Low-dimensional lattices. VI. Voronoi reduction of three-dimensional lattices

1992 ◽  
Vol 436 (1896) ◽  
pp. 55-68 ◽  
Keyword(s):  
Simple Formula ◽  
Three Dimensional ◽  
Voronoi Cell ◽  
Dimensional Lattice ◽  
Usual Treatment ◽  
Low Dimensional

The aim of this paper is to describe how the Voronoi cell of a lattice changes as that lattice is continuously varied. The usual treatment is simplified by the introduction of new parameters called the vonorms and conorms of the lattice. The present paper deals with dimensions n ≼ 3; a sequel will treat four-dimensional lattices. An elegant algorithm is given for the Voronoi reduction of a three-dimensional lattice, leading to a new proof of Voronoi’s theorem that every lattice of dimension n ≼ 3 is of the first kind, and of Fedorov’s classification of the three-dimensional lattices into five types. There is a very simple formula for the determinant of a three-dimensional lattice in terms of its conorms.

The classification of three-dimensional Lie algeb­ras on complex field

2020 ◽  
pp. 143-155
Author(s):  
E. R. Shamardina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Structural Equations ◽  
Complex Numbers ◽  
Complex Field ◽  
Low Dimensional

In this paper, we study the classification of three-dimensional Lie al­gebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with re­spect to isomorphism, namely such quantities as the derivative of a subal­gebra and the center of a Lie algebra. The above classification is distin­guished from others by a more detailed and simple presentation. Any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie alge­bras. One of the reasons for considering these low dimensional Lie alge­bras that they often occur as subalgebras of large Lie algebras

scholarly journals Continued Classification of 3D Lattice Models in the Positive Octant

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Axel Bacher ◽  
Manuel Kauers ◽  
Rika Yatchak
Keyword(s):  
Asymptotic Behaviour ◽  
Generating Function ◽  
Asymptotic Form ◽  
Three Dimensional ◽  
Lattice Models ◽  
Dimensional Lattice ◽  
Lattice Walks ◽  
Small Set ◽  
International Audience

International audience We continue the investigations of lattice walks in the three-dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that their generating function is D-finite, and we isolate a small set of models whose nature remains unclear and requires further investigation. For these, we give some experimental results about their asymptotic behaviour, based on the inspection of a large number of initial terms. At least for some of them, the guessed asymptotic form seems to tip the balance towards non-D-finiteness.

scholarly journals Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1455
Author(s):  
Alina Dobrogowska ◽  
Karolina Wojciechowicz
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Three Dimensional ◽  
Poisson Structures ◽  
Triangular Matrices ◽  
Dual Spaces ◽  
Upper Triangular Matrices ◽  
Low Dimensional ◽  
Algebra Level

We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

scholarly journals Classification of real low-dimensional Jacobi (generalized)–Lie bialgebras

2016 ◽  
Vol 14 (01) ◽  
pp. 1750007 ◽  
Author(s):  
A. Rezaei-Aghdam ◽  
M. Sephid
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Lie Bialgebras ◽  
Structure Constants ◽  
Definition Of ◽  
Low Dimensional

We describe the definition of Jacobi (generalized)–Lie bialgebras [Formula: see text] in terms of structure constants of the Lie algebras [Formula: see text] and [Formula: see text] and components of their 1-cocycles [Formula: see text] and [Formula: see text] in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low-dimensional Jacobi–Lie bialgebras. In this way, we obtain and classify real two- and three-dimensional Jacobi–Lie bialgebras.

Phonetic iconicity in the evaluative morphology of a sample of Indo-European, Niger-Congo and Austronesian languages

2010 ◽  
Vol 3 (2) ◽  
pp. 156-180 ◽  
Author(s):  
Renáta Gregová ◽  
Lívia Körtvélyessy ◽  
Július Zimmermann
Keyword(s):  
Three Dimensional ◽  
Sound Symbolism ◽  
Three Dimensional Model ◽  
Austronesian Languages ◽  
Place Of Articulation ◽  
European Languages ◽  
Phonetic Symbolism ◽  
The Individual ◽  
Core Vocabulary

Universals Archive (Universal #1926) indicates a universal tendency for sound symbolism in reference to the expression of diminutives and augmentatives. The research ( Štekauer et al. 2009 ) carried out on European languages has not proved the tendency at all. Therefore, our research was extended to cover three language families – Indo-European, Niger-Congo and Austronesian. A three-step analysis examining different aspects of phonetic symbolism was carried out on a core vocabulary of 35 lexical items. A research sample was selected out of 60 languages. The evaluative markers were analyzed according to both phonetic classification of vowels and consonants and Ultan's and Niewenhuis' conclusions on the dominance of palatal and post-alveolar consonants in diminutive markers. Finally, the data obtained in our sample languages was evaluated by means of a three-dimensional model illustrating the place of articulation of the individual segments.

