Related papers: Fitness, Apprenticeship, and Polynomials
In this paper we introduce the concept of polynomial diagrams and its area for special polynomials.We study the properties of polynomial area diagrams. The formula for the area of an arbitrary polynomial diagram.
This paper is primarily intended as an introduction for the mathematically inclined to some of the rich algebraic combinatorics arising in for instance CFT. It is essentially self-contained, apart from some of the background motivation and…
Results of Koebe (1936), Schramm (1992), and Springborn (2005) yield realizations of $3$-polytopes with edges tangent to the unit sphere. Here we study the algebraic degrees of such realizations. This initiates the research on constrained…
Several aspects of fusion rings and fusion rule algebras, and of their manifestations in twodimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of…
We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…
The article presents an algebra to represent two dimensional patterns using reciprocals of polynomials. Such a representation will be useful in neural network training and it provides a method of training patterns that is much more…
We study multivariate polynomials over `structured' grids. We begin by proposing an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend…
The three key documents for study geometry are: 1) "The Elements" of Euclid, 2) the lecture by B. Riemann at G\"ottingen in 1854 entitled "\"Uber die Hypothesen welche der Geometrie zu Grunde liegen" (On the hypotheses which underlie…
The subject matter of this work is quadratic and cubic polynomial functions with integer coefficients;and all of whose roots are integers. The material of this work is directed primarily at educators,students,and teachers of…
This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with…
Combinatorial optimization problems (COPs) are an important research topic in various fields. In recent times, there have been many attempts to solve COPs using deep learning-based approaches. We propose a novel neural network model that…
This chapter narrates the journey of developing and integrating computing into the physics curriculum through three consecutive courses, each tailored to the learners' level. It starts with the entry-level "Physics Playground in Python" for…
We embark on a tour that takes us through four closely related topics: the dual concepts of independence and spanning in finite abelian groups and the analogous dual concepts of designs and distance sets on spheres. We review some of the…
The aim of this study is to construct and compose an instructional design in combinatorial learning, particularly in the concept of counting. A composed design is expected to optimize students' combinatorial-thinking skill. This research…
This note discusses recent new approaches to studying flopping curves on 3-folds. In a joint paper, the author and Michael Wemyss introduced a 3-fold invariant, the contraction algebra, which may be associated to such curves. It…
This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…
These are lecture notes from a course given at the summer school "Heat kernels and spectral geometry: from manifolds to graphs" in Bregenz, Austria, 2022. They are designed to be accessible to doctoral level students, and include background…
This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…
The polynomial affine model of gravity was proposed as an alternative to metric and metric-affine gravitational models. What at the beginning was thought as a source of unpredictability, the presence of many terms in the action, turned out…
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of…