Academic literature on the topic 'Arithmetical hyperplanes'

AbstractWe study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

In this paper we produce a few continuations of our previous work on partitions into fractions. Specifically, we study strictly increasing sequences of positive integers such that there are partitions for all natural numbers less than the floor of the sum of the first j terms divided by j, where j is greater than two. We also require that all summands be distinct terms drawn from this series of fractions. We call such sequences “semicomplete”. We find that there are only three semicomplete arithmetic sequences. We also study sequences that give the maximum number of pieces that an M dimensional hypercube can be cut into using N – 1 hyperplanes. We find that these are semicomplete in one, two, three, and four dimensions. As an aside, we use one of our generating functions to produce what appears to be a new identity for the Pell constant, a number which is closely connected to the density of solutions to the negative Pell equation.

AbstractWe provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions (APs). More precisely, we say $F$ uniformly avoids APs of length $k \geq 3$ if there is an $\epsilon>0$ such that one cannot find an AP of length $k$ and gap length $\Delta>0$ inside the $\epsilon \Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $\epsilon$. In the other direction, we provide examples of sets which uniformly avoid APs of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where APs are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretized version of a “reverse Kakeya problem:” we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates APs in every direction.

Abstract Consider the space M = O(p, q)/O(p) × O(q) of positive p-dimensional subspaces in a pseudo-Euclidean space V of signature (p, q), where p > 0, q > 1 and $(p,q)\neq (1,2)$, with integral structure: $V = V_{\mathbb{Z}} \otimes \mathbb{Z}$. Let Γ be an arithmetic subgroup in $G = O(V_{\mathbb{Z}})$, and $R \subset V_{\mathbb{Z}}$ a Γ-invariant set of vectors with negative square. Denote by R⊥ the set of all positive p-planes W ⊂ V such that the orthogonal complement W⊥ contains some r ∈ R. We prove that either R⊥ is dense in M or Γ acts on R with finitely many orbits. This is used to prove that the squares of primitive classes giving the rational boundary of the Kähler cone (i.e., the classes of “negative” minimal rational curves) on a hyperkähler manifold X are bounded by a number which depends only on the deformation class of X. We also state and prove the density of orbits in a more general situation when M is the space of maximal compact subgroups in a simple real Lie group.

Word embeddings, which represent words as dense feature vectors, are widely used in natural language processing. In their seminal paper on word2vec, Mikolov and colleagues showed that a feature space created by training a word prediction network on a large text corpus will encode semantic information that supports analogy by vector arithmetic, e.g., "king" minus "man" plus "woman" equals "queen". To help novices appreciate this idea, people have sought effective graphical representations of word embeddings. We describe a new interactive tool for visually exploring word embeddings. Our tool allows users to define semantic dimensions by specifying opposed word pairs, e.g., gender is defined by pairs such as boy/girl and father/mother, and age by pairs such as father/son and mother/daughter. Words are plotted as points in a zoomable and rotatable 3D space, where the third ”residual” dimension encodes distance from the hyperplane defined by all the opposed word vectors with age and gender subtracted out. Our tool allows users to visualize vector analogies, drawing the vector from “king” to “man” and a parallel vector from “woman” to “king-man+woman”, which is closest to “queen”. Visually browsing the embedding space and experimenting with this tool can make word embeddings more intuitive. We include a series of experiments teachers can use to help K-12 students appreciate the strengths and limitations of this representation.

Abstract Severi varieties and Brill–Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type {(1,n)} , we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system {|L|} for {0\leq\delta\leq n-1=p-2} (here p is the arithmetic genus of any curve in {|L|} ). We also show that a general genus g curve having as nodal model a hyperplane section of some {(1,n)} -polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many {(1,n)} -polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in {|L|} . It turns out that a general curve in {|L|} is Brill–Noether general. However, as soon as the Brill–Noether number is negative and some other inequalities are satisfied, the locus {|L|^{r}_{d}} of smooth curves in {|L|} possessing a {g^{r}_{d}} is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus {{\mathcal{M}}^{r}_{p,d}} having the expected codimension in the moduli space of curves {{\mathcal{M}}_{p}} . For {r=1} , the results are generalized to nodal curves.

