Quadratic Convexity and Sums of Squares Martin Ames Harrison The length of a sum of squares σ in a ring R is the smallest natural k such that σ can be realized as a sum of k squares in R. For a set S ⊆ R, the pythagoras number of S, denoted by P(S), is the maximum value of length over all σ ∈ S. This dissertation is motivated by the following simple question: if R = R[x1, . . . , xn] and S = R[...

There are many good references for this material including [EKM], [L], [Pf] and [S]. 1 Quadratic forms Let k be a field with char k = 2. Definition 1.1. A quadratic form q : V → k on a finite-dimensional vector space V over k is a map satisfying: 1. q(λv) = λ 2 q(v) for v ∈ V , λ ∈ k. 2. The map b q : V × V → k, defined by b q (v, w) = 1 2 [q(v + w) − q(v) − q(w)] is bilinear. We denote a quadr...

Let V be a renk N vector bundle on a d-dimensional complex projective scheme X; assume that V is equipped with a quadratic form with values in a line bundle L and that SV ∗ ⊗ L is ample. Suppose that the maximum rank of the quadratic form at any point of X is r > 0. The main result of this paper is that if d > N − r, the locus of points where the rank of the quadratic form is at most r − 1 is n...

Abstract In this article, we prove that every quadratic rational map whose multipliers all lie in the ring of integers a given imaginary field is power map, Chebyshev or Lattès map. particular, provides some evidence support conjecture by Milnor concerning maps have an integer multiplier at each cycle.

This paper gives some criteria for the existence and the non existence of chaotic attractors in the general 2-D quadratic map. 2000 Mathematics Subject Classification: 37C05

Universal regularities of the Fourier spectrum of signal, generated by complex analytic map at the period-tripling bifurcations accumulation point are considered. The difference between intensities of the subharmonics at the values of frequency corresponding to the neighbor hierarchical levels of the spectrum is characterized by a constant γ = 21.9 dB?, which is an analogue of the known value γ...

We study bifurcations of a three-dimensional diffeomorphism, g 0 , that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers (λe iϕ , λe −iϕ , γ), where 0 < λ < 1 < |γ| and |λ 2 γ| = 1. We show that in a three-parameter family, g ε , of dif-feomorphisms close to g 0 , there exist infinitely many open regions near ε = 0 where the corresponding normal form of the fir...

Using the symplectic tomography map, both for the probability distributions in classical phase space and for the Wigner functions of its quantum counterpart, we discuss a notion of Lyapunov exponent for quantum dynamics. Because the marginal distributions, obtained by the tomography map, are always well defined probabilities, the correspondence between classical and quantum notions is very clea...

We establish basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and explain consequences for geometric complexity theory. This includes quadratic set-theoretic equations, a description of the ideal in terms of the kernel of a linear map that generalizes the Foulkes-Howe map, and an explicit description of the coordinate ring of the normalization. We ...

Correlation dimension is attained from the correlations between random points on the attractor. For an one dimensional discrete dynamical system with initial condition, x0 Lyapunov exponent (x0) is defined. The Henon map is a discrete time dynamical system. In logistic map is quadratic. The Kaplan-Yorke map is a discrete -time dynamical system. A nnr ( ) subroutine is written to find all mean n...