Symbolic computation with finite biquandles

A method of computing a basis for the second Yang–Baxter cohomology of a finite biquandle with coefficients in Q$\mathbb{Q}$ and _Zp${\mathbb{Z}}_p$ from a matrix presentation of the finite biquandle is described. We also describe a method for computing the Yang–Baxter cocycle invariants of an oriented knot or link represented as a signed Gauss code. We provide a URL for our Maple implementations of these algorithms.     

An extension of Kedlaya’s algorithm for hyperelliptic curves

In this paper we describe a generalisation and adaptation of Kedlaya’s algorithm for computing the zeta-function of a hyperelliptic curve over a finite field of odd characteristic that the author used for the implementation of the algorithm in the Magma library. We generalise the algorithm to the case of an even degree model. We also analyse the adaptation of working with the xⁱdx/y³ rather than the xⁱdx/y differential basis. This basis has the computational advantage of always leading to an integral transformation matrix whereas the latter fails to in small genus cases. There are some theoretical subtleties that arise in the even degree case where the two differential bases actually lead to different redundant eigenvalues that must be discarded.     

Physics inspired algorithms for (co)homology computations of three-dimensional combinatorial manifolds with boundary

The issue of computing (co)homology generators of a cell complex is gaining a pivotal role in various branches of science. While this issue may be rigorously solved in polynomial time, it is still overly demanding for large scale problems. Drawing inspiration from low-frequency electrodynamics, this paper presents a physics inspired algorithm for first cohomology group computations on three-dimensional complexes. The algorithm is general and exhibits orders of magnitude speed up with respect to competing ones, allowing to handle problems not addressable before. In particular, when generators are employed in the physical modeling of magneto-quasistatic problems, this algorithm solves one of the most long-lasting problems in low-frequency computational electromagnetics. In this case, the effectiveness of the algorithm and its ease of implementation may be even improved by introducing the novel concept of lazy cohomology generators.     

Structures of systems 1. Cohomology of manufacturing and supply network-like systems

Understanding the relation between large-scale and small-scale functionality is an acute problem in large systems. This paper presents possible structures that capture those relations. Here “structure” means a functor on a category, in this case the category of “combinatorial systems” that model manufacturing systems and supply networks. The details of the dynamics of these systems are frequently the domain of specialists and are ill understood outside such areas. Clustering activities in subsystems introduces covers as scales of analysis. This allows the development of sheaves of schedules and their properties which are analyzed by an adaptation of ?ech cohomology. These are applied in supply networks to characterize the fragmentation of these networks and to characterize the conditions for local changes in schedules to extend to the whole system. The intent of this paper is to see sheaves and their cohomology as analytical probes that can enhance our understanding of large-scale systems.     

广义路代数及其Hochschild上同调

为了研究quiver △上的A-广义路代数R=k(△,A),基于本原正交幂等元完全集,给出了广义路代数R=k(△,A)的不可分解投射模与内射模以及单模的构造形式。基于遗传代数性质得到了广义路代数是遗传代数的充要条件,并进一步在同调理论和有限维代数的Hochschild上同调的基础上得到了广义路代数的Hochschild上同调。     

Annihilating ideals for an algebraic local cohomology class

Hypersurface isolated singularities are considered in the context of algebraic analysis. A method for computing relative ?ech cohomology representations of algebraic local cohomology classes supported at the isolated singular point is described. An effective method to test membership and a method to compute the normal form for the Jacobi ideal are presented. The main purpose of this paper is to provide an effective algorithm for computing annihilating ideals, in the ring of partial differential operators, of the algebraic local cohomology class that generates the dual vector space to the local Milnor algebra.     

Formal Normal Forms for the Perturbations of a Quasi-Homogeneous Hamiltonian Vector Field

We classify up to a formal change of variable the vector fields of two complex variables together with the foliations that they de.ne which are a perturbation of a quasi-homogeneous Hamiltonian vector field X₀by terms of higher degree of quasi-homogeneity. We do not require anything about the degree ₀of the initial vector field X₀, but we assume that the perturbed vector field still keeps invariant the separatrices of X₀. We obtain formal normal forms which extend those obtained in the case of an initial vector field with a semisimple or nilpotent linear part. We give an interpretation of the dual version of these normal forms through the relative cohomology with respect to the dual initial di.erential form ₀.     

Structures in systems 2: viewpoint cohomology of general systems

This paper introduces new homology and cohomology functors designed to analyze a broad range of logical systems including representations of enterprise information systems. Such systems contain nested relations and are a class of higher order logic. The homology and cohomology augment simplex constructions with ideas of type extensions to capture the logic of the system. Here logical systems, including large-scale information systems, have a class of derived objects called “scenarios” that give the connective logic of the system. Scenarios have a “secondary” logic derived from the underlying business rules of the enterprise. A “viewpoint” is a flexible classification logic which creates “proxy” variables so that the (co)homology functors can be applied to the secondary (and higher) scenario logics but with different (but functionally related) variables. These viewpoint (co)homologies can then analyze structures in emergent phenomena that reflect structures in the logic of the original system.     

Quasi-quadratic elliptic curve point counting using rigid cohomology

Let E$E$ be a nonsupersingular elliptic curve over the finite field with pⁿ$p^n$ elements. We present a deterministic algorithm that computes the zeta function and hence the number of points of such a curve E$E$ in time quasi-quadratic in n$n$. An older algorithm having the same time complexity uses the canonical lift of E$E$, whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.     

路代数和路余代数弱entwining结构的Hochschild上同调

为了研究弱entwining结构的性质,利用同调代数方法讨论了路代数和路余代数的弱entwining结构,研究了此弱entwining结构的Hochschild上同调及其性质,得到了此弱entwining结构中的路余代数所在基础图为树的等价条件.