scholarly journals FRACTAL ANALYSIS OF HOPF BIFURCATION AT INFINITY

2012 ◽  
Vol 22 (12) ◽  
pp. 1230043 ◽  
Author(s):  
GORAN RADUNOVIĆ ◽  
DARKO ŽUBRINIĆ ◽  
VESNA ŽUPANOVIĆ
Keyword(s):  
Hopf Bifurcation ◽  
Fractal Analysis ◽  
Stereographic Projection ◽  
Riemann Sphere ◽  
Box Dimension ◽  
Euclidean Spaces ◽  
Equivalent Definition ◽  
Basic Properties ◽  
Definition Of

Using geometric inversion with respect to the origin, we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the Riemann sphere. We study its basic properties, and apply it to the study of the Hopf–Takens bifurcation at infinity.

New definition of MV-semimodules

2021 ◽  
pp. 1-12
Author(s):  
Simin Saidi Goraghani ◽  
Rajab Ali Borzooei ◽  
Sun Shin Ahn
Keyword(s):  
Basic Properties ◽  
Mv Algebra ◽  
Definition Of ◽  
Torsion Free

In recent years, A. Di Nola et al. studied the notions of MV-semiring and semimodules and investigated related results [9, 10, 12, 26]. Now in this paper, by using an MV-semiring and an MV-algebra, we introduce the new definition of MV-semimodule, study basic properties and find some examples. Then we study A-ideals on MV-semimodules and Q-ideals on MV-semirings, and by using them, we study the quotient structures of MV-semimodule. Finally, we present the notions of prime A-ideal, torsion free MV-semimodule and annihilator on MV-semimodule and we study the relations among them.

scholarly journals Strong submeasures and applications to non-compact dynamical systems

2020 ◽  
pp. 1-23
Author(s):  
TUYEN TRUNG TRUONG
Keyword(s):  
Metric Space ◽  
Bounded Operator ◽  
Continuous Functions ◽  
Open Dense Subset ◽  
Compact Metric Space ◽  
Basic Properties ◽  
Banach Theorem ◽  
Complex Varieties ◽  
Definition Of ◽  
Meromorphic Maps

Abstract A strong submeasure on a compact metric space X is a sub-linear and bounded operator on the space of continuous functions on X. A strong submeasure is positive if it is non-decreasing. By the Hahn–Banach theorem, a positive strong submeasure is the supremum of a non-empty collection of measures whose masses are uniformly bounded from above. There are many natural examples of continuous maps of the form $f:U\rightarrow X$ , where X is a compact metric space and $U\subset X$ is an open-dense subset, where f cannot extend to a reasonable function on X. We can mention cases such as transcendental maps of $\mathbb {C}$ , meromorphic maps on compact complex varieties, or continuous self-maps $f:U\rightarrow U$ of a dense open subset $U\subset X$ where X is a compact metric space. For the aforementioned mentioned the use of measures is not sufficient to establish the basic properties of ergodic theory, such as the existence of invariant measures or a reasonable definition of measure-theoretic entropy and topological entropy. In this paper we show that strong submeasures can be used to completely resolve the issue and establish these basic properties. In another paper we apply strong submeasures to the intersection of positive closed $(1,1)$ currents on compact Kähler manifolds.

scholarly journals Conditional Rényi Entropy and the Relationships between Rényi Capacities

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 526
Author(s):  
Gautam Aishwarya ◽  
Mokshay Madiman
Keyword(s):  
Mutual Information ◽  
Renyi Entropy ◽  
Rényi Entropy ◽  
Basic Properties ◽  
Reference Measure ◽  
Definition Of ◽  
Discrete Setting

The analogues of Arimoto’s definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored.

