Related papers: Vector Partition Identities for $2$D, $3$D and $n$…
Given a positive integer $n$ and a partition $(n_1,\ldots,n_r)$ of $n$, one can consider the associated $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$. These multiprojective spaces are…
We provide a simple derivation of particle-vortex duality in d=2+1 dimensions. Our starting point is a relativistic form of flux attachment, designed to transmute the statistics of particles. From this seed, we derive a web of new…
We study the field of rational first integrals of distributions. We show that for a distribution on 3 dimensional manifolds there exists a tangent vector field with the same field of first integrals. We also show a similar result for…
We investigate an integrable property and observables of 2 dimensional N=(4,4) topological field theory defined on a discrete lattice by using the "orbifolding" and "deconstruction" methods. We show that our lattice model possesses the…
We Define moments of partitions of integers, and show that they appear in higher order derivatives of certain combinations of functions.
We construct a class of lattices in three and higher dimensions for which the number of dimer coverings can be determined exactly using elementary arguments. These lattices are a generalization of the two-dimensional kagome lattice, and the…
We study the bounded regions in a generic slice of the hyperplane arrangement in $\mathbb{R}^n$ consisting of the hyperplanes defined by $x_i$ and $x_i+x_j$. The bounded regions are in bijection with several classes of combinatorial…
Reasoning 3D shapes from 2D images is an essential yet challenging task, especially when only single-view images are at our disposal. While an object can have a complicated shape, individual parts are usually close to geometric primitives…
Many polytopes arising in polyhedral combinatorics are linear projections of higher-dimensional polytopes with significantly fewer facets. Such lifts may yield compressed representations of polytopes, which are typically used to construct…
We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of…
The main objective of this thesis is a classification project for integral lattices. Using Kneser's neighbour method we have developed the computer program tn to classify complete genera of integral lattices. Main results are detailed…
For each positive integer $n$, we define the divisibility relation graph $D_n$ whose vertex set is the set of divisors of $n$, and in which two vertices are adjacent if one is a divisor of the other. This type of graph is a special case of…
Point defects govern many important functional properties of two-dimensional (2D) materials. However, resolving the three-dimensional (3D) arrangement of these defects in multi-layer 2D materials remains a fundamental challenge, hindering…
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level…
Using shift vector method we obtain a large class of self-dual lattices of dimension $(l,l)$, which has a one to one correspondence with modular invariants of free bosonic theory compactified on co-root lattice of a rank $l$ Lie group. Then…
The aim of this paper is to investigate an attempt to build a binary classification algorithm using principles of geometry such as vectors, planes, and vector algebra. The basic idea behind the proposed algorithm is that a hyperplane can be…
Extending the method of the paper [FS3] we prove three structure theorems for vector valued modular forms, where two correspond to 4-dimensional cases (two hermitian modular groups, one belonging to the field of Eisenstein numbers, the…
The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes…
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 give a series of recursive identities for the number of partitions with exactly $k$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of…