scholarly journals Translation, modulation and dilation systems in set-valued signal processing

2018 ◽  
Vol 10 (1) ◽  
pp. 143-164 ◽  
Author(s):  
H. Levent ◽  
Y. Yilmaz
Keyword(s):  
Signal Processing ◽  
Linear Space ◽  
Algebraic Structure ◽  
Compact Convex ◽  
Inner Product ◽  
Complex Numbers ◽  
Real Numbers ◽  
Aumann Integral ◽  
Definition Of ◽  
Interval Valued

In this paper, we investigate a very important function space consists of set-valued functions defined on the set of real numbers with values on the space of all compact-convex subsets of complex numbers for which the $p$th power of their norm is integrable. In general, this space is denoted by $L^{p}% (\mathbb{R},\Omega(\mathbb{C}))$ for $1\leq p<\infty$ and it has an algebraic structure named as a quasilinear space which is a generalization of a classical linear space. Further, we introduce an inner-product (set-valued inner product) on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ and we think it is especially important to manage interval-valued data and interval-based signal processing. This also can be used in imprecise expectations. The definition of inner-product on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is a Hilbert quasilinear space. Finally, we give translation, modulation and dilation operators which are three fundational set-valued operators on Hilbert quasilinear space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$.

A Reproducing Kernel Hilbert Space Framework for Spike Train Signal Processing

2009 ◽  
Vol 21 (2) ◽  
pp. 424-449 ◽  
Author(s):  
António R. C. Paiva ◽  
Il Park ◽  
José C. Príncipe
Keyword(s):  
Signal Processing ◽  
Spike Train ◽  
Point Processes ◽  
Clustering Algorithm ◽  
Reproducing Kernel ◽  
Spike Trains ◽  
Inner Product ◽  
Inner Products ◽  
Definition Of ◽  
Intensity Functions

This letter presents a general framework based on reproducing kernel Hilbert spaces (RKHS) to mathematically describe and manipulate spike trains. The main idea is the definition of inner products to allow spike train signal processing from basic principles while incorporating their statistical description as point processes. Moreover, because many inner products can be formulated, a particular definition can be crafted to best fit an application. These ideas are illustrated by the definition of a number of spike train inner products. To further elicit the advantages of the RKHS framework, a family of these inner products, the cross-intensity (CI) kernels, is analyzed in detail. This inner product family encapsulates the statistical description from the conditional intensity functions of spike trains. The problem of their estimation is also addressed. The simplest of the spike train kernels in this family provide an interesting perspective to others' work, as will be demonstrated in terms of spike train distance measures. Finally, as an application example, the RKHS framework is used to derive a clustering algorithm for spike trains from simple principles.

scholarly journals Some Properties of Fuzzy Anti-Inner Product Spaces

2020 ◽  
pp. 3042-3047
Author(s):  
Radhi I. M. Ali ◽  
Esraa A. Hussein
Keyword(s):  
Linear Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Definition Of

In this paper, the definition of fuzzy anti-inner product in a linear space is introduced. Some results of fuzzy anti-inner product spaces are given, such as the relation between fuzzy inner product space and fuzzy anti-inner product. The notion of minimizing vector is introduced in fuzzy anti-inner product settings.

Thinking the Unthinkable: The Story of Complex Numbers (with a Moral)

1988 ◽  
Vol 81 (7) ◽  
pp. 583-592 ◽  
Author(s):  
Israel Kleiner
Keyword(s):  
History Of Mathematics ◽  
Complex Numbers ◽  
Real Numbers ◽  
Usual Definition ◽  
History Of ◽  
Definition Of ◽  
The Impact ◽  
Ordered Pairs

The usual definition of complex numbers, either as ordered pairs (a, b) of real numbers or as “numbers” of the form a + bi, does not give any indication of their long and tortuous evolution, which lasted about three hundred years. I want to describe this evolution very briefly because I think some lessons can be learned from this story, just as from many other such stories concerning the evolution of a concept, result, or theory. These lessons have to do with the impact of the history of mathematics on our understanding of mathematics and on our effectiveness in teaching it. But more about the moral of this story later.

Generalized interval-valued hesitant intuitionistic fuzzy soft sets

2021 ◽  
pp. 1-12
Author(s):  
Admi Nazra ◽  
Yudiantri Asdi ◽  
Sisri Wahyuni ◽  
Hafizah Ramadhani ◽  
Zulvera
Keyword(s):  
Hesitant Fuzzy Set ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Binary Operations ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Soft Sets ◽  
Comparison Table ◽  
Definition Of ◽  
Interval Valued ◽  
Intuitionistic Fuzzy Soft Sets

This paper aims to extend the Interval-valued Intuitionistic Hesitant Fuzzy Set to a Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS). Definition of a GIVHIFSS and some of their operations are defined, and some of their properties are studied. In these GIVHIFSSs, the authors have defined complement, null, and absolute. Soft binary operations like operations union, intersection, a subset are also defined. Here is also verified De Morgan’s laws and the algebraic structure of GIVHIFSSs. Finally, by using the comparison table, a different approach to GIVHIFSS based decision-making is presented.

