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 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
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
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
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
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
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
In the paper, the determinate atlecation decision model and the probabilistic allocation decision model of a kind of renewable resource are separatly studied by means of dynamic programming, and the optimal allocation...In the paper, the determinate atlecation decision model and the probabilistic allocation decision model of a kind of renewable resource are separatly studied by means of dynamic programming, and the optimal allocation policy is given under some special conditions.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
In a prior paper, the d = 1 to d = 7 sense of AdS/CFT solutions were described in general whereas we did not introduce commentary as to GW polarization of gravitational radiation from a worm hole. We will discuss GW p...In a prior paper, the d = 1 to d = 7 sense of AdS/CFT solutions were described in general whereas we did not introduce commentary as to GW polarization of gravitational radiation from a worm hole. We will discuss GW polarization, for d = 1 and in addition say concrete facts as to the strength of the GW radiation, and admissible frequencies. First off, the term Δt is for the smallest unit of time step. Note that in the small Δt limit for d = 1 we avoid any imaginary time no matter what the sign of Ttemp is. And when d = 1 in order to have any solvability one would need X = Δt assumed to be infinitesimal. To first approximation, we set X = Δt as being of Planck time, 10-31 or so seconds, in duration.展开更多
In this paper,a C\|K model with solvable endogenous fertility under the strongly additive utility function is presented.The discrimination conditions of the existence of the nonzero steady states are given.Under a kin...In this paper,a C\|K model with solvable endogenous fertility under the strongly additive utility function is presented.The discrimination conditions of the existence of the nonzero steady states are given.Under a kind of utility function and production function,we prove that these conditions are satisfied and the economy at least has an optimal growth path.The positional relationship of the multiple steady states on the plane is discussed when multiple steady states and multiple growth paths exist.By numerical analysis,the fertility decreases with the per capita capital and per capita consumption increasing and increases with the per capita capital and per capita consumption decreasing on the economic growth path are obtained.展开更多
We propose an algebraic model, presenting individual contributions separately in the system of interest, for the exact solutions of one-dimensional Poisson-Schr?dinger equations used generally in semiconductor device ...We propose an algebraic model, presenting individual contributions separately in the system of interest, for the exact solutions of one-dimensional Poisson-Schr?dinger equations used generally in semiconductor device simulations. The model presented here reveals an interesting relation between the corresponding Poisson and Schr?dinger equation for the physical structure considered, which leads to closed solutions without solving the required electrostatic equation.展开更多
A new example of PT-symmetric quasi-exactly solvable (QES) 22×-matrix Hamiltonian which is associated to a trigonometric Razhavi potential is con-sidered. Like the QES analytic method considered in the Ref. [1] [...A new example of PT-symmetric quasi-exactly solvable (QES) 22×-matrix Hamiltonian which is associated to a trigonometric Razhavi potential is con-sidered. Like the QES analytic method considered in the Ref. [1] [2], we es-tablish three necessary and sufficient algebraic conditions for this Hamilto-nian to have a finite-dimensional invariant vector space whose generic ele-ment is polynomial. This non hermitian matrix Hamiltonian is called qua-si-exactly solvable [3].展开更多
The non PT-symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction is considered. We poin...The non PT-symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction is considered. We point out all properties of both of the original Mandal and the original Jaynes-Cummings Hamitonians. It is shown that these Hamiltonians are respectively pseudo-hermitian and hermitian REF _Ref536606452 \r \h \* MERGEFORMAT [1] REF _Ref536606454 \r \h [2]. Like the direct approach to invariant vector spaces used in Refs. REF _Ref536606456 \r \h [3] REF _Ref536606457 \r \h [4], we reveal the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator.展开更多
Bozek(1980)has introduced a class of solvable Lie groups Gn with arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is s...Bozek(1980)has introduced a class of solvable Lie groups Gn with arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is solvable.In this article,the authors provide the set of all left-invariant minimal unit vector fields on the solvable Lie group Gn,and give the relationships between the minimal unit vector fields and the geodesic vector fields,the strongly normal unit vectors respectively.展开更多
Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In...Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.展开更多
Let S be a finite linear space, and let G be a group of automorphisms of S. If G is soluble and line-transitive, then for a given k but a finite number of pairs of ( S, G), S has v= pn points and G≤AΓ L(1,pn).
The possible derived lengths and structures of solvable subgroups of SL(2,C) are given in Theorem 1,and the structures of the Riemann surfaces of solutions of the Fuchsian equation on torus with a solvable monodromy g...The possible derived lengths and structures of solvable subgroups of SL(2,C) are given in Theorem 1,and the structures of the Riemann surfaces of solutions of the Fuchsian equation on torus with a solvable monodromy group are discussed.Examples in verifying the solvability of monodromy groups for some Fuchsian equations on T2 are given.展开更多
Thompson's theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson's theorem, a finite group is solvable if it has an abelian maximal...Thompson's theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson's theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.展开更多
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.
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.
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.
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.
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].
