The Explicit Solution of the Matrix Equation AX-XB=C

Almost all of the existing results on the explicit solutions of the matrix equationAX-XB= C are obtained under the condition that A and B have no eigenvalues incommon For both symmetric or skewsymmetric matrices A and B. we shall give outthe explicit general solutions of this equation by using the notions of eigenprojectionsThe results we obtained are applicable not only to any cases of eigenvalues regardlessof their multiplicities but also to the discussion of the general case of this equation

A class of exact solutions for N-dimensional incompressible magnetohydrodynamic equations

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic （MHD） equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations axe directly constructed.

Numerical Solution for Fractional Partial Differential Equation with Bernstein Polynomials

A framework to obtain numerical solution of the fractional partial differential equation using Bernstein polynomials is presented. The main characteristic behind this approach is that a fractional order operational matrix of Bernstein polynomials is derived. With the operational matrix, the equation is transformed into the products of several dependent matrixes which can also be regarded as the system of linear equations after dispersing the variable. By solving the linear equations, the numerical solutions are acquired. Only a small number of Bernstein polynomials are needed to obtain a satisfactory result. Numerical examples are provided to show that the method is computationally efficient.

A Note on Exact Solutions to Linear Differential Equations by the Matrix Exponential

It is known that the solution to a Cauchy problem of linear differential equations:x'(t)=A(t)x(t),with x(t0)=x0,can be presented by the matrix exponential as exp(∫_(t0)^(t)A(s)ds)x0,if the commutativity condition for the coefficient matrix A(t)holds:[∫_(t0)^(t)A(s)ds,A(t)]=0.A natural question is whether this is true without the commutativity condition.To give a definite answer to this question,we present two classes of illustrative examples of coefficient matrices,which satisfy the chain rule d/dt exp(∫_(t0)^(t)A(s)ds)=A(t)exp(∫_(t0)^(t)A(s)ds),but do not possess the commutativity condition.The presented matrices consist of finite-times continuously differentiable entries or smooth entries.

The GI/M/1 Queue in a Multi-phase Service Environment with Working Vacations and Bernoulli Vacation Interruption

In this paper,we consider a GI/M/1 queue operating in a multi-phase service environment with working vacations and Bernoulli vacation interruption.Whenever the queue becomes empty,the server begins a working vacation of random length,causing the system to move to vacation phase 0.During phase 0,the server takes service for the customers at a lower rate rather than stopping completely.When a vacation ends,if the queue is non-empty,the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N.Moreover,we assume Bernoulli vacation interruption can happen.At a service completion instant,if there are customers in a working vacation period,vacation interruption happens with probability p,then the system switches from the phase 0 to some normal service phase i with probability qi,i=1,2,⋯,N,or the server continues the vacation with probability 1−p.Using the matrix geometric solution method,we obtain the stationary distributions for queue length at both arrival epochs and arbitrary epochs.The waiting time of an arbitrary customer is also derived.Finally,several numerical examples are presented.

Ergodicity of Quasi-birth and Death Processes(Ⅰ)

Quasi-birth and death processes with block tridiagonal matrices find many applications in various areas. Neuts gave the necessary and sufficient conditions for the ordinary ergodicity and found an expression of the stationary distribution for a class of quasi-birth and death processes. In this paper we obtain the explicit necessary and sufficient conditions for/-ergodicity and geometric ergodicity for the class of quasi-birth and death processes, and prove that they are not strongly ergodic. Keywords ergodicity, quasi-birth and death process.

Charting the ‘composition–strength’ space for novel austenitic,martensitic and ferritic creep resistant steels

We report results of a large computational ＇alloy by design＇ study, in which the ＇chemical composition-mechanical strength＇ space is explored for austenitic, ferritic and martensitic creep resistant steels. The approach used allows simultaneously optimization of alloy composition and processing parameters based on the integration of thermodynamic, thermo-kinetics and a genetic algorithm optimization route. The nature of the optimisation depends on both the intended matrix（ferritic, martensitic or austenitic） and the desired precipitation family. The models are validated by analysing reported strengths of existing steels. All newly designed alloys are predicted to outperform existing high end reference grades.