Related papers: The complete classification of empty lattice $4$-s…
Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice simplices defining weighted projective spaces. We investigate the question of…
Provided n points in an (n-1)-dimensional affine space, and one ordering of the points for each coordinate, we address the problem of testing whether these orderings determine if the points are the vertices of a simplex (i.e. are affinely…
In this study, we investigate the lattice angle, which is defined as the angle between two vectors whose components are integers. We focus on the set of angles between a fixed integer vector and other integer vectors. For…
We construct several families of perfect sublattices with minimum $4$ of $\mathbb Z^d$. In particular, the number of $d-$dimensional perfect integral lattices with minimum $4$ grows faster than $d^k$ for every exponent $k$.
In this paper we construct a family of simply connected, spin, non-complex, symplectic 4-manifolds which cover all but finitely many allowed lattice points $(\chi, c)$ lying in $0 \leq c \leq 8.76\chi$. Furthermore, as a corollary, we prove…
Flat plumbing basket surfaces of links were introduced to study the geometry of the complement of the links. In present article, we study links of the flat plumbing basket numbers $4$ or less using a special presentation of the flat…
Maximally supersymmetric Yang--Mills theory (N=4 SYM) is conformal for any value of the coupling. Lattice regularization breaks conformality through the introduction of a non-zero lattice spacing and a finite lattice volume. This…
Let $\mathcal{T}^d(1)$ be the set of all $d$-dimensional simplices $T$ in $\real^d$ with integer vertices and a single integer point in the interior of $T$. It follows from a result of Hensley that $\mathcal{T}^d(1)$ is finite up to affine…
In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…
We construct a torsion-free arithmetic lattice in $\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))\times\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))$ arising from a quaternion algebra over $\mathbb{F}_2(z)$. It is the fundamental group of a square complex…
An integral lattice which is generated by some vectors of norm $q$ is called $q$-lattice. Classification of 3-lattices of dimension at most four is given by Mimura (On 3-lattice, 2006). As a expansion, we give a classification of 3-lattices…
We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and…
We study certain lattices constructed from finite abelian groups. We show that such a lattice is eutactic, thereby confirming a conjecture by B\"ottcher, Eisenbarth, Fukshansky, Garcia, Maharaj. Our methods also yield simpler proofs of two…
We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope…
In 1982, Zaks, Perles and Wills discovered a d-dimensional lattice simplex S_{d,k} with k interior lattice points, whose volume is linear in k and doubly exponential in the dimension d. It is conjectured that, for all d \ge 3 and k \ge 1,…
We present slight refinements of known general lower and upper bounds on sizes of extended formulations for polytopes. With these observations we are able to compute the extension complexities of all 0/1-polytopes up to dimension 4. We…
A standard barely set-valued tableau of shape $\lambda$ is a filling of the Young diagram $\lambda$ with integers $1,2,\dots,|\lambda|+1$ such that the integers are increasing in each row and column, and every cell contains one integer…
We study flat metrics arising from right regular $n$-prisms by viewing them as $n$-differentials and analyzing their associated unfoldings. We show that the unfolding of a right regular $n$-prism is never a lattice surface unless $n=4$, in…
We classify the four dimensional perfect non-simple evolution algebras over a field having characteristic different from 2 and in which there are roots of orders 2, 3 and 7.
We begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the following question: given $k\in \mathbb{N}$, what is the maximum number of points in a plane that determine at most $k$ distinct distances, and can such optimal…