Related papers: Univalent Foundations and the UniMath Library
We lay down the foundations for a systematic study of differentiable and algebraic supervarieties, with a special attention to supergroups.
Akin to the idea of complete sets of Mutually Unbiased Bases for prime dimensional Hilbert spaces, $\mathcal{H}_d$, we study its analogue for a $d$ dimensional subspace of $M (d,\mathbb{C})$, i.e. Mutually Unbiased Unitary Bases (MUUBs)…
Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived since a quantum logic…
This book invites readers to see mathematics not just as formulas and rules, but as the deepest expression of human thought. It begins by exploring the timeless idea of mathematics as a universal language, contrasting its precision with the…
Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of…
Qubits are a great way to build a quantum computer, but a limited way to program one. We replace the usual "states and gates" formalism with a "props and ops" (propositions and operators) model in which (a) the C*-algebra of observables…
In this expositional paper, we discuss commutative algebra -- a study inspired by the properties of integers, rational numbers, and real numbers. In particular, we investigate rings and ideals, and their various properties. After, we…
In this tutorial, we provide an overview of many of the established combinatorial and algebraic tools of Schubert calculus, the modern area of enumerative geometry that encapsulates a wide variety of topics involving intersections of linear…
By a series of simple examples, we illustrate how the lack of mathematical concern can readily lead to surprising mathematical contradictions in wave mechanics. The basic mathematical notions allowing for a precise formulation of the theory…
This paper presents an expository reverse-mathematical analysis of two fundamental theorems in commutative algebra: Hilbert's Nullstellensatz and Basis Theorem. In addition to its profound significance in commutative algebra and algebraic…
Despite several (accepted) standards, core notions typically employed in information technology or system engineering architectures lack the precise and exact foundations encountered in logic, algebra, and other branches of mathematics. In…
We establish a close connection between a reversible programming language based on type isomorphisms and a formally presented univalent universe. The correspondence relates combinators witnessing type isomorphisms in the programming…
Starting from the observation that distinct notions of copying have arisen in different categorical fields (logic and computation, contrasted with quantum mechanics) this paper addresses the question of when, or whether, they may coincide.…
A unifying approach to software and hardware design generated by ideas of Idempotent Mathematics is discussed. The so-called idempotent correspondence principle for algorithms, programs and hardware units is described. A software project…
Various problems of mathematical physics consider octonions and split-octonions as a mathematical structure, which underpins the eight-dimensional nature of these problems. Therefore, it is not surprising that octonionic analysis has become…
Fusion frames are a very active area of research today because of their myriad of applications in pure mathematics, applied mathematics, engineering, medicine, signal and image processing and much more. They provide a great flexibility for…
There is a problem with the foundations of classical mathematics, and potentially even with the foundations of computer science, that mathematicians have by-and-large ignored. This essay is a call for practicing mathematicians who have been…
We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.
We give a categorification of the notion of a mathematical structure originally given by Bourbaki in their set theory textbook. We show that any isomorphism-invariant property of a finite structure can be computed by counting the number of…
We consider the problem of learning a function respecting a symmetry from among a class of symmetries. We develop a unified framework that enables symmetry discovery across a broad range of subgroups including locally symmetric, dihedral…