Let G be a finite group.A generating set X of G is said to be minimal if no proper subset of X generates G.Let d(G)and m(G)denote the smallest size and the largest size of a minimal generating set of G,respectively.In...Let G be a finite group.A generating set X of G is said to be minimal if no proper subset of X generates G.Let d(G)and m(G)denote the smallest size and the largest size of a minimal generating set of G,respectively.In this paper we present a characterization for finite solvable groups G such that m(G)-d(G)=1 and m(G)≥m(G/N)+m(N)for any non-trivial normal subgroup N of G.展开更多
In this paper,we study the Hom-structures of a special class of solvable Lie algebras with naturally graded filiform nilradical n_(n,1).Over an algebraically closed field F of zero characteristic,we calculate the Hom-...In this paper,we study the Hom-structures of a special class of solvable Lie algebras with naturally graded filiform nilradical n_(n,1).Over an algebraically closed field F of zero characteristic,we calculate the Hom-structures of these solvable Lie algebras using the Hom-Jacobi identity,obtain the bases of these Hom-structures and observe that there are certain similarities among these bases.展开更多
In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification...In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification theorems, we prove some sufficient conditions of finite solvable groups. Finally, we provide a supplement of Doerks Theorem.展开更多
The two-dimensional gravity model with a coupling constant and a vanishing cosmological constant coupled to a nonlinear matter field is investigated. We found that the classical equations of motion are exactly solvab...The two-dimensional gravity model with a coupling constant and a vanishing cosmological constant coupled to a nonlinear matter field is investigated. We found that the classical equations of motion are exactly solvable and the static solutions of the induced metric and scalar curvature can be obtained analytically. These solutions may be used to describe the naked singularity at the origin.展开更多
The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvab...The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.展开更多
Structures of two classes of solvable subgroups in SL（3, C） are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is di...Structures of two classes of solvable subgroups in SL（3, C） are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is discussed.展开更多
The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinet...The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.展开更多
We study the quasi-exactly solvable problems in relativistic quantum mechanics. We consider the problems for the two-dimensional Klein–Gordon and Dirac equations with equal vector and scalar potentials, and try to fi...We study the quasi-exactly solvable problems in relativistic quantum mechanics. We consider the problems for the two-dimensional Klein–Gordon and Dirac equations with equal vector and scalar potentials, and try to find the general form of the quasi-exactly solvable potential. After obtaining the general form of the potential, we present several examples to give the specific forms. In the examples, we show for special parameters the harmonic potential plus Coulomb potential, Killingbeck potential and a quartic potential plus Cornell potential are quasi-exactly solvable potentials.展开更多
The quantum phase properties of the generalized squeezed vacuum states associated with solvable quantum systems are studied by using the Pegg-Barnett formalism.Then,two nonclassical features,i.e.,squeezing in the numb...The quantum phase properties of the generalized squeezed vacuum states associated with solvable quantum systems are studied by using the Pegg-Barnett formalism.Then,two nonclassical features,i.e.,squeezing in the number and phase operators,as well as the number-phase Wigner function of the generalized squeezed states are investigated.Due to some actual physical situations,the present approach is applied to two classes of generalized squeezed states：solvable quantum systems with discrete spectra and nonlinear squeezed states with particular nonlinear functions.Finally,the time evolution of the nonclassical properties of the considered systems has been numerically investigated.展开更多
We propose a new method to obtain the correlation length of gapped XXZ spin 1/2 antiferromagnetic chains. Following the relativistic quantum field theory in space-time dimensions, we use the exact dispersion of massi...We propose a new method to obtain the correlation length of gapped XXZ spin 1/2 antiferromagnetic chains. Following the relativistic quantum field theory in space-time dimensions, we use the exact dispersion of massive spinon to calculate the correlation length for XXZ spin 1/2 chain. We conjecture that the correlation length for other 1D lattice models can be obtained in the same way. Relation between dispersion and the oscillated correlation of gapped incommensurate lattice models is also discussed.展开更多
