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This paper surveys and illustrates geometric methods for constructing normal bases allowing efficient finite field arithmetic. These bases are constructed using the additive group, the multiplicative group and the Lucas torus. We describe…
We develop a functional calculus for $d$-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is nc functions that are bounded on certain polynomial polyhedra.
In this paper we study dual bases functions in subspaces. These are bases which are dual to functionals on larger linear space. Our goal is construct and derive properties of certain bases obtained from the construction, with primary focus…
In this article is studied the construction of free operads functor, for the symmetric and non-symmetric case. In order to do this, the operads are seen as monoids on the differential graded modules category. In the last part we show some…
We prove that monoids $\mathrm{Mon}\langle a,b,c,d : a^nb=0, ac=1, db=1, dc=1, dab=1, da^2b=1, \ldots, da^{n-1}b=1\rangle$ are congruence-free for all $n\geq 1$. This provides a new countable family of finitely presented congruence-free…
Sorting algorithms are fundamental to computer science, and their correctness criteria are well understood as rearranging elements of a list according to a specified total order on the underlying set of elements. As mathematical functions,…
The main objective of the paper is to define the category of monoids as a weighted limit. We also define the category of actions of monoids along the action of a monoidal category as a weighted limit.
In this note we give two proofs of Brooks' Theorem. The first is obtained by modifying an earlier proof and the second by combining two earlier proofs. We believe these proofs are easier to teach in Computer Science courses.
We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including…
This article is an overview of the vanishing cycles method in number theory over function fields. We first explain how this works in detail in a toy example, and then give three examples which are relevant to current research. The focus…
We prove that the free Fock space ${\F}(\R^+;\C)$, which is very commonly used in Free Probability Theory, is the continuous free product of copies of the space $\C^2$. We describe an explicit embedding and approximation of this continuous…
This paper is devoted to the investigation of the property of order separability for free products of groups.
We study the functorial and growth properties of closed orbits for maps. By viewing an arbitrary sequence as the orbit-counting function for a map, iterates and Cartesian products of maps define new transformations between integer…
The purpose of this paper is to give new examples of families of free singularities. We first show that a generic equidimensional subspace arrangement is free. Furthermore, we show that a product of two reduced Cohen-Macaulay subspaces is…
In this paper, we study connections between the structure of a group and the structure of the group (under pointwise product) of its polynomial functions.
The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set theoretical functions defined on, and with values in, a suitable ring of scalars and…
In this manuscript, we investigate some properties of certain counting functions, associated to the ergodic sums computed along the periodic orbits of the skew-product map, related to a finitely generated rational semigroup. To be precise,…
We present a general method for proving that a semigroup is non-finitely based. The method is strong enough to cover the non-finite basis arguments in articles [1,3,4,5,7,8, 11,14,16,21,27,31,36,37]. In particular, the method allows to…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
We develop a constructive theory of finite multisets in Homotopy Type Theory, defining them as free commutative monoids. After recalling basic structural properties of the free commutative-monoid construction, we formalise and establish the…