Related papers: Computing Arakelov class groups
Based on the multiplicative rule of Schubert classes obtained in [Du3], we present an algorithm computing the product of two arbitrary Schubert classes. As a result, the algorithm gives also a method to compute the integral cohomology ring…
An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest…
We give a new proof of the known Shunkov's Theorem on locally finite groups with the minimal condition for nonabelian subgroups and also an extension of the known Suchkova-Shunkov Theorem on Shunkov groups with the minimal condition for…
Although Buchberger's algorithm, in theory, allows us to compute Gr\"obner bases over any field, in practice, however, the computational efficiency depends on the arithmetic of the ground field. Consider a field $K = \mathbb{Q}(\alpha)$, a…
We study recursive cubes of rings as models for interconnection networks. We first redefine each of them as a Cayley graph on the semidirect product of an elementary abelian group by a cyclic group in order to facilitate the study of them…
We introduce and study kernel algebras, i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting…
We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we…
We generalize Hrushovski's Group Configuration Theorem to quasiminimal classes. As an application, we present Zariski-like structures, a generalization of Zariski geometries, and show that a group can be found there if the pregeometry…
In this paper we present the first-ever computer formalization of the theory of Gr\"obner bases in reduction rings, which is an important theory in computational commutative algebra, in Theorema. Not only the formalization, but also the…
Boolean calculus has been studied extensively in the past in the context of switching circuits, error-correcting codes etc. This work generalizes several approaches to defining a differential calculus for Boolean functions. A unified theory…
A brief review of the construction and classifiaction of the bicovariant differential calculi on quantum groups is given.
We present a convenient trick for computing the sizes of clusters within a network. The rationale relies on the mathematics of the geometric series and the fundamental matrix of a Markov Chain.
We discuss an algorithm computing the push-forward to projective space of several classes associated to a (possibly singular, reducible, nonreduced) projective scheme. For example, the algorithm yields the topological Euler characteristic…
We formulate generalizations of Pauli's theorem on the cases of real and complex Clifford algebras of even and odd dimensions. We give analogues of these theorems in matrix formalism. Using these theorems we present an algorithm for…
We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we…
We first introduce global arithmetic cohomology groups for quasi-coherent sheaves on arithmetic varieties, adopting an adelic approach. Then, we establish fundamental properties, such as topological duality and inductive long exact…
We describe a group theoretic analysis of Shor's algorithm and other related hidden subgroup problems in mathematics and relate these to symmetries of molecular and condensed phase assemblies. By recasting Shor's algorithm through the lens…
Covariant first order differential calculus over quantum complex Grassmann manifolds is considered. It is shown by a Pusz-Woronowicz type argument that under restriction to calculi close to classical Kaehler differentials there exist…
We study equivariant Arakelov-Euler characteristics of hermitian sheaves on arithmetic varieties which support a tame action by a finite group G. The tameness of the group action allows us to produce an equivariant Arakelov-Euler…
In his thesis, N. Durov develops a theory of algebraic geometry in which schemes are locally determined by commutative algebraic monads. In this setting, one is able to construct the Arakelov geometric compactification of the spectrum of…