Stable Border Bases for Ideals of Numerical Cartesian Sets

In this paper, we discuss a special class of sets of bivariate empirical points, namely, numerical cartesian sets. We find that the stable quotient bases for numerical cartesian sets are unique if they exist. Furthermore, the corresponding border bases are the unique stable border bases for the vanishing ideals of numerical cartesian sets.     

Cartesian closed categories of FƵ-domains

A subset system Z assigns to each partially ordered set P a certain collection Z（P） of subsets. In this paper, a new kind of subset systems called directable subset systems is introduced. For a directable subset system Z, the concepts of FZ-way-below relation and FZ-domain are introduced. The well-known Scott topology is naturally generalized to the Z-level and the resulting topology is called FZ-Scott topology, and the continuous functions with respect to this topology are characterized by preserving the suprema of directed Z-sets. Then, we mainly consider a generalization of the cartesian closedness of the categories DCPO of directed complete posets, BF of bifinite domains and FS of FS-domains to the Z-level. Corresponding to them, it is proved that, for a suitable subset system Z, the categories FZCPO of Z-complete posets, FSFZ of finitely separated FZ-domains and BFFZ of bifinite FZ-domains are all cartesian closed. Some examples of these categories are given.     

A fast algorithm for multivariate Hermite interpolation

Multivariate Hermite interpolation is widely applied in many fields, such as finite element construction, inverse engineering, CAD etc.. For arbitrarily given Hermite interpolation conditions, the typical method is to compute the vanishing ideal I （the set of polynomials satisfying all the homogeneous interpolation conditions are zero） and then use a complete residue system modulo I as the interpolation basis. Thus the interpolation problem can be converted into solving a linear equation system. A generic algorithm was presented in [18], which is a generalization of BM algorithm [22] and the complexity is O（τ^3） where r represents the number of the interpolation conditions. In this paper we derive a method to obtain the residue system directly from the relative position of the points and the corresponding derivative conditions （presented by lower sets） and then use fast GEPP to solve the linear system with O（（τ ＋ 3）τ^2） operations, where τ is the displacement-rank of the coefficient matrix. In the best case τ = 1 and in the worst case τ = [τ/n], where n is the number of variables.     

A generalization of Simpson's Rule

Let and let , where P _c ⁿ denoles the Taylor polynomial to f at c of order n, where n is even. TA and TM are reach generalizations of the Trapezoidal rule and the midpoint rule, respectively, and are each exact for all polynomials of degree ≤n+1. We let L(f)=αTM(f)+(1−α)TA(f), where , to obtain a numerical integration rule L which is exact for all polynomials of degree≤n+3 (see Theorem 1). The case n=0 is just the classical Simpson's rule. We analyze in some detail the case n=2, where our formulae appear to be new. By replacing P _(a+b) ^2/n+1 (x) by the Hermite cubic interpolant at a and b, we obtain some known formulae by a different approach (see [1] and [2]). Finally we discuss some nonlinear numerical integration rules obtained by taking piecewise polynomials of odd degree, each piece being the Taylor polynomial of f at a and b, respectively. Of course all of our formulae can be compounded over subintervals of [a,b].     

When the cartesian product of directed cycles is Hamiltonian

The cartesian product of two hamiltonian graphs is always hamiltonian. For directed graphs, the analogous statement is false. We show that the cartesian product C_n1 × C_n2 of directed cycles is hamiltonian if and only if the greatest common divisor (g.c.d.) d of n₁ and n₂ is at least two and there exist positive integers d₁, d₂ so that d₁ + d₂ = d and g.c.d. (n₁, d₁) = g.c.d. (n₂, d₂) = 1. We also discuss some number-theoretic problems motivated by this result.     

On graded generalized local cohomology

Let be a homogeneous Noetherian ring with local base ring (R₀,m₀) and let M,N be two finitely generated graded R-modules. Let denote the i-th graded generalized local cohomology of N relative to M with support in . We study the vanishing, tameness and asymptotical stability of the homogeneous components of . Received: 22 March 2005; revised: 25 June 2005     

A Search for Best Constants in the Hardy-Littlewood Maximal Theorem

Let be the centered maximal operator on the line. Through a numerical search procedure, we have conjectural best constants for the weak-type 1-1 estimate (3/2) and the L^p estimate (the constant B(p,1) such that We prove that these constants are lower bounds for the best constants and discuss the numerical evidence for the conjectures.     

Vanishing Cycles of Irregular D-Modules

Considering a holonomic -module and a hypersurface, we define a finite family of -modules on the hypersurface which we call modules of vanishing cycles. The first one had been previously defined and corresponds to formal solutions. The last one corresponds, via Riemann-Hilbert, to the geometric vanishing cycles of Grothendieck-Deligne. For regular holonomic -modules there is only one sheaf and for non regular modules the sheaves of vanishing cycles control the growth and the index of solutions. Our results extend to non holonomic modules under some hypothesis.     

Numerical procedure for solving a minimization eigenvalue problem

A numerical procedure for solving the left eigenvalue problem

Nonlinear Elliptic Problems Above Criticality

We consider the elliptic problem Δu + u^p = 0, u > 0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay at infinity. In addition, a fast decay solution exists if p is close enough to the critical exponent. If p differs from certain sequence of resonant values which tends to infinity, then the Dirichlet problem is also solvabe in a bounded domain Ω with a sufficiently small spherical hole. Lecture held in the Seminario Matematico e Fisico on Januay 23, 2006 Received: July 2006     

BOUNDARY LAYER AND VANISHING DIFFUSION LIMIT FOR NONLINEAR EVOLUTION EQUATIONS

In this paper, we consider an initial-boundary value problem for some nonlinear evolution equations with damping and diffusion. The main purpose is to investigate the boundary layer effect and the convergence rates as the diffusion parameter α goes to zero.     

Uniqueness and Sharp Estimates on Solutions to Hyperbolic Systems with Dissipative Source

Global weak solutions of a strictly hyperbolic system of balance laws in one-space dimension were constructed (cf. Christoforou, 2006 Christoforou , C. ( 2006 ). Hyperbolic systems of balance laws via vanishing viscosity . J. Diff. Eq. 221 ( 2 ): 470 – 541 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) via the vanishing viscosity method under the assumption that the source term g is dissipative. In this article, we establish sharp estimates on the uniformly Lipschitz semigroup 𝒫 generated by the vanishing viscosity limit in the general case which includes also nonconservative systems. Furthermore, we prove uniqueness of solutions by means of local integral estimates and show that every “viscosity solution” can be constructed as a limit of vanishing viscosity approximations.     

Entire functions that share a set with their derivatives

Let f be a nonconstant entire function and let S = {a, b, c}, where a, b and c are distinct complex numbers. If E(S, f) = E(S, f′), then either (i) f = Ce ^z ; or (ii) ; or (iii) , where C is a nonzero constant. Received: 22 May 2007     

The Aharonov-Bohm Solenoids in a Constant Magnetic Field

We study the spectral properties of a two-dimensional magnetic Schrödinger operator The magnetic field is given by where B > 0 is a constant, and the points are uniformly separated. We give an upper bound for the number of eigenvalues of H_N between two Landau levels or below the lowest Landau level, when N is finite. We prove the spectral localization of H_N near the spectrum of the single solenoid operator, when are far from each other, all the values are the same, and the boundary conditions at z_j are uniform. We determine the deficiency indices of the minimal operator and give a characterization of self-adjoint extensions of the minimal operator.submitted 28/05/04, accepted 23/07/04     

Analytic first integrals of the FitzHugh – Nagumo systems

We study the analytical integrability of the FitzHugh–Nagumo systems in with parameters      

Global Holder Continuous Solutions of Nonstrictly Hyperbolic Systems

This paper considers the Cauchy problem for nonstrictly hyperbolic systems u_t + \frac{1}{2}(au² + v²)_x = 0, v_t + (uv)_x = 0 and gives the Hölder continuous solutions under some stronger restrictions of data by applying the method of vanishing viscosity, where a > 2 is a constant.     

A method for solving d.c. programming problems. Application to fuel mixture nonconvex optimization problem

We present a numerical method for solving the d.c. programming problem

Numerical bifurcation and stability analysis for steady-states of reaction diffusion equations

Solutions of a semilinear elliptic boundary value problem, (with bounded below) can be put into a one-to-one correspondence with zeros of a function . Often d is small. The function is called the bifurcation function. It can also be shown that the eigenvalues of the matrix characterize the stability properties of the solutions of the elliptic problem as rest points of . A finite element method that can be used for computing B and B _c has recently been proposed. An overview of these results and the finite element method is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Optimal Value Function of a Linearly Perturbed Quadratic Program

The optimal value function of the quadratic program , where is a given symmetric matrix, a given matrix, and are the linear perturbations, is considered. It is proved that is directionally differentiable at any point in its effective domain . Formulae for computing the directional derivative of at in a direction are obtained. We also present an example showing that, in general, is not piecewise linear-quadratic on W. The preceding (unpublished) example of Klatte is also discussed.     

Some boundary-value problems for linear multidimensional second-order hyperbolic equations

For the linear hyperbolic equations