Search results for "rational function"
showing 8 items of 18 documents
Invariant Jordan curves of Sierpinski carpet rational maps
2015
In this paper, we prove that if $R\colon\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ is a postcritically finite rational map with Julia set homeomorphic to the Sierpi\'nski carpet, then there is an integer $n_0$, such that, for any $n\ge n_0$, there exists an $R^n$-invariant Jordan curve $\Gamma$ containing the postcritical set of $R$.
Analytic form of the full two-loop five-gluon all-plus helicity amplitude
2019
We compute the full-color two-loop five-gluon amplitude for the all-plus helicity configuration. In order to achieve this, we calculate the required master integrals for all permutations of the external legs, in the physical scattering region. We verify the expected divergence structure of the amplitude, and extract the finite hard function. We further validate our result by checking the factorization properties in the collinear limit. Our result is fully analytic and valid in the physical scattering region. We express it in a compact form containing logarithms, dilogarithms and rational functions.
On the evaluation of sunset-type Feynman diagrams
1999
We introduce an efficient configuration space technique which allows one to compute a class of Feynman diagrams which generalize the scalar sunset topology to any number of massive internal lines. General tensor vertex structures and modifications of the propagators due to particle emission with vanishing momenta can be included with only a little change of the basic technique described for the scalar case. We discuss applications to the computation of $n$-body phase space in $D$-dimensional space-time. Substantial simplifications occur for odd space-time dimensions where the final results can be expressed in closed form through rational functions. We present explicit analytical formulas fo…
Vanishing Abelian integrals on zero-dimensional cycles
2011
In this paper we study conditions for the vanishing of Abelian integrals on families of zero-dimensional cycles. That is, for any rational function $f(z)$, characterize all rational functions $g(z)$ and zero-sum integers $\{n_i\}$ such that the function $t\mapsto\sum n_ig(z_i(t))$ vanishes identically. Here $z_i(t)$ are continuously depending roots of $f(z)-t$. We introduce a notion of (un)balanced cycles. Our main result is an inductive solution of the problem of vanishing of Abelian integrals when $f,g$ are polynomials on a family of zero-dimensional cycles under the assumption that the family of cycles we consider is unbalanced as well as all the cycles encountered in the inductive proce…
The differential Galois group of the rational function field
2020
We determine the absolute differential Galois group of the field $\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem posed by B.H. Matzat. For the proof we develop a new characterization of free proalgebraic groups in terms of split embedding problems, and we use patching techniques in order to solve a very general class of differential embedding problems. Our result about $\mathbb{C}(x)$ also applies to rational function fields over more general fields of coefficients.
A Note on the Density of Rational Functions in A ∞(Ω)
2018
We present a sufficient condition to ensure the density of the set of rational functions with prescribed poles in the algebra A ∞ (Ω).
Analysis of multi degree of freedom systems with fractional derivative elements of rational order
2014
In this paper a novel method based on complex eigenanalysis in the state variables domain is proposed to uncouple the set of rational order fractional differential equations governing the dynamics of multi-degree-of-freedom system. The traditional complex eigenanalysis is appropriately modified to be applicable to the coupled fractional differential equations. This is done by expanding the dimension of the problem and solving the system in the state variable domain. Examples of applications are given pertaining to multi-degree-of-freedom systems under both deterministic and stochastic loads.
Wybrane zagadnienia związane z dydaktyką całkowania funkcji wymiernych
2018
Jednym z rodzajów całek przedstawianych w procesie dydaktycznym są całki wyrażeń wymiernych. Techniki rozwiązywania tego rodzaju całek są wielorakie, niemniej w literaturze i dydaktyce wykorzystywana jest zazwyczaj jedna metoda. Metoda ta jest uniwersalna ale czasochłonna i nie lubiana przez studentów. Poniższy artykuł stanowi kompendium metod umożliwiających rozwiązywanie całek funkcji wymiernych.