The two-dimensional hydrogen with a linear potential in a magnetic field is solved by two different methods.Furthermore the connection between the model and an anharmonic oscillator is investigated by methods of KStra...The two-dimensional hydrogen with a linear potential in a magnetic field is solved by two different methods.Furthermore the connection between the model and an anharmonic oscillator is investigated by methods of KStransformation.展开更多
The dislocation equations of a simple cubic lattice have been obtained by using Green＇s function method based on the discrete lattice theory with the coefficients of the secondorder differential terms and the integra...The dislocation equations of a simple cubic lattice have been obtained by using Green＇s function method based on the discrete lattice theory with the coefficients of the secondorder differential terms and the integral terms have been given explicitly in advance. The simple cubic lattice we have discussed is a solvable model, which is obtained according to the lattice statics and the symmetry principle and can verify and validate the dislocation lattice theory. It can present unified dislocation equations which are suitable for most of metals with arbitral lattice structures. Through comparing the results of the present solvable model with the dislocation lattice theory, it can be seen that, the coefficients of integral terms of the edge and screw components we obtain are in accordance with the results of the dislocation lattice theory, however, the coefficient of the second-order differential term of the screw component is not in agreement with the result of the dislocation lattice theory. This is mainly caused by the reduced dynamical matrix of the surface term, which is the essence to obtain the dislocation equation. According to the simple cubic solvable model, not only the straight dislocations but also the curved dislocations, such as the kink, can be investigated further.展开更多
The filtered-graded transfer of SAGBI bases computation in solvable polynomial algebras was considered. The relations among the SAGBI bases of a subalgebra B, its associated graded algebra G(B) and Rees algebra B were...The filtered-graded transfer of SAGBI bases computation in solvable polynomial algebras was considered. The relations among the SAGBI bases of a subalgebra B, its associated graded algebra G(B) and Rees algebra B were got. These relations solve a natural question: how to determine the generating set of G(B) and B from any given generating set of B. Based on these some equivalent conditions for the existence of finite SAGBI bases can be got.展开更多
A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with （｜H｜, ｜K｜） = 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） ...A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with （｜H｜, ｜K｜） = 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） = 1. In this paper, some sufficient conditions for a group to be solvable are obtained in terms of s-semipermutability.展开更多
In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on...In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on each of the separate intervals with the cases of one and two singular end-points and when all solutions of the equation and its adjoint are in (the limit circle case) that all well-posed extensions of the minimal operator have resolvents which are HilbertSchmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression studied in [1-10] and those of general quasi-differential expressions in [11-19].展开更多
The finite-dimensional indecomposable solvable Lie algebras s with Q_(2n+1) as their nilradical are studied and classified and their Casimir invariants are calculated. It turns out that the dimension of s is at most d...The finite-dimensional indecomposable solvable Lie algebras s with Q_(2n+1) as their nilradical are studied and classified and their Casimir invariants are calculated. It turns out that the dimension of s is at most dim Q_(2n+1)+2.展开更多
A quasi-exactly solvable model refers to any second order differential equation with polynomial coefficients of the form A(x)y’’(x)+B(x)y’(x)+C(x)y(x)=0 where a pair of exact polynomials {y(x), C(x)} with respectiv...A quasi-exactly solvable model refers to any second order differential equation with polynomial coefficients of the form A(x)y’’(x)+B(x)y’(x)+C(x)y(x)=0 where a pair of exact polynomials {y(x), C(x)} with respective degrees {deg[y]=n, deg[C]=p} are to be found simultaneously in terms of the coefficients of two given polynomials {A(x), B(x)}. The existing methods for solving quasi-exactly solvable models require the solution of a system of nonlinear algebraic equations of which the dimensions depend on n, the degree of the exact polynomial solution y(x). In this paper, a new method employing a set of polynomials, called canonical polynomials, is proposed. This method requires solving a system of nonlinear algebraic equations of which the dimensions depend only on p, the degree of C(x), and do not vary with n. Several examples are implemented to testify the efficiency of the proposed method.展开更多
