scholarly journals Regular nonlinear generalized functions and applications

2006 ◽  
Vol 133 (31) ◽  
pp. 163-174 ◽  
Author(s):  
A. Delcroix
Keyword(s):  
Fourier Transform ◽  
Integral Operators ◽  
Generalized Functions ◽  
Regular Growth ◽  
Simplified Model ◽  
Local Analysis ◽  
Schwartz Kernel ◽  
Linear Maps ◽  
Nonlinear Generalized Functions ◽  
Kernel Theorem

We present new types of regularity for Colombeau nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the simplified model. This generalizes the notion of G8-regularity introduced by M. Oberguggenberger. As a first application we show that these new spaces are useful in a problem of representation of linear maps by integral operators, giving an analogon to Schwartz kernel theorem in the framework of nonlinear generalized functions. Secondly, we remark that these new regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to micro local analysis of singularities of generalized functions, with respect to these regularities. AMS Mathematics Subject Classification (2000): 35A18, 35A27, 42B10, 46E10, 46F30.

Vanishing Theorems in Colombeau Algebras of Generalized Functions

2008 ◽  
Vol 51 (4) ◽  
pp. 618-626 ◽  
Author(s):  
V. Valmorin
Keyword(s):  
Integral Operators ◽  
Generalized Functions ◽  
Vanishing Theorems ◽  
Smooth Functions ◽  
Linear Maps ◽  
Colombeau Algebras ◽  
Algebras Of Generalized Functions ◽  
Linear Embedding ◽  
Linear Integral ◽  
Linear Integral Operators

AbstractUsing a canonical linear embedding of the algebra of Colombeau generalized functions in the space of -valued ℂ-linear maps on the space of smooth functions with compact support, we give vanishing conditions for functions and linear integral operators of class . These results are then applied to the zeros of holomorphic generalized functions in dimension greater than one.

scholarly journals Some spectral formulas for functions generated by differential and integral operators in Orlicz spaces

2021 ◽  
Vol 13 (2) ◽  
pp. 326-339
Author(s):  
H.H. Bang ◽  
V.N. Huy
Keyword(s):  
Fourier Transform ◽  
Orlicz Space ◽  
Differential Operators ◽  
Orlicz Spaces ◽  
Integral Operators ◽  
Young Function ◽  
The Fourier Transform

In this paper, we investigate the behavior of the sequence of $L^\Phi$-norm of functions, which are generated by differential and integral operators through their spectra (the support of the Fourier transform of a function $f$ is called its spectrum and denoted by sp$(f)$). With $Q$ being a polynomial, we introduce the notion of $Q$-primitives, which will return to the notion of primitives if ${Q}(x)= x$, and study the behavior of the sequence of norm of $Q$-primitives of functions in Orlicz space $L^\Phi(\mathbb R^n)$. We have the following main result: let $\Phi $ be an arbitrary Young function, ${Q}({\bf x} )$ be a polynomial and $(\mathcal{Q}^mf)_{m=0}^\infty \subset L^\Phi(\mathbb R^n)$ satisfies $\mathcal{Q}^0f=f, {Q}(D)\mathcal{Q}^{m+1}f=\mathcal{Q}^mf$ for $m\in\mathbb{Z}_+$. Assume that sp$(f)$ is compact and $sp(\mathcal{Q}^{m}f)= sp(f)$ for all $m\in \mathbb{Z}_+.$ Then $$ \lim\limits_{m\to \infty } \|\mathcal{Q}^m f\|_{\Phi}^{1/m}= \sup\limits_{{\bf x} \in sp(f)} \bigl|1/ {Q}({\bf x}) \bigl|. $$ The corresponding results for functions generated by differential operators and integral operators are also given.

On the foundations of nonlinear generalized functions I and II

2001 ◽  
Vol 153 (729) ◽  
pp. 0-0 ◽  
Author(s):  
Michael Grosser ◽  
Eva Farkas ◽  
Michael Kunzinger ◽  
Roland Steinbauer
Keyword(s):  
Generalized Functions ◽  
Nonlinear Generalized Functions

GENERALISED PSEUDO-RIEMANNIAN GEOMETRY FOR GENERAL RELATIVITY

2002 ◽  
Vol 17 (20) ◽  
pp. 2776-2776
Author(s):  
R. STEINBAUER ◽  
M. KUNZINGER
Keyword(s):  
General Relativity ◽  
Riemannian Geometry ◽  
Rapid Development ◽  
Generalized Functions ◽  
Field Equations ◽  
Mathematical Methods ◽  
Colombeau Algebras ◽  
Rigorous Treatment ◽  
General Relativistic ◽  
Nonlinear Generalized Functions

The study of singular spacetimes by distributional methods faces the fundamental obstacle of the inherent nonlinearity of the field equations. Staying strictly within the distributional (in particular: linear) regime, as determined by Geroch and Traschen2 excludes a number of physically interesting examples (e.g., cosmic strings). In recent years, several authors have therefore employed nonlinear theories of generalized functions (Colombeau algebras, in particular) to tackle general relativistic problems1,5,8. Under the influence of these applications in general relativity the nonlinear theory of generalized functions itself has undergone a rapid development lately, resulting in a diffeomorphism invariant global theory of nonlinear generalized functions on manifolds3,4,6. In particular, a generalized pseudo-Riemannian geometry allowing for a rigorous treatment of generalized (distributional) spacetime metrics has been developed7. It is the purpose of this talk to present these new mathematical methods themselves as well as a number of applications in mathematical relativity.

Fourier Analysis Guided Cable Actuator Design for Coordinated Walking Assistance

2021 ◽  
Author(s):  
Chong Liu ◽  
Rand Hidayah ◽  
Sunil Agrawal
Keyword(s):  
Fourier Transform ◽  
Curve Fitting ◽  
Degrees Of Freedom ◽  
State Of The Art ◽  
Simplified Model ◽  
Forward Kinematics ◽  
Joint Angles ◽  
Sine Curve ◽  
Design And Manufacture ◽  
Actuator Design

Abstract Cable-driven exoskeletons add minimal inertia and restrictions to the user’s leg while still providing feedback and quantitative measures of the user’s performance. However, cable robots require at least n + 1 cables to control n degrees-of-freedom, i.e., they require more actuators than the leg’s degrees-of-freedom, challenging their widespread adoption as wearable technology. The state-of-the-art in this field aims to reduce the number of actuated motors. In this paper, we design and evaluate a “single motor-driven” leg exoskeleton prototype based on the Cable-driven Active Leg EXoskleton (C-ALEX). The prototype consists of four crank-spring mechanisms and a crankshaft designed using epicycle analysis. The epicycle analysis is performed using discrete Fourier transform (DFT) and sine curve fitting (SCF). While DFT suggests the maximum number of epicycles to imitate the target waveform, a large number of nested epicycles is challenging to design and manufacture for implementation. To validate the epicycle-guided design, we constructed a simple crankshaft model using one epicycle. Our proposed simplified model predicted and produced the joint angles calculated from the inverse and forward kinematics of a cable-driven leg exoskeleton with multiple motors. To our knowledge, this is the first multi-cable driven exoskeleton powered by a single actuator that is designed to provide continuous assistance to the user.

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