Related papers: Alternative elements in the Cayley-Dickson algebra…

We formulate a relative analogue of the Clemens conjectures for 1/2-log Calabi-Yau threefold pairs (X,Y) (where K_X+2Y is isomorphic to O_X). This framework rests on the restoration of a perfect deformation/obstruction duality specific to…

Line integration of generalized functions is studied. Second order partial differential equations with piecewise continuous and generalized variable coefficients over Cayley-Dickson algebras are investigated. Formulas for integrations of…

Let $A$ be a pseudocompact (or profinite) algebra, so $A=C^*$ where $C$ is a coalgebra. We show that the if the semiartinian part (the "Dickson" part) of every $A$-module $M$ splits off in $M$, then $A$ is semiartinian, also giving a…

A subset $X$ of a groupoid is said to be deficient if $|X \cdot X|\leq |X|$. It is well-known that the probability that a random groupoid has a deficient $t$-element set with $t\geq 3$ is zero. However, as conjectured in [4], we show that…

We study the quotient Q_i(A) of a free algebra A by the ideal M_i(A) generated by relation that the i-th commutator of any elements is zero. In particular, we completely describe such quotient for i=4 (for i<=3 this was done previously by…

We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian with $1<q<p$. The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive…

In this article, we study the test for independence of two random elements $X$ and $Y$ lying in an infinite dimensional space ${\cal{H}}$ (specifically, a real separable Hilbert space equipped with the inner product $\langle .,…

We prove that for any positive integer c there are at least N(c), $1\leq N(c) < \phi(c)/2$ representations of c as a sum of two positive integers a, b, with no common divisor, such that the N(c) radicals R(abc) are all greater than kc,…

We prove the rationality of the K\"ahler cone and the positivity of $c_2(X)$, if $X$ is a Calabi-Yau-threefold with $\rho(X)=2$ and has an embedding into a ${\bb P}^n$-bundle over ${\bb P}^m$ in the cases $(n,m)=(1,3),(3,1)$. The case…

In this paper I consider locally finite Lie algebras of characteristic zero satisfying the condition that for every finite number of elements $x_{1}, x_{2},..., x_{k}$ of such an algebra $L$ there is finite-dimensional subalgebra $A$ which…

Another counter-example to Dirac's Conjecture is presented, which resembles the Cawley model but is so modified as to include second class constraints. The arbitrary function in the general solution to the defining equations of momenta…

In this paper, we prove some relations between Fibonacci elements in an arbitrary algebra. Moreover, we define imaginary Fibonacci quaternions and imaginary Fibonacci octonions and we prove that always three arbitrary imaginary Fibonacci…

The general Nullstellensatz states that if $A$ is a Jacobson ring, $A[X]$ is Jacobson. We introduce the notion of an $\alpha$-Jacobson ring for an ordinal $\alpha$ and prove a quantitative version of the general Nullstellensatz: if $A$ is…

Recently, motivated by supersymmetric gauge theory, Cachazo, Douglas, Seiberg, and Witten proposed a conjecture about finite dimensional simple Lie algebras, and checked it in the classical cases. Later V. Kac and the author proposed a…

Certain reduced free products of C*-algebras, (A,phi)=(A_1,phi_1)*(A_2,\phi_2), taken with respect to faithful states, at least one of which is not a trace, are shown to be purely infinite and simple. It is assumed that one of the A_i…

The Cayley-Dickson algebra has long been a challenge due to the lack of an explicit multiplication table. Despite being constructible through inductive construction, its explicit structure has remained elusive until now. In this article, we…

Over a composition algebra $A$, a polynomial $f(x) \in A[x]$ has a root $\alpha$ if and only $f(x)=g(x)\cdot (x-\alpha)$ for some $g(x) \in A[x]$. We examine whether this is true for general Cayley-Dickson algebras. The conclusion is that…

Suppose that $A \subset \{1,\dots, N\}$ has no two elements differing by $p-1$, $p$ prime. Then $|A| \ll N^{1 - c}$.

We introduce the notions of alternating roots of polynomials and alternating polynomials over a Cayley-Dickson algebra, and prove a connection between the alternating roots of a given polynomial and the roots of the corresponding…

Jacobson's commutativity theorem says that a ring is commutative if, for each $x$, $x^n = x$ for some $n > 1$. Herstein's generalization says that the condition can be weakened to $x^n-x$ being central. In both theorems, $n$ may depend on…