Milnor numbers, spanning trees, and the Alexander–Conway polynomial

We study relations between the Alexander–Conway polynomial _L and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of _L of an m-component link L all of whose Milnor numbers μ_i1…ip vanish for pn. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spanning trees. Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods.     

Asymptotic convergence of degree‐raising

It is well known that the degree‐raised Bernstein–Bézier coefficients of degree n of a polynomial g converge to g at the rate 1/n. In this paper we consider the polynomial A _n(g) of degree ⩼ n interpolating the coefficients. We show how A _n can be viewed as an inverse to the Bernstein polynomial operator and that the derivatives A _n(g)(^r) converge uniformly to g(^r) at the rate 1/n for all r. We also give an asymptotic expansion of Voronovskaya type for A _n(g) and discuss some shape preserving properties of this polynomial. This revised version was published online in June 2006 with corrections to the Cover Date.     

An extremal property of Hermite polynomials

Let H_n be the nth Hermite polynomial, i.e., the nth orthogonal on polynomial with respect to the weight w(x)=exp(−x²). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||H_n| at the zeros of H_n+1, then for k=1,…,n we have f^(k)H_n^(k), where · is the norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the norm, and estimates for the expansion coefficients in the basis of Hermite polynomials.     

Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R 2

Let E\subset \Bbb R ^s be compact and let d _n ^E denote the dimension of the space of polynomials of degree at most n in s variables restricted to E . We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists , describes the asymptotic behavior of any scheme τ _n ={ \bf x _k,n } _k=1 ^dnE , n=1,2,\ldots , of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n . We demonstrate the existence of AIMs for the finite union of compact subsets of certain algebraic curves in R ² . It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. Furthermore, for the sets mentioned above, we give a computationally simple construction for ``good' interpolation schemes. November 9, 2000. Date revised: August 4, 2001. Date accepted: September 14, 2001.     

On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q _n (x)} _n=0 ^∞, define another polynomial system by where α _n are complex numbers and t is a positive integer. We find conditions for {P _n (x)} _n=0 ^∞ to be an orthogonal polynomial system. When t=1 and α₁≠0, it turns out that {Q _n (x)} _n=0 ^∞ must be kernel polynomials for {P _n (x)} _n=0 ^∞ for which we study, in detail, the location of zeros and semi-classical character. Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001     

Hermite—Fejér Interpolation for Rational Systems

This paper considers Hermite—Fejér and Grünwald interpolation based on the zeros of the Chebyshev polynomials for the real rational system P _n (a ₁ , . . . , a _n ) with the nonreal poles in {a}ⁿ _k=1 C\[-1,1] paired by complex conjugation. This extends some well-known results of Fejér and Grünwald for the classical polynomial case. July 11, 1996. Dates revised: January 6, 1997 and July 30, 1997.     

On Singular Artin Monoids and Contributions to Birman's Conjecture

ABSTRACT

Baez and Birman introduced the singular braid monoid on n + 1 strings, 𝒮B _n+1, which Birman uses in understanding knot invariants. 𝒮? _n+1 is the type A _n case of an infinite class of monoids known as singular Artin monoids and denoted by 𝒮G _M for a Coxeter matrix M. Birman conjectured, and Paris proved, that 𝒮B _n+1 embeds in the complex algebra of the braid group under the desingularisation map or Vassiliev homomorphism, η. In effect, Birman's conjecture generalizes to arbitrary types since, as noted by Corran, the Vassiliev homomorphism from 𝒮G _M to the algebra of the corresponding Artin group is well defined. We deduce general combinatorial results regarding divisibility in positive singular Artin monoids, and when M is of finite type, a well-defined positive form for 𝒮G _M is produced. These facts are then invoked to infer that, when M is of finite type, η is injective on pairs of words such that a common multiple exists for their positive form.     

ANEW RELATION-COMBINING THEOREM AND ITS APPLICATION

Let ?ⁿ denote the set of all formulas ?x₁…?x_n[P(x₁, …,x_n) = 0], where P is a polynomial with integer coefficients. We prove a new relation-combining theorem from which it follows that if ?ⁿ is undecidable over N, then ?²ⁿ⁺² is undecidable over Z.     

Vassiliev invariants and finite-dimensional approximations of the euler equation in magnetohydrodynamics

We consider Hamiltonian systems that correspond to Vassiliev invariants defined by Chen’s iterated integrals of logarithmic differential forms. We show that Hamiltonian systems generated by first-order Vassiliev invariants are related to the classical problem of motion of vortices on the plane. Using second-order Vassiliev invariants, we construct perturbations of Hamiltonian systems for the classical problem of n vortices on the plane. We study some dynamical properties of these systems.     

