Related papers: Unit equations on quaternions

To construct ternary "quaternions" following Hamilton we must introduce two "imaginary "units, $q_1$ and $q_2$ with propeties $q_1^n=1$ and $q_2^m=1$. The general is enough difficult, and we consider the $m=n=3$. This case gives us the…

For a finitely generated group $G$, the \emph{Diophantine problem} over $G$ is the algorithmic problem of deciding whether a given equation $W(z_1,z_2,\ldots,z_k) = 1$ (perhaps restricted to a fixed subclass of equations) has a solution in…

Thompson's theorem stated that a finite group $G$ is solvable if and only if every $2$-generated subgroup of $G$ is solvable. In this paper, we prove some new criteria for both solvability and nilpotency of a finite group using certain…

The subgroup commutativity degree of a group G has been defined in [6] as the probability that two subgroups of G commute, or equivalently that the product of two subgroups is again a subgroup. Problem 4.3 of [6] asks whether there exist…

Representable implication algebras are known to be axiomatised by a finite number of equations (making the representation and finite representation problems decidable here). We show that this also holds in the context of unary (and binary)…

We introduce the semiring of values $\Gamma$ with respect to the tropical operations associated to an algebroid curve. As a set, $\Gamma$ determines and is determined by the well known semigroup of values $S$ and we prove that $\Gamma$ is…

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…

Let $\S$ be a commutative semigroup with identity $e$ and let $\Gamma$ be a compact subset in the pointwise convergence topology of the space $\S'$ of all non-zero multiplicative functions on $\S.$ Given a continuous function $F: \Gamma \to…

In 1904, Issai Schur proved the following result. If $G$ is an arbitrary group such that $G/\Z(G)$ is finite, where $\Z(G)$ denotes the center of the group $G$, then the commutator subgroup of $G$ is finite. A partial converse of this…

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$…

It is shown how to construct *-homomorphic quantum stochastic Feller cocycles for certain unbounded generators, and so obtain dilations of strongly continuous quantum dynamical semigroups on C* algebras; this generalises the construction of…

We study the von Neumann algebra generated by q--deformed Gaussian elements l_i+l_i^* where operators l_i fulfill the q--deformed canonical commutation relations l_i l_j^*-q l_j^* l_i=delta_{ij} for -1<q<1. We show that if the number of…

Let $G$ be a finite group and $G'$ its commutator subgroup. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered…

We construct six unitary trace invariants for 2 by 2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We prove two quaternionic versions of a well known…

Quantum symmetry of graph $C^{*}$-algebras has been studied, under the consideration of different formulations, in the past few years. It is already known that the compact quantum group $(\underbrace{C(S^{1})*C(S^{1})*\cdots…

In this paper, we generalize the Mandelbrot set using quaternions and spherical coordinates. In particular, we use pure quaternions to define a spherical product. This product, which is inspired by the product of complex numbers, add the…

In this work, we analyze two models beyond the Standard Models descriptions that make ad hoc hypotheses of three point-like lepton and quark generations without explanations of their physical origins. Instead of using the same Dirac…

A conjecture of Coleman implies that only finitely many quaternion algebras over the rational numbers can be the endomorphism $\mathbf{Q}$-algebras of abelian surfaces over the complex numbers which can be defined over $\mathbf{Q}$. One may…

Let R be a Dedekind domain. Enochs' solution of the Flat Cover Conjecture was extended as follows: (*) If C is a cotorsion pair generated by a class of cotorsion modules, then C is cogenerated by a set. We show that (*) is the best result…

Let R be a unital commutative ring and let $M$ be an $R$-module that is generated by $k$ elements but not less. Let $E_n(R)$ be the subgroup of $GL_n(R)$ generated by the elementary matrices. In this paper we study the action of $E_n(R)$ by…