Intrinsic definition of the Colombeau algebra of generalized functions

1991 ◽  
Vol 17 (2) ◽  
pp. 75-132 ◽  
Author(s):  
J. Aragona ◽  
H. A. Biagioni
Keyword(s):  
Generalized Functions ◽  
Colombeau Algebra ◽  
Definition Of ◽  
Intrinsic Definition

scholarly journals Certain fundamental properties of generalized natural transform in generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari ◽  
Serkan Araci
Keyword(s):  
Inversion Formula ◽  
Acta Math ◽  
Generalized Functions ◽  
General Nature ◽  
Qualitative Behavior ◽  
Fundamental Properties ◽  
Integrable Functions ◽  
Extended Operator ◽  
Definition Of ◽  
Space Of Integrable Functions

AbstractThis paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

scholarly journals A generalized inversion formula for the continuous Jacobi transform

1987 ◽  
Vol 10 (4) ◽  
pp. 671-692 ◽  
Author(s):  
Ahmed I. Zayed
Keyword(s):  
Analytic Function ◽  
Inversion Formula ◽  
Generalized Functions ◽  
Generalized Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Inversion ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Jacobi Transform

In this paper we extend the definition of the continuous Jacobi transform to a class of generalized functions and obtain a generalized inversion formula for it. As a by-product of our technique we obtain a necessary and sufficient condition for an analytic functionF(λ)inReλ>0to be the continuous Jacobi transform of a generalized function.

scholarly journals A remark on convergence of test functions

1975 ◽  
Vol 20 (1) ◽  
pp. 73-76 ◽  
Author(s):  
W. F. Moss
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Generalized Functions ◽  
Test Functions ◽  
Definition Of ◽  
Theory Of Generalized Functions

In this note it is shown in the most frequently encountered spaces of test functions in the theory of generalized functions that the customary definitions of convergence are equivalent to apparently much weaker definitions. For example, in the space g the condition of uniform convergence of the functions together with all derivatives (which appears in the definition of convergence) is equivalent to the condition of pointwise convergence of the functions alone. Thus verification of convergence is simplified somewhat.

REVISITING Xor-IMPLICATIONS: CLASSES OF FUZZY (CO)IMPLICATIONS BASED ON f-Xor (f-XNor) CONNECTIVES

2013 ◽  
Vol 21 (06) ◽  
pp. 899-925 ◽  
Author(s):  
BENJAMÍN BEDREGAL ◽  
RENATA HAX SANDER REISER ◽  
GRAÇALIZ PEREIRA DIMURO
Keyword(s):  
Boundary Conditions ◽  
Unit Interval ◽  
End Points ◽  
Triangular Norms ◽  
Fuzzy Implications ◽  
Classical Definition ◽  
Definition Of ◽  
Exclusive Or ◽  
Intrinsic Definition ◽  
Fuzzy Connectives

The main contribution of this paper is the introduction of an intrinsic definition of the connective “fuzzy exclusive or” E (f-Xor E), based only on the properties of boundary conditions, commutativity and partial isotonicity-antitonicity on the the end-points of the unit interval U = [0,1], in a way that the classical definition of the boolean Xor is preserved. We show three classes of the f-Xor E that can be also obtained from the composition of fuzzy connectives, namely, triangular norms, triangular conorms and fuzzy negations. A discussion about extra properties satisfied by the f-Xor E is presented. Additionally, the paper introduces a class of fuzzy equivalences that generalizes the Fodor and Roubens's fuzzy equivalence, and four classes of fuzzy implications induced by the f-Xor E, discussing their main properties. The relationships between those classes of fuzzy implications and automorphisms are explored. The action of automorphisms on f-Xor E is analyzed.

Singular fractional evolution differential equations

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Mirjana Stojanovic
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Models ◽  
Uniqueness Result ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Fractional Evolution Equations ◽  
Linear And Nonlinear ◽  
Duhamel Principle

AbstractWe give an existence-uniqueness result for linear and nonlinear time fractional evolution equations with singularities in corresponding norm in extended Colombeau algebra of generalized functions using fractional analog for Duhamel principle. Paper deals with some nonlinear models with singularities appearing in viscoelasticity and in anomalous processes which have met great interest among researchers who consider them as a challenge in recent years.

scholarly journals On Products of Distributions in Colombeau Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Marija Miteva ◽  
Biljana Jolevska-Tuneska ◽  
Tatjana Atanasova-Pacemska
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Schwartz Distributions ◽  
Products Of Distributions ◽  
Renormalization Theory