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

scholarly journals Some Inequalities for LR-$$\left({h}_{1}, {h}_{2}\right)$$-Convex Interval-Valued Functions by Means of Pseudo Order Relation

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Kottakkaran Sooppy Nisar ◽  
Khadiga Ahmed Ismail ◽  
...  
Keyword(s):  
Applied Mathematics ◽  
Critical Role ◽  
Order Relation ◽  
Strong Relationship ◽  
New Class ◽  
Starting Point ◽  
Interval Valued Function ◽  
Fejér Inequality ◽  
Definition Of ◽  
Interval Valued

AbstractIn both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and symmetry is strong. Whichever one we work on, we introduced the new class of generalized convex function is known as LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued function (LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -IVF) by means of pseudo order relation. Then, we established its strong relationship between Hermite–Hadamard inequality (HH-inequality)) and their variant forms. Besides, we derive the Hermite–Hadamard–Fejér inequality (HH–Fejér inequality)) for LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued functions. Several exceptional cases are also obtained which can be viewed as its applications of this new concept of convexity. Useful examples are given that verify the validity of the theory established in this research. This paper’s concepts and techniques may be the starting point for further research in this field.

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 On the Geometrical Representation of Classical Statistical Mechanics

2018 ◽  
Author(s):  
Georgios C. Boulougouris
Keyword(s):  
Statistical Mechanics ◽  
Parametric Representation ◽  
Gauss Map ◽  
Inner Product ◽  
Necessary And Sufficient Condition ◽  
Thermodynamic State ◽  
Classical Statistical Mechanics ◽  
Geometrical Representation ◽  
Definition Of ◽  
Necessary And Sufficient

In this work a geometrical representation of equilibrium and near equilibrium statistical mechanics is proposed. Using a formalism consistent with the Bra-Ket notation and the definition of inner product as a Lebasque integral, we describe the macroscopic equilibrium states in classical statistical mechanics by “properly transformed probability Euclidian vectors” that point on a manifold of spherical symmetry. Furthermore, any macroscopic thermodynamic state “close” to equilibrium is described by a triplet that represent the “infinitesimal volume” of the points, the Euclidian probability vector at equilibrium that points on a hypersphere of equilibrium thermodynamic state and a Euclidian vector a vector on the tangent bundle of the hypersphere. The necessary and sufficient condition for such representation is expressed as an invertibility condition on the proposed transformation. Finally, the relation of the proposed geometric representation, to similar approaches introduced under the context of differential geometry, information geometry, and finally the Ruppeiner and the Weinhold geometries, is discussed. It turns out that in the case of thermodynamic equilibrium, the proposed representation can be considered as a Gauss map of a parametric representation of statistical mechanics.

Encircling Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Nineteenth Century ◽  
Real Number ◽  
Traveling Salesman Problem ◽  
Complex Number ◽  
Traveling Salesman ◽  
Complex Numbers ◽  
Real Numbers ◽  
Hamiltonian Graphs ◽  
The World ◽  
The Traveling Salesman Problem

This chapter considers Hamiltonian graphs, a class of graphs named for nineteenth-century physicist and mathematician Sir William Rowan Hamilton. In 1835 Hamilton discovered that complex numbers could be represented as ordered pairs of real numbers. That is, a complex number a + b i (where a and b are real numbers) could be treated as the ordered pair (a, b). Here the number i has the property that i² = -1. Consequently, while the equation x² = -1 has no real number solutions, this equation has two solutions that are complex numbers, namely i and -i. The chapter first examines Hamilton's icosian calculus and Icosian Game, which has a version called Traveller's Dodecahedron or Voyage Round the World, before concluding with an analysis of the Knight's Tour Puzzle, the conditions that make a given graph Hamiltonian, and the Traveling Salesman Problem.

The EXPRESS-I Language

1994 ◽  
Author(s):  
Douglas Schenck ◽  
Peter Wilson
Keyword(s):  
Information Modeling ◽  
Real Numbers ◽  
Geometric Point ◽  
Computer Automation ◽  
Definition Of ◽  
Data Elements ◽  
Specific Implementation ◽  
And Behavior ◽  
Specification Techniques ◽  
Test Suites

Now we turn to the question: ‘Once I have created an abstract declaration in EXPRESS, what would an instance of that thing look like?’ EXPRESS-I allows you to create instances of EXPRESS things that have values in place of references to datatypes. The main reason for doing this is to study some realistic examples of things that otherwise might be difficult to understand. After all, it is one thing to describe a tree and quite another to actually see one. Some of the design goals of EXPRESS-I are based on these requirements: • Major information modeling projects are large and complex. Managing them without appropriate tools based on formal languages and methods is a risky proposition. Informal specification techniques eliminate the possibility of employing computer automation in checking for inconsistencies in presentation or specification. • The language should focus on the display of the realization of the properties of entities, which are the things of interest. The definition of entities is in terms of data and behavior. Data represents the properties by which an entity is realized and behavior is represented by constraints. • The language should seek to avoid, as far as possible, specific implementation views. That is, EXPRESS-I models do not suggest the structure of databases, object bases, or of information bases in general. • The language should provide a means for displaying small populated models of EXPRESS schemas as examples for design reviews. • The language should provide a means for supporting the specification of test suites for information model processors. EXPRESS-I represents entity instances in terms of the values of its attributes (attributes are the traits or characteristics considered important for use and understanding). These values have a representation which might be considered simple (an integer value) or something more complex (an entity value). A geometric point might be defined in terms of three real numbers named x, y and z, and the actual values associated with those attributes might be 1.0, 2.5 and 7.9. The EXPRESS-I instance language provides a means of displaying instantiations of EXPRESS data elements. The language is designed principally for human readability and for ease of generating EXPRESS-I element instances from definitions in an EXPRESS schema.

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