Related papers: Dimension expanders

Given a finite-dimensional faithful representation $V$ of a linearly reductive group $G$ over a field $K=\bar K$, we consider the growth of the number of irreducible factors of $V^{\otimes n}$ when $n$ is large. We prove that there exist…

Let $F$ be a field of characteristic not $2$ with finitely many square classes. Using combinatorial arguments applied to objects related to vector spaces over finite fields, we deduce an upper bound for the number of Pfister forms over $F$.…

Let $T$ be a complete theory of fields, possibly with extra structure. Suppose that model-theoretic algebraic closure agrees with field-theoretic algebraic closure, or more generally that model-theoretic algebraic closure has the exchange…

Associative algebras with involution over a field of zero characteristic are considered. It is proved that in this case for any finitely generated associative algebra with involution there exists a finite dimensional algebra with involution…

In this manuscript we generalize Ref. [1] and derive a complete set of local consistency conditions for bulk fields in braneworld scenarios with an arbitrary number of dimensions. This provides the first fully local and…

For each integer k >= 2, let F(k) denote the largest n for which there exists a permutation \sigma \in S_n, all of whose patterns of length k are distinct. We prove that F(k) = k + \lfloor \sqrt{2k-3} \rfloor + e_k, where e_k \in {-1,0} for…

We prove several rigidity results on multiplier spectrum and length spectrum. For example, we show that for every non-exceptional rational map $f:\mathbb{P}^1(\mathbb{C})\to\mathbb{P}^1(\mathbb{C})$ of degree $d\geq2$, the…

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. Let $End(V)$ denote the $K$-algebra consisting of all $K$-linear transformations from $V$ to $V$. We consider a pair $A,A^* \in End(V)$ that…

A Gross space is a vector space E of infinite dimension over some field F, which is endowed with a symmetric bilinear form Phi:E^2 -> F and has the property that every infinite dimensional subspace U subseteq E satisfies dim U^perp < dim E.…

We give a pedagogical introduction to the physics of large extra dimensions. We focus our discussion on minimal extensions of the Standard Model in which gauge fields may propagate in a single, compact extra dimension while the fermions are…

We show that there exists a sequence $\{n_k, k\ge 1\}$ growing at least geometrically such that for any finite non-negative measure $\nu$ such that $\hat \nu\ge 0$, any $T>0$, $$ \int_{-2^{n_k} T}^{2^{n_k} T} \hat \nu(x) \dd x \ll_\e…

We show that if $B \subset \mathbb{R}^n$ and $E \subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $\mathbb{R}^n$ such that every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least $\alpha$ with…

We prove the existence of similar and multi-similar point configurations (or simplexes) in sets of fractional Hausdorff measure in Euclidean space. These results can be viewed as variants, for thin sets, of theorems for sets of positive…

We show that there exist constants $\alpha,\epsilon>0$ such that for every positive integer $n$ there is a continuous odd function $f:S^m\to S^n$, with $m\geq \alpha n$, such that the $\epsilon$-expansion of the image of $f$ does not…

For a totally positive definite quadratic form over the ring of integers of a totally real number field $K$, we show that there are only finitely many totally real field extensions of $K$ of a fixed degree over which the form is universal…

Let $\mathbb{F}$ be a field, and $n \geq p \geq r>0$ be integers. In a recent article, Rubei has determined, when $\mathbb{F}$ is the field of real numbers, the greatest possible dimension for an affine subspace of $n$--by--$p$ matrices…

We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of $\mathbb N$ with positive upper density to properties of…

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We…

Given varieties $X, Y, W$ and dominant morphisms $\phi:X\to Y$ and $f:X\to W$ such that $f$ is constant on fibres of $\phi$ , we give sufficient conditions to guarantee that $f$ descends to a rational map or a morphism $Y\to W.$ We pay…

Van der Waerden's theorem states that for any positive integers $k$ and $r$, there exists a smallest value $n = w(k,r)$, called the van der Waerden number, such that every $r$-coloring of $\{1,\dots,n\}$ contains a monochromatic $k$-term…