216 research outputs found
The inverse problem for representation functions for general linear forms
The inverse problem for representation functions takes as input a triple
(X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a
function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A
\subseteq X such that there are f(x) solutions (counted appropriately) to
L(x_1,...,x_h) = x for every x \in X, or a proof that no such subset exists.
This paper represents the first systematic study of this problem for
arbitrary linear forms when X = Z, the setting which in many respects is the
most natural one. Having first settled on the "right" way to count
representations, we prove that every primitive form has a unique representation
basis, i.e.: a set A which represents the function f \equiv 1. We also prove
that a partition regular form (i.e.: one for which no non-empty subset of the
coefficients sums to zero) represents any function f for which {f^{-1}(0)} has
zero asymptotic density. These two results answer questions recently posed by
Nathanson.
The inverse problem for partition irregular forms seems to be more
complicated. The simplest example of such a form is x_1 - x_2, and for this
form we provide some partial results. Several remaining open problems are
discussed.Comment: 15 pages, no figure
Phase transitions in the Ramsey-Turán theory
Let f(n) be a function and L be a graph. Denote by RT(n, L, f(n)) the maximum number of edges of an L-free graph on n vertices with independence number less than f(n). Erdos and Sós asked if RT (n, K5, c√
n) = o (n2) for some constant c. We answer this question by proving the stronger RT(n, K5, o (√n log n)) = o(n2). It is known that RT (n, K5, c√n log n
)= n2/4 + o (n2) for c > 1, so one can say that K5 has a Ramsey-Turán-phase transition at c√n log n. We extend this result to several other Kp's and functions f(n), determining many more phase transitions. We shall formulate
several open problems, in particular, whether variants of the Bollobás-Erdos graph, which is a geometric construction, exist to give good lower bounds
on RT (n, Kp, f(n)) for various pairs of p and f(n). These problems are studied in depth by Balogh-HuSimonovits, where among others, the Szemerédi's Regularity Lemma and the Hypergraph Dependent
Random Choice Lemma are used.National Science Foundatio
Tur\'{a}n's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials
An elegant and fruitful way to bring harmonic analysis into the theory of
orthogonal polynomials and special functions, or to associate certain Banach
algebras with orthogonal polynomials satisfying a specific but frequently
satisfied nonnegative linearization property, is the concept of a polynomial
hypergroup. Polynomial hypergroups (or the underlying polynomials,
respectively) are accompanied by -algebras and a rich, well-developed and
unified harmonic analysis. However, the individual behavior strongly depends on
the underlying polynomials. We study the associated symmetric Pollaczek
polynomials, which are a two-parameter generalization of the ultraspherical
polynomials. Considering the associated -algebras, we will provide
complete characterizations of weak amenability and point amenability by
specifying the corresponding parameter regions. In particular, we shall see
that there is a large parameter region for which none of these amenability
properties holds (which is very different to -algebras of locally compact
groups). Moreover, we will rule out right character amenability. The crucial
underlying nonnegative linearization property will be established, too, which
particularly establishes a conjecture of R. Lasser (1994). Furthermore, we
shall prove Tur\'{a}n's inequality for associated symmetric Pollaczek
polynomials. Our strategy relies on chain sequences, asymptotic behavior,
further Tur\'{a}n type inequalities and transformations into more convenient
orthogonal polynomial systems.Comment: Main changes towards first version: The part on associated symmetric
Pollaczek polynomials was extended (with more emphasis on Tur\'{a}n's
inequality and including a larger parameter region), and the part on little
-Legendre polynomials became a separate paper. We added several references
and corrected a few typos. Title, abstract and MSC class were change
Taylor Domination, Difference Equations, and Bautin Ideals
We compare three approaches to studying the behavior of an analytic function
from its Taylor coefficients. The first is
"Taylor domination" property for in the complex disk , which is an
inequality of the form The second approach is based on a possibility to generate
via recurrence relations. Specifically, we consider linear non-stationary
recurrences of the form with uniformly bounded coefficients.
In the third approach we assume that are polynomials in a
finite-dimensional parameter We study "Bautin
ideals" generated by in the ring
of polynomials in .
\smallskip
These three approaches turn out to be closely related. We present some
results and questions in this direction.Comment: arXiv admin note: substantial text overlap with arXiv:1301.603
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