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We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces…

Reflection and braid equations for rank two $q$-tensors are derived from the covariance properties of quantum vectors by using the $R$-matrix formalism.

We define a group of extended non-Abelian gauge transformations for tensor gauge fields. On this group one can define generalized field strength tensors, which are transforming homogeneously with respect to the extended gauge…

Let T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and…

Let $E/F$ be an extension of number fields with $\mathrm{Gal}(E/F)$ simple and nonabelian. In [G] the first named author suggested an approach to nonsolvable base change and descent of automorphic representations of $\mathrm{GL}_2$ along…

We present novel equivalences in random matrix and tensor models between complex and self-adjoint theories with nontrivial quadratic terms in the action, established through an intermediate field representation. More precisely, we show that…

Here, we suggest a method to represent general directed uniform and non-uniform hypergraphs by different connectivity tensors. We show many results on spectral properties of undirected hypergraphs also hold for general directed uniform…

We propose a novel matrix regularization for tensor fields. In this regularization, tensor fields are described as rectangular matrices and both area-preserving diffeomorphisms and local rotations of the orthonormal frame are realized as…

We give a detailed analysis of the semisimple elements, in the sense of Vinberg, of the third exterior power of a 9-dimensional vector space over an algebraically closed field of characteristic different from 2 and 3. To a general such…

Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of…

A non associative, noncommutative algebra is defined that may be interpreted as a set of vector modules over a noncommutative surface of rotation. Two of these vector modules are identified with the analogues of the tangent and cotangent…

We show that a three rank Lanczos type tensor field is an appropriate choice to describe relativistic electromagnetic and gravitational effects. More precisely, we identify the irreducible field-decompositions of this tensor as…

These lecture notes are intended as an introduction to several notions of tensor rank and their connections to the asymptotic complexity of matrix multiplication. The latter is studied with the exponent of matrix multiplication, which will…

A conducting two-dimensional periodic composite of two anisotropic phases with anisotropic, not necessarily symmetric, conductivity tensors is considered. By finding approximate representations for the relevant operators, an approximation…

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}\gamma(y)\exp(2\pi i a \lambda^{xy}/p)\Big|, \end{eqnarray*} where $\lambda$ is an element of the residue ring modulo a…

Surfaces of amplitude 1 in ordinary projective space are of general type, but this need not be the case in weighted projective spaces. Indeed, there are 4 classes of quasi-smooth weighted hypersurfaces in $\mathbf{P}(1,2,a,b)$ of amplitude…

Tensor models are measures for random tensors. They generalise matrix models and were developed to study random geometry in arbitrary dimension. Moreover, they are strongly connected to quantum gravity theories as additionally to the…

Over a real field which is an extension of transcendence degree 1 of a hereditarily pythagorean base field, every quadratic form which is torsion decomposes into an orthogonal sum of 2-dimensional torsion forms. This is obtained from a more…

We introduce a notion of $n$-commutativity ($0\le n\le \infty$) for cosimplicial monoids in a symmetric monoidal category ${\bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${\bf V,}$ while $n=\infty$ corresponds to…

The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…