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The Desarguesian ovoids in the orthogonal polar space $Q^+(7,q)$ with $q$ even have first been introduced by Kantor by examining the $8$-dimensional absolutely irreducible modular representations of $\text{PGL}(2,q^3)$. We investigate this…

We study the set $\mathcal{S}$ of odd positive integers $n$ with the property ${2n}/{\sigma(n)} - 1 = 1/x$, for positive integer $x$, i.e., the set that relates to odd perfect and odd "spoof perfect" numbers. As a consequence, we find that…

Effective Lagrangians with dimension-six operators are widely used to analyse Higgs and other electroweak data. We show how to build a basis of operators such that each operator corresponds to a coupling which is well measured or will be in…

(1) Let 1\leq k\leq \omega. Call an atom structure \alpha weakly k neat representable, the term algebra is in \RCA_n\cap \Nr_n\CA_{n+k}, but the complex algebra is not representable. Call an atom structure neat if there is an atomic algebra…

For a fixed positive integer $n$ and an $r$-uniform hypergraph $H$, the Tur\'an number $ex(n,H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices, and the Lagrangian density of $H$ is defined as…

Extending the notion of sunflowers, we call a family of at least two sets an odd-sunflower if every element of the underlying set is contained in an odd number of sets or in none of them. It follows from the Erd\H os--Szemer\'edi…

Let p be a prime and F(p) the maximal p-extension of a field F containing a primitive p-th root of unity. We give a new characterization of Demuskin groups among Galois groups Gal(F(p)/F) when p=2, and, assuming the Elementary Type…

Let $E_1,E_2\subset \mathbb{R}^n$ be two homothetic solid ellipsoids, $n\geq 3$, with center at the origin $O$ of a system coordinates of $\mathbb{R}^n$, and $E_1\subset E_2$. Then there exists a $O$-symmetric ellipsoid $E_3$ such that…

A digraph $D=(V, A)$ has a good pair at a vertex $r$ if $D$ has a pair of arc-disjoint in- and out-branchings rooted at $r$. Let $T$ be a digraph with $t$ vertices $u_1,\dots , u_t$ and let $H_1,\dots H_t$ be digraphs such that $H_i$ has…

For a prime $p>2$, let $G$ be a semi-simple, simply connected, split Chevalley group over $\mathbb{Z}_p$, $G(1)$ be the first congruence kernel of $G$ and $\Omega_{G(1)}$ be the mod-$p$ Iwasawa algebra defined over the finite field…

A generalised hexagon of order $(s,t)$ is said to be \emph{extremal} if $t$ meets the Haemers-Roos bound, that is, $t=s^3$. The \emph{dual twisted triality hexagons} associated to the exceptional Lie type groups $\,^3D_4(s)$ have these…

We prove the Tits-Weiss conjecture for Albert division algebras over fields of arbitrary characteristics in the affirmative. The conjecture predicts that every norm similarity of an Albert division algebra is a product of a scalar homothety…

Let $J_G$ be the binomial edge ideal of a graph $G$. We characterize all graphs whose binomial edge ideals, as well as their initial ideals, have regularity $3$. Consequently we characterize all graphs $G$ such that $J_G$ is extremal…

Hermitian positive definite, totally positive, and nonsingular M-matrices enjoy many common properties, in particular: (A) positivity of all principal minors, (B) weak sign symmetry, (C) eigenvalue monotonicity, (D) positive stability. The…

We give a proof of the Universality Conjecture for orthogonal (beta=1) and symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the spectrum as well as at the hard and soft spectral edges. Our results are stated…

We consider the variation of spherical characters in families. We formulate conjectures for the rationality and meromorphic property of spherical characters. As an example, we establish these conjectures in the unitary Gan-Gross-Prasad…

The exceptional Dehn filling conjecture of the second author concerning the relationship between exceptional slopes $\alpha, \beta$ on the boundary of a hyperbolic knot manifold $M$ has been verified in all cases other than small Seifert…

In characteristic two, it is shown that a central simple algebra of degree equal to a power of two with anisotropic orthogonal involution is totally decomposable, if it becomes either anisotropic or metabolic over all extensions of the…

We extend the jigsaw percolation model to analyze graphs where both underlying people and puzzle graphs are Erd\H{o}s-R\'enyi random graphs. Let $p_{\text{ppl}}$ and $p_{\text{puz}}$ denote the probability that an edge exists in the…

Exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian systems. Second-order EPs are by far the most studied due to their abundance, requiring only the tuning of…