In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the cano...In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.展开更多
A generalized method which helps to find a time-dependent SchrÖdinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-...A generalized method which helps to find a time-dependent SchrÖdinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-dependent Hamiltonian of quasi-exactly solvable Lamé equation and to construct the matrix 2 × 2 time-dependent polynomial Hamiltonian.展开更多
A suit of lacustrine source rocks are developed in volcanic deposits in the Shangkuli Formation of the Cretaceous. However, it is poor to understand the characteristics of source rocks due to the low degree of explora...A suit of lacustrine source rocks are developed in volcanic deposits in the Shangkuli Formation of the Cretaceous. However, it is poor to understand the characteristics of source rocks due to the low degree of exploration, thus the exploration is severely constrained in this area. Based on the geochemical analysis, the analytic technique of GC and GC-MS, combined with the characteristics of solvable organic matter and biomarkers of the source rocks, the authors discussed the depositional environment, the derivation of the matrix and the maturity characteristics of the organic material of the Shangkuli Formation in Cretaceous. The results show that the organic matter mainly belongs to type Ⅱ1 kerogen, whose abundance is relatively high; it was formed in reductive surrounding where was deep-lake; the hydrocarbon is characterized by mixed-source of organic matter. The thermal evolution of source rocks had reached maturation stage.展开更多
基金Supported by China Scholarship Council(Grant No.202208360148)the National Natural Science Foundation of China(Grant Nos.12126415,12261042,12301026)the Natural Science Foundation of Jiangxi Province(Grant No.20232BAB211006).
文摘Let G be a finite group.A generating set X of G is said to be minimal if no proper subset of X generates G.Let d(G)and m(G)denote the smallest size and the largest size of a minimal generating set of G,respectively.In this paper we present a characterization for finite solvable groups G such that m(G)-d(G)=1 and m(G)≥m(G/N)+m(N)for any non-trivial normal subgroup N of G.
基金Supported by National Natural Science Foundation of China(12271085)Supported by National Natural Science Foundation of Heilongjiang Province(LH2022A019)+3 种基金Basic Scientic Research Operating Funds for Provincial Universities in Heilongjiang Province(2020 KYYWF 1018)Heilongjiang University Outstanding Youth Science Foundation(JCL202103)Heilongjiang University Educational and Teaching Reform Research Project(2024C43)Heilongjiang University Postgraduate Education Reform Project(JGXM_YJS_2024010).
文摘In this paper,we study the Hom-structures of a special class of solvable Lie algebras with naturally graded filiform nilradical n_(n,1).Over an algebraically closed field F of zero characteristic,we calculate the Hom-structures of these solvable Lie algebras using the Hom-Jacobi identity,obtain the bases of these Hom-structures and observe that there are certain similarities among these bases.
文摘In this paper, we obtain some classification theorems of finite simple groups with two subgroups of coprime indices which are both supersolvable or one supersolvable and the other nilpotent. Using these classification theorems, we prove some sufficient conditions of finite solvable groups. Finally, we provide a supplement of Doerks Theorem.
文摘The two-dimensional gravity model with a coupling constant and a vanishing cosmological constant coupled to a nonlinear matter field is investigated. We found that the classical equations of motion are exactly solvable and the static solutions of the induced metric and scalar curvature can be obtained analytically. These solutions may be used to describe the naked singularity at the origin.
基金Foundation item： Supported by the National Natural Science Foundation of China（11071187） Supported by the Natural Science Foundation of Henan Province（13A110785）
文摘The finite-dimensional indecomposable solvable Lie algebras s with Q2n+1as their nilradical are studied and classified, it turns out that the dimension of s is dim Q2n+1+1.Then the Hom-Lie algebra structures on solvable Lie algebras s are calculated.
