Related papers: Embeddable partial Hadamard matrices related to th…

We define and study embeddings of cycles in finite affine and projective planes. We show that for all $k$, $3\le k\le q^2$, a $k$-cycle can be embedded in any affine plane of order $q$. We also prove a similar result for finite projective…

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that…

We revisit the coordinatisation method for projective planes. First, we discuss how the behaviour of the additive and multiplicative loops can be described in terms of its action on the "vertical" line, and how this means one can…

We define a combinatorial object that can be associated with any conic-line arrangement with ordinary singularities, which we call the combinatorial Poincar\'e polynomial. We prove a Terao-type factorization statement on the splitting of…

A $3$-Prismatoid is the convex hull of two convex polygons $A$ and $B$ which lie in parallel planes $H_A, H_B\subset\mathbb{R}^3$. Let $A'$ be the orthogonal projection of $A$ onto $H_B$. A prismatoid is called nested if $A'$ is properly…

A square matrix of order n with $n\geq 2$ is called permutative matrix when all its rows (up to the frst one) are permutations of precisely its frst row. In this paper recalling spectral results for partitioned into $2$-by-$2$ symmetric…

We consider three forms of composition of matroids, each of which extends the category of bimatroids to a rigid monoidal category. Many well-known constructions are functorial or defined by morphisms in these categories. Motivating examples…

The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic…

In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\prod_{1\le i,j\le(p-1)/2\atop…

We describe the space of projective planes of complex skew-symmetric matrices of order six and constant rank four. We prove that it has four connected components, all of dimension 26 and homogeneous under the action of PGL_6.

We show that the systems of prime-to-$p$ Hecke eigenvalues arising from automorphic forms$\pmod p$ for a good prime $p$ associated to an algebraic group $G/\mathbb Q$ of Hodge type are the same as those arising from algebraic modular…

We define formal vector bundles with marked sections on Hilbert modular schemes and we show how to use them to construct modular sheaves with an integrable meromorphic connection and a filtration which, in degree 0, gives to us a $p$-adic…

Let $p$ be an odd prime. In this paper we investigate quadratic residues modulo $p$ and related permutations, congruences and identities. If $a_1<\ldots<a_{(p-1)/2}$ are all the quadratic residues modulo $p$ among $1,\ldots,p-1$, then the…

In this paper we introduce modular symmetric designs and use them to study the existence of Hadamard matrices modulo 5. We prove that there exist 5-modular Hadamard matrices of order n if and only if n != 3, 7 (mod 10) or n != 6, 11. In…

We characterize integral homology classes of the product of two projective planes which are representable by a subvariety.

It was shown in \cite{FPV} that the classification of $n$-component systems of conservation laws possessing a third-order Hamiltonian structure reduces to the following algebraic problem: classify $n$-planes $H$ in $\wedge^2(V_{n+2})$ such…

A $\{1,-1\}$-matrix $H$ of order $m$ is a Hadamard matrix if $HH^T=mI_m$, where $T$ is the transposition operator and $I_m$ the identity matrix of order $m$. J. Hadamard published his paper on Hadamard matrices in 1893. Five years later,…

We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular…

A matrix (and any associated linear system) will be referred to as structured if it has a small displacement rank. It is known that the inverse of a structured matrix is structured, which allows fast inversion (or solution), and reduced…

We consider an interesting class of braidings defined by a combinatorial property in an earlier paper. We show that it consists exactly of those braidings that come from certain Yetter-Drinfeld module structures over pointed Hopf algebras…