continuous maps
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Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 32
Author(s):  
Zachary McGuirk ◽  
Byungdo Park

In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give a construction of a cofiber of a directed graph map whose image is contractible in the cofiber. Our work reveals that a category-theoretically correct construction in continuous setup is no longer correct when it is discretized and hence leads to look at canonical constructions in category theory in a different perspective.


2021 ◽  
pp. 63-98
Author(s):  
V. Baladze ◽  
A. Beridze ◽  
R. Tsinaridze
Keyword(s):  

Author(s):  
R. Kazemi ◽  
M.R. Miri ◽  
G.R.M. Borzadaran

The category of metric spaces is a subcategory of quasi-metric spaces. It is shown that the entropy of a map when symmetric properties is included is greater or equal to the entropy in the case that the symmetric property of the space is not considered. The topological entropy and Shannon entropy have similar properties such as nonnegativity, subadditivity and conditioning reduces entropy. In other words, topological entropy is supposed as the extension of classical entropy in dynamical systems. In the recent decade, different extensions of Shannon entropy have been introduced. One of them which generalizes many classical entropies is unified $(r,s)$-entropy. In this paper, we extend the notion of unified $(r, s)$-entropy for the continuous maps of a quasi-metric space via spanning and separated sets. Moreover, we survey unified $(r, s)$-entropy of a map for two metric spaces that are associated with a given quasi-metric space and compare unified $(r, s)$-entropy of a map of a given quasi-metric space and the maps of its associated metric spaces. Finally we define Tsallis topological entropy for the continuous map on a quasi-metric space via Bowen's definition and analyze some properties such as chain rule.


2021 ◽  
Vol 2070 (1) ◽  
pp. 012024
Author(s):  
A. Mughil ◽  
A. Vadivel ◽  
O. Uma Maheswari ◽  
G. Saravanakumar

Abstract We introduce fuzzy θ*-semicontinuous mappings and relate with fuzzy continuity, fuzzy θ-continuity, fuzzy a-continuity, fuzzy semicontinuity, fuzzy θ-semicontinuity, fuzzy Y-continuity, fuzzy Z-continuity and fuzzy γ-continuity in Ŝostak sense.


2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Hariwan Z. Ibrahim ◽  
Tareq M. Al-shami ◽  
O. G. Elbarbary

The purpose of this paper is to define the concept of (3, 2)-fuzzy sets and discuss their relationship with other kinds of fuzzy sets. We describe some of the basic set operations on (3, 2)-fuzzy sets. (3, 2)-Fuzzy sets can deal with more uncertain situations than Pythagorean and intuitionistic fuzzy sets because of their larger range of describing the membership grades. Furthermore, we familiarize the notion of (3, 2)-fuzzy topological space and discuss the master properties of (3, 2)-fuzzy continuous maps. Then, we introduce the concept of (3, 2)-fuzzy points and study some types of separation axioms in (3, 2)-fuzzy topological space. Moreover, we establish the idea of relation in (3, 2)-fuzzy set and present some properties. Ultimately, on the basis of academic performance, the decision-making approach of student placement is presented via the proposed (3, 2)-fuzzy relation to ascertain the suitability of colleges to applicants.


2021 ◽  
pp. 209-224
Author(s):  
Nazir Ahmad Ahengar ◽  
Jitendra Kumar Maitra ◽  
Roshani Sharma ◽  
Sujeet Chaturvedi ◽  
Mudasir Ahmad
Keyword(s):  

2021 ◽  
Vol 40 (5) ◽  
pp. 1249-1266
Author(s):  
Boulbaba Ghanmi ◽  
Rim Messaoud ◽  
Amira Missaoui

In this paper, we introduce Tαm-Super-Spaces, αm-contra-closed maps, αm-contra-open maps, αm-contra-continuous maps, αm-contrairresolute maps, b-ω-open sets and Continuity via b-ω-open sets and studied some of their properties


2021 ◽  
Vol 76 (5) ◽  
pp. 821-881
Author(s):  
L. S. Efremova ◽  
E. N. Makhrova

Abstract The survey is devoted to the topological dynamics of maps defined on one-dimensional continua such as a closed interval, a circle, finite graphs (for instance, finite trees), or dendrites (locally connected continua without subsets homeomorphic to a circle). Connections between the periodic behaviour of trajectories, the existence of a horseshoe and homoclinic trajectories, and the positivity of topological entropy are investigated. Necessary and sufficient conditions for entropy chaos in continuous maps of an interval, a circle, or a finite graph, and sufficient conditions for entropy chaos in continuous maps of dendrites are presented. Reasons for similarities and differences between the properties of maps defined on the continua under consideration are analyzed. Extensions of Sharkovsky’s theorem to certain discontinuous maps of a line or an interval and continuous maps on a plane are considered. Bibliography: 207 titles.


2021 ◽  
Author(s):  
Stephen Michael Town ◽  
Jennifer Kim Bizley

The location of sounds can be described in multiple coordinate systems that are defined relative to ourselves, or the world around us. World-centered hearing is critical for stable understanding of sound scenes, yet it is unclear whether this ability is unique to human listeners or generalizes to other species. Here, we establish novel behavioral tests to determine the coordinate systems in which non-human listeners (ferrets) can localize sounds. We found that ferrets could learn to discriminate sounds using either world-centered or head-centered sound location, as evidenced by their ability to discriminate locations in one space across wide variations in sound location in the alternative coordinate system. Using infrequent probe sounds to assess broader generalization of spatial hearing, we demonstrated that in both head and world-centered localization, animals used continuous maps of auditory space to guide behavior. Single trial responses of individual animals were sufficiently informative that we could then model sound localization using speaker position in specific coordinate systems and accurately predict ferrets' actions in held-out data. Our results demonstrate that non-human listeners can thus localize sounds in multiple spaces, including those defined by the world that require abstraction across traditional, head-centered sound localization cues.


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