The reverse order law for {1, 3, 4}-inverse of the product of two matrices

The relationship between {1, 3, 4}-inverses of AB and the product of {1, 3, 4}-inverses of A and B have been studied in this paper. The necessary and sufficient conditions for B{1,3,4}A{1,3,4}⊆(AB){1,3,4}, B{1,3,4}A{1,3,4}⊇(AB){1,3,4} and B{1,3,4}A{1,3,4}=(AB){1,3,4} are given.     

Comments on some recent results concerning {2, 3} and {2, 4}-generalized inverses

We give a comment on some recent results concerning the representations of generalized {2, 3} and {2, 4}-inverses. Shorter proofs of some previous results are presented.     

Further results on the reverse order law for the Moore-Penrose inverse in rings with involution

We present some equivalent conditions of the reverse order law for the Moore-Penrose inverse in rings with involution, extending some well-known results to more general settings. Then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted Moore-Penrose inverse in C^∗-algebras.     

Reverse order law for the Moore-Penrose inverse

In this paper we present new results related to the reverse order law for the Moore-Penrose inverse of operators on Hilbert spaces. Some finite-dimensional results are extended to infinite-dimensional settings.     

Full-rank representations of {2, 4}, {2, 3}-inverses and successive matrix squaring algorithm

We present the full-rank representations of {2, 4} and {2, 3}-inverses (with given rank as well as with prescribed range and null space) as particular cases of the full-rank representation of outer inverses. As a consequence, two applications of the successive matrix squaring (SMS) algorithm from [P.S. Stanimirovi?, D.S. Cvetkovi?-Ili?, Successive matrix squaring algorithm for computing outer inverses, Appl. Math. Comput. 203 (2008) 19-29] are defined using the full-rank representations of {2, 4} and {2, 3}-inverses. The first application is used to approximate {2, 4}-inverses. The second application, after appropriate modifications of the SMS iterative procedure, computes {2, 3}-inverses of a given matrix. Presented numerical examples clarify the purpose of the introduced methods.     

Triple reverse order law for the Drazin inverse

In this paper,we investigate the reverse order law for Drazin inverse of three bound-ed linear operators under some commutation relations.Moreover,the Drazin invertibility of sum is also obtained for two bounded linear operators and its expression is presented.     

Integrating partial optimization with scatter search for solving bi-criteria {0, 1}-knapsack problems

This paper deals with the problem of inaccuracy of the solutions generated by metaheuristic approaches for combinatorial optimization bi-criteria {0, 1}-knapsack problems. A hybrid approach which combines systematic and heuristic searches is proposed to reduce that inaccuracy in the context of a scatter search method. The components of this method are used to determine regions in the decision space to be systematically searched.     

Sequential Determination of the {1, 4}-Inverse of a Matrix

In this paper, we provide a set of results for the sequential determination of the {1, 4}-generalized inverse of a matrix. This inverse is of importance in areas where the minimal norm solution of a system of algebraic equations is desired.     

A note on the stability of a pth order iteration for finding generalized inverses

A number of papers intended for the numerical computation of generalized inverses by means of higher-order iterative methods have been published recently. This note investigates the numerical stability of a general family of iterative methods in order to complete the previous studies in this trend of research.     

The estimate for mean values on prime numbers relative to \frac{4} {p} = \frac{1} {{n_1 }} + \frac{1} {{n_2 }} + \frac{1} {{n_3 }}

If n is a positive integer,let f （n） denote the number of positive integer solutions （n 1,n 2,n 3） of the Diophantine equation 4/n=1/n₁ + 1/n₂ + 1/n₃.For the prime number p,f （p） can be split into f 1 （p） + f 2 （p）,where f i （p） （i=1,2） counts those solutions with exactly i of denominators n 1,n 2,n 3 divisible by p.In this paper,we shall study the estimate for mean values ∑ p相似文献   

Soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m, n) equations

We consider the nonlinear dispersive K(m,n) equation with the generalized evolution term and derive analytical expressions for some conserved quantities. By using a solitary wave ansatz in the form of sech^p function, we obtain exact bright soliton solutions for (2 + 1)-dimensional and (3 + 1)-dimensional K(m,n) equations with the generalized evolution terms. The results are then generalized to multi-dimensional K(m,n) equations in the presence of the generalized evolution term. An extended form of the K(m,n) equation with perturbation term is investigated. Exact bright soliton solution for the proposed K(m,n) equation having higher-order nonlinear term is determined. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients.     

