17,777 research outputs found
The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications
We completely describe the functional graph associated to iterations of
Chebyshev polynomials over finite fields. Then, we use our structural results
to obtain estimates for the average rho length, average number of connected
components and the expected value for the period and preperiod of iterating
Chebyshev polynomials
Tame Decompositions and Collisions
A univariate polynomial f over a field is decomposable if f = g o h = g(h)
for nonlinear polynomials g and h. It is intuitively clear that the
decomposable polynomials form a small minority among all polynomials over a
finite field. The tame case, where the characteristic p of Fq does not divide n
= deg f, is fairly well-understood, and we have reasonable bounds on the number
of decomposables of degree n. Nevertheless, no exact formula is known if
has more than two prime factors. In order to count the decomposables, one wants
to know, under a suitable normalization, the number of collisions, where
essentially different (g, h) yield the same f. In the tame case, Ritt's Second
Theorem classifies all 2-collisions.
We introduce a normal form for multi-collisions of decompositions of
arbitrary length with exact description of the (non)uniqueness of the
parameters. We obtain an efficiently computable formula for the exact number of
such collisions at degree n over a finite field of characteristic coprime to p.
This leads to an algorithm for the exact number of decomposable polynomials at
degree n over a finite field Fq in the tame case
The Tensor Track, III
We provide an informal up-to-date review of the tensor track approach to
quantum gravity. In a long introduction we describe in simple terms the
motivations for this approach. Then the many recent advances are summarized,
with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion,
Osterwalder-Schrader positivity...) which, while important for the tensor track
program, are not detailed in the usual quantum gravity literature. We list open
questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure
Nonlocal, noncommutative diagrammatics and the linked cluster Theorems
Recent developments in quantum chemistry, perturbative quantum field theory,
statistical physics or stochastic differential equations require the
introduction of new families of Feynman-type diagrams. These new families arise
in various ways. In some generalizations of the classical diagrams, the notion
of Feynman propagator is extended to generalized propagators connecting more
than two vertices of the graphs. In some others (introduced in the present
article), the diagrams, associated to noncommuting product of operators inherit
from the noncommutativity of the products extra graphical properties. The
purpose of the present article is to introduce a general way of dealing with
such diagrams. We prove in particular a "universal" linked cluster theorem and
introduce, in the process, a Feynman-type "diagrammatics" that allows to handle
simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams
arising from the study of interacting systems (such as the ones where the
ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or
electric field, by impurities...) or Wightman fields (that is, expectation
values of products of interacting fields). Our diagrammatics seems to be the
first attempt to encode in a unified algebraic framework such a wide variety of
situations. In the process, we promote two ideas. First, Feynman-type
diagrammatics belong mathematically to the theory of linear forms on
combinatorial Hopf algebras. Second, linked cluster-type theorems rely
ultimately on M\"obius inversion on the partition lattice. The two theories
should therefore be introduced and presented accordingl
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