文摘Structures of two classes of solvable subgroups in SL（3, C） are given in this paper, and the integrability of the 3-order Fuchsian equation which is integrable in the sense that its monodromy group is solvable is discussed.
基金The authors gratefully acknowledge Qassim University,represented by the Deanship of Scienti c Research,on the material support for this research under the number(1671-ALRASSCAC-2016-1-12-S)during the academic year 1437 AH/2016 AD.
文摘The study of physical systems endowed with a position-dependent mass (PDM) remains a fundamental issue of quantum mechanics. In this paper we use a new approach, recently developed by us for building the quantum kinetic energy operator (KEO) within the Schrodinger equation, in order to construct a new class of exactly solvable models with a position varying mass, presenting a harmonic-oscillator-like spectrum. To do so we utilize the formalism of supersymmetric quantum mechanics (SUSY QM) along with the shape invariance condition. Recent outcomes of non-Hermitian quantum mechanics are also taken into account.
基金Supported in part by National Natural Science Foundation of China under Grant Nos.11247274 and 11075115supported by Fundamental Research Funds for the Central Universities under Grant No.3122013k003
文摘We study the quasi-exactly solvable problems in relativistic quantum mechanics. We consider the problems for the two-dimensional Klein–Gordon and Dirac equations with equal vector and scalar potentials, and try to find the general form of the quasi-exactly solvable potential. After obtaining the general form of the potential, we present several examples to give the specific forms. In the examples, we show for special parameters the harmonic potential plus Coulomb potential, Killingbeck potential and a quartic potential plus Cornell potential are quasi-exactly solvable potentials.
文摘The quantum phase properties of the generalized squeezed vacuum states associated with solvable quantum systems are studied by using the Pegg-Barnett formalism.Then,two nonclassical features,i.e.,squeezing in the number and phase operators,as well as the number-phase Wigner function of the generalized squeezed states are investigated.Due to some actual physical situations,the present approach is applied to two classes of generalized squeezed states：solvable quantum systems with discrete spectra and nonlinear squeezed states with particular nonlinear functions.Finally,the time evolution of the nonclassical properties of the considered systems has been numerically investigated.
文摘We propose a new method to obtain the correlation length of gapped XXZ spin 1/2 antiferromagnetic chains. Following the relativistic quantum field theory in space-time dimensions, we use the exact dispersion of massive spinon to calculate the correlation length for XXZ spin 1/2 chain. We conjecture that the correlation length for other 1D lattice models can be obtained in the same way. Relation between dispersion and the oscillated correlation of gapped incommensurate lattice models is also discussed.
基金Supported in part by National Natural Science Foundation of China under Grant Nos.10605013 and 10975075 the Fundamental Research Funds for the Central Universities
文摘The two-dimensional hydrogen with a linear potential in a magnetic field is solved by two different methods.Furthermore the connection between the model and an anharmonic oscillator is investigated by methods of KStransformation.
基金supported by the National Natural Science Foundation of China (No. 11074313)
文摘The dislocation equations of a simple cubic lattice have been obtained by using Green＇s function method based on the discrete lattice theory with the coefficients of the secondorder differential terms and the integral terms have been given explicitly in advance. The simple cubic lattice we have discussed is a solvable model, which is obtained according to the lattice statics and the symmetry principle and can verify and validate the dislocation lattice theory. It can present unified dislocation equations which are suitable for most of metals with arbitral lattice structures. Through comparing the results of the present solvable model with the dislocation lattice theory, it can be seen that, the coefficients of integral terms of the edge and screw components we obtain are in accordance with the results of the dislocation lattice theory, however, the coefficient of the second-order differential term of the screw component is not in agreement with the result of the dislocation lattice theory. This is mainly caused by the reduced dynamical matrix of the surface term, which is the essence to obtain the dislocation equation. According to the simple cubic solvable model, not only the straight dislocations but also the curved dislocations, such as the kink, can be investigated further.
