A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications

A square matrix A of order n is said to be tripotent if A ³?=?A. In this note, we give a nine-term disjoint idempotent decomposition for the linear combination of two commutative tripotent matrices and their products. Using the decomposition, we derive some closed-form formulae for the eigenvalues, determinant, rank, trace, power, inverse and group inverse of the linear combinations. In particular, we show that the linear combinations of two commutative tripotent elements and their products can produce 3⁹?=?19,683 tripotent elements.     

Determinants and Algebraic Identities Associated with the Root Systems of Classical Lie Algebras

Burnside asked questions about periodic groups in his influential paper of 1902. The study of groups with exponent six is a special case of the study of the Burnside questions on which there has been significant progress. It has contributed a number of worthwhile aspects to the theory of groups and in particular to computation related to groups. Finitely generated groups with exponent six are finite. We investigate the nature of relations required to provide proofs of finiteness for some groups with exponent six. We give upper and lower bounds for the number of sixth powers needed to define the largest 2-generator group with exponent six. We solve related questions about other groups with exponent six using substantial computations which we explain.     

Generalized Wronskian and Grammian Solutions to a Isospectral B-type Kadomtsev-Petviashvili equation

Generally speaking, the BKP hierarchy which only has Pfaffian solutions. In this paper, based on the Grammian and Wronskian derivative formulae, generalized Wronskian and Grammian determinant solutions are obtained for the isospectral BKP equation (the second member on the BKP hierarchy) in the Hirota bilinear form. Especially, with the help of the properties of the computing of Young diagram, we have first applied Young diagram proved the proposition of this paper. Moreover, by considering the different combinations of the entries in Wronskian, we obtain various types of Wronskian solutions.     

The Inverse Resonance Problem for Hermite Operators

In this paper the inverse resonance problem for the Hermite operator is investigated. The Hermite operator with the creation operator , the annihilation operator , and a finitely supported multiplication operator b, is an unbounded operator on ℓ ²(ℕ₀) having finitely many eigenvalues and infinitely many resonances (except for b=0, when there are no eigenvalues or resonances). It is shown that knowing the location of eigenvalues and resonances determines the potential b uniquely.      

The number of eigenvalues of the two-particle discrete Schrödinger operator

We consider the two-particle discrete Schrödinger operator associated with the Hamiltonian of a system of two particles (fermions) interacting only at the nearest neighbor sites. We find the number and the location of the eigenvalues of this operator depending on the particle interaction energy, the system quasimomentum, and the dimension of the lattice ? ^ν , ν ≥ 1.     

运用SD枝向量行列式反馈环计算分析新兴产业协同创新障碍——以江西省绿色光源产业为例

通过组织管理系统动力学(SD)流率基本入树建模法建立江西省绿色光源产业协同创新系统的4棵入树,利用枝向量行列式反馈环算法计算出系统的18条反馈环,然后紧密联系江西省绿色光源产业协同创新增效缓慢的问题,对全部反馈环逐条进行关键因果链分析,揭示产生制约绿色光源产业协同创新增效缓慢问题的16条关键因果链,基于关键因果链所指问题的共性,将16条关键因果链分成4类,再基于4类关键因果链生成系统发展的4条管理对策,为破解江西省绿色光源产业创新增效缓慢的问题提供决策依据。更多还原     

Hungry Lotka–Volterra lattice under nonzero boundaries,block-Hankel determinant solution,and biorthogonal polynomials

In this paper, we first extend the hungry Lotka–Volterra lattice to a case of nonzero boundary conditions and present its corresponding exact solution expressed in terms of a block-Hankel determinant. Then, we establish a connection between this hungry Lotka–Volterra lattice under nonzero boundary conditions and a set of biorthogonal polynomials. It turns out that the hungry Lotka–Volterra lattice under nonzero boundary conditions possesses a Lax pair expressed in terms of the biorthogonal polynomials. Moreover, we consider two special cases of the hungry Lotka–Volterra lattice. For the case $M=1$, it reduces to the Lotka–Volterra lattice under nonzero boundary condition, which has been discussed in literature. We also present the result for $M=2$ in detail, which extends a known result to a case of nonzero boundary functions. All these results are obtained by virtue of Hirota's bilinear method and determinant techniques.