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Concept

Support (mathematics)

In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname{supp}(f), is the set of points in X where f is non-zero: \operatorname{supp}(f) = { x \in X ,:, f(x) \neq 0}. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then

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The aim of this course is to provide a solid foundation of theory of distributions, Sobolev spaces and an introduction to the more general theory of interpolation spaces.

This course gives an introduction to the fundamental concepts and methods of the Digital Humanities, both from a theoretical and applied point of view. The course introduces the Digital Humanities circle of processing and interpretation, from data acquisition to new understandings.

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.

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Scattering of wave maps from $\mathbb{R}^{2+1}$ to general targets

Joules Elie Nahas

We show that smooth, radially symmetric wave maps

U

from

\mathbb{R}^{2+1}

to a compact target manifold

N

, where

\partial_r U

and

\partial_t U

have compact support for any fixed time, scatter. The result will follow from the work of Christodoulou and Tahvildar-Zadeh, and Struwe, upon proving that for

\lambda' \in (0,1)

, energy does not concentrate in the set

K_\frac{5}{8}T,\frac{7}{8}T^{\lambda'} = \{(x,t) \in \mathbb R^{2+1} \vert \hspace{5pt} |x| \leq \lambda' t, t \in [(5/8)T,(7/8)T] \}.

Springer-Verlag2013

Optimal Spline Generators for Derivative Sampling

Shayan Aziznejad, Alireza Naderi, Michaël Unser

The goal of derivative sampling is to reconstruct a signal from the samples of the function and of its first-order derivative. In this paper, we consider this problem over a shift-invariant reconstruction subspace generated by two compact-support functions. We assume that the reconstruction subspace reproduces polynomials up to a certain degree. We then derive a lower bound on the sum of supports of its generators. Finally, we illustrate the tightness of our bound with some examples.

IEEE2019

An Authoring Model For Interactive 360 Videos

Roberto Gerson De Albuquerque Azevedo

The recent availability of consumer-level head-mounted displays and omnidirectional cameras has been driving an explosion of 360 video content. Transforming the original recorded 360 video content in meaningful interactive multimedia presentations that support viewers in tasks such as learning, entertainment, and telepresence, however, is not trivial and requires new tools. Such tools must provide an easy-to-use authoring model for the integration of different media objects and active user interface elements.In this paper, based on real-world scenarios for interactive 360 videos, we gather a set of requirements and propose a declarative authoring model to support authors in the process of designing and creating 360-degree interactive video presentations. As a case study, we implement different applications showing the expressiveness and completeness of the model in the scope of the target scenarios.

IEEE2020

Distribution (mathematics)

Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to

Dirac delta function

In mathematical physics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is

Convolution

In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (f*g) th

The aim of this course is to provide a solid foundation of theory of distributions, Sobolev spaces and an introduction to the more general theory of interpolation spaces.

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables.