A sufficient condition for the order of approximation of a continuous 2π periodic function with a given majorant for the modulus of continuity by the [F, d_n] means of its Fourier serier to be of Jackson order is obt...A sufficient condition for the order of approximation of a continuous 2π periodic function with a given majorant for the modulus of continuity by the [F, d_n] means of its Fourier serier to be of Jackson order is obtained. This sufficient condition is shown to be not enough for the order of approximation by partial sums of their Fourier series to be of Jackson order. The error estimate is shown to be the best possible.展开更多
This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts servi...This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts serving, it serves all customers in the queue in a single batch, which is the so-called batch service. If a new customer or a retrial customer finds all the customers’ rooms are occupied, he will decide whether or not to join the retrial orbit. By using the censoring technique and the matrix analysis method, we first obtain the decay function of the stationary distribution for the quantity of customers in the retrial orbit and the quantity of customers in the queue. Then based on the form of decay rate function and the Karamata Tauberian theorem, we finally get the exact tail asymptotics of the stationary distribution.展开更多
We consider the random difference equations S =_d(X + S)Y and T =_dX + TY, where =_ddenotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right hand side are independent of...We consider the random difference equations S =_d(X + S)Y and T =_dX + TY, where =_ddenotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right hand side are independent of(X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that(X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by products which are interesting in their own right.展开更多
In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6w...In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6where 3 ≤ a 〈 4, D^ is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f（x, t） = -p（x）t^σ, with cr ~ （-1, 1） and p being a nonnegative continuous function that may be singular at x - 0 or x - 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch^uder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem （P0,0）. Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a ＋ b 〉 0 and f（x, t） -= tφ（x, t）, with φ（x, t） being a nonnegative continuous function in （0, 1） × [0, ∞） that is required to satisfy some suitable integrability condition. Using estimates on the Green＇s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem （Pa,b）, which behaves like the unique solution of the homogeneous problem corresponding the existence results. to （Pa,b）. Some examples are given to illustrate the existence results.,展开更多
文摘A sufficient condition for the order of approximation of a continuous 2π periodic function with a given majorant for the modulus of continuity by the [F, d_n] means of its Fourier serier to be of Jackson order is obtained. This sufficient condition is shown to be not enough for the order of approximation by partial sums of their Fourier series to be of Jackson order. The error estimate is shown to be the best possible.
文摘This paper discusses a queueing system with a retrial orbit and batch service, in which the quantity of customers’ rooms in the queue is finite and the space of retrial orbit is infinite. When the server starts serving, it serves all customers in the queue in a single batch, which is the so-called batch service. If a new customer or a retrial customer finds all the customers’ rooms are occupied, he will decide whether or not to join the retrial orbit. By using the censoring technique and the matrix analysis method, we first obtain the decay function of the stationary distribution for the quantity of customers in the retrial orbit and the quantity of customers in the queue. Then based on the form of decay rate function and the Karamata Tauberian theorem, we finally get the exact tail asymptotics of the stationary distribution.
基金supported by the National Science Foundation of the United States (Grant No. CMMI-1435864)
文摘We consider the random difference equations S =_d(X + S)Y and T =_dX + TY, where =_ddenotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right hand side are independent of(X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that(X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by products which are interesting in their own right.
基金funded by the National Plan for Science,Technology and Innovation(MAARIFAH),King Abdulaziz City for Science and Technology,Kingdom of Saudi Arabia,Award Number(No.13-MAT2137-02)
文摘In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6where 3 ≤ a 〈 4, D^ is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f（x, t） = -p（x）t^σ, with cr ~ （-1, 1） and p being a nonnegative continuous function that may be singular at x - 0 or x - 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch^uder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem （P0,0）. Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a ＋ b 〉 0 and f（x, t） -= tφ（x, t）, with φ（x, t） being a nonnegative continuous function in （0, 1） × [0, ∞） that is required to satisfy some suitable integrability condition. Using estimates on the Green＇s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem （Pa,b）, which behaves like the unique solution of the homogeneous problem corresponding the existence results. to （Pa,b）. Some examples are given to illustrate the existence results.,
