An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

Pathwise Estimation of Stochastic Differential Equations with Unbounded Delay and Its Application to Stochastic Pantograph Equations

The existence and uniqueness of the global solution of stochastic differential equations with discrete variable delay is investigated in this paper, and the pathwise estimation is also done by using Lyapunov function method and exponential martingale inequality. The results can be used not only in the case of bounded delay but also in the case of unbounded delay. As the applications, this paper considers the pathwise estimation of solutions of stochastic pantograph equations.     

A new method of moving asymptotes for large-scale linearly equality-constrained minimization

A new method of moving asymptotes for large-scale minimization subject to linear equality constraints is discussed.In this method,linear equality constraints are deleted with null space technique and the descending direction is obtained by solving a convex separable subproblem of moving asymptotes in each iteration. New rules for controlling the asymptotes parameters are designed and the global convergence of the method under some reasonable conditions is established and proved.The numerical results show that the new method may be capable of processing some large scale problems.     

Fluid approximation and its convergence Rate for GI/G/1 queue with vacations

A GI/G/1 queue with vacations is considered in this paper.We develop an approximating technique on max function of independent and identically distributed(i.i.d.) random variables,that is max{ηi,1 ≤ i ≤ n}.The approximating technique is used to obtain the fluid approximation for the queue length,workload and busy time processes.Furthermore,under uniform topology,if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate,we prove by the...     

Positive-Part Moments via the Fourier–Laplace Transform

Integral expressions for positive-part moments E X₊^pmathsf{E},X_{+}^{p} (p>0) of random variables X are presented, in terms of the Fourier–Laplace or Fourier transforms of the distribution of X. A necessary and sufficient condition for the validity of such an expression is given. This study was motivated by extremal problems in probability and statistics, where one needs to evaluate such positive-part moments.     

Branching on hyperplane methods for mixed integer linear and convex programming using adjoint lattices

We present branching-on-hyperplane methods for solving mixed integer linear and mixed integer convex programs. In particular, we formulate the problem of finding a good branching hyperplane using a novel concept of adjoint lattice. We also reformulate the problem of rounding a continuous solution to a mixed integer solution. A worst case complexity of a Lenstra-type algorithm is established using an approximate log-barrier center to obtain an ellipsoidal rounding of the feasible set. The results for the mixed integer convex programming also establish a complexity result for the mixed integer second order cone programming and mixed integer semidefinite programming feasibility problems as a special case. Our results motivate an alternative reformulation technique and a branching heuristic using a generalized (e.g., ellipsoidal) norm reduced basis available at the root node.     

On complex matrices with simple spectrum that are unitarily similar to real matrices

Suppose that one should verify whether a given complex n × n matrix can be converted into a real matrix by a unitary similarity transformation. Sufficient conditions for this property to hold were found in an earlier publication of this author. These conditions are relaxed in the following way: as before, the spectrum is required to be simple, but pairs of complex conjugate eigenvalues $ \lambda ,\bar \lambda $ \lambda ,\bar \lambda are now allowed. However, the eigenvectors corresponding to such eigenvalues must not be orthogonal.     

On HSS-based constraint preconditioners for generalized saddle-point problems

For the generalized saddle-point problems with non-Hermitian (1,1) blocks, we present an HSS-based constraint preconditioner, in which the (1,1) block of the preconditioner is constructed by the HSS method for solving the non-Hermitian positive definite linear systems. We analyze the invertibility of the HSS-based constraint preconditioner and prove the convergence of the preconditioned iteration method. Numerical experiments are used to demonstrate the efficiency of the preconditioner as well as the corresponding preconditioned iteration method, especially when the (1,1) block of the saddle-point matrix is essentially non-Hermitian.     

Singular values of some modular functions

For an integer N greater than 5 and a triple \({\mathfrak{a}}=[a_{1},a_{2},a_{3}]\) of integers with the properties 0<a _i ≤N/2 and a _i ≠a _j for i≠j, we consider a modular function \(W_{\mathfrak{a}}(\tau)=\frac{\wp (a_{1}/N;L_{\tau})-\wp (a_{3}/N;L_{\tau})}{\wp (a_{2}/N;L_{\tau})-\wp(a_{3}/N;L_{\tau})}\) for the modular group Γ ₁(N), where ?(z;L _τ ) is the Weierstrass ?-function relative to the lattice L _τ generated by 1 and a complex number τ with positive imaginary part. For a pair of such triples \({\mathfrak{A}}=[{\mathfrak{a}},{\mathfrak{b}}]\) and a pair of non-negative integers F=[m,n], we define a modular function \(T_{{\mathfrak{A}},F}\) for the group Γ ₀(N) as the trace of the product \(W_{\mathfrak{a}}^{m}W_{\mathfrak{b}}^{n}\) to the modular function field of Γ ₀(N). In this article, we study the integrality of singular values of the functions \(W_{\mathfrak{a}}\) and \(T_{{\mathfrak{A}},F}\) by using their modular equations. We prove that the functions \(T_{{\mathfrak{A}},F}\) for suitably chosen \({\mathfrak{A}}\) and F generate the modular function field of Γ ₀(N), and from Shimura reciprocity and Gee–Stevenhagen method we obtain that singular values \(T_{{\mathfrak{A}},F}(\tau)\) for suitably chosen \({\mathfrak{A}}\) and F generate ring class fields. Further, we study the class polynomial of \(T_{{\mathfrak{A}},F}\) for Schertz N-system.