Markov-type Inequalities for Surface Gradients of Multivariate Polynomials

Let K ^d be a compact set with a smooth boundary and consider a polynomial p of total degree n such that p_C(K)1. Then we show that D_Tp(x)=o(n²) for any x Bd K and T a tangential direction at x. Moreover, the o(n²) term is given in terms of the modulus of smoothness of Bd K.     

Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies

For a compact set K\subset R ^d with nonempty interior, the Markov constants M _n (K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K . It is known that for convex, symmetric bodies M _n (K) = n ² /r(K) , where r(K) is the ``half-width' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B ^d (\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1] ^d the extremal polynomials are essentially unique. September 9, 1999. Date revised: September 28, 2000. Date accepted: November 14, 2000.     

On the Convergence and Iterates of q-Bernstein Polynomials

The convergence properties of q-Bernstein polynomials are investigated. When q1 is fixed the generalized Bernstein polynomials _nf of f, a one parameter family of Bernstein polynomials, converge to f as n→∞ if f is a polynomial. It is proved that, if the parameter 0<q<1 is fixed, then _nf→f if and only if f is linear. The iterates of _nf are also considered. It is shown that _n^Mf converges to the linear interpolating polynomial for f at the endpoints of [0,1], for any fixed q>0, as the number of iterates M→∞. Moreover, the iterates of the Boolean sum of _nf converge to the interpolating polynomial for f at n+1 geometrically spaced nodes on [0,1].     

Semigroup varieties with a small number of term operations

We give an equational characterization of (varieties of) semigroups having a p_n-sequence bounded above by a polynomial function of n. This is achieved by studying the syntactical connections between certain semigroup identities and their equational consequences.     

Separations of first and second order theories in bounded arithmetic

We prove that PTC_N(n) (the polynomial time closure of the nonstandard natural number n in the model N of S₂.) cannot be a model of U¹₂. This implies that there exists a first order sentence of bounded arithmetic which is provable in U¹₂ but does not hold in PTC_N(n).     

Les invariants rationnels de type fini ne distinguent pas les nœuds dans SS2×SS1

We introduce geometric sequences of knots and establish the following criterion: if v is a rational invariant of degree ≤m in the sense of Vassiliev, then v is a polynomial of degree ≤m on every geometric sequence of knots. The torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: we construct knots in SS²×SS¹ which cannot be distinguished by rational invariants of finite type. They can, however, be distinguished by invariants of finite type with values in a finite abelian group.     

Deterministic approximation algorithms for sphere constrained homogeneous polynomial optimization problems

Due to their fundamental nature and numerous applications, sphere constrained polynomial optimization problems have received a lot of attention lately. In this paper, we consider three such problems: (i) maximizing a homogeneous polynomial over the sphere; (ii) maximizing a multilinear form over a Cartesian product of spheres; and (iii) maximizing a multiquadratic form over a Cartesian product of spheres. Since these problems are generally intractable, our focus is on designing polynomial-time approximation algorithms for them. By reducing the above problems to that of determining the L ₂-diameters of certain convex bodies, we show that they can all be approximated to within a factor of Ω((log n/n) ^d/2–1) deterministically, where n is the number of variables and d is the degree of the polynomial. This improves upon the currently best known approximation bound of Ω((1/n) ^d/2–1) in the literature. We believe that our approach will find further applications in the design of approximation algorithms for polynomial optimization problems with provable guarantees.     

On approximation properties of Balázs-Szabados operators their Kantorovich extension

In this paper we deal with a sequence of positive linear operatorsR _n^[β] approximating functions on the unbounded interval [0, t8) which were firstly used by K. Balázs and J. Szabados. We give pointwise estimates in the framework of polynomial weighted function spaces. Also we establish a Voronovskaja type theorem in the same weighted spaces for K_n^[β] operators, representing the integral generalization in Kantorovich sense of the R_n^[β].     

Smoothing noisy data with spline functions

Summary A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m–1 fitted ton distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires orderm ² n operations and therefore permits efficient orderm ² n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires ordern ³ operations.     

On the Divergence of Polynomial Interpolation in the Complex Plane

We extend the results in [1] and [2] from the divergence of Hermite—Fejér interpolation in the complex plane to the divergence of arbitrary polynomial interpolation in the complex plane. In particular, we prove the following theorem: Let \D _n =-1≤ t ₁ ⁽ⁿ⁾ <⋅s<t _n ⁽ⁿ⁾ <1 . Let \v _k ⁽ⁿ⁾ be polynomials of arbitrary degree such that \v _k ⁽ⁿ⁾ (t _j ⁽ⁿ⁾ )=\d _kj . Then the Lebesgue function tends to infinity at every complex neighborhood of some point in [-1,1] . March 23, 2000. Date revised: September 28, 2000. Date accepted: October 10, 2000.