Primitive normal polynomials with the specified last two coefficients

We get that for any element a∈F_q and any primitive element , there exists a primitive normal polynomial f(x)=xⁿ−σ₁xⁿ⁻¹+?+(−1)ⁿ⁻¹σ_n−1x+(−1)ⁿσ_n with σ_n−1=a, σ_n=b for n≥5. The estimates of hybrid extended Kloostermann sums over finite fields and some new variations of Cohen’s sieve techniques help to get the above result.     

Algorithm for calculating the roots of polynomials with coefficients in the ring of polynomials over an arbitrary integral domain

A deterministic algorithm for calculating the roots of polynomials in one variable with coefficients in the ring of polynomials in several variables over an arbitrary integral domain is constructed. An estimate for the arithmetic complexity of the algorithm in the worst case is obtained.     

A projection algorithm on the set of polynomials with two bounds

Numerical Algorithms - The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the...     

An enhanced finite difference time domain method for two dimensional Maxwell's equations

An enhanced finite-difference time-domain (FDTD) algorithm is built to solve the transverse electric two-dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across the interface. To resolve the instability issue of Yee's scheme (staircasing) caused by discontinuous permittivity across the interface, our algorithm revises the permittivities and makes some corrections to the scheme for the cells around the interface. It is also an improvement over the contour-path effective permittivity algorithm by including some extra terms in the formulas. The scheme is validated in solving the scattering of a dielectric cylinder with exact solution from Mie theory and is then compared with the above contour-path method, the usual staircasing and the volume-average method. The numerical results demonstrate that the new algorithm has achieved significant improvement in accuracy over other methods. Furthermore, the algorithm has a simple structure and can be merged into current FDTD software packages easily. The C++ source code for this paper is provided as supporting information for public access.     

An algorithm for calculating one subgradient of a convex function of two variables

Summary We present an algorithm which enables us to calculate one particular subgradient of a convex functionf: ² at a given point. Such a calculation is required in many existing numerical methods for convex nondifferentiable optimization. The novelty of our approach lies in the assumption that only the values off are computable and no analytical formula for the subdifferential is known. We include some numerical examples.     

Valuation ideals of order one in 2-dimensional regular local rings

Let υ be a prime divisor of a 2-dimensional regular local ring (R m) with algebraically closed residue field k. Zariski showed that a prime divisor υ of R is uniquely associated to a simple m-primary integrally closed ideal I of R, there exist finitely many simple υ-ideals including I, and all the other υ-ideals can be uniquely factored into products of simple υ-ideals. It is known that such an m-primary ideal I of R can be minimally generated by o(I) + 1 elements.Given a simple integrally closed ideal I of order one with arbitrary rank and its associated prime divisor υ, we find minimal generating sets of all the simple υ-ideals and describe factorizations of all the composite υ-ideals in terms of power products of simple υideals as explicitly as possible.     

An analysis of the solution set to a homotopy equation between polynomials with real coefficients

We investigate the structure of the solution setS to a homotopy equationH(Z,t)=0 between two polynomialsF andG with real coefficients in one complex variableZ. The mapH is represented asH(x+iy, t)=h ₁(x, y, t)+ih ₂(x, y, t), whereh ₁ andh ₂ are polynomials from ℝ² × [0,1] into ℝ and i is the imaginary unit. Since all the coefficients ofF andG are real, there is a polynomialh ₃ such thath ₂(x, y, t)=yh₃(x, y, t). Then the solution setS is divided into two sets {(x, t)∶h ₁(x, 0, t)=0} and {(x+iy, t)∶h ₁(x, y, t)=0,h ₃(x, y, t)=0}. Using this division, we make the structure ofS clear. Finally we briefly explain the structure of the solution set to a homotopy equation between polynomial systems with real coefficients in several variables.     

Time dependent critical fluctuations of a one dimensional local mean field model

Summary One-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.     

Identification of two coefficients with data of one component for a nonlinear parabolic system

In this article, we consider a nonlinear parabolic system with two components and prove a stability estimate of Lipschitz type in determining two coefficients of the system by data of only one component. The main idea for the proof is a Carleman estimate.     

A remark on roots of polynomials with positive coefficients

A short proof of a theorem of Dubickas on roots of polynomials with positive rational coefficients is presented.     

Counting irreducible polynomials with prescribed coefficients over a finite field

We continue our study on counting irreducible polynomials over a finite field with prescribed coefficients. We set up a general combinatorial framework using generating functions with coefficients from a group algebra which is generated by equivalence classes of polynomials with prescribed coefficients. Simplified expressions are derived for some special cases. Our results extend some earlier results.     

一种新的局部时间积分的区域分解波形松弛算法

提出一种新的区域分解波形松弛算法, 使得可以在不同的子域采用不同的时间步长来并行求解线性抛物方程的初边值问题. 与传统的区域分解波形松弛算法相比, 该算法可以通过预条件子来加快收敛速度, 并且对内存的需求大大降低. 给出了局部时间步长一种具体的实现方法, 证明了离散解的存在唯一性, 并在时间连续水平分析了预条件系统. 数值实验显示了新算法的有效性.     

Approximation properties of a new type Bernstein-Stancu polynomials of one and two variables

A new generalization of Bernstein-Stancu type polynomials for one and two variables are constructed and the theorems on convergence and the degree of convergence are established. In addition some numerical examples, corresponding to obtaining results are given.     

Periodic solutions to nonlinear one dimensional wave equation with -dependent coefficients

This paper deals with -periodicity and regularity of solutions to the one dimensional nonlinear wave equation with -dependent coefficients