Remarks on some generalization of the notion of microscopic sets

2020 ◽  
Vol 70 (6) ◽  
pp. 1349-1356
Author(s):  
Aleksandra Karasińska
Keyword(s):  
General Sense ◽  
Equivalent Definition ◽  
Definition Of ◽  
Null Set

AbstractWe consider properties of defined earlier families of sets which are microscopic (small) in some sense. An equivalent definition of considered families is given, which is helpful in simplifying a proof of the fact that each Lebesgue null set belongs to one of these families. It is shown that families of sets microscopic in more general sense have properties analogous to the properties of the σ-ideal of classic microscopic sets.

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

scholarly journals From Euclidean Spaces to Metric Spaces

2019 ◽  
Vol 8 (1) ◽  
Author(s):  
Ryan Joseph Rogers ◽  
Ning Zhong
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Euclidean Spaces ◽  
Definition Of

In this note, we provide the definition of a metric space and establish that, while all Euclidean spaces are metric spaces, not all metric spaces are Euclidean spaces. It is then natural and interesting to ask which theorems that hold in Euclidean spaces can be extended to general metric spaces and which ones cannot be extended. We survey this topic by considering six well-known theorems which hold in Euclidean spaces and rigorously exploring their validities in general metric spaces.

scholarly journals Some fixed point theorems in n-normed spaces

2020 ◽  
Vol 25 (3) ◽  
pp. 1-15 ◽  
Author(s):  
Hanan Sabah Lazam ◽  
Salwa Salman Abed
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Normed Space ◽  
Fixed Point Theorems ◽  
Normed Spaces ◽  
Weak Contraction ◽  
Contraction Mappings ◽  
Basic Properties ◽  
Definition Of

In this article, we recall the definition of a real n-normed space and some basic properties. fixed point theorems for types of Kannan, Chatterge, Zamfirescu, -Weak contraction and  - (,)-Weak contraction mappings in  Banach spaces.

scholarly journals Fuzzy Soft Multiset Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Shawkat Alkhazaleh ◽  
Abdul Razak Salleh
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Fuzzy Set ◽  
Soft Set ◽  
Mathematical Tool ◽  
Basic Properties ◽  
Definition Of

In 1999 Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Alkhazaleh et al. in 2011 introduced the definition of a soft multiset as a generalization of Molodtsov's soft set. In this paper we give the definition of fuzzy soft multiset as a combination of soft multiset and fuzzy set and study its properties and operations. We give examples for these concepts. Basic properties of the operations are also given. An application of this theory in decision-making problems is shown.

On EMV-Semirings

2019 ◽  
Vol 69 (4) ◽  
pp. 739-752 ◽  
Author(s):  
R. A. Borzooei ◽  
M. Shenavaei ◽  
A. Di Nola ◽  
O. Zahiri
Keyword(s):  
Distributive Lattice ◽  
Algebraic Structure ◽  
Maximal Ideal ◽  
Boolean Algebras ◽  
Algebraic Extension ◽  
Theoretic Approach ◽  
Basic Properties ◽  
Mv Algebra ◽  
Definition Of ◽  
Generalized Boolean Algebras

Abstract The paper deals with an algebraic extension of MV-semirings based on the definition of generalized Boolean algebras. We propose a semiring-theoretic approach to EMV-algebras based on the connections between such algebras and idempotent semirings. We introduce a new algebraic structure, not necessarily with a top element, which is called an EMV-semiring and we get some examples and basic properties of EMV-semiring. We show that every EMV-semiring is an EMV-algebra and every EMV-semiring contains an MV-semiring and an MV-algebra. Then, we study EMV-semiring as a lattice and prove that any EMV-semiring is a distributive lattice. Moreover, we define an EMV-semiring homomorphism and show that the categories of EMV-semirings and the category of EMV-algebras are isomorphic. We also define the concepts of GI-simple and DLO-semiring and prove that every EMV-semiring is a GI-simple and a DLO-semiring. Finally, we propose a representation for EMV-semirings, which proves that any EMV-semiring is either an MV-semiring or can be embedded into an MV-semiring as a maximal ideal.

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