Le monde numérique est parsemé de structures mathématiques discrètes, destinées à être facilement manipulables par un ordinateur tout en donnant à notre cerveau l'impression d'être de belles formes réelles continues. Les images numériques peuvent ainsi être vues comme des sous-ensembles de Z^2. En géométrie discrète, nous nous intéressons aux structures de Z^d et cherchons à établir des propriétés géométriques ou topologiques sur ces objets. Si les questions que nous nous posons sont relativement simples en géométrie euclidienne, elles deviennent beaucoup plus difficiles en géométrie discrète : plus de division, adieu les limites, tout n'est plus qu'arithmétique. Cette thèse est également l'occasion de jongler avec de nombreuses notions élémentaires de mathématiques et d'informatique (algèbre linéaire, anneaux, automates, analyse réelle, arithmétique, combinatoire) pour résoudre des questions de géométrie discrète. Nous nous intéressons à des structures fondamentales de cette géométrie : les hyperplans arithmétiques. Ceux-ci ont en effet une définition très simple et purement arithmétique : un hyperplan arithmétique est l'ensemble des points entiers situés entre deux hyperplans (réels) affines parallèles. Nous parlons dans cette thèse de trois problèmes portant sur les hyperplans arithmétiques : - la connexité : un hyperplan arithmétique est-il composé d'un seul morceau ou de plusieurs ? Apport principal de ce manuscrit, nous étendons des résultats déjà connus pour la connexité par faces pour des voisinages quelconques. Si certains phénomènes demeurent dans le cas général, l'explosion combinatoire rend difficile l'adaptation des algorithmes connus pour résoudre le problème. Nous adoptons donc une approche analytique et prouvons des propriétés de connexité en étudiant la régularité d'une fonction. - la reconnaissance : comment connaître les caractéristiques d'un hyperplan arithmétique ? Problème plus classique de géométrie discrète, avec une littérature très riche, nous proposons pour le résoudre un algorithme de reconnaissance reposant sur l'arbre de Stern-Brocot généralisé. Nous introduisons notamment la notion de corde séparante qui caractérise géométriquement les zones auxquelles appartiennent les paramètres d'un hyperplan arithmétique. - les transformations douces : comment transformer continûment un hyperplan arithmétique via des translations ou rotations ? Approche discrète des transformations homotopiques, nous caractérisons les mouvements de pixels possibles dans une structure discrète tout en préservant ses propriétés géométriques. Au-delà de l'étude de ces problèmes et des résultats que nous avons pu obtenir, cette thèse montre l'intérêt d'utiliser des réels, et notamment de l'analyse réelle, pour mieux comprendre les hyperplans arithmétiques. Ces derniers sont en effet caractérisés en grande partie par leur vecteur normal, souvent considéré entier pour obtenir des propriétés de périodicité. Considérer des vecteurs normaux réels quelconques permet de gagner en souplesse, et de faire disparaître les phénomènes de bruit induits par les relations arithmétiques du vecteur. S'ouvrir de nouveau au réel est enfin un moyen de créer des ponts vers d'autres branches des mathématiques, comme la combinatoire des mots ou les systèmes de numération
The digital world is littered with discrete mathematical structures, designed to be easily manipulated by a computer while giving our brains the impression of beautiful continuous real shapes. Digital images can thus be seen as subsets of Z^2. In discrete geometry, we are interested in the structures of Z^d and seek to establish geometric or topological properties on these objects. While the questions we ask are relatively simple in Euclidean geometry, they become much more difficult in discrete geometry: no more division, goodbye to limits, everything is just arithmetic. This thesis is also an opportunity to juggle many elementary notions of mathematics and computer science (linear algebra, rings, automata, real analysis, arithmetic, combinatorics) to solve discrete geometry questions. We are interested in the fundamental structures of this geometry: arithmetic hyperplanes. These have a very simple and purely arithmetical definition: an arithmetical hyperplane is the set of integer points lying between two parallel (real) affine hyperplanes. In this thesis, we discuss three problems involving arithmetic hyperplanes:- connectedness: is an arithmetic hyperplane composed of a single piece or of several pieces? The main contribution of this manuscript is to extend results already known for facewise connectedness for any neighbourhood. While certain phenomena remain in the general case, the combinatorial explosion makes it difficult to adapt known algorithms to solve the problem. We therefore adopt an analytical approach and prove connectivity properties by studying the regularity of a function. - recognition: how can we find out the characteristics of an arithmetic hyperplane? This is a more traditional problem in discrete geometry, with a very rich literature. To solve it, we propose a recognition algorithm based on the generalised Stern-Brocot tree. In particular, we introduce the notion of separating chord, which geometrically characterises the zones to which the parameters of an arithmetic hyperplane belong. - soft transformations: how can an arithmetic hyperplane be continuously transformed using translations or rotations? A discrete approach to homotopic transformations, we characterise the possible pixel movements in a discrete structure while preserving its geometric properties. Beyond the study of these problems and the results we were able to obtain, this thesis shows the interest of using the reals, and in particular real analysis, to better understand arithmetic hyperplanes. Arithmetic hyperplanes are largely characterised by their normal vector, which is often considered integer to obtain periodicity properties. Considering any real normal vectors provides greater flexibility and eliminates the noise induced by the arithmetic relationships of the vector. Finally, opening up to the real again is a way of building bridges to other branches of mathematics, such as word combinatorics or numbering systems