scholarly journals Estimates of antipodal sets in oriented real Grassmann manifolds

2015 ◽  
Vol 26 (06) ◽  
pp. 1541008 ◽  
Author(s):  
Hiroyuki Tasaki
Keyword(s):  
Combinatorial Problem ◽  
Grassmann Manifolds ◽  
Combinatorial Objects

We estimate the cardinalities of antipodal sets in oriented real Grassmann manifolds of low ranks. The author reduced the classification of antipodal sets in oriented real Grassmann manifolds to a certain combinatorial problem in a previous paper. So we can reduce estimates of the antipodal sets to those of certain combinatorial objects. The sequences of antipodal sets we obtained in previous papers show that the estimates we obtained in this paper are the best.

The Social Spiders in the Clustering of Texts

2012 ◽  
Vol 3 (3) ◽  
pp. 1-14 ◽  
Author(s):  
Reda Mohamed Hamou ◽  
Abdelmalek Amine ◽  
Ahmed Chaouki Lokbani
Keyword(s):  
Data Stream ◽  
Combinatorial Problem ◽  
Large Data ◽  
Social Spiders ◽  
Stream Flows ◽  
The Social ◽  
Textual Data ◽  
Biomimetic Approach ◽  
N Gram

In this paper the authors experiment and test a new biomimetic approach based on social spiders to solve a combinatorial problem ie the automatic classification of texts because a very large data stream flows and particularly on the web. Representation of textual data was performed by a method independent of the language ie n-gram characters and words because there is currently no method of learning that can directly represent unstructured data (text). To validate the classification, the authors used a measure of evaluation based on recall and precision (F-measure). During the experiment, the authors found a powerful visualization tool in social spiders that they exploit to make visual classification.

scholarly journals Complex manifolds with maximal torus actions

2019 ◽  
Vol 2019 (751) ◽  
pp. 121-184 ◽  
Author(s):  
Hiroaki Ishida
Keyword(s):  
Complete Classification ◽  
Vector Subspace ◽  
Toric Geometry ◽  
Complex Manifolds ◽  
Combinatorial Objects ◽  
Complex Dimension ◽  
Compact Torus ◽  
Equivalence Of Categories ◽  
Compact Tori

AbstractIn this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples {(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in {\mathfrak{g}}, complex vector subspace {\mathfrak{h}} of {\mathfrak{g}^{\mathbb{C}}} and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate {\mathbb{C}^{n}}-actions on compact connected complex manifolds of complex dimension n, and construction of concrete examples of non-Kähler manifolds.

Parametric Classification of Bingham Distributions Based on Grassmann Manifolds

2019 ◽  
Vol 28 (12) ◽  
pp. 5771-5784
Author(s):  
Muhammad Ali ◽  
Junbin Gao ◽  
Michael Antolovich
Keyword(s):  
Grassmann Manifolds ◽  
Parametric Classification

scholarly journals Structural Theory and Classification of 2D Adinkras

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Kevin Iga ◽  
Yan X. Zhang
Keyword(s):  
Structural Theory ◽  
Combinatorial Objects ◽  
Simple Characterization ◽  
The Way

Adinkras are combinatorial objects developed to study (1-dimensional) supersymmetry representations. Recently,2D Adinkrashave been developed to study2-dimensional supersymmetry. In this paper, we classify all2D Adinkras, confirming a conjecture of T. Hübsch. Along the way, we obtain other structural results, including a simple characterization of Hübsch’seven-split doubly even codes.

scholarly journals A classification of all 1-Salem graphs

2014 ◽  
Vol 17 (1) ◽  
pp. 582-594 ◽  
Author(s):  
Lee Gumbrell ◽  
James McKee
Keyword(s):  
New York ◽  
Minimal Polynomial ◽  
Cyclotomic Polynomials ◽  
Critical Interval ◽  
Combinatorial Structures ◽  
Induced Subgraph ◽  
Combinatorial Objects ◽  
Salem Number ◽  
Bipartite Case

AbstractOne way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph $G$ has an associated reciprocal polynomial $R_{G}$, and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is $R_{G}$ a product of cyclotomic polynomials (giving the cyclotomic graphs)? (b) when does $R_{G}$ have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trivial Salem graphs)? Cyclotomic graphs were classified by Smith (Combinatorial structures and their applications, Proceedings of Calgary International Conference, Calgary, AB, 1969 (eds R. Guy, H. Hanani, H. Saver and J. Schönheim; Gordon and Breach, New York, 1970) 403–406); the maximal connected ones are known as Smith graphs. Salem graphs are ‘spectrally close’ to being cyclotomic, in that nearly all their eigenvalues are in the critical interval $[-2,2]$. On the other hand, Salem graphs do not need to be ‘combinatorially close’ to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny.We define an $m$-Salem graph to be a connected Salem graph $G$ for which $m$ is minimal such that there exists an induced cyclotomic subgraph of $G$ that has $m$ fewer vertices than $G$. The $1$-Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a $1$-Salem graph as an induced subgraph, so these $1$-Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all $1$-Salem graphs: in the non-bipartite case there are $25$ infinite families and $383$ sporadic examples.

scholarly journals Wedge Operations and Torus Symmetries II

2017 ◽  
Vol 69 (4) ◽  
pp. 767-789 ◽  
Author(s):  
Suyoung Choi ◽  
Hanchul Park
Keyword(s):  
Positive Integer ◽  
Simplicial Complex ◽  
Main Idea ◽  
Picard Number ◽  
Fundamental Idea ◽  
Combinatorial Objects ◽  
Toric Topology ◽  
Simplicial Sphere ◽  
Fixed Positive Integer

AbstractA fundamental idea in toric topology is that classes of manifolds with well-behaved torus actions (simply, toric spaces) are classified by pairs of simplicial complexes and (non-singular) characteristic maps. In a previous paper, the authors provided a new way to find all characteristic maps on a simplicial complex K(J) obtainable by a sequence of wedgings from K.The main idea was that characteristic maps on K theoretically determine all possible characteristic maps on a wedge of K.We further develop our previous work for classification of toric spaces. For a star-shaped simplicial sphere K of dimension n-1 with m vertices, the Picard number Pic(K) of K is m-n. We call K a seed if K cannot be obtained by wedgings. First, we show that for a fixed positive integer 𝓁, there are at most finitely many seeds of Picard 𝓁 number supporting characteristic maps. As a corollary, the conjecture proposed by V. V. Batyrev in is solved affirmatively.Secondly, we investigate a systematicmethod to find all characteristic maps on K(J) using combinatorial objects called (realizable) puzzles that only depend on a seed K. These two facts lead to a practical way to classify the toric spaces of fixed Picard number.

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