Related papers: ELT Linear Algebra II
We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra…
Slicing a module into semisimple ones is useful to study modules. Loewy structures provide a means of doing so. To establish the Loewy structures of projective modules over a finite dimensional symmetric algebra over a field $F$, the…
In this note we prove an integral formula for the bare one-leg PT vertex with descendents. The formula follows from the PT version of Ellingsrud-G\"ottsche-Lehn formula that is explained here. We apply the integral formula to obtain an…
The aim of this paper is to prove a version of Lie's theorem for the supertropical algebra.
We develop the basic theory of projective modules and splitting in the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically…
Extreme value statistics is the max analogue of classical statistics, while tropical geometry is the max analogue of classical geometry. In this paper, we review recent work where insights from tropical geometry were used to develop new,…
In general, quantum matrix algebras are associated with a couple of compatible braidings. A particular example of such an algebra is the so-called Reflection Equation algebra. In this paper we analyse its specific properties, which…
We introduce and investigate the concept of Stratified Algebra, a new algebraic framework equipped with a layer-based structure on a vector space. We formalize a set of axioms governing intra-layer and inter-layer interactions, study their…
We introduce a framework for expanding residual computational graphs using jets, operators that generalize truncated Taylor series. Our method provides a systematic approach to disentangle contributions of different computational paths to…
Enhancing the adaptive capabilities of large language models is a critical pursuit in both research and application. Traditional fine-tuning methods require substantial data and computational resources, especially for enhancing specific…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
The construction of the Leavitt path algebra associated to a directed graph $E$ is extended to incorporate a family $C$ consisting of partitions of the sets of edges emanating from the vertices of $E$. The new algebras, $L_K(E,C)$, are…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit…
In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in…
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proofs, and topological proofs. Algebraic proofs make use of the…
In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not…
In this paper we introduce the concept of L-algebras, which can be seen as a generalization of the structure determined by the Eilenberg-Mac lane transformation and Alexander-Whitney diagonal in chain complexes. In this sense, our main…
This paper introduces a new category, Edgl, of enriched differential graded Lie algebras (edgl), directly related to the topology of all connected CW complexes and simplicial sets. It is equipped with a homotopy theory analogous to that…
Exponential family distributions are highly useful in machine learning since their calculation can be performed efficiently through natural parameters. The exponential family has recently been extended to the t-exponential family, which…
We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(\beta, \alpha) introduced by Martin and Saleur, and…