Related papers: The Programming of Algebra
With recent dramatic increases in AI system capabilities, there has been growing interest in utilizing machine learning for reasoning-heavy, quantitative tasks, particularly mathematics. While there are many resources capturing mathematics…
Many recursive functions can be defined elegantly as the unique homomorphisms, between two algebras, two coalgebras, or one each, that are induced by some universal property of a distinguished structure. Besides the well-known applications…
In this paper we study multiserial and special multiserial algebras. These algebras are a natural generalization of biserial and special biserial algebras to algebras of wild representation type. We define a module to be multiserial if its…
Although database systems perform well in data access and manipulation, their relational model hinders data scientists from formulating machine learning algorithms in SQL. Nevertheless, we argue that modern database systems perform well for…
Many problems in robotics require reasoning over a mix of continuous dynamics and discrete events, such as making and breaking contact in manipulation and locomotion. These problems are locally well modeled by linear complementarity…
Multi-model databases are designed to store, manage, and query data in various models, such as relational, hierarchical, and graph data, simultaneously. In this paper, we provide a theoretical basis for querying categorical databases. We…
We reduce the set of classic relational algebra operators to two binary operations: natural join and generalized union. We further demonstrate that this set of operators is relationally complete and honors lattice axioms.
The aim of this paper is to show that even if the natural algebraic semantic for modal (normal) logic is modal algebra, the more general class of subordination algebras (roughly speaking, the non symmetric contact algebras) is adequate too…
The common in ring, module and algebra is that they are Abelian group with respect to addition. This property is enough to study integration. I treat integral of measurable map into normed Abelian $\Omega$-group. Theory of integration of…
For any finite dimensional basic associative algebra, we study the presentation spaces and their relation with the representation spaces. We prove two propositions about a general presentation, one on its subrepresentations and the other on…
The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra $\Lambda$ are explored, with regard to both their geometry and the structure of the modules they encode. Provided…
We characterize completey (give a necessary and suffcient condition using special neat embeddings)for a relation algebra to belong to the amalgamation, strong amalgamation, and superamalgamation base of the class of representable algebras.…
Systems of cooperation and interaction are usually studied in the context of real or complex vector spaces. Additional insight, however, is gained when such systems are represented in vector spaces with multiplicative structures, i.e., in…
The vast quantity of data generated and captured every day has led to a pressing need for tools and processes to organize, analyze and interrelate this data. Automated reasoning and optimization tools with inherent support for data could…
In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module M may be computed in terms of a "maximal" map f from M to…
We propose a new framework that generalizes the parameters of neural network models to $C^*$-algebra-valued ones. $C^*$-algebra is a generalization of the space of complex numbers. A typical example is the space of continuous functions on a…
Representations are essential to mathematically model phenomena, but there are many options available. While each of those options provides useful properties with which to solve problems related to the phenomena in study, comparing results…
We give new positive and negative results (some conditional) on speeding up computational algebraic geometry over the reals: (1) A new and sharper upper bound on the number of connected components of a semialgebraic set. Our bound is novel…
The image of a polynomial map is a constructible set. While computing its closure is standard in computer algebra systems, a procedure for computing the constructible set itself is not. We provide a new algorithm, based on algebro-geometric…
In this paper, we present a generalized version of the matrix chain algorithm to generate efficient code for linear algebra problems, a task for which human experts often invest days or even weeks of works. The standard matrix chain problem…