scholarly journals Counting and equidistribution in quaternionic Heisenberg groups

2021 ◽  
pp. 1-38
Author(s):  
JOUNI PARKKONEN ◽  
FRÉDÉRIC PAULIN
Keyword(s):  
Hyperbolic Geometry ◽  
Quaternion Algebra ◽  
Rational Points ◽  
Hyperbolic Spaces ◽  
Heisenberg Groups ◽  
Arithmetic Groups ◽  
Hermitian Forms ◽  
Dimension 2 ◽  
Counting Formula ◽  
The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

On the Hyperplane Sections Through two Given Points of an Algebraic Variety

1970 ◽  
Vol 22 (1) ◽  
pp. 128-133 ◽  
Author(s):  
Wei-Eihn Kuan
Keyword(s):  
Simple Proof ◽  
Hyperplane Section ◽  
Rational Points ◽  
Infinite Field ◽  
Important Special Case ◽  
Irreducible Curve ◽  
Irreducible Variety ◽  
Dimension 2 ◽  
Hyperplane Sections ◽  
Special Case

1. Let k be an infinite field and let V/k be an irreducible variety of dimension ≧ 2 in a projective n-space Pn over k. Let P and Q be two k-rational points on V In this paper, we describe ideal-theoretically the generic hyperplane section of V through P and Q (Theorem 1) and prove that the section is almost always an absolutely irreducible variety over k1/pe if V/k is absolutely irreducible (Theorem 3). As an application (Theorem 4), we give a new simple proof of an important special case of the existence of a curve connecting two rational points of an absolutely irreducible variety [4], namely any two k-rational points on V/k can be connected by an irreducible curve.I wish to thank Professor A. Seidenberg for his continued advice and encouragement on my thesis research.

Introduction to the Schwartz Space of T\G

1989 ◽  
Vol 41 (2) ◽  
pp. 285-320 ◽  
Author(s):  
W. Casselman
Keyword(s):  
Reductive Group ◽  
Rational Points ◽  
Schwartz Space ◽  
Parabolic Subgroups ◽  
Arithmetic Groups ◽  
Similar Question ◽  
Arithmetic Subgroup ◽  
Cohomology Of Arithmetic Groups ◽  
Image Position

Let G be the group of R-rational points on a reductive group defined over Q and T an arithmetic subgroup. The aim of this paper is to describe in some detail the Schwartz space (whose definition I recall in Section 1) and in particular to explain a decomposition of this space into constituents parametrized by the T-associate classes of rational parabolic subgroups of G. This is analogous to the more elementary of the two well known decompositions of L2 (T\G) in [20](or [17]), and a proof of something equivalent was first sketched by Langlands himself in correspondence with A. Borel in 1972. (Borel has given an account of this in [8].)Langlands’ letter was in response to a question posed by Borel concerning a decomposition of the cohomology of arithmetic groups, and the decomposition I obtain here was motivated by a similar question, which is dealt with at the end of the paper.

scholarly journals ON GENERATORS OF ARITHMETIC GROUPS OVER FUNCTION FIELDS

2011 ◽  
Vol 07 (06) ◽  
pp. 1573-1587
Author(s):  
MIHRAN PAPIKIAN
Keyword(s):  
Quaternion Algebra ◽  
Function Fields ◽  
Rational Functions ◽  
Arithmetic Groups ◽  
Finite Sets

Let F = 𝔽q(T) be the field of rational functions with 𝔽q-coefficients, and A = 𝔽q[T] be the subring of polynomials. Let D be a division quaternion algebra over F which is split at 1/T. For certain A-orders in D we find explicit finite sets generating their groups of units.

scholarly journals Factor Influencing Vaccine Rejection of Complete Basic Immunization in Indonesia

2021 ◽  
Vol 9 (E) ◽  
pp. 1300-1306
Author(s):  
Agustina Setyaningsih ◽  
Kemal N Siregar
Keyword(s):  
Social Media ◽  
Factor Analysis ◽  
Vaccine Efficacy ◽  
Side Effect ◽  
Psychological Factors ◽  
Secondary Data ◽  
Analysis Method ◽  
Dimension 2 ◽  
Childhood Vaccines ◽  
The Relationship

