scholarly journals A model of quotient spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 13-18
Author(s):  
Hawete Hattab
Keyword(s):  
Simple Model ◽  
Topological Space ◽  
Equivalence Relation ◽  
Quotient Space ◽  
Finite Model ◽  
Orbit Spaces ◽  
Quotient Spaces

AbstractLet R be an open equivalence relation on a topological space E. We define on E a new equivalence relation ̃ℜ̅ by x̃ ̃ℜ̅y if the closure of the R-trajectory of x is equal to the closure of the R-trajectory of y. The quotient space E/̃ ̃ℜ̅ is called the trajectory class space. In this paper, we show that the space E/̃ ̃ℜ̅ is a simple model of the quotient space E/R. This model can provide a finite model. Some applications to orbit spaces of groups of homeomorphisms and leaf spaces are given.

A quotient ordered space

1968 ◽  
Vol 64 (2) ◽  
pp. 317-322 ◽  
Author(s):  
S. D. McCartan
Keyword(s):  
Topological Space ◽  
Partial Order ◽  
Equivalence Relation ◽  
Quotient Space ◽  
Partial Orders ◽  
Projection Mapping ◽  
Quotient Spaces ◽  
The Family ◽  
Ordered Space ◽  
Image Position

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the propertya partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

scholarly journals On the Homomorphisms of the Lie Groups and

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Fatma Özdemir ◽  
Hasan Özekes
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Heisenberg Group ◽  
Lie Groups ◽  
Equivalence Relation ◽  
Quotient Space

We first construct all the homomorphisms from the Heisenberg group to the 3-sphere. Also, defining a topology on these homomorphisms, we regard the set of these homomorphisms as a topological space. Next, using the kernels of homomorphisms, we define an equivalence relation on this topological space. We finally show that the quotient space is a topological group which is isomorphic to .

On simple, weak and strong models of propositional calculi

1973 ◽  
Vol 74 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Ronald Harrop
Keyword(s):  
Simple Model ◽  
Propositional Calculus ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
Finite Structure ◽  
Propositional Calculi ◽  
Simple Models

In this paper we will be concerned primarily with weak, strong and simple models of a propositional calculus, simple models being structures of a certain type in which all provable formulae of the calculus are valid. It is shown that the finite model property defined in terms of simple models holds for all calculi. This leads to a new proof of the fact that there is no general effective method for testing, given a finite structure and a calculus, whether or not the structure is a simple model of the calculus.

Classification Algorithm of Case Retrieval Based on Granularity Calculation of Quotient Space

2020 ◽  
pp. 2150003
Author(s):  
Jiaxing Lu ◽  
Jiang Qing ◽  
Huang He ◽  
Zhang Zhengyong ◽  
Wang Rujing
Keyword(s):  
Quotient Space ◽  
Inner Product ◽  
Retrieval Algorithm ◽  
Case Based Reasoning ◽  
Case Retrieval ◽  
Current Case ◽  
Quotient Spaces ◽  
Similarity Calculation ◽  
Key Steps ◽  
Case Based

Case retrieval is one of the key steps of case-based reasoning. The quality of case retrieval determines the effectiveness of the system. The common similarity calculation methods based on attributes include distance and inner product. Different similarity calculations have different influences on the effect of case retrieval. How to combine different similarity calculation results to get a more widely used and better retrieval algorithm is a hot issue in the current case-based reasoning research. In this paper, the granularity of quotient space is introduced into the similarity calculation based on attribute, and a case retrieval algorithm based on granularity synthesis theory is proposed. This method first uses similarity calculation of different attributes to get different results of case retrieval, and considers that these classification results constitute different quotient spaces, and then organizes these quotient spaces according to granularity synthesis theory to get the classification results of case retrieval. The experimental results verify the validity and correctness of this method and the application potential of granularity calculation of quotient space in case-based reasoning.

scholarly journals On theS-Invariance Property forS-Flows

2010 ◽  
Vol 2010 ◽  
pp. 1-5
Author(s):  
Amin Saif ◽  
Adem Kılıçman
Keyword(s):  
Topological Space ◽  
Equivalence Relation ◽  
Transformation Semigroup ◽  
Invariance Property ◽  
Topological Monoid

We define an equivalence relation on a topological space which is acted by topological monoidSas a transformation semigroup. Then, we give some results about theS-invariant classes for this relation. We also provide a condition for the existence of relativeS-invariant classes.

scholarly journals The Mathematics of Compositional Analysis

2016 ◽  
Vol 45 (4) ◽  
pp. 57-71 ◽  
Author(s):  
Carles Barcelo-Vidal ◽  
Josep-Antoni Martín-Fernández
Keyword(s):  
Scale Invariance ◽  
Compositional Data ◽  
Quotient Space ◽  
Compositional Analysis ◽  
Space Structure ◽  
Compositional Data Analysis ◽  
New Developments ◽  
Quotient Spaces ◽  
Subcompositional Coherence ◽  
Logratio Transformations

The term compositional data analysis is historically associated to the approach based on the logratio transformations introduced in the eighties. Two main principles of this methodology are scale invariance and subcompositional coherence. New developments and concepts emerged in the last decade revealed the need to clarify the concepts of compositions, compositional sample space and subcomposition. In this work the mathematics of compositional analysis based on equivalence relation is presented. The two principles are essential attributes of the corresponding quotient space. A logarithmic isomorphism between quotient spaces induces a metric space structure for compositions. Using this structure, the statistical analysis of compositions consists of analysing logratio coordinates.

The Position of Rough Set in Soft Set

2012 ◽  
Vol 3 (3) ◽  
pp. 33-48
Author(s):  
Tutut Herawan
Keyword(s):  
Topological Space ◽  
Set Theory ◽  
Rough Set ◽  
Equivalence Relation ◽  
Rough Set Theory ◽  
Soft Set ◽  
General Topology ◽  
Discrete Topology ◽  
Special Case

In this paper, the author presents the concept of topological space that must be used to show a relation between rough set and soft set. There are two main results presented; firstly, a construction of a quasi-discrete topology using indiscernibility (equivalence) relation in rough set theory is described. Secondly, the paper describes that a “general” topology is a special case of soft set. Hence, it is concluded that every rough set can be considered as a soft set.

The Application of Fuzzy Equivalence Relation Based on the Quotient Space in Cluster Analysis

2014 ◽  
Vol 686 ◽  
pp. 333-339
Author(s):  
Qin Wei Shen ◽  
Yuan Zhang
Keyword(s):  
Cluster Analysis ◽  
Equivalence Relation ◽  
Quotient Space ◽  
Analysis Model ◽  
Dynamic Selection ◽  
Fuzzy Clustering Analysis ◽  
The Hierarchical Structure ◽  
Coarse To Fine ◽  
Selection Of ◽  
Relation Matrix

A fuzzy clustering analysis model based on the quotient space is proposed. Firstly, the conversion from coarse to fine granularity and the hierarchical structure are used to reduce the multidimensional samples. Secondly, the fuzzy compatibility relation matrix of the model is converted into fuzzy equivalence relation matrix. Finally, the diagram of clustering genealogy is generated according to the fuzzy equivalence relation matrix, which enables the dynamic selection of different thresholds to effectively solve the problem of cluster analysis of the samples with multi-dimensional attributes.

scholarly journals Semi-quotient mappings and spaces

2016 ◽  
Vol 14 (1) ◽  
pp. 1014-1022 ◽  
Author(s):  
Moiz ud Din Khan ◽  
Rafaqat Noreen ◽  
Muhammad Siddique Bosan
Keyword(s):  
Quotient Space ◽  
Topological Groups ◽  
Quotient Spaces ◽  
Continuous Mappings ◽  
Quotient Mappings

AbstractIn this paper, we continue the study of s-topological and irresolute-topological groups. We define semi-quotient mappings which are stronger than semi-continuous mappings, and then consider semi-quotient spaces and groups. It is proved that for some classes of irresolute-topological groups (G, *, τ) the semi-quotient space G/H is regular. Semi-isomorphisms of s-topological groups are also discussed.

scholarly journals The space of Keplerian orbits and a family of its quotient spaces

2021 ◽  
Vol 8 (2) ◽  
pp. 359-369
Author(s):  
Konstantin V. Kholshevnikov ◽  
◽  
Danila V. Milanov ◽  
Anastasia S. Shchepalova ◽  
◽  
...  
Keyword(s):  
Triangle Inequality ◽  
Quotient Space ◽  
Special Kind ◽  
Distance Functions ◽  
Future Research ◽  
Meteoroid Streams ◽  
C Language ◽  
Quotient Spaces ◽  
Keplerian Orbits ◽  
Elliptic Orbits

Distance functions on the set of Keplerian orbits play an important role in solving problems of searching for parent bodies of meteoroid streams. A special kind of such functions are distances in the quotient spaces of orbits. Three metrics of this type were developed earlier. These metrics allow to disregard the longitude of ascending node or the argument of pericenter or both. Here we introduce one more quotient space, where two orbits are considered identical if they differ only in their longitudes of nodes and arguments of pericenters, but have the same sum of these elements (the longitude of pericenter). The function q is defined to calculate distance between two equivalence classes of orbits. The algorithm for calculation of ̺6 value is provided along with a reference to the corresponding program, written in C++ language. Unfortunately, ̺6 is not a full-fledged metric. We proved that it satisfies first two axioms of metric space, but not the third one: the triangle inequality does not hold, at least in the case of large eccentricities. However there are two important particular cases when the triangle axiom is satisfied: one of three orbits is circular, longitudes of pericenters of all three orbits coincide. Perhaps the inequality holds for all elliptic orbits, but this is a matter of future research.

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