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## ‣ Aproximação de funções contínuas e de funções diferenciáveis; Approximation of continuous functions and of differentiable functions

Fonte: Biblioteca Digital da Unicamp
Publicador: Biblioteca Digital da Unicamp

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 13/06/2014
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#Funções contínuas#Funções diferenciais#Teoria da aproximação#Continuous functions#Differentiable functions#Approximation theory

O objetivo desta dissertação é apresentar e demonstