Special bases for solutions of a generalized Maxwell system in 3‐dimensional space

The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L² on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2.     

Newton's method for overdetermined systems of equations

Complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors. In this paper we consider overdetermined systems where there are more equations than unknowns. We study Newton's method for such a system.

     

On the calculation of the Kauffman bracket polynomial

In this paper, we present a new algorithm to evaluate the Kauffman bracket polynomial. The algorithm uses cyclic permutations to count the number of states obtained by the application of ‘A’ and ‘B’ type smoothings to the each crossing of the knot. We show that our algorithm can be implemented easily by computer programming.     

Decomposition and approximation of multivariate functions on the cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jφjψj , where each φj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each φjψj is a product of separated variable type and its smoothness is same as f . Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.     

信号处理中一类非线性方程组的快速求解

在声纳和雷达信号处理中,需要求解一类维数可变的非线性方程组,这类方程组具有混合三角多项式方程组形式.由于该问题有很多解,且其对应的最小二乘问题有很多局部极小点,用牛顿法等传统的迭代法很难找到有物理意义的解.若把它化为多项式方程组,再用解多项式方程组的符号计算方法或现有的同伦方法求解,由于该问题规模太大而不能在规定的时间内求解,而当考虑的问题维数较大时,利用已有的方法甚至根本无法求解.综合利用我们提出的解混合三角多项式方程组的混合同伦方法和保对称的系数参数同伦方法,我们给出该类问题一种有效的求解方法.利用这种方法,可以达到实时求解的目的,满足实际问题的需要.     

Dynamic frictionless contact in linear viscoelasticity

David E. Stewart Department of Mathematics, University of Iowa, Iowa, IA 52242, USA ^{In this work, we formulate a dynamic frictionless contact problemwith linear viscoelasticity of Kelvin–Voigt type, basedon the Signorini contact conditions. We show existence of solutions,and investigate the possibility for obtaining an energy balance.Employing time discretization and the finite-element method,we compute numerical solutions. Our numerical scheme is implementedwith non-smooth Newton's method which solves the complementarityproblem. The numerical results support the idea that the energylosses in the limit of the numerical solution are equal to thelosses due to viscosity.}     

Newton–Hensel Interpolation Lifting

The main result of this paper is a new version of Newton-Hensel lifting that relates to interpolation questions. It allows one to lift polynomials in ℤ[x] from information modulo a prime number p ≠ 2 to a power p^k for any k, and its originality is that it is a mixed version that not only lifts the coefficients of the polynomial but also its exponents. We show that this result corresponds exactly to a Newton--Hensel lifting of a system of 2t generalized equations in 2t unknowns in the ring of p-adic integers ℤ_p. Finally, we apply our results to sparse polynomial interpolation in ℤ[x].     

The Dirichlet problem on quadratic surfaces

We give a fast, exact algorithm for solving Dirichlet problems with polynomial boundary functions on quadratic surfaces in such as ellipsoids, elliptic cylinders, and paraboloids. To produce this algorithm, first we show that every polynomial in can be uniquely written as the sum of a harmonic function and a polynomial multiple of a quadratic function, thus extending a theorem of Ernst Fischer. We then use this decomposition to reduce the Dirichlet problem to a manageable system of linear equations. The algorithm requires differentiation of the boundary function, but no integration. We also show that the polynomial solution produced by our algorithm is the unique polynomial solution, even on unbounded domains such as elliptic cylinders and paraboloids.

     

A recursive algorithm for constructing complicated Dixon matrices

Whether the determinant of the Dixon matrix equals zero or not is used for determining if a system of n + 1 polynomial equations in n variables has a common root, and is a very efficient quantifier elimination approach too. But for a complicated polynomial system, it is not easy to construct the Dixon matrix. In this paper, a recursive algorithm to construct the Dixon matrix is proposed by which some problems that cannot be tackled by other methods can be solved on the same computer platform. A dynamic programming algorithm based on the recursive formula is developed and compared for speed and efficiency to the recursive algorithm.     

Global convergence of SSM for minimizing a quadratic over a sphere

In an earlier paper [Minimizing a quadratic over a sphere, SIAM J. Optim., 12 (2001), 188-208], we presented the sequential subspace method (SSM) for minimizing a quadratic over a sphere. This method generates approximations to a minimizer by carrying out the minimization over a sequence of subspaces that are adjusted after each iterate is computed. We showed in this earlier paper that when the subspace contains a vector obtained by applying one step of Newton's method to the first-order optimality system, SSM is locally, quadratically convergent, even when the original problem is degenerate with multiple solutions and with a singular Jacobian in the optimality system. In this paper, we prove (nonlocal) convergence of SSM to a global minimizer whenever each SSM subspace contains the following three vectors: (i) the current iterate, (ii) the gradient of the cost function evaluated at the current iterate, and (iii) an eigenvector associated with the smallest eigenvalue of the cost function Hessian. For nondegenerate problems, the convergence rate is at least linear when vectors (i)-(iii) are included in the SSM subspace.

     

正规升列在参数代数方程组求解中的应用

本文提出一种利用多项式系统的正规零点分解的算法来求解代数方程组以及带有参数的代数方程组的方法．对于给定的的代数方城组,通过正规分解,可以得到一组具有三角形式的分解．根据这种三角形式,我们可以给出代数方程组的所有解．而对于带有参数方程组,将给出方程组有解时参数需满足的条件．进一步,对于给定的参数值,正规分解中得到三角形式仍然保持,通过求解三角形式的方程组从而得出原参数方程组的解．     

Parametrizing Over ℤ Integral Values of Polynomials Over ℚ

Given a polynomial f ∈ ?[X] such that f(?) ? ?, we investigate whether the set f(?) can be parametrized by a multivariate polynomial with integer coefficients, that is, the existence of g ∈ ?[X ₁,…, X _m ] such that f(?) = g(? ^m ). We offer a necessary and sufficient condition on f for this to be possible. In particular, it turns out that some power of 2 is a common denominator of the coefficients of f, and there exists a rational β with odd numerator and odd prime-power denominator such that f(X) = f(β ?X). Moreover, if f(?) is likewise parametrizable, then this can be done by a polynomial in one or two variables.     