AIM: This study aims to identify psychological factors against vaccine rejection in Indonesia. The study also provides a review of the group of different factors on psychological factors in social media. METHODS: This study uses secondary data sourced from Facebook, Twitter, YouTube and Instagram about vaccines rejection from 2018 to 2019. That text is labeled based on seven psychological factors that influence vaccine rejection. The factor analysis method is used to determine the relationship between vaccine rejection and psychological factors. RESULTS: Dimension 1 focused on individual and group influences, where the correlation value between factors such as vaccine misinformation, health worker trust, perception of side effect is 0.906 (>0.5). Dimension 2 used different factors such as trust in the goverment, negative opinion about vaccine efficacy, and social influence as contextual/environmental influencers,with a correlation value of 0.866 (>0.5). Meanwhile, Dimension 3 with general perception is a factor in vaccine and vaccination specific problems with a correlation value of 0.940 (>0.5). CONCLUSION: Psychological factors are mainly associated with vaccine rejection. Stakeholders need to observe these factors in identifying conditions for childhood vaccines rejection posted on social media in Indonesia.

scholarly journals Visualization of Escher-like Spiral Patterns in Hyperbolic Space

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 134
Author(s):  
Chongyang Qiu ◽  
Xinfei Li ◽  
Jianhua Pang ◽  
Peichang Ouyang
Keyword(s):  
Hyperbolic Space ◽  
Hyperbolic Geometry ◽  
Fast Algorithm ◽  
Hyperbolic Spaces ◽  
Cyclic Symmetry ◽  
Simple Method ◽  
One To One

Spirals, tilings, and hyperbolic geometry are important mathematical topics with outstanding aesthetic elements. Nonetheless, research on their aesthetic visualization is extremely limited. In this paper, we give a simple method for creating Escher-like hyperbolic spiral patterns. To this end, we first present a fast algorithm to construct Euclidean spiral tilings with cyclic symmetry. Then, based on a one-to-one mapping between Euclidean and hyperbolic spaces, we establish two simple approaches for constructing spiral tilings in hyperbolic models. Finally, we use wallpaper templates to render such tilings, which results in the desired Escher-like hyperbolic spiral patterns. The method proposed is able to generate a great variety of visually appealing patterns.

scholarly journals Small zeros of hermitian forms over a quaternion algebra

2010 ◽  
Vol 142 (3) ◽  
pp. 251-266
Author(s):  
Wai Kiu Chan ◽  
Lenny Fukshansky
Keyword(s):  
Quaternion Algebra ◽  
Hermitian Forms

scholarly journals Hyperbolic Heterogeneous Information Network Embedding

2019 ◽  
Vol 33 ◽  
pp. 5337-5344 ◽  
Author(s):  
Xiao Wang ◽  
Yiding Zhang ◽  
Chuan Shi
Keyword(s):  
Complex Network ◽  
Power Law ◽  
Hyperbolic Geometry ◽  
Hyperbolic Spaces ◽  
Superior Performance ◽  
Information Network ◽  
Network Embedding ◽  
Euclidean Spaces ◽  
Heterogeneous Information Network ◽  
Heterogeneous Information

Heterogeneous information network (HIN) embedding, aiming to project HIN into a low-dimensional space, has attracted considerable research attention. Most of the exiting HIN embedding methods focus on preserving the inherent network structure and semantic correlations in Euclidean spaces. However, one fundamental problem is that whether the Euclidean spaces are the appropriate or intrinsic isometric spaces of HIN? Recent researches argue that the complex network may have the hyperbolic geometry underneath, because the underlying hyperbolic geometry can naturally reflect some properties of complex network, e.g., hierarchical and power-law structure. In this paper, we make the first effort toward HIN embedding in hyperbolic spaces. We analyze the structures of two real-world HINs and discover some properties, e.g., the power-law distribution, also exist in HIN. Therefore, we propose a novel hyperbolic heterogeneous information network embedding model. Specifically, to capture the structure and semantic relations between nodes, we employ the meta-path guided random walk to sample the sequences for each node. Then we exploit the distance in hyperbolic spaces as the proximity measurement. The hyperbolic distance is able to meet the triangle inequality and well preserve the transitivity in HIN. Our model enables the nodes and their neighborhoods have small hyperbolic distances. We further derive the effective optimization strategy to update the hyperbolic embeddings iteratively. The experimental results, in comparison with the state-of-the-art, demonstrate that our proposed model not only has superior performance on network reconstruction and link prediction tasks but also shows its ability of capture hierarchy structure in HIN via visualization.

