相关论文: Intersection of subspaces in $A^2$ for a three-dim…
Given subvarieties $X, Y$ of a complex algebraic variety $S$ of complementary dimension, must they intersect? When $S$ is projective space, this is a consequence of the classical B\'ezout theorem, and an analogue for simple abelian…
A double algebra is a linear space $V$ equipped with linear map $V\otimes V\to V\otimes V$. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double…
Two hexagons in the space are said to intersect heavily if their intersection consists of at least one common vertex as well as an interior point. We show that the number of hexagons on n points in 3-space without heavy intersections is…
Two subspaces of a vector space are here called ``nonintersecting'' if they meet only in the zero vector. The following problem arises in the design of noncoherent multiple-antenna communications systems. How many pairwise nonintersecting…
We define a variant of intersection space theory that applies to many compact complex and real analytic spaces $X$, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to…
Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman…
Let $A$ be a finitely generated $K$-algebra that is a domain of GK dimension less than 3, and let $Q(A)$ denote the quotient division algebra of $A$. We show that if $D$ is a division subalgebra of $Q(A)$ of GK dimension at least 2 then…
Let F be an infinite field. We prove that the right zero divisors of a three-dimensional associative F-algebra A must form the union of at most finitely many linear subspaces of A. The proof is elementary and written with students as the…
In this note a simple extension of the complex algebra to higher dimension is proposed. Using the postulated algebra a two dimensional Dirac equation is formulated and its solution is calculated. It is found that there is a sub-algebra…
Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is…
This paper consists of a description of the variety of two dimensional associative algebras within the framework of Nonstandard Analysis. By decomposing each algebra in A^2 as sum of a Jordan algebra and a Lie algebra, we calculate the…
We study the set of intersection sizes of a k-dimensional affine subspace and a point set of size m \in [0, 2^n] of the n-dimensional binary affine space AG(n,2). Following the theme of Erd\H{o}s, F\"uredi, Rothschild and T. S\'os, we…
Let $n,n'$ be positive integers and let $V$ be an $(n+n')$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs $(U, U')$,…
For a vector space $V$ the \emph{intersection graph of subspaces} of $V$, denoted by $G(V)$, is the graph whose vertices are in a one-to-one correspondence with proper nontrivial subspaces of $V$ and two distinct vertices are adjacent if…
Let $\mathbb F$ denote a field and let $V$ denote a vector space over $\mathbb F$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfies the following conditions: (i) each of…
We generalize Albert's twisted field construction, applying it to unital division algebras with a multiplicative norm. We give conditions for the resulting algebras to be division algebras.Four- and eight-dimensional real unital and…
We prove a geometric property of the set A^{-1} of inverses of the nonzero elements of an F_q-subspace A of a finite field involving the size of its intersection with two-dimensional F_q-subspaces. We give some applications, including a new…
We introduce a new multiplication for the polytope algebra, defined via the intersection of polytopes. After establishing the foundational properties of this intersection product, we investigate finite-dimensional subalgebras that arise…
Supersymmetry is deeply related to division algebras. Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz…
In this paper, we study a problem posed by Furstenberg on intersections between $\times 2, \times 3$ invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used…