scholarly journals The Partial Inner Product Space Method: A Quick Overview

2010 ◽  
Vol 2010 ◽  
pp. 1-37 ◽  
Author(s):  
Jean-Pierre Antoine ◽  
Camillo Trapani
Keyword(s):  
Modulation Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Gabor Analysis ◽  
Decay Properties ◽  
Special Cases ◽  
The Family ◽  
Definition Of ◽  
Amalgam Spaces ◽  
Space Method

Many families of function spaces play a central role in analysis, in particular, in signal processing (e.g., wavelet or Gabor analysis). Typical are spaces, Besov spaces, amalgam spaces, or modulation spaces. In all these cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases ofpartial inner product spaces(PIP-spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, and frames on them should be defined globally, for the whole family, instead of individual spaces. In this paper, we will give an overview of PIP-spaces and operators on them, illustrating the results by space families of interest in mathematical physics and signal analysis. The interesting fact is that they allow a global definition of operators, and various operator classes on them have been defined.

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.

scholarly journals On the Influences of Variables on Boolean Functions in Product Spaces

2010 ◽  
Vol 20 (1) ◽  
pp. 83-102 ◽  
Author(s):  
NATHAN KELLER
Keyword(s):  
Boolean Functions ◽  
Clear Definition ◽  
Product Spaces ◽  
Special Cases ◽  
Definition Of ◽  
General Product

In this paper we consider the influences of variables on Boolean functions in general product spaces. Unlike the case of functions on the discrete cube, where there is a clear definition of influence, in the general case several definitions have been presented in different papers. We propose a family of definitions for the influence that contains all the known definitions, as well as other natural definitions, as special cases. We show that the proofs of the BKKKL theorem and of other results can be adapted to our new definition. The adaptation leads to generalizations of these theorems, which are tight in terms of the definition of influence used in the assertion.

A Common Generalization of Functional Equations Characterizing Normed and Quasi-Inner-Product Spaces

1992 ◽  
Vol 35 (3) ◽  
pp. 321-327 ◽  
Author(s):  
B. R. Ebanks ◽  
PL. Kannappan ◽  
P. K. Sahoo
Keyword(s):  
Functional Equation ◽  
Characterization Theorem ◽  
Inner Product ◽  
Quadratic Functional ◽  
Product Spaces ◽  
Commutative Field ◽  
Von Neumann ◽  
Inner Product Spaces ◽  
Special Cases ◽  
Image Position

AbstractWe determine the general solutions of the functional equation for ƒi: G → F (i = 1,2,3,4), where G is a 2-divisible group and F is a commutative field of characteristic different from 2. The motivation for studying this equation came from a result due to Dry gas [4] where he proved a Jordan and von Neumann type characterization theorem for quasi-inner products. Also, this equation is a generalization of the quadratic functional equation investigated by several authors in connection with inner product spaces and their generalizations. Special cases of this equation include the Cauchy equation, the Jensen equation, the Pexider equation and many more. Here, we determine the general solution of this equation without any regularity assumptions on ƒi.

scholarly journals NONPERTURBATIVE QUANTIZATION OF PHANTOM AND GHOST THEORIES: RELATING DEFINITE AND INDEFINITE REPRESENTATIONS

2007 ◽  
Vol 22 (14n15) ◽  
pp. 2563-2608 ◽  
Author(s):  
ANDRÉ VAN TONDER ◽  
MIQUEL DORCA
Keyword(s):  
Path Integral ◽  
Degrees Of Freedom ◽  
Ad Hoc ◽  
Inner Product ◽  
Product Spaces ◽  
Indefinite Inner Product ◽  
Inner Product Spaces ◽  
Physical Information ◽  
Definition Of ◽  
Nonperturbative Quantization

We investigate the nonperturbative quantization of phantom and ghost degrees of freedom by relating their representations in definite and indefinite inner product spaces. For a large class of potentials, we argue that the same physical information can be extracted from either representation. We provide a definition of the path integral for these theories, even in cases where the integrand may be exponentially unbounded, thereby removing some previous obstacles to their nonperturbative study. We apply our results to the study of ghost fields of Pauli–Villars and Lee–Wick type, and we show in the context of a toy model how to derive, from an exact nonperturbative path integral calculation, previously ad hoc prescriptions for Feynman diagram contour integrals in the presence of complex energies. We point out that the pole prescriptions obtained in ghost theories are opposite to what would have been expected if one had added conventional i∊ convergence factors in the path integral.

