In this paper we derive perturbation theorems for the LU and QR factors. Moreover, bounds for κL(A)/κL′(A) and κU(A)/κ′U(A) are given for the LU factorization of a nonsingular matrix. By applying pivoting strategies in the LU factorization, estimates for κL(PAQ)/κL′(PAQ) and κU(PAQ)/κ′U(PAQ) are also obtained.

In this paper, we investigate computable lower bounds for the best strongly ergodic rate of convergence of the transient probability distribution to the stationary distribution for stochastically monotone continuous-time Markov chains and reversible continuous-time Markov chains, using a drift function and the expectation of the first hitting time on some state. We apply these results to birth–death processes, branching processes and population processes.

The aim of this article is to prove a symmetry result for several overdetermined boundary value problems. For the two first problems, our method combines the maximum principle with the monotonicity of the mean curvature. For the others, we use essentially the compatibility condition of the Neumann problem.

The total claim amount for a fixed period of time is, by definition, a sum of a random number of claims of random size. In this paper we explore the probabilistic distribution of the total claim amount for claims that follow a Weibull distribution, which can serve as a satisfactory model for both small and large claims. As models for the number of claims we use the geometric, Poisson, logarithmic and negative binomial distributions. In all these cases, the densities of the total claim amount are obtained via Laplace transform of a density function, an expansion in Bell polynomials of a convolution and a subsequent Laplace inversion.

This paper presents a thermodynamic model for a heat engine based on evaporative cooling of unsaturated air at reduced pressure. Also analysed is a related heat pump based on condensation of water vapour in moist air at reduced pressure. These devices operate as two-stroke reciprocating engines, which are their simplest possible embodiments. The mathematical models for the two devices are based on conservation of mass for both air and water vapour, ideal gas laws, constant specific heats, and, as appropriate, either constant entropy processes or cooling/heating by evaporation/condensation. Both models take the form of coupled algebraic systems in six variables, which require numerical solution for certain stages of the cycle. The specific work output of the heat engine increases as the inlet air becomes hotter and as the expansion ratio of the engine increases. The engine provides evaporative cooling of air from inlet to outlet. The heat pump has a good coefficient of performance, which decreases as the expansion ratio increases. The heat pump also has the effect of drying the air from inlet to outlet, producing distilled water as a by-product.

In this paper, we introduce and consider a new class of complementarity problems, which are called the generalized mixed quasi-complementarity problems in a topological vector space. We show that the generalized mixed quasi-complementarity problems are equivalent to the generalized mixed quasi variational inequalities. Using a new type of KKM mapping theorem, we study the existence of a solution of the generalized mixed quasi-variational inequalities and generalized mixed quasi-complementarity problems. Several special cases are also discussed. The results obtained in this paper can be viewed as extension and generalization of the previously known results.

For a linear random field (linear p-parameter stochastic process) generated by a dependent random field with zero mean and finite qth moments (q>2p), we give sufficient conditions that the linear random field converges weakly to a multiparameter standard Brownian motion if the corresponding dependent random field does so.

In this paper an expansion method, based on Legendre or any orthogonal polynomials, is developed to find numerical solutions of two-dimensional linear Fredholm integral equations. We estimate the error of the method, and present some numerical examples to demonstrate the accuracy of the method.

In this paper, we study the existence of positive solutions for the one-dimensional p-Laplacian differential equation, subject to the multipoint boundary condition by applying a monotone iterative method.

Hu et al. [“A boundary problem for group testing”, SIAM J. Algebraic Discrete Meth.2 (1981), 81–87] conjectured that the minimax test number to find d defectives in 3d items is 3d−1, a surprisingly difficult combinatorial problem about which very little is known. In this article we state three more conjectures and prove that they are all equivalent to the conjecture of Hu et al. Notably, as a byproduct, we also obtain an interesting upper bound for M(d,n).

To ensure that the elevator of a cruise missile is operating within the design specification in high-attitude flight, we present a design method for the construction of a sliding mode recursive variable structure controller. In this design method, a target sliding mode surface is first designed without considering the engineering specification of the elevator. Secondly, by using this specification, the critical state is solved. Then, the transitional sliding mode surfaces are designed recursively by using the critical state of the previous sliding mode surface so that the state will move smoothly from one transitional sliding mode surface to the next until the target sliding mode surface. This design method is based on linear sliding mode variable structure theory. Thus, the controller obtained is simple in structure and practical. Furthermore, the elevator will operate within the engineering specification. The simulation results show the effectiveness of the proposed method.