Related papers: Hadamard matrices modulo p and small modular Hadam…

A real symmetric matrix $M$ is completely positive semidefinite if it admits a Gram representation by (Hermitian) positive semidefinite matrices of any size $d$. The smallest such $d$ is called the (complex) completely positive semidefinite…

We have extended the Paley constructions for Hadamard matrices and obtained some series of Hadamard matrices. Especially Paley construction-II is applicable for odd prime power q is congruent to 1(mod 4) however our method is applicable for…

There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that,…

Consider the set of scalars $\alpha$ for which the $\alpha$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots, n-3\}\cup…

Computing modular coincidences can show whether a given substitution system, which is supported on a point lattice in R^d, consists of model sets or not. We prove the computatibility of this problem and determine an upper bound for the…

This paper studies the possible Hodge groups of simple polarizable $\mathbb{Q}$-Hodge structures with Hodge numbers $(n,0,\ldots,0,n)$. In particular, it generalizes earlier work of Ribet and Moonen-Zarhin to completely determine the…

For finitely generated subgroups $W_1, \ldots , W_t$ of $\mathbb{Q}^{\times}$, integers $k_1, \ldots , k_t$, a Galois extension $F$ of $\mathbb{Q}$ and a union of conjugacy classes $C \subset \text{Gal}(F/\mathbb{Q})$, we develop methods…

If $p$ is an odd prime, then we prove that $\e(H_2(G,\mathbb{Z})) \mid p\ \e(G)$ for $p$ groups of class 7. We prove the same for $p$ groups of class at most $p+1$ with $\e(Z(G))=p$. We also prove Schurs conjecture if $\e(G/Z(G))$ is $2,3$…

In this short note we construct a spectral, non-tiling set of size $2p$ in $(\mathbb{Z}/p\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$.…

We classify all the cocyclic Butson Hadamard matrices $\mathrm{BH}(n,p)$ of order $n$ over the $p$th roots of unity for an odd prime $p$ and $np\leq 100$. That is, we compile a list of matrices such that any cocyclic $\mathrm{BH}(n,p)$ for…

In the present paper we obtain new upper bound estimates for the number of solutions of the congruence $$ x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in\cU, $$ for certain ranges of $H$ and $|\cU|$, where $\cU$ is a…

We study the index of nilpotency relative to certain Hecke operators in spaces of modular forms with integer weight and level $N$ with integer coefficients modulo primes $p$ for $(p, N) \in \{(3, 1), (5, 1), (7, 1), (3, 4)\}$. In these…

In this article, we consider a special class of Williamson type matrices which we call them near Williamson matrices. They are in fact four $n\times n$ $(-1, 1)$-matrices $A, B, C, D$ so that $A$ is circulant, $B,C,D$ are symmetric…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

Given a finite dimensional algebra $\Lambda$, we show that a frequently satisfied finiteness condition for the category ${\cal P}^{\infty}(\Lambda\rm{-mod})$ of all finitely generated (left) $\Lambda$-modules of finite projective dimension,…

Given a primitive, non-CM, holomorphic cusp form $f$ with normalized Fourier coefficients $a(n)$ and given an interval $I\subset [-2, 2]$, we study the least prime $p$ such that $a(p)\in I$ . This can be viewed as a modular form analogue of…

We give general lower bounds on the maximal determinant of n by n {+1,-1}-matrices, both with and without the assumption of the Hadamard conjecture. Our bounds improve on earlier results of de Launey and Levin (2010) and, for certain…

Solving a problem of Diestel and Pott, we construct a large class of infinite matroids. These can be used to provide counterexamples against the natural extension of the Well-quasi-ordering-Conjecture to infinite matroids and to show that…

Around 2016, Calinescu, Milas and Penn conjectured that the rank $r$ Nahm sum associated with the $r\times r$ tadpole Cartan matrix is modular, and they provided a proof for $r=2$. The $r=3$ case was recently resolved by Milas and Wang. We…

Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate…