The combinatorial power of the companion matrix

This paper's goal is to determine which complete theories of perfect, e-free Ax fields are model-complete. A field K is e-free for a positive integer e if the Galois group g(KS∣K), where Ks is the separable closure of K, is an e-free, profinite group. A perfect field K is pseudo-algebraically closed if each nonvoid, absolutely irreducible variety defined over K has a K-rational point. A perfect, pseudo-algebraically closed field is called an Ax field. The main theorem isA complete theory of e-free Ax fields is model-complete if and only if its field of absolute numbers is e-free.The sufficiency of the latter condition is an easy consequence of a result of Moshe Jarden and Ursel Kiehne [10] and has been noted independently by A. Macintyre and K. McKenna and undoubtedly by others as well. Consequently the necessity of the latter condition is the interesting part of this paper.James Ax [3] initiated the investigation of 1-free Ax fields. He proved that these fields, which he called pseudo-finite fields, are precisely the infinite models of the theory of finite fields. He [3] also presented examples of perfect, 1-free fields which are not pseudo-algebraically closed and an example of a 1-free Ax field whose complete theory is not model-complete. Moshe Jarden [5] showed that the first examples are isolated cases in that almost all, perfect, 1-free, algebraic extensions of a denumerable, Hilbertian field are pseudo-algebraically closed. The results in this paper show that the second example is also an isolated case in that almost all complete theories of 1-free Ax fields are model-complete.

Cellular automata and finite fields

The matrices corresponding to the Walsh transforms (see chapter 23on page 459) are special cases of Hadamard matrices. Such matrices also exist for certain sizes N × N for N not a power of 2. We give construction schemes for Hadamard matrices that come from the theory of finite fields.

A new efficient hierarchical multi-secret sharing scheme based on linear homogeneous recurrence relations

In the following chapter we address the techniques for the resolution of some celebrated recurrence relations. We will discuss in detail the linear recurrences with constant coefficients. Our emphasis goes to the application of the theory: the proofs, though elementary, are relegated to the end of the chapter. We proceed step by step in showing first how to solve just homogeneous recurrences, then how to find a particular solution in some special cases and only finally how to obtain all the solutions to the original problem. We also consider linear recurrences with variable coefficients and the divide and conquer recurrences: here we focus on the order of magnitude of the solutions, a fact which has an impact in the analysis of algorithms. There are about 40 examples and 50 classified problems.

Abstract. Let p(z) be a complex polynomial of degree n. To each complex number a we associate a sequence called the Basic Sequence {Bm(a) = a − p(a)Dm−2(a)/Dm−1(a)}, where Dm(a) is defined via a homogeneous linear recurrence relation and depends only on the normalized derivatives p (i) (a)/i!. Each Dm(a) is also representable as a Toeplitz determinant. Except possibly for the locus of points equidistant to two distinct roots, given any input a, the Basic Sequence converges to a root of p. The roots of p partition the Euclidean plane into Voronoi regions. Under some regularity assumption (e.g. simplicity of the roots), for almost all inputs within the Voronoi polygon of a root, the corresponding Basic Sequence converges to that root. The discovery of the Basic Sequence, its error estimates, and several of its properties are consequences of our previous analysis of a fundamental family of iteration functions {Bm(z)}, called the Basic Family. Given any fixed m ≥ 2 and an appropriate input a0, the fixed-point iterates ak+1 = Bm(ak) converge to a simple root having order m. The Basic Sequence corresponds to the pointwise evaluation of the Basic Family. We present several fractal images that confirm the theoretical convergence results: as m increases, the basins of attractions to the roots, as computed with respect to the iteration function Bm(z), rapidly converge to the Voronoi regions. Thus the regions with chaotic behavior rapidly shrink to the boundaries of the Voronoi regions. The interplay between the Basic Sequence and the Basic Family result in new algorithms for approximation of polynomial roots. Also as a byproduct of the analysis we give determinantal representation of solution of homogeneous linear recurrence relations, easily computable bounds on the modulus of terms of such recurrence relations, and fast computation of determinant of certain Toeplitz matrices. Finally, we prove a relationship between the Basic Sequence and the iterates of Bernoulli Method for approximation of extreme roots.

