Related papers: Algebra and logic. Some problems
This book is a continuation of the book n-linear algebra of type I and its applications. Most of the properties that could not be derived or defined for n-linear algebra of type I is made possible in this new structure: n-linear algebra of…
Algebraic effects are computational effects that can be described with a set of basic operations and equations between them. As many interesting effect handlers do not respect these equations, most approaches assume a trivial theory,…
Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly…
Bases, mappings, projections and metrics, natural for Neural network training, are introduced. Graph-theoretical interpretation is offered. Non-Gaussianity naturally emerges, even in relatively simple datasets. Training statistics,…
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
Chapters 1 to 4 are the lecture notes of my course "Real Algebraic Geometry I" from the winter term 2020/2021. Chapters 5 to 8 are the lecture notes of its continuation "Real Algebraic Geometry II" from the summer term 2021. Chapters 9 and…
This paper is devoted to the classification and studying properties of complex unital $3$-dimensional structurable algebras. We provide a complete list of non-isomorphic classes, identifying five algebras for type $(2, 1)$ and two algebras…
This is a survey paper on Alegbraic Geometry over Lie Algebras
In this Master of Science Thesis I introduce geometric algebra both from the traditional geometric setting of vector spaces, and also from a more combinatorial view which simplifies common relations and operations. This view enables us to…
A classical logic exhibits a threefold inner structure comprising an algebra of propositions `A', a space of ``truth values'' `V', and a distinguished family of mappings `phi' from propositions to truth values. Classically A is a Boolean…
Although contemporary model theory has been called "algebraic geometry minus fields", the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes,…
This paper studies the loop algebras that arise from pairs consisting of a symmetrizable Kac-Moody Lie algebra $\g$ and a finite order automorphism $\sigma$ of $\g$. We obtain necessary and sufficient conditions for two such loop algebras…
What is the best representation for doing euclidean geometry on computers? These notes from a SIGGRAPH 2019 short course entitled "Geometric algebra for computer graphics" introduce projective geometric algebra (PGA) as a modern framework…
Hom-algebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist. We enumerate here all the possible choices in the Lie and associative categories and study the…
Starting from classical algebraic geometry over the complex numbers (as it can be found for example in Griffiths and Harris it was the goal of these lectures to introduce some concepts of the modern point of view in algebraic geometry. Of…
Paravectors just like integers have a ring structure. By introducing an integrated product we get geometric properties which make paravectors similar to vectors. The concepts of parallelism, perpendicularity and the angle are conceptually…
Models of complex systems are widely used in the physical and social sciences, and the concept of layering, typically building upon graph-theoretic structure, is a common feature. We describe an intuitionistic substructural logic called…
Geometric algebra is the natural outgrowth of the concept of a vector and the addition of vectors. After reviewing the properties of the addition of vectors, a multiplication of vectors is introduced in such a way that it encodes the famous…
The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical…
This survey is meant to provide an introduction to the fundamental theorem of linear algebra and the theories behind them. Our goal is to give a rigorous introduction to the readers with prior exposure to linear algebra. Specifically, we…