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A residue-theoretic representation is given for massless matter fields in (quotients) of (weighted) \CY\ complete intersection models and the corresponding chiral operators in \LGO{s}. The well known polynomial deformations are thus…
We study Grothendieck rings (in the sense of logic) of fields. We prove the triviality of the Grothendieck rings of certain fields by constructing definable bijections which imply the triviality. More precisely, we consider valued fields,…
We present a unified framework for representing commutative rings through affine algebraic theories and Boolean rings through hyperaffine algebraic theories. This yields categorical equivalences between these theories and, respectively,…
Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear…
For von Neumann *-regular rings R of endomorphisms (the involution given by taking adjoints) of inner product spaces we provide a condition on r in R (in terms of action of r on finite dimensional subspaces) for r being a unit. It remains…
This paper introduces the notion of referring forms as a new metric for analyzing sequential circuits from a functional perspective. Sequential circuits are modeled as causal stream functions, the outputs of which depend solely on the past…
We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the…
Linear finite dynamical systems play an important role, for example, in coding theory and simulations. Methods for analyzing such systems are often restricted to cases in which the system is defined over a field %and usually strive to…
Permutation polynomials with explicit constructions over finite fields have long been a topic of great interest in number theory. In recent years, by applying linear translators of functions from $\mathbb{F}_{q^n}$ to $\mathbb{F}_q$, many…
The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of…
The aim of this paper is to study the product of $n$ linear forms over function fields. We calculate the maximum value of the minima of the forms with determinant one when $n$ is small. The value is equal to the natural bound given by…
We construct a realizability model of linear dependent type theory from a linear combinatory algebra. Our model motivates a number of additions to the type theory. In particular, we add a universe with two decoding operations: one takes…
We exhibit a set of generating relations for the modular invariant ring of a vector and a covector for the two-dimensional general linear group over a finite field.
We address the problems of constructing quantum convolutional codes (QCCs) and of encoding them. The first construction is a CSS-type construction which allows us to find QCCs of rate 2/4. The second construction yields a quantum…
We define basic notions in the category of conic representations of a topological group and prove elementary facts about them. We show that a conic representation determines an ordinary dynamical system of the group together with a…
Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$.…
Linear type systems have a long and storied history, but not a clear path forward to integrate with existing languages such as OCaml or Haskell. In this paper, we study a linear type system designed with two crucial properties in mind:…
Generalizations of linear numeration systems in which the set of natural numbers is recognizable by finite automata are obtained by describing an arbitrary infinite regular language following the lexicographic ordering. For these systems of…
We present a formal language with expressions denoting general symbol structures and queries which access information in those structures. A sequence-to-sequence network processing this language learns to encode symbol structures and query…
In this paper we give the generalization of lifted codes over any finite chain ring. This has been done by using the construction of finite chain rings from $p$-adic fields. Further we propose a lattice construction from linear codes over…