scholarly journals Free and Forced Vibrations of Elastically Connected Structures

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
S. Graham Kelly
Keyword(s):  
Natural Frequencies ◽  
Iterative Solution ◽  
Forced Response ◽  
Inner Product ◽  
Mode Shapes ◽  
Square Roots ◽  
Parallel Structures ◽  
Special Cases ◽  
Definition Of ◽  
The Individual

A general theory for the free and forced responses of elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for , then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on . This leads to the definition of energy inner products defined on . When a normal mode solution is used to develop the free response, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is self-adjoint with respect to the energy inner product. The completeness of the eigenvectors in is used to develop a forced response. Special cases are considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapes reduces to a matrix eigenvalue problem, and it is shown that for each spatial mode there is a set of intramodal mode shapes. When the structures are identical, uniform, or nonuniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.

REGULATION OF MARRIAGE RELATIONS IN THE TURKESTAN REGION

2019 ◽  
Vol 22 (2) ◽  
pp. 27-32
Author(s):  
Khurshida Tillahodjaeva ◽  
Keyword(s):  
The Family ◽  
Role Of Women ◽  
Family And Marriage ◽  
Material Conditions ◽  
Definition Of

In this article we will talk about the scale of family and marriage relations in the early XX century in the Turkestan region, their regulation, legislation. Clearly reveals the role of women and men in the family, the definition of which is based on the material conditions of society, equality of rights and freedoms and its features.

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 Fuzzy Inner Product Space: Literature Review and a New Approach

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 765
Author(s):  
Lorena Popa ◽  
Lavinia Sida
Keyword(s):  
Literature Review ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Comprehensive Overview ◽  
New Approach ◽  
Inner Product Spaces ◽  
Product Function ◽  
Suitable Definition

The aim of this paper is to provide a suitable definition for the concept of fuzzy inner product space. In order to achieve this, we firstly focused on various approaches from the already-existent literature. Due to the emergence of various studies on fuzzy inner product spaces, it is necessary to make a comprehensive overview of the published papers on the aforementioned subject in order to facilitate subsequent research. Then we considered another approach to the notion of fuzzy inner product starting from P. Majundar and S.K. Samanta’s definition. In fact, we changed their definition and we proved some new properties of the fuzzy inner product function. We also proved that this fuzzy inner product generates a fuzzy norm of the type Nădăban-Dzitac. Finally, some challenges are given.

scholarly journals Hexokinase 2 in Cancer: A Prima Donna Playing Multiple Characters

2021 ◽  
Vol 22 (9) ◽  
pp. 4716
Author(s):  
Francesco Ciscato ◽  
Lavinia Ferrone ◽  
Ionica Masgras ◽  
Claudio Laquatra ◽  
Andrea Rasola
Keyword(s):  
Endoplasmic Reticulum ◽  
Aerobic Glycolysis ◽  
Neoplastic Progression ◽  
Malignant Growth ◽  
Hexokinase 2 ◽  
Genetic Ablation ◽  
Contact Points ◽  
Survival Pathways ◽  
The Family ◽  
Definition Of

Hexokinases are a family of ubiquitous exose-phosphorylating enzymes that prime glucose for intracellular utilization. Hexokinase 2 (HK2) is the most active isozyme of the family, mainly expressed in insulin-sensitive tissues. HK2 induction in most neoplastic cells contributes to their metabolic rewiring towards aerobic glycolysis, and its genetic ablation inhibits malignant growth in mouse models. HK2 can dock to mitochondria, where it performs additional functions in autophagy regulation and cell death inhibition that are independent of its enzymatic activity. The recent definition of HK2 localization to contact points between mitochondria and endoplasmic reticulum called Mitochondria Associated Membranes (MAMs) has unveiled a novel HK2 role in regulating intracellular Ca2+ fluxes. Here, we propose that HK2 localization in MAMs of tumor cells is key in sustaining neoplastic progression, as it acts as an intersection node between metabolic and survival pathways. Disrupting these functions by targeting HK2 subcellular localization can constitute a promising anti-tumor strategy.

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