A variation of parameters formula and Gronwall type integral inequality are proved for a differential equation involving general piecewise alternately advanced and retarded argument.
In the present paper we introduce the q analogue of the Baskakov Beta operators. We establish some direct results in the polynomial weighted space of continuous functions defined on the interval [0, ∞). Then we obtain point-wise estimate, using the Lipschitz type maximal function.
In this paper, we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions. By using some classical inequalities, we obtain the regularity of strong solutions to the three-dimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.
We give more efficient criteria to characterise prime ideal or primary ideal. Further, we obtain the necessary and sufficient conditions that an ideal is prime or primary in real field from the Gröbner bases directly.
Sampling from a truncated multivariate normal distribution (TMVND) constitutes the core computational module in fitting many statistical and econometric models. We propose two efficient methods, an iterative data augmentation (DA) algorithm and a non-iterative inverse Bayes formulae (IBF) sampler, to simulate TMVND and generalize them to multivariate normal distributions with linear inequality constraints. By creating a Bayesian incomplete-data structure, the posterior step of the DA algorithm directly generates random vector draws as opposed to single element draws, resulting obvious computational advantage and easy coding with common statistical software packages such as S-PLUS, MATLAB and GAUSS. Furthermore, the DA provides a ready structure for implementing a fast EM algorithm to identify the mode of TMVND, which has many potential applications in statistical inference of constrained parameter problems. In addition, utilizing this mode as an intermediate result, the IBF sampling provides a novel alternative to Gibbs sampling and eliminates problems with convergence and possible slow convergence due to the high correlation between components of a TMVND. The DA algorithm is applied to a linear regression model with constrained parameters and is illustrated with a published data set. Numerical comparisons show that the proposed DA algorithm and IBF sampler are more efficient than the Gibbs sampler and the accept-reject algorithm.
In this paper we study the average sample-path cost (ASPC) problem for continuous-time Markov decision processes in Polish spaces. To the best of our knowledge, this paper is a first attempt to study the ASPC criterion on continuous-time MDPs with Polish state and action spaces. The corresponding transition rates are allowed to be unbounded, and the cost rates may have neither upper nor lower bounds. Under some mild hypotheses, we prove the existence of ε (ε ≥ 0)-ASPC optimal stationary policies based on two different approaches: one is the “optimality equation” approach and the other is the “two optimality inequalities” approach.
In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process (BMAP) and Markovian arrival process (MAP) respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated (multiple) vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.
In this paper, we define a class of domains in R^{n}. Using the synchronous coupling of reflecting Brownian motion, we obtain the monotonicity property of the solution of the heat equation with the Neumann boundary conditions. We then show that the hot spots conjecture holds for this class of domains.
In this paper we first investigate zero-sum two-player stochastic differential games with reflection, with the help of theory of Reflected Backward Stochastic Differential Equations (RBSDEs). We will establish the dynamic programming principle for the upper and the lower value functions of this kind of stochastic differential games with reflection in a straightforward way. Then the upper and the lower value functions are proved to be the unique viscosity solutions to the associated upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations with obstacles, respectively. The method differs significantly from those used for control problems with reflection, with new techniques developed of interest on its own. Further, we also prove a new estimate for RBSDEs being sharper than that in the paper of El Karoui, Kapoudjian, Pardoux, Peng and Quenez (1997), which turns out to be very useful because it allows us to estimate the L^{p}-distance of the solutions of two different RBSDEs by the p-th power of the distance of the initial values of the driving forward equations. We also show that the unique viscosity solution to the approximating Isaacs equation constructed by the penalization method converges to the viscosity solution of the Isaacs equation with obstacle.
In this paper, we consider the dividend problem in a two-state Markov-modulated dual risk model, in which the gain arrivals, gain sizes and expenses are influenced by a Markov process. A system of integrodifferential equations for the expected value of the discounted dividends until ruin is derived. In the case of exponential gain sizes, the equations are solved and the best barrier is obtained via numerical example. Finally, using numerical example, we compare the best barrier and the expected discounted dividends in the two-state Markov-modulated dual risk model with those in an associated averaged compound Poisson risk model. Numerical results suggest that one could use the results of the associated averaged compound Poisson risk model to approximate those for the two-state Markov-modulated dual risk model.
Let {Y_{i};-∞ < i < ∞} be a doubly infinite sequence of identically distributed φ-mixing random variables and let {a_{i};-∞ < i < ∞} be an absolutely summable sequence of real numbers. In this paper we study the moments of (1 ≤ r < 2, p > 0) under the conditions of some moments.
In this paper, we are concerned with the asymptotic behaviour of a weak solution to the Navier-Stokes equations for compressible barotropic flow in two space dimensions with the pressure function satisfying p(ρ) = aρ log^{d}(ρ) for large ρ. Here d > 2, a>0. We introduce useful tools from the theory of Orlicz spaces and construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is also briefly discussed.
In this paper, based on the recent results of Gozlan and Lé onard we give optimal transportationentropy inequalities for several usual distributions on R, such as Bernoulli, Binomial, Poisson, Gamma distributions and infinitely divisible distributions with positive or negative jumps.
In this paper, we study the controllability of the nonlinear evolution systems. We establish the controllability results by using the monotone operator theory. No compactness assumptions are imposed in the main results. We present an example to illustrate our results.
In this paper, we introduce a modified Landweber iteration to solve the sideways parabolic equation, which is an inverse heat conduction problem (IHCP) in the quarter plane and is severely ill-posed. We shall show that our method is of optimal order under both a priori and a posteriori stopping rule. Furthermore, if we use the discrepancy principle we can avoid the selection of the a priori bound. Numerical examples show that the computation effect is satisfactory.
The main objective of this paper is to present a new rectangular nonconforming finite element scheme with the second order convergence behavior for approximation of Maxwell's equations. Then the corresponding optimal error estimates are derived. The difficulty in construction of this finite element scheme is how to choose a compatible pair of degrees of freedom and shape function space so as to make the consistency error due to the nonconformity of the element being of order O(h^{3}), properly one order higher than that of its interpolation error O(h^{2}) in the broken energy norm, where h is the subdivision parameter tending to zero.
Our aim is to present some limit theorems for capacities. We consider a sequence of pairwise negatively correlated random variables. We obtain laws of large numbers for upper probabilities and 2-alternating capacities, using some results in the classical probability theory and a non-additive version of Chebyshev's inequality and Boral-Contelli lemma for capacities.
In this paper, the fundamental solution of rotating generalized Stokes problem in R^{3} is established. To obtain it, some fundamental solutions of other problems also are established, such as generalized Laplace problem, generalized Stokes problem and rotating Stokes problem.