In an earlier paper [1] we show that if a family of oriented links obey an iterative pattern, their HOMFLY polynomials satisfy a homogeneous linear recurrence relation. This also holds for Kauffman polynomials. Here we prove the result using a different construction which in some cases yields a recurrence relation of much lower order, thereby drastically reducing the amount of computation required to obtain explicit polynomials for individual members or a general formula for the entire family.

Recursive families of graphs

Recent work in perturbative quantum field theory has led to much study of the Connes-Kreimer Hopf algebra. Its (graded) dual, the Grossman-Larson Hopf algebra of rooted trees, had already been studied by algebraists. L. Foissy introduced a noncommutative version of the Connes-Kreimer Hopf algebra, which turns out to be self-dual. Using some homomorphisms defined by the author and W. Zhao, we describe a commutative diagram that relates the aforementioned Hopf algebras to each other and to the Hopf algebras of symmetric functions, noncommutative symmetric functions, and quasi-symmetric functions.

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

Weighted First-Order Model Counting (WFOMC) computes the weighted sum of the models of a first-order logic theory on a given finite domain. First-Order Logic theories that admit polynomial-time WFOMC w.r.t domain cardinality are called domain liftable. We introduce the concept of lifted interpretations as a tool for formulating closed forms for WFOMC. Using lifted interpretations, we reconstruct the closed-form formula for polynomial-time FOMC in the universally quantified fragment of FO2, earlier proposed by Beame et al. We then expand this closed-form to incorporate cardinality constraints, existential quantifiers, and counting quantifiers (a.k.a C2) without losing domain-liftability. Finally, we show that the obtained closed-form motivates a natural definition of a family of weight functions strictly larger than symmetric weight functions.

In this paper, we study the problem of transforming to normality. We propose to estimate the transformation parameter by minimizing a weighted squared distance between the empirical characteristic function of transformed data and the characteristic function of the normal distribution. Our approach also allows for other symmetric target characteristic functions. Asymptotics are established for a random sample selected from an unknown distribution. The proofs show that the weight function <TEX>$t^{-2}$</TEX> needs to be modified to have thinner tails. We also propose the method to compute the influence function for M-equation taking the form of U-statistics. The influence function calculations and a small Monte Carlo simulation show that our estimates are less sensitive to a few outliers than the maximum likelihood estimates.

We study one-loop corrections to retarded and symmetric hydrostatic correlation functions within the Schwinger-Keldysh effective field theory framework for relativistic hydrodynamics, focusing on charge diffusion. We first consider the simplified setup with only diffusive charge density fluctuations, and then augment it with momentum fluctuations in a model where the sound modes can be ignored. We show that the loop corrections, which generically induce non-analyticities and long-range effects at finite frequency, non-trivially preserve analyticity of retarded correlation functions in spatial momentum due to the KMS constraint, as a manifestation of thermal screening. For the purposes of this analysis, we develop an interacting field theory for diffusive hydrodynamics, seen as a limit of relativistic hydrodynamics in the absence of temperature and longitudinal velocity fluctuations.

The aim of this paper is to investigate the behavior of quotients of sequences w n defined by the linear homogeneous recurrence relation $$w_{n + 2} = pw_{n + 1} - qw_n$$ (1) with arbitrary initial values w 0 = a, w 1 = b belonging to a normed field K, and p, q ∈ K. We shall write w n (a,b; p,q) when it swill be necessary to specify the initial data and parameters. In [3], Horadam has studied the properties of these sequences. If w n ≠ 0, ∀n ∈ N, then the sequence of quotien1ts $$Q_n (a,b;p,q) = \frac{{w_{n + 1} }} {{w_n }}$$ (2.1) is defined in K. If the field K is normed and α, β are the roots of the polynomial x 2 – px + q, with ∥ α ∥ > ∥ β ∥ then the sequences of Q n ’s, where it is defined, converges, and $$ \mathop{{\lim }}\limits_{{n \to \infty }} {Q_n} = \alpha. $$ (3)

In the present paper, we study triangular numbers. We focus on the linear homogeneous recurrence relation of degree 3 with constant coefficients for triangular numbers. Then we deal with the relationship between generalized Fibonacci polynomials and triangular numbers. We show that different properties of triangular numbers can be obtained by using this relationship. Finally, we examine the properties of the sequence A052529 that has strong relationships with triangular numbers.