A distance-regular graph with the intersection array {8, 7, 5; 1, 1, 4} and its automorphisms

Possible orders and subgraphs of the fixed points of a distance-regular graph with the intersection array {8, 7, 5; 1, 1, 4} are found. It is shown that such a graph is not vertex-transitive.     

Lower bound for matrix operators on the Euler weighted sequence space e_{w,p}^{\theta} (0<p<1)

Let A=(a _n,k ) _n,k≥0 be a non-negative matrix. Denote by \(L_{l_{p} (w),~e_{w,q}^{\theta}}(A)\) the supremum of those L, satisfying the following inequality:

     

17.

Kepler's area law in the Principia: filling in some details in Newton's proof of Proposition 1     

Michael Nauenberg 《Historia Mathematica》2003,30(4):441

During the past 30 years there has been controversy regarding the adequacy of Newton's proof of Prop. 1 in Book 1 of the Principia. This proposition is of central importance because its proof of Kepler's area law allowed Newton to introduce a geometric measure for time to solve problems in orbital dynamics in the Principia. It is shown here that the critics of Prop. 1 have misunderstood Newton's continuum limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3. We clarify the proof of Prop. 1 by filling in some details left out by Newton which show that his proof of this proposition was adequate and well-grounded.     

18.

A series of exact solutions for (2 + 1)-dimensional Wick-type stochastic generalized Broer-Kaup system via a modified variable-coefficient projective Riccati equation mapping method     

Qing Liu  Jia-Cheng Lan  Zi-Hua Wang 《Applied mathematics and computation》2010,217(2):629-638

A modified variable-coefficient projective Riccati equation mapping method is applied to (2 + 1)-dimensional Wick-type stochastic generalized Broer-Kaup system. With the help of Hermit transformation, we obtain a series of new exact stochastic solutions to the stochastic Broer-Kaup system in the white noise environment.     

19.

A difference analogue of the imbedding theorem for the anisotropic Sobolev space\mathop {W_{p_1 , \ldots,p_n }^1 }\limits^0     

I. M. Kolodii  I. I. Verba 《Ukrainian Mathematical Journal》1989,41(7):837-841

     

20.

A note on “New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation”     

Nikolay A. Kudryashov  Pavel N. Ryabov  Dmitry I. Sinelshchikov 《Applied mathematics and computation》2010,216(8):2479-4515

Exact solutions of the Nizhnik-Novikov-Veselov equation by Li [New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation, Appl. Math. Comput. 215 (2009) 3777-3781] are analyzed. We have observed that fourteen solutions by Li from 30 do not satisfy the equation. The other 16 exact solutions by Li can be found from the general solutions of the well-known solution of the equation for the Weierstrass elliptic function.     

Kepler's area law in the Principia: filling in some details in Newton's proof of Proposition 1

During the past 30 years there has been controversy regarding the adequacy of Newton's proof of Prop. 1 in Book 1 of the Principia. This proposition is of central importance because its proof of Kepler's area law allowed Newton to introduce a geometric measure for time to solve problems in orbital dynamics in the Principia. It is shown here that the critics of Prop. 1 have misunderstood Newton's continuum limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3. We clarify the proof of Prop. 1 by filling in some details left out by Newton which show that his proof of this proposition was adequate and well-grounded.     

A series of exact solutions for (2 + 1)-dimensional Wick-type stochastic generalized Broer-Kaup system via a modified variable-coefficient projective Riccati equation mapping method

A modified variable-coefficient projective Riccati equation mapping method is applied to (2 + 1)-dimensional Wick-type stochastic generalized Broer-Kaup system. With the help of Hermit transformation, we obtain a series of new exact stochastic solutions to the stochastic Broer-Kaup system in the white noise environment.     

A note on “New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation”

Exact solutions of the Nizhnik-Novikov-Veselov equation by Li [New kink-shaped solutions and periodic wave solutions for the (2 + 1)-dimensional Sine-Gordon equation, Appl. Math. Comput. 215 (2009) 3777-3781] are analyzed. We have observed that fourteen solutions by Li from 30 do not satisfy the equation. The other 16 exact solutions by Li can be found from the general solutions of the well-known solution of the equation for the Weierstrass elliptic function.