相关论文: Singularity patterns and dynamical degrees

We describe an algorithm that, given a k-tuple of permutations representing the monodromy of a rational map, constructs an arbitrarily precise floating-point complex approximation of that map. We then explain how it has been used to study a…

Two well studied invariants of a complex projective variety are the unit Euclidean distance degree and the generic Euclidean distance degree. These numbers give a measure of the algebraic complexity for "nearest" point problems of the…

We give a direct and simple proof of Touchard's continued fraction, provide an extension of it, and transform it into similar expansions related to Motzkin and Schroeder numbers. Another proof is then given that uses only induction. We use…

A rather complete phenomenology of the singularities is developed according to a new algebraic point of view in the frame of Langlands global correspondences. That is to say,a process of: -singularizations and versal deformations of these,…

In this note we give an overview of various quantities that are used to measure the complexity of an algebraic dynamical system f:X-->X, including the dynamical degree d(f), which gives a coarse measure of the geometric complexity of the…

The path integral formulation of singular systems with second order Lagrangian is studied by using the canonical path integral method. The path integral of Podolsky electrodynamics is studied.

We discuss the effective computation of geometric singularities of implicit ordinary differential equations over the real numbers using methods from logic. Via the Vessiot theory of differential equations, geometric singularities can be…

We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space…

In this text I present some problems which led to the introduction of special kinds of graphs as tools for studying singular points of algebraic surfaces. I explain how such graphs were first described using words, and how several…

A set of algorithms is presented for efficient numerical calculation of the time evolution of classical dynamical systems. Starting with a first approximation for solving the differential equations that has a "reversible" character, we show…

In this short note, we provide an elementary complex analytic method for converting known real integrals into numerous strange and interesting looking real integrals.

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate…

We give a definition of integration by quadratures of first-order ordinary differential equations, and recover a little known result by Maximovic which states that a first-order ordinary differential equation can be integrated by…

We present simple examples of rational maps of the complex projective plane with equal first and second dynamical degrees and no invariant foliation.

We study a general model of random dynamical simplicial complexes and derive a formula for the asymptotic degree distribution. This asymptotic formula encompasses results for a number of existing models, including random Apollonian networks…

This work deals with planar dynamical systems with and without noise. In the first part, we seek to gain a refined understanding of such systems by studying their differential-geometric transformation properties under an arbitrary smooth…

We consider the notion of mixed multiplicities for multigraded modules by using Hilbert series, and this is later applied to study the projective degrees of rational maps. We use a general framework to determine the projective degrees of a…

We apply methods of dynamical systems to study the behaviour of the Randall-Sundrum models. We determine evolutionary paths for all possible initial conditions in a 2-dimensional phase space and we investigate the set of accelerated models.…

This work establishes simple criteria for detecting higher rational singularities via the intersection Du Bois complex and the irrationality complex of a normal variety over the complex numbers.

In this paper we prove strong unique continuation principle and unique continuation from sets of positive measure for solutions of a higher order fractional Laplace equation in an open domain. Our proofs are based on the…