Related papers: Pasting and Reversing Operations over some Vector …

In this paper we aim to generalize results obtained in the framework of fractional calculus by the way of reformulating them in terms of operator theory. In its own turn, the achieved generalization allows us to spread the obtained…

Transformers are a type of neural network that have demonstrated remarkable performance across various domains, particularly in natural language processing tasks. Motivated by this success, research on the theoretical understanding of…

We study the reproducing kernel for weighted polynomial Bergman spaces and consider applications to the Berezin transform. Some of our results have applications in random matrix theory, a topic which we discuss in a separate (companion)…

Vector spaces over finite fields and Anzahl formulas of subspaces were studied by Wan (Geometry of Classical Groups over Finite Fields, Science Press, 2002). As a generalization, we study vector spaces and singular linear spaces over…

The seminal work by Mackey et al. in 2006 (reference [21] of the article) introduced vector spaces of matrix pencils, with the property that almost all the pencils in the spaces are strong linearizations of a given square regular matrix…

Vector space word representations are learned from distributional information of words in large corpora. Although such statistics are semantically informative, they disregard the valuable information that is contained in semantic lexicons…

In this paper we study the action of the Binomial and Invert (interpolated) operators on the set of linear recurrent sequences. We prove that these operators preserve this set, and we determine how they change the characteristic…

We develop the theory of frames and Parseval frames for finite-dimensional vector spaces over the binary numbers. This includes characterizations which are similar to frames and Parseval frames for real or complex Hilbert spaces, and the…

We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…

Petri nets are a well-known model of concurrency and provide an ideal setting for the study of fundamental aspects in concurrent systems. Despite their simplicity, they still lack a satisfactory causally reversible semantics. We develop…

Matrix coordinate transformations are defined as substitution operators without requiring an ordering prescription or an inclusion function from the Abelian coordinate transformations. We construct transforming objects mimicking most of the…

Motivated by some recent developments in abstract theories of quadratic forms, we start to develop in this work an expansion of Linear Algebra to multivalued structures (a multialgebraic structure is essentially an algebraic structure but…

A notable property of word embeddings is that word relationships can exist as linear substructures in the embedding space. For example, $\textit{gender}$ corresponds to $\vec{\textit{woman}} - \vec{\textit{man}}$ and $\vec{\textit{queen}} -…

A random number of items each independently marked with one of a collection of colours gives rise to the multinomial marking, which generalises binomial thinning. A multivariate version, where previously marked items are then re-marked, has…

There is strong interest in developing mathematical methods that can be used to understand complex neural networks used in image analysis. In this paper, we introduce techniques from Linear Algebra to model neural network layers as maps…

Obtaining meaningful solutions for inverse problems has been a major challenge with many applications in science and engineering. Recent machine learning techniques based on proximal and diffusion-based methods have shown promising results.…

Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These…

We prove that linearizing certain families of polynomial optimization problems leads to new functorial operations in real convex sets. We show that under some conditions these operations can be computed or approximated in ways amenable to…

To study operator algebras with symmetries in a wide sense we introduce a notion of {\em relative convolution operators} induced by a Lie algebra. Relative convolutions recover many important classes of operators, which have been already…

We consider space-saving versions of several important operations on univariate polynomials, namely power series inversion and division, division with remainder, multi-point evaluation, and interpolation. Now-classical results show that…