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In the paper, the determinate atlecation decision model and the probabilistic allocation decision model of a kind of renewable resource are separatly studied by means of dynamic programming, and the optimal allocation policy is given under some special conditions.
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In a prior paper, the d = 1 to d = 7 sense of AdS/CFT solutions were described in general whereas we did not introduce commentary as to GW polarization of gravitational radiation from a worm hole. We will discuss GW polarization, for d = 1 and in addition say concrete facts as to the strength of the GW radiation, and admissible frequencies. First off, the term Δt is for the smallest unit of time step. Note that in the small Δt limit for d = 1 we avoid any imaginary time no matter what the sign of Ttemp is. And when d = 1 in order to have any solvability one would need X = Δt assumed to be infinitesimal. To first approximation, we set X = Δt as being of Planck time, 10-31 or so seconds, in duration.
基金Supported by National Natural Science Foundation of China!(79970104)
文摘In this paper,a C\|K model with solvable endogenous fertility under the strongly additive utility function is presented.The discrimination conditions of the existence of the nonzero steady states are given.Under a kind of utility function and production function,we prove that these conditions are satisfied and the economy at least has an optimal growth path.The positional relationship of the multiple steady states on the plane is discussed when multiple steady states and multiple growth paths exist.By numerical analysis,the fertility decreases with the per capita capital and per capita consumption increasing and increases with the per capita capital and per capita consumption decreasing on the economic growth path are obtained.
文摘We propose an algebraic model, presenting individual contributions separately in the system of interest, for the exact solutions of one-dimensional Poisson-Schr?dinger equations used generally in semiconductor device simulations. The model presented here reveals an interesting relation between the corresponding Poisson and Schr?dinger equation for the physical structure considered, which leads to closed solutions without solving the required electrostatic equation.
文摘A new example of PT-symmetric quasi-exactly solvable (QES) 22×-matrix Hamiltonian which is associated to a trigonometric Razhavi potential is con-sidered. Like the QES analytic method considered in the Ref. [1] [2], we es-tablish three necessary and sufficient algebraic conditions for this Hamilto-nian to have a finite-dimensional invariant vector space whose generic ele-ment is polynomial. This non hermitian matrix Hamiltonian is called qua-si-exactly solvable [3].
文摘The non PT-symmetric exactly solvable Hamiltonian describing a system of a fermion in the external magnetic field which couples to a harmonic oscillator through some pseudo-hermitian interaction is considered. We point out all properties of both of the original Mandal and the original Jaynes-Cummings Hamitonians. It is shown that these Hamiltonians are respectively pseudo-hermitian and hermitian REF _Ref536606452 \r \h \* MERGEFORMAT [1] REF _Ref536606454 \r \h [2]. Like the direct approach to invariant vector spaces used in Refs. REF _Ref536606456 \r \h [3] REF _Ref536606457 \r \h [4], we reveal the exact solvability of both the Mandal and Jaynes-Cummings Hamiltonians after expressing them in the position operator and the impulsion operator.
基金supported by the National Natural Science Foundation of China (Nos. 12001007,12201358)the Natural Science Foundation of Shandong Province (No. ZR2021QA051)+1 种基金the Natural Science Foundation of Anhui Province (No. 1908085QA03)Starting Research Funds of Shandong University of Science and Technology (Nos. 0104060511817, 0104060540626)
文摘Bozek(1980)has introduced a class of solvable Lie groups Gn with arbitrary odd dimension to construct irreducible generalized symmetric Riemannian space such that the identity component of its full isometry group is solvable.In this article,the authors provide the set of all left-invariant minimal unit vector fields on the solvable Lie group Gn,and give the relationships between the minimal unit vector fields and the geodesic vector fields,the strongly normal unit vectors respectively.
基金supported by NSFC(11671063)a grant from the Simons Foundation(#499532 to Yong Yang)a grant from the Simons Foundation(#280770 to Thomas M.Keller).
文摘Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.
文摘Let S be a finite linear space, and let G be a group of automorphisms of S. If G is soluble and line-transitive, then for a given k but a finite number of pairs of ( S, G), S has v= pn points and G≤AΓ L(1,pn).
基金Project supported by the National Natural Science Foundation of China.
文摘The possible derived lengths and structures of solvable subgroups of SL(2,C) are given in Theorem 1,and the structures of the Riemann surfaces of solutions of the Fuchsian equation on torus with a solvable monodromy group are discussed.Examples in verifying the solvability of monodromy groups for some Fuchsian equations on T2 are given.
基金Supported by National Natural Science Foundation of China (Grant No. 10871032), China Postdoctoral Science Foundation (Grant No. 20100470136) the second author is supported in part by "Agencija za raziskovalno dejavnost Republike Slovenije", proj. mladi raziskovalci, "Agencija za raziskovalno dejavnost Republike Slovenije", Research Program P1-0285
文摘Thompson's theorem indicates that a finite group with a nilpotent maximal subgroup of odd order is solvable. As an important application of Thompson's theorem, a finite group is solvable if it has an abelian maximal subgroup. In this paper, we give some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable.