arithmetic and developed sequential binary decision diagrams method is proposed to evaluate system fuzzy reliability. Compared with the existing approach,the proposed method can effectively reduce fuzzy cumulative and be applicable to any time-tofailure distribution type for system components. Finally, a case study is presented to illustrate the application and advantages of the proposed approach.展开更多
In this paper, inspired by the multiplicative generators of overlap functions, we mainly propose the concepts of multiplicative generator pairs of n-dimensional overlap functions, in order to extend the dimensionality...In this paper, inspired by the multiplicative generators of overlap functions, we mainly propose the concepts of multiplicative generator pairs of n-dimensional overlap functions, in order to extend the dimensionality of overlap functions from 2 to n. We present the condition under which the pair (g, h) can multiplicatively generate an n-dimensional overlap function O<sub>g,h</sub>. we focus on the homogeneity and idempotency property on multiplicatively generated n-dimensional overlap functions.展开更多
In this paper, the general calculation formulas of radial matrix elements for relativistic n-dimensional hydrogen atom of spin S=0 are obtained, and the recurrence relation of different power order radial matrix eleme...In this paper, the general calculation formulas of radial matrix elements for relativistic n-dimensional hydrogen atom of spin S=0 are obtained, and the recurrence relation of different power order radial matrix elements are also derived.展开更多
A series of liquid crystalline polymers with two-dimensional mesogenic units were synthesized by solution polycondensation at low temperature. All the polymers were liquid crystalline. The melting temperature T;（exce...A series of liquid crystalline polymers with two-dimensional mesogenic units were synthesized by solution polycondensation at low temperature. All the polymers were liquid crystalline. The melting temperature T;（except that with substituent of methoxy） and the clearing temperature T;of the polymers change regularly with varying of the length of the alkyl substituent groups.展开更多
A theory of（N＋1）-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity（TEGR）.The fundamental gravitational field variables are the（N＋1）-dimensional vector fields,de...A theory of（N＋1）-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity（TEGR）.The fundamental gravitational field variables are the（N＋1）-dimensional vector fields,defined globally on a manifold M,and the gravitational field is attributed to the torsion.The form of Lagrangian density is quadratic in torsion tensor.We then give an exact five-dimensional spherically symmetric solution（Schwarzschild（4＋1）-dimensions）.Finally,we calculate energy and spatial momentum using gravitational energy-momentum tensor and superpotential 2-form.展开更多
A hierarchical scheme for clustering data is presented which applies to spaces with a high number of dimensions (). The data set is first reduced to a smaller set of partitions (multi-dimensional bins). Multiple clust...A hierarchical scheme for clustering data is presented which applies to spaces with a high number of dimensions (). The data set is first reduced to a smaller set of partitions (multi-dimensional bins). Multiple clustering techniques are used, including spectral clustering;however, new techniques are also introduced based on the path length between partitions that are connected to one another. A Line-of-Sight algorithm is also developed for clustering. A test bank of 12 data sets with varying properties is used to expose the strengths and weaknesses of each technique. Finally, a robust clustering technique is discussed based on reaching a consensus among the multiple approaches, overcoming the weaknesses found individually.展开更多
The title complex [Mn（μ1,5-dca）2（POM）2] （C16H12MnN10O6,Mr = 495.30） has been prepared and structurally characterized. It crystallizes in triclinic,space group P1 with a = 7.4685（9）,b = 7.5406（9）,c = 9.8646...The title complex [Mn（μ1,5-dca）2（POM）2] （C16H12MnN10O6,Mr = 495.30） has been prepared and structurally characterized. It crystallizes in triclinic,space group P1 with a = 7.4685（9）,b = 7.5406（9）,c = 9.8646（12）A,α = 83.202（2）,β = 69.1200（10）,γ = 79.4130（10）o,V = 509.38（11） A^3,Z = 1,Dc = 1.602 g/cm^3,F（000） = 249,μ（MoKa） = 0.704 mm^-1,the final R = 0.0701 and wR = 0.2071 for 1615 observed reflections with I 〉 2σ（I）. Each Mn（Ⅱ） exhibits a slightly distorted octahedral environment and connects with each other to form a one-dimensional complex by double bridging μ1,5-N≡C-N-C≡N ligands. The magnetic susceptibility of the title complex has been measured,and it displays weak antiferromagnetic interactions.展开更多
In this paper, two recurrence formulas for radial average values of N-dimensional hydrogen atom are derived. Explicit expressions for <n rJ N-2 |r s|n rJ N-2 > are given for 3≥s≥-6. These results can be applie...In this paper, two recurrence formulas for radial average values of N-dimensional hydrogen atom are derived. Explicit expressions for <n rJ N-2 |r s|n rJ N-2 > are given for 3≥s≥-6. These results can be applied to discuss average value of centrifugal potential energy and other physical quantities. The relevant results of the usual hydrogen atom are contained in more general conclusion of this paper as special cases.展开更多