A Primer on Mapping Class Groups (PMS-49)

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Graduate Students ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Surface Bundles ◽  
Surface Homeomorphisms ◽  
Low Dimensional

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. It begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn–Nielsen–Baer–theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

scholarly journals Classification of Brainwaves for Sleep Stages by High-Dimensional FFT Features from EEG Signals

2020 ◽  
Vol 10 (5) ◽  
pp. 1797 ◽  
Author(s):  
Mera Kartika Delimayanti ◽  
Bedy Purnama ◽  
Ngoc Giang Nguyen ◽  
Mohammad Reza Faisal ◽  
Kunti Robiatul Mahmudah ◽  
...  
Keyword(s):  
Machine Learning ◽  
Sleep Stage ◽  
Machine Learning Algorithms ◽  
High Dimensional ◽  
Sleep Stages ◽  
Eeg Signals ◽  
Stage Classification ◽  
Sleep Stage Classification ◽  
Low Dimensional

Manual classification of sleep stage is a time-consuming but necessary step in the diagnosis and treatment of sleep disorders, and its automation has been an area of active study. The previous works have shown that low dimensional fast Fourier transform (FFT) features and many machine learning algorithms have been applied. In this paper, we demonstrate utilization of features extracted from EEG signals via FFT to improve the performance of automated sleep stage classification through machine learning methods. Unlike previous works using FFT, we incorporated thousands of FFT features in order to classify the sleep stages into 2–6 classes. Using the expanded version of Sleep-EDF dataset with 61 recordings, our method outperformed other state-of-the art methods. This result indicates that high dimensional FFT features in combination with a simple feature selection is effective for the improvement of automated sleep stage classification.

scholarly journals DISCRETE AND CONTINUUM APPROACHES TO THREE-DIMENSIONAL QUANTUM GRAVITY

1991 ◽  
Vol 06 (39) ◽  
pp. 3591-3600 ◽  
Author(s):  
HIROSI OOGURI ◽  
NAOKI SASAKURA
Keyword(s):  
Gauge Theory ◽  
Moduli Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Analogue Model ◽  
Gauge Invariant ◽  
Invariant Functions ◽  
Dimensional Lattice ◽  
Chern Simons ◽  
Dimensional Surface

It is shown that, in the three-dimensional lattice gravity defined by Ponzano and Regge, the space of physical states is isomorphic to the space of gauge-invariant functions on the moduli space of flat SU(2) connections over a two-dimensional surface, which gives physical states in the ISO(3) Chern–Simons gauge theory. To prove this, we employ the q-analogue of this model defined by Turaev and Viro as a regularization to sum over states. A recent work by Turaev suggests that the q-analogue model itself may be related to an Euclidean gravity with a cosmological constant proportional to 1/k2, where q=e2πi/(k+2).

scholarly journals How Can I Grab That?

i-com ◽  
2020 ◽  
Vol 19 (2) ◽  
pp. 67-85
Author(s):  
Matthias Weise ◽  
Raphael Zender ◽  
Ulrike Lucke
Keyword(s):  
Virtual Reality ◽  
User Experience ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Advantages And Disadvantages ◽  
Multiple Dimensions ◽  
Application Developers ◽  
Three Dimensional Space

AbstractThe selection and manipulation of objects in Virtual Reality face application developers with a substantial challenge as they need to ensure a seamless interaction in three-dimensional space. Assessing the advantages and disadvantages of selection and manipulation techniques in specific scenarios and regarding usability and user experience is a mandatory task to find suitable forms of interaction. In this article, we take a look at the most common issues arising in the interaction with objects in VR. We present a taxonomy allowing the classification of techniques regarding multiple dimensions. The issues are then associated with these dimensions. Furthermore, we analyze the results of a study comparing multiple selection techniques and present a tool allowing developers of VR applications to search for appropriate selection and manipulation techniques and to get scenario dependent suggestions based on the data of the executed study.

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