Related papers: Norms, Normsets, and Factorization
Humans have the innate capability to answer diverse questions, which is rooted in the natural ability to correlate different concepts based on their semantic relationships and decompose difficult problems into sub-tasks. On the contrary,…
Motivated by the application problem of sensor fusion the author introduced the concept of graded set. It is reasoned that in classification problem arising in an information system (represented by information table), a novel set called…
Univalent homotopy type theory (HoTT) may be seen as a language for the category of $\infty$-groupoids. It is being developed as a new foundation for mathematics and as an internal language for (elementary) higher toposes. We develop the…
Named entity classification is the task of classifying text-based elements into various categories, including places, names, dates, times, and monetary values. A bottleneck in named entity classification, however, is the data problem of…
The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…
Matrix factorization is an important mathematical problem encountered in the context of dictionary learning, recommendation systems and machine learning. We introduce a new `decimation' scheme that maps it to neural network models of…
We have introduced and studied in [3] the class of Globalized multiplicatively pinched-Dedekind domains (GMPD domains). This class of domains could be characterized by a certain factorization property of the non-invertible ideals, (see [3,…
Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we…
Let $\mathcal S \subseteq \mathbb Z^m \oplus T$ be a finitely generated and reduced monoid. In this paper we develop a general strategy to study the set of elements in $\mathcal S$ having at least two factorizations of the same length,…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
A central problem in the study of resource theories is to find functions that are nonincreasing under resource conversions - termed monotones - in order to quantify resourcefulness. Various constructions of monotones appear in many…
A factorization of an element $x$ in a monoid $(M, \cdot)$ is an expression of the form $x = u_1^{z_1} \cdots u_k^{z_k}$ for irreducible elements $u_1, \ldots, u_k \in M$, and the length of such a factorization is $z_1 + \cdots + z_k$. We…
In this note one tries to venture into a study of some notions, in the context of a (unital) normed algebra, in particular the algebra of operators on a Hilbert space. Namely, one considers ``moving norms'', i.e.\ norming an element minus a…
We study regularization in the context of small sample-size learning with over-parameterized neural networks. Specifically, we shift focus from architectural properties, such as norms on the network weights, to properties of the internal…
We study links between first-order formulas and arbitrary properties for families of theories, classes of structures and their isomorphism types. Possibilities for ranks and degrees for formulas and theories with respect to given properties…
There are different notions of computation, the most popular being monads, applicative functors, and arrows. In this article we show that these three notions can be seen as monoids in a monoidal category. We demonstrate that at this level…
The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally…
Monads are a useful tool for structuring effectful features of computation such as state, non-determinism, and continuations. In the last decade, several generalisations of monads have been suggested which provide a more fine-grained model…
We present a systematic study of the method of "norms on possibilities" of building forcing notions with keeping their properties under full control. This technique allows us to answer several open problems, but on our way to get the…
In this paper, we introduce novel characterizations of the classical concept of majorization in terms of upper triangular (resp., lower triangular) row-stochastic matrices, and in terms of sequences of linear transforms on vectors. We used…