The preparation and crystal structure of complex Co（Hsae）2·2H2O （1, H2sae = N-salicylidene-2-iminoethanol） are reported. X- ray analysis revealed that every six Co（Hsae）2 forms a cyclic chip and every 12 wa...The preparation and crystal structure of complex Co（Hsae）2·2H2O （1, H2sae = N-salicylidene-2-iminoethanol） are reported. X- ray analysis revealed that every six Co（Hsae）2 forms a cyclic chip and every 12 water forms a novel gear-like cluster. Acting as building blocks, the gear-like water cluster and complex chip are connected in A-B fashion and extend into one-dimensional supramolecular chain. Hydrogen bond is the primary bridging force in the formation of supramolecular framework.展开更多
In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and ...In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.展开更多
The generalized (n + 1)-dimensional KP equation with variable coefficients is investigated in this paper. The bilinear form of the equation has been obtained by the Hirota direct method. In addition, with the help of ...The generalized (n + 1)-dimensional KP equation with variable coefficients is investigated in this paper. The bilinear form of the equation has been obtained by the Hirota direct method. In addition, with the help of Wronskian technique and the Pfaffian properties, Wronskian and Grammian solutions have been generated.展开更多
Two series of new liquid crystalline polymers with T-shaped two-dimensional mesogenic units were synthesized by low temperature solution polycondensations of 4-substituted N-2, 5-dihydroxybenzylidene aniline monomers ...Two series of new liquid crystalline polymers with T-shaped two-dimensional mesogenic units were synthesized by low temperature solution polycondensations of 4-substituted N-2, 5-dihydroxybenzylidene aniline monomers with different diacyl dichlorides. The polymers were found to be nematic and shown liquid crystalline behavior. The melting temperature T-m and the clearing temperature T-i of the polymers change regularly with varying of the monomer structures.展开更多
In this paper, we firstly introduce some new results on overlap functions and <em>n</em>-dimensional overlap functions. On the other hand, in a previous study, Gómez <em>et al</em>. presen...In this paper, we firstly introduce some new results on overlap functions and <em>n</em>-dimensional overlap functions. On the other hand, in a previous study, Gómez <em>et al</em>. presented some open problems. One of these open problems is “to search the construction of <em>n</em>-dimensional overlapping functions based on bi-dimensional overlapping functions”. To answer this open problem, in this paper, we mainly introduce one construction method of <em>n</em>-dimensional overlap functions based on bivariate overlap functions. We mainly use the conjunction operator ∧ to construct <em>n</em>-dimensional overlap functions <img src="Edit_0e82dd84-0f25-4b14-8f26-ae9532b10190.bmp" alt="" /> based on bivariate overlap functions and study their basic properties.展开更多
We introduce and develop a novel approach to outlier detection based on adaptation of random subspace learning. Our proposed method handles both high-dimension low-sample size and traditional low-dimensional high-samp...We introduce and develop a novel approach to outlier detection based on adaptation of random subspace learning. Our proposed method handles both high-dimension low-sample size and traditional low-dimensional high-sample size datasets. Essentially, we avoid the computational bottleneck of techniques like Minimum Covariance Determinant (MCD) by computing the needed determinants and associated measures in much lower dimensional subspaces. Both theoretical and computational development of our approach reveal that it is computationally more efficient than the regularized methods in high-dimensional low-sample size, and often competes favorably with existing methods as far as the percentage of correct outlier detection are concerned.展开更多
In this paper, we introduce the concepts of additive generators and additive generator pair of <em>n</em>-dimensional overlap functions, in order to extend the dimensionality of overlap functions from 2 to...In this paper, we introduce the concepts of additive generators and additive generator pair of <em>n</em>-dimensional overlap functions, in order to extend the dimensionality of overlap functions from 2 to <em>n</em>. We mainly discuss the conditions under which an <em>n</em>-dimensional overlap function can be expressed in terms of its generator pair.展开更多
文摘In this article, I consider the right triangle as the simplex in the Euclidean plane, and extend this definition to higher dimensions. The n-dimensional simplex has one hypotenuse and (n−1)legs (catheti). The (n−1)legs define an orthogonal path of edges in the solid with perpendicular adjacent edges along the path. The length of the hypotenuse and the volume of the solid can be calculated without the Cayley-Menger determinant, by direct extension of the corresponding right triangle formulas. I give a proof of the existence of these shapes, describe the distribution of right angles in them, give an algebraic proof of the Coxeter trisection of a right tetrahedron into three smaller right tetrahedra, and generalize this construction to n-dimensional spaces. Finally, I investigate the connection between the Coxeter partition and the Hadwiger conjecture on the partition of the simplex into orthoschemes, which I call Pythagorean simplexes.