Extrema of a Real Polynomial

In this paper, we investigate critical point and extrema structure of a multivariate real polynomial. We classify critical surfaces of a real polynomial f into three classes: repeated, intersected and primal critical surfaces. These different critical surfaces are defined by some essential factors of f, where an essential factor of f means a polynomial factor of f–c ₀, for some constant c ₀. We show that the degree sum of repeated critical surfaces is at most d–1, where d is the degree of f. When a real polynomial f has only two variables, we give the minimum upper bound for the number of other isolated critical points even when there are nondegenerate critical curves, and the minimum upper bound of isolated local extrema even when there are saddle curves. We show that a normal polynomial has no odd degree essential factors, and all of its even degree essential factors are normal polynomials, up to a sign change. We show that if a normal quartic polynomial f has a normal quadratic essential factor, a global minimum of f can be either easily found, or located within the interior(s) of one or two ellipsoids. We also show that a normal quartic polynomial can have at most one local maximum.     

Quadratic Newton Iteration for Systems with Multiplicity

Newton's iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton's iterator is still an open problem: How to design an efficient iterator with a quadratic convergence even in degenerate cases? We propose an answer for an m -adic topology when the ideal m can be chosen generic enough: compared to a smooth case we prove quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.     

A Tau Method with Perturbed Boundary Conditions for Certain Ordinary Differential Equations

Ortiz recursive formulation of the Lanczos Tau method (TM) is a powerful and efficient technique for producing polynomial approximations for initial or boundary value problems. The method consists in obtaining a polynomial which satisfies (i) a perturbed version of the given differential equation, and (ii) the imposed supplementary conditions exactly. This paper introduces a new form of the TM, (denoted by PTM), for a restricted class of differential equations, in which the differential equations as well as the supplementary conditions are perturbed simultaneously. PTM is compared to the classical TM from the point of view of their errors: it is found that the PTM error is smaller and more oscillatory than that of the TM; we further find that approximations nearly as accurate as minimax polynomial approximations can be constructed by means of the PTM. Detailed formulae are derived for the polynomial approximations in TM and PTM, based on Canonical Polynomials. Moreover, various limiting properties of Tau coefficients are established and it is shown that the perturbation in PTM behaves asymptotically proportional to a Chebyshev polynomial.     

Knot adjacency and fibering

It is known that the Alexander polynomial detects fibered knots and 3-manifolds that fiber over the circle. In this note, we show that when the Alexander polynomial becomes inconclusive, the notion of knot adjacency can be used to obtain obstructions to the fibering of knots and of 3-manifolds. As an application, given a fibered knot , we construct infinitely many non-fibered knots that share the same Alexander module with . Our construction also provides, for every , examples of irreducible 3-manifolds that cannot be distinguished by the Cochran-Melvin finite type invariants of order .

     

Graph polynomials from principal pivoting

The recursive computation of the interlace polynomial introduced by Arratia, Bollobás and Sorkin is defined in terms of a new pivoting operation on undirected simple graphs. In this paper, we interpret the new pivoting operation on graphs in terms of standard pivoting (on matrices). Specifically, we show that, up to swapping vertex labels, Arratia et al.'s pivoting operation on a graph is equivalent to a principal pivot transform on the graph's adjacency matrix, provided that all computations are performed in the Galois field F₂. Principal pivoting on adjacency matrices over F₂ has a natural counterpart on isotropic systems. Thus, our view of the interlace polynomial is closely related to the one by Aigner and van der Holst.The observations that adjacency matrices of undirected simple graphs are skew-symmetric in F₂ and that principal pivoting preserves skew-symmetry in all fields suggest to extend Arratia et al.'s pivoting operation to fields other than F₂. Thus, the interlace polynomial extends to polynomials on gain graphs, namely bidirected edge-weighted graphs whereby reversed edges carry non-zero weights that differ only by their sign. Extending a proof by Aigner and van der Holst, we show that the extended interlace polynomial can be represented in a non-recursive form analogous to the non-recursive form of the original interlace polynomial, i.e., the Martin polynomial.For infinite fields it is shown that the extended interlace polynomial does not depend on the (non-zero) gains, as long as they obey a non-singularity condition. These gain graphs are all supported by a single undirected simple graph. Thus, a new graph polynomial is defined for undirected simple graphs. The recursive computation of the new polynomial can be done such that all ends of the recursion correspond to independent sets. Moreover, its degree equals the independence number. However, the new graph polynomial is different from the independence polynomial.     

Phase-locking for dynamical systems on the torus and perturbation theory for mathieu-type problems

Summary When several oscillators are coupled together and the parameters of their coupling are varied, the oscillators pass through so-called phase-locked regimes. In physical terms this means that the oscillators tend to synchronize their motion. To describe this phenomenon, we frame the concepts ofpartial phase andphase-locking. A partial phase of a toral flow puts emphasis on how orbits of the flow drift around the torus in some fixed direction. The partial phase is locked if it grows in time along some orbit slower than any linear function. When a toral flow is given by a trigonometric polynomial, its phase-locked regions are quite narrow. With the coupling amplitude increasing, each region grows in width as some power of the amplitude. That power can be calculated in terms of both the partial phase and degree of the trigonometric polynomial.