Related papers: Complexity and Algorithms for Euler Characteristic…
We survey work on the paradigm called "computing by observing." Its central feature is that one considers the behavior of an evolving system as the result of a computation. To this end an observer records this behavior. It has turned out…
We address the question of computing one selected term of an algebraic power series. In characteristic zero, the best algorithm currently known for computing the $N$th coefficient of an algebraic series uses differential equations and has…
A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic r\'egime when such singularities are very far from the real domain; this allows an exact…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
We introduce a definition of algorithmic symmetry able to capture essential aspects of geometric symmetry. We review, study and apply a method for approximating the algorithmic complexity (also known as Kolmogorov-Chaitin complexity) of…
This paper introduces the theory and hardware implementation of two new algorithms for computing a single component of the discrete Fourier transform. In terms of multiplicative complexity, both algorithms are more efficient, in general,…
We give a diagrammatic summary of the connections between various theorems and conjectures about the vanishing of the Euler characteristic.
We present an algorithm that computes Friedl and L\"uck's twisted $L^2$-Euler characteristic for a suitable regular CW complex, employing Oki's matrix expansion algorithm to indirectly evaluate the Dieudonn\'e determinant. The algorithm…
Simple Hamiltonian systems, such as mathematical pendulum or Euler equations for rigid body, are solved without computation. It is nothing but a joke but maybe you will find it nice.
A new method of algebraic nature is proposed for the study of the asymptotic properties of special polynomials. The technique we foresee is based on the use of umbral operators, allowing a unified treatment of a large body of polynomial…
We compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion…
Two of the pillars of combinatorics are the notion of choosing an arbitrary subset of a set with $n$ elements (which can be done in $2^n$ ways), and the notion of choosing a $k$-element subset of a set with $n$ elements (which can be done…
Decomposing an Eulerian graph into a minimum respectively maximum number of edge disjoint cycles is an NP-complete problem. We prove that an Eulerian graph decomposes into a unique number of cycles if and only if it does not contain two…
In this paper we give a framework for describing how abstract systems can be used to compute if no randomness or error is involved. Using this we describe a class of classical "physical" computation systems whose computational capabilities…
Higher-dimensional orthogonal packing problems have a wide range of practical applications, including packing, cutting, and scheduling. Previous efforts for exact algorithms have been unable to avoid structural problems that appear for…
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer…
We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring N_*(X) of a topological space X. This homology theory Eh_* has coefficients Z/2 in every nonnegative…
We introduce a new combinatorial abstraction for the graphs of polyhedra. The new abstraction is a flexible framework defined by combinatorial properties, with each collection of properties taken providing a variant for studying the…
We stratify families of projective and very affine hypersurfaces according to their topological Euler characteristic. Our new algorithms compute all strata using algebro-geometric techniques. For very affine hypersurfaces, we investigate…
We study the Euler characteristic of the real Milnor fibres of a real analytic map, using a relation between complex monodromy and complex conjugation. We deduce the result of Coste and Kurdyka that the Euler characteristic of the link of…