基金Supported by the Technological Project of Jiangxi Province Education Department(GJJ 08389)
文摘In this paper, the concept of the equidistant conjugate points of a triangle to the n-dimensional Euclidean space is extended. The concept of equidistant conjugate point in high dimensional simplex is defined, and the property of the equidistant conjugate points of a triangle is generalized to high dimensional simplex.
基金The NSF (Q2008A01) of Shandong,Chinathe NSF (10871024) of China
文摘In this paper, it is proved that the commutator Hβ,b which is generated by the n-dimensional fractional Hardy operator Hβ and b ∈λα （R^n） is bounded from L^p（R^n） to L^q（R^n）, where 0 〈 α 〈 1, 1 〈 p, q 〈 ∞ and 1/P - 1/q = （α＋β）/n. Furthermore, the boundedness of Hβ,b on the homogenous Herz space Kq^α,p（R^n） is obtained.
基金supported by the National Defense Basic Scientific Research program of China (No.61325102)
文摘Dynamic fault tree analysis is widely used for the reliability analysis of the complex system with dynamic failure characteristics. In many circumstances, the exact value of system reliability is difficult to obtain due to absent or insufficient data for failure probabilities or failure rates of components. The traditional fuzzy operation arithmetic based on extension principle or interval theory may lead to fuzzy accumulations. Moreover, the existing fuzzy dynamic fault tree analysis methods are restricted to the case that all system components follow exponential time-to-failure distributions. To overcome these problems, a new fuzzy dynamic fault tree analysis approach based on the weakest n-dimensional t-norm arithmetic and developed sequential binary decision diagrams method is proposed to evaluate system fuzzy reliability. Compared with the existing approach,the proposed method can effectively reduce fuzzy cumulative and be applicable to any time-tofailure distribution type for system components. Finally, a case study is presented to illustrate the application and advantages of the proposed approach.
文摘In this paper, inspired by the multiplicative generators of overlap functions, we mainly propose the concepts of multiplicative generator pairs of n-dimensional overlap functions, in order to extend the dimensionality of overlap functions from 2 to n. We present the condition under which the pair (g, h) can multiplicatively generate an n-dimensional overlap function O<sub>g,h</sub>. we focus on the homogeneity and idempotency property on multiplicatively generated n-dimensional overlap functions.
文摘In this paper, the general calculation formulas of radial matrix elements for relativistic n-dimensional hydrogen atom of spin S=0 are obtained, and the recurrence relation of different power order radial matrix elements are also derived.
基金This work was supported by the FEYUT, SEDC, CHINA and by the National Natural Science Foundation of China.
文摘A series of liquid crystalline polymers with two-dimensional mesogenic units were synthesized by solution polycondensation at low temperature. All the polymers were liquid crystalline. The melting temperature T;（except that with substituent of methoxy） and the clearing temperature T;of the polymers change regularly with varying of the length of the alkyl substituent groups.
文摘A theory of（N＋1）-dimensional gravity is developed on the basis of the teleparallel equivalent of general relativity（TEGR）.The fundamental gravitational field variables are the（N＋1）-dimensional vector fields,defined globally on a manifold M,and the gravitational field is attributed to the torsion.The form of Lagrangian density is quadratic in torsion tensor.We then give an exact five-dimensional spherically symmetric solution（Schwarzschild（4＋1）-dimensions）.Finally,we calculate energy and spatial momentum using gravitational energy-momentum tensor and superpotential 2-form.
文摘A hierarchical scheme for clustering data is presented which applies to spaces with a high number of dimensions (). The data set is first reduced to a smaller set of partitions (multi-dimensional bins). Multiple clustering techniques are used, including spectral clustering;however, new techniques are also introduced based on the path length between partitions that are connected to one another. A Line-of-Sight algorithm is also developed for clustering. A test bank of 12 data sets with varying properties is used to expose the strengths and weaknesses of each technique. Finally, a robust clustering technique is discussed based on reaching a consensus among the multiple approaches, overcoming the weaknesses found individually.