scholarly journals A WEIGHT INDEPENDENCE RESULT FOR QUATERNIONIC HECKE ALGEBRAS

2013 ◽  
Vol 09 (08) ◽  
pp. 1895-1922
Author(s):  
LEA TERRACINI
Keyword(s):  
Modular Forms ◽  
Quaternion Algebra ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Hecke Operators ◽  
Langlands Correspondence ◽  
Independence Result ◽  
Relationship Of ◽  
The Relationship

Let p be a prime and B be a quaternion algebra indefinite over Q and ramified at p. We consider the space of quaternionic modular forms of weight k and level p∞, endowed with the action of Hecke operators. By using cohomological methods, we show that the p-adic topological Hecke algebra does not depend on the weight k. This result provides a quaternionic version of a theorem proved by Hida for classical modular forms; we discuss the relationship of our result to Hida's theorem in terms of Jacquet–Langlands correspondence.

scholarly journals Combinatorial Modulus on Boundary of Right-Angled Hyperbolic Buildings

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Antoine Clais
Keyword(s):  
Hyperbolic Spaces ◽  
Weak Version ◽  
Dimension 2

AbstractIn this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular, we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner property. This is motivated by the fact that the quasiconformal structure of the boundary led to many results of rigidity in hyperbolic spaces since G.D.Mostow. In the case of buildings of dimension 2, many work have been done by M. Bourdon and H. Pajot. In particular, the Loewner property on the boundary permitted them to prove the quasi-isometry rigidity of right-angled Fuchsian buildings.

scholarly journals A Multiple Correspondence Analysis of Patterns of CBD Use in Hemp and Marijuana Users

2021 ◽  
Vol 11 ◽  
Author(s):  
Joseph R. Vilches ◽  
Mackenzie B. Taylor ◽  
Francesca M. Filbey
Keyword(s):  
Correspondence Analysis ◽  
Online Survey ◽  
Multiple Correspondence Analysis ◽  
Marijuana Use ◽  
Chi Square ◽  
Dimension 2 ◽  
Antecedents And Consequences ◽  
Survey Responses ◽  
The Relationship ◽  
Lower Educational Attainment

Background: With the passing of the 2018 Agriculture Improvement Act that legalized hemp-derived products, i.e., cannabidiol (CBD), the use of CBD has increased exponentially. To date, the few studies that have characterized individuals who use CBD suggest that co-use of CBD and tetrahydrocannabinol (THC)-dominant cannabis, i.e., marijuana, is highly prevalent. It is, therefore, important to investigate the relationship between CBD use and marijuana use to understand the antecedents and consequences of co-use of these two cannabis products.Methods: We conducted an online survey using structured questionnaires to determine differences in CBD users with (CBD+MJ) and without co-morbid marijuana use. Group comparisons were carried out using chi-square tests and ANOVA. Multiple correspondence analysis (MCA) with bootstrap ratio testing was performed to examine the relationship between the categorical data.Results: We received 182 survey responses from current CBD users. CBD+MJ had more types of CBD administration (F = 17.07, p < 0.001) and longer lifetime duration of CBD use (χ2 = 12.85, p < 0.05). Results from the MCA yielded two statistically significant dimensions that accounted for 77% of the total variance. Dimension 1 (representing 57% of the variance) associated CBD+MJ with indication of CBD use for medical ailments, use of CBD for more than once a day for longer than 2 years, applying CBD topically or consuming it via vaping or edibles, being female, and, having lower educational attainment. Dimension 2 (representing 20% of the variance) separated the groups primarily on smoking-related behaviors where CBD+MJ was associated with smoking CBD and nicotine.Conclusions: Identifying the factors that influence use of CBD and marijuana can inform future studies on the risks and benefits associated with each substance as well as the impacts of policies related to cannabis-based products.

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