文摘The filtered-graded transfer of SAGBI bases computation in solvable polynomial algebras was considered. The relations among the SAGBI bases of a subalgebra B, its associated graded algebra G(B) and Rees algebra B were got. These relations solve a natural question: how to determine the generating set of G(B) and B from any given generating set of B. Based on these some equivalent conditions for the existence of finite SAGBI bases can be got.
基金Supported by the NSF of China(10471085) Supported by the Shanxi Province(20051007) Supported by the Returned Chinese Students Found of Shanxi Province(Jinliuguanban [2004]7)
文摘A subgroup H of a group G is called semipermutable if it is permutable with every subgroup K of G with （｜H｜, ｜K｜） = 1, and s-semipermutable if it is permutable with every Sylow p-subgroup of G with （p, ｜H｜） = 1. In this paper, some sufficient conditions for a group to be solvable are obtained in terms of s-semipermutability.
文摘In this paper, we consider the general quasi-differential expressions each of order n with complex coefficients and their formal adjoints on the interval (a,b). It is shown in direct sum spaces of functions defined on each of the separate intervals with the cases of one and two singular end-points and when all solutions of the equation and its adjoint are in (the limit circle case) that all well-posed extensions of the minimal operator have resolvents which are HilbertSchmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. These results extend those of formally symmetric expression studied in [1-10] and those of general quasi-differential expressions in [11-19].
基金Supported by the National Natural Science Foundation of China(11071187)Supported by the Basic and Advanced Technology Research Project of Henan Province(142300410449)Supported by the Natural Science Foundation of Education Department of Henan Province(16A110035)
文摘The finite-dimensional indecomposable solvable Lie algebras s with Q_(2n+1) as their nilradical are studied and classified and their Casimir invariants are calculated. It turns out that the dimension of s is at most dim Q_(2n+1)+2.
文摘A quasi-exactly solvable model refers to any second order differential equation with polynomial coefficients of the form A(x)y’’(x)+B(x)y’(x)+C(x)y(x)=0 where a pair of exact polynomials {y(x), C(x)} with respective degrees {deg[y]=n, deg[C]=p} are to be found simultaneously in terms of the coefficients of two given polynomials {A(x), B(x)}. The existing methods for solving quasi-exactly solvable models require the solution of a system of nonlinear algebraic equations of which the dimensions depend on n, the degree of the exact polynomial solution y(x). In this paper, a new method employing a set of polynomials, called canonical polynomials, is proposed. This method requires solving a system of nonlinear algebraic equations of which the dimensions depend only on p, the degree of C(x), and do not vary with n. Several examples are implemented to testify the efficiency of the proposed method.
文摘In this paper, we present a new method for solving a class of high-order quasi exactly solvable ordinary differential equations. With this method, the computed solution is expressed as a linear combination of the canonical polynomials associated with the given differential operator. An iterative algorithm summarizing the procedure is presented and its efficiency is demonstrated through considering two applied problems.
文摘A generalized method which helps to find a time-dependent SchrÖdinger equation for any static potential is established. We illustrate this method with two examples. Indeed, we use this method to find the time-dependent Hamiltonian of quasi-exactly solvable Lamé equation and to construct the matrix 2 × 2 time-dependent polynomial Hamiltonian.
文摘A suit of lacustrine source rocks are developed in volcanic deposits in the Shangkuli Formation of the Cretaceous. However, it is poor to understand the characteristics of source rocks due to the low degree of exploration, thus the exploration is severely constrained in this area. Based on the geochemical analysis, the analytic technique of GC and GC-MS, combined with the characteristics of solvable organic matter and biomarkers of the source rocks, the authors discussed the depositional environment, the derivation of the matrix and the maturity characteristics of the organic material of the Shangkuli Formation in Cretaceous. The results show that the organic matter mainly belongs to type Ⅱ1 kerogen, whose abundance is relatively high; it was formed in reductive surrounding where was deep-lake; the hydrocarbon is characterized by mixed-source of organic matter. The thermal evolution of source rocks had reached maturation stage.