文摘The title complex [Mn（μ1,5-dca）2（POM）2] （C16H12MnN10O6,Mr = 495.30） has been prepared and structurally characterized. It crystallizes in triclinic,space group P1 with a = 7.4685（9）,b = 7.5406（9）,c = 9.8646（12）A,α = 83.202（2）,β = 69.1200（10）,γ = 79.4130（10）o,V = 509.38（11） A^3,Z = 1,Dc = 1.602 g/cm^3,F（000） = 249,μ（MoKa） = 0.704 mm^-1,the final R = 0.0701 and wR = 0.2071 for 1615 observed reflections with I 〉 2σ（I）. Each Mn（Ⅱ） exhibits a slightly distorted octahedral environment and connects with each other to form a one-dimensional complex by double bridging μ1,5-N≡C-N-C≡N ligands. The magnetic susceptibility of the title complex has been measured,and it displays weak antiferromagnetic interactions.
文摘In this paper, two recurrence formulas for radial average values of N-dimensional hydrogen atom are derived. Explicit expressions for <n rJ N-2 |r s|n rJ N-2 > are given for 3≥s≥-6. These results can be applied to discuss average value of centrifugal potential energy and other physical quantities. The relevant results of the usual hydrogen atom are contained in more general conclusion of this paper as special cases.
基金supported by the Natural Science Foundation of Education Bureau of Liaoning Province,China(No.05L159).
文摘The preparation and crystal structure of complex Co（Hsae）2·2H2O （1, H2sae = N-salicylidene-2-iminoethanol） are reported. X- ray analysis revealed that every six Co（Hsae）2 forms a cyclic chip and every 12 water forms a novel gear-like cluster. Acting as building blocks, the gear-like water cluster and complex chip are connected in A-B fashion and extend into one-dimensional supramolecular chain. Hydrogen bond is the primary bridging force in the formation of supramolecular framework.
文摘In this paper, we mainly investigate the realization of 3-Lie algebras from a family of Lie algebras. We prove the realization theorem, offer a concrete example realizing all type of 4-dimensional 3-Lie algebras, and also give some properties about semi-simple n-Lie algebras.
文摘The generalized (n + 1)-dimensional KP equation with variable coefficients is investigated in this paper. The bilinear form of the equation has been obtained by the Hirota direct method. In addition, with the help of Wronskian technique and the Pfaffian properties, Wronskian and Grammian solutions have been generated.
基金Tins work was supported by the National Natural Science Foundation of China.
文摘Two series of new liquid crystalline polymers with T-shaped two-dimensional mesogenic units were synthesized by low temperature solution polycondensations of 4-substituted N-2, 5-dihydroxybenzylidene aniline monomers with different diacyl dichlorides. The polymers were found to be nematic and shown liquid crystalline behavior. The melting temperature T-m and the clearing temperature T-i of the polymers change regularly with varying of the monomer structures.
文摘In this paper, we firstly introduce some new results on overlap functions and <em>n</em>-dimensional overlap functions. On the other hand, in a previous study, Gómez <em>et al</em>. presented some open problems. One of these open problems is “to search the construction of <em>n</em>-dimensional overlapping functions based on bi-dimensional overlapping functions”. To answer this open problem, in this paper, we mainly introduce one construction method of <em>n</em>-dimensional overlap functions based on bivariate overlap functions. We mainly use the conjunction operator ∧ to construct <em>n</em>-dimensional overlap functions <img src="Edit_0e82dd84-0f25-4b14-8f26-ae9532b10190.bmp" alt="" /> based on bivariate overlap functions and study their basic properties.
文摘We introduce and develop a novel approach to outlier detection based on adaptation of random subspace learning. Our proposed method handles both high-dimension low-sample size and traditional low-dimensional high-sample size datasets. Essentially, we avoid the computational bottleneck of techniques like Minimum Covariance Determinant (MCD) by computing the needed determinants and associated measures in much lower dimensional subspaces. Both theoretical and computational development of our approach reveal that it is computationally more efficient than the regularized methods in high-dimensional low-sample size, and often competes favorably with existing methods as far as the percentage of correct outlier detection are concerned.
文摘In this paper, we introduce the concepts of additive generators and additive generator pair of <em>n</em>-dimensional overlap functions, in order to extend the dimensionality of overlap functions from 2 to <em>n</em>. We mainly discuss the conditions under which an <em>n</em>-dimensional overlap function can be expressed in terms of its generator pair.