Results on products of distributionsx+-kandδ(p)(x)are derived. They are obtained in Colombeau differential algebra𝒢(R)of generalized functions that contains the space𝒟'(R)of Schwartz distributions as a subspace. Products of this form are useful in quantum renormalization theory in Physics.

scholarly journals Amplituhedron-like geometries

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Gabriele Dian ◽  
Paul Heslop
Keyword(s):  
Canonical Form ◽  
Star Product ◽  
Natural Extension ◽  
Tree Level ◽  
Winding Number ◽  
Alternative Definition ◽  
Standard Definition ◽  
Definition Of ◽  
Intrinsic Definition ◽  
Simple Definition

Abstract We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical form of these geometries corresponds to the product of parity conjugate amplitudes at tree as well as loop level. The product of amplitudes in superspace lifts to a star product in bosonised superspace which we give a precise definition of. We give an alternative definition of amplituhedron-like geometries, analogous to the original amplituhedron definition, and also a characterisation as a sum over pairs of on-shell diagrams that we use to prove the conjecture at tree level. The union of all amplituhedron-like geometries has a very simple definition given by only physical inequalities. Although such a union does not give a positive geometry, a natural extension of the standard definition of canonical form, the globally oriented canonical form, acts on this union and gives the square of the amplitude.

scholarly journals Active faults sources for the Pátzcuaro–Acambay fault system (Mexico): fractal analysis of slip rates and magnitudes <i>M</i><sub>w</sub> estimated from fault length

2018 ◽  
Vol 18 (11) ◽  
pp. 3121-3135
Author(s):  
Avith Mendoza-Ponce ◽  
Angel Figueroa-Soto ◽  
Diana Soria-Caballero ◽  
Víctor Hugo Garduño-Monroy
Keyword(s):  
Time Domain ◽  
Fractal Analysis ◽  
Active Faults ◽  
Slip Rate ◽  
Spatial Domain ◽  
Fault System ◽  
Slip Rates ◽  
Oblique Convergence ◽  
Definition Of ◽  
Intrinsic Definition

Abstract. The Pátzcuaro–Acambay fault system (PAFS), located in the central part of the Trans-Mexican Volcanic Belt (TMVB), is delimited by an active transtensive deformation area associated with the oblique subduction zone between the Cocos and North American plates, with a convergence speed of 55 mm yr−1 at the latitude of the state of Michoacán, Mexico. Part of the oblique convergence is transferred to this fault system, where the slip rates range from 0.009 to 2.78 mm yr−1. This has caused historic earthquakes in Central Mexico, such as the Acambay quake (Ms=6.9) on 19 November 1912 with surface rupture, and another in Maravatío in 1979 with Ms=5.6. Also, paleoseismic analyses are showing Quaternary movements in some faults, with moderate to large magnitudes. Notably, this zone is seismically active, but lacks a dense local seismic network, and more importantly, its neotectonic movements have received very little attention. The present research encompasses three investigations carried out in the PAFS. First, the estimation of the maximum possible earthquake magnitudes, based on 316 fault lengths mapped on a 15 m digital elevation model, by means of three empirical relationships. In addition, the Hurst exponent Hw and its persistence, estimated for magnitudes Mw (spatial domain) and for 32 slip-rate data (time domain) by the wavelet variance analysis. Finally, the validity of the intrinsic definition of active fault proposed here. The average results for the estimation of the maximum and minimum magnitudes expected for this fault population are 5.5≤Mw≤7. Also, supported by the results of H at the spatial domain, this paper strongly suggests that the PAFS is classified in three different zones (western PAFS, central PAFS, and eastern PAFS) in terms of their roughness (Hw=0.7,Hw=0.5,Hw=0.8 respectively), showing different dynamics in seismotectonic activity and; the time domain, with a strong persistence Hw=0.949, suggests that the periodicities of slip rates are close in time (process with memory). The fractal capacity dimension (Db) is also estimated for the slip-rate series using the box-counting method. Inverse correlation between Db and low slip-rate concentration was observed. The resulting Db=1.86 is related to a lesser concentration of low slip-rates in the PAFS, suggesting that larger faults accommodate the strain more efficiently (length ≥3 km). Thus, in terms of fractal analysis, we can conclude that these 316 faults are seismically active, because they fulfill the intrinsic definition of active faults for the PAFS.

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