arXiv subjects

David Hobby

Publications and source records attributed to David Hobby.

9 recordsLinked to original sources

arXiv2025

Blocks in Finite Hyperfields

This paper studies the structure of finite hyperfields $H$, and finds a subtle pattern in their addition operation. Consider the class $\mathcal{H}$ of all hyperfields with a given multiplicative group on $H^\times = H - \{0\}$ and given value of $-1$. Then the addition of hyperfields in this class is determined by the set of pairs $(x,y)$ with $y \in x+1$ for $x,y \in H^\times$. There are blocks of such pairs, where $(x_0,y_0)$ and $(x_1,y_1)$ are in the same block iff every hyperfield with $y_0 \in x_0 + 1$ also has $y_1 \in x_1 + 1$. The theory of these blocks is developed, they can easily be computed without using hyperfields. Exploiting this theory of blocks would greatly speed up future computer searches for small hyperfields. The theory of blocks is then used to show that the number of nonquotient hyperfields of size $n$ grows exponentially with $n$, and that for even $n$ most hyperfields are nonquotient. A final application shows that a large class of finite hyperfields has the FETVINS property, meaning that systems of linear homogeneous equations with fewer equations than variables always have nontrivial solutions.

math.RA↗

arXiv2023

Nontrivial solutions for homogeneous linear equations over some non-quotient hyperfields

We introduce a class of hyperfields which includes several constructions of non-quotient hyperfields. We then use it to partially answer a question posed by M. Baker and T. Zhang: Does a system of homogeneous linear equations with more unknowns than equations always have a nonzero solution? We also consider a class of hyperfields that was claimed in the literature to be non-quotient, and show that this is false.

math.RA↗

arXiv2020

Laplace invariants of differential operators

We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential invariants of partial differential operators under gauge transformations and introduce a sufficient condition for a set of invariants to be complete. We also give a some mild conditions that guarantee the existence of such a set. The proof is constructive. The method gives many examples of invariants previously known in the literature as well as many new examples including multidimensional.

math-ph↗

arXiv2017

Sums of finitely many distinct rationals

${\cal E}$ denotes the family of all finite nonempty $S\subseteq{\mathbb N}:=\{1,2,\ldots\}$, and ${\cal E}(X):={\cal E}\cap\{S:S\subseteq X\}$ when $X\subseteq{\mathbb N}$. Similarly, ${\cal F}$ denotes the family of all finite nonempty $T\subseteq{\mathbb Q}^+$, and ${\cal F}(Y) := {\cal F}\cap\{T:T\subseteq Y\}$ where ${\mathbb Q}^+$ is the set of all positive rationals and $Y\subseteq{\mathbb Q}^+$. This paper treats the functions $\sigma:{\cal E}\rightarrow{\mathbb Q}^+$ given by $\sigma:S\mapsto\sigma S :=\sum\{1/x:x\in S\}$, the function $\delta:{\cal E}\rightarrow{\mathbb N}$ defined by $\sigma S = \nu S/\delta S$ where the integers $\nu S$ and $\delta S$ are coprime, and the more general function $\Sigma:{\cal F}\rightarrow{\mathbb Q}^+$ where $\Sigma T$ denotes the sum of the elements in $T$ for $T\in{\cal F}$. Theorem 1.1. For each $r\in{\mathbb Q}^+$, there exists an infinite pairwise disjoint subfamily ${\cal H}_r\subseteq{\cal E}$ such that $r=\sigma S$ for all $S\in{\cal H}_r$. Theorem 1.2. Let $X$ be a pairwise coprime set of positive integers. Then $\sigma$ restricted to ${\cal E}(X)$ and $\delta$ restricted to ${\cal E}(X)$ are injective. Also, $\sigma C\in{\mathbb N}$ for $C\in{\cal E}(X)$ only if $C=\{1\}$. Theorem 6.5. There is a set $X$ of positive rational numbers for which $\Sigma:{\cal F}(X)\rightarrow{\mathbb Q}^+$ is a surjection, but for which $1\in X$ and the only $S\in{\cal F}(X)$ with $\Sigma S = 1$ is $S = \{1\}$.

math.NT↗

arXiv2016

Classification of Multidimensional Darboux Transformations: First Order and Continued Type

We analyze Darboux transformations in very general settings for multidimensional linear partial differential operators. We consider all known types of Darboux transformations, and present a new type. We obtain a full classification of all operators that admit Wronskian type Darboux transformations of first order and a complete description of all possible first-order Darboux transformations. We introduce a large class of invertible Darboux transformations of higher order, which we call Darboux transformations of continued Type I. This generalizes the class of Darboux transformations of Type I, which was previously introduced. There is also a modification of this type of Darboux transformations, continued Wronskian type, which generalize Wronskian type Darboux transformations.

math.DG↗

arXiv2016

Term inequalities in finite algebras

Given an algebra $\mathbf{A}$, and terms $s(x_{1},x_{2},\dots x_{k})$ and $t(x_{1},x_{2},\dots x_{k})$ of the language of ${\mathbf A}$, we say that $s$ and $t$ are {\em separated} in ${\mathbf A}$ iff for all $a_{1},a_{2}\dots a_{k}\in A$, $s(a_{1},a_{2},\dots a_{k})$ and $t(a_{1},a_{2},\dots a_{k})$ are never equal. We prove that given two terms that are separated in any algebra, there exists a finite algebra in which they are separated. As a corollary, we obtain that whenever the sentence $\sigma$ is a universally quantified conjunction of negated atomic formulas, $\sigma$ is consistent iff it has a finite model.

math.LO↗

arXiv2015

Roots with common tails

Some cubic polynomials over the integers have three distinct real roots with continued fractions that all have the same common tail. We characterize the polynomials for which this happens, and then investigate the situation for other polynomials of low degree.

math.NT↗

arXiv2014

Antiassociative Groupoids

Given a groupoid $< G, \star >$, and $k \geq 3$, we say that $G$ is antiassociative iff for all $x_1, x_2, x_3 \in G$, $(x_1 \star x_2) \star x_3$ and $x_1 \star (x_2 \star x_3)$ are never equal. Generalizing this, $< G, \star >$ is $k$-antiassociative iff for all $x_1, x_2, ... x_k \in G$, any two distinct expressions made by putting parentheses in $x_1 \star x_2 \star x_3 \star ...x_k$ are never equal. We prove that for every $k \geq 3$, there exist finite groupoids that are $k$-antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.

math.RA↗

arXiv2011

Completely dissociative groupoids

Consider arbitrarily parenthesized expressions on the $k$ variables $x_0, x_1, ..., x_{k-1}$, where each $x_i$ appears exactly once and in the order of their indices. We call these expressions {\em formal $k$--products}. $F^σ(k)$ denotes the set of formal $k$--products. For ${{\bf u},{\bf v}}\subseteq F^σ(k)$, the claim, that ${\bf u}$ and ${\bf v}$ produce equal elements in a groupoid $G$ for all values assumed in $G$ by the variables $x_i$, attributes to $G$ a {\em generalized associative law}. Many groupoids are {\em completely dissociative}; i.e., no generalized associative law holds for them; two examples are the groupoids on ${0,1}$ whose binary operations are implication and NAND. We prove a variety of results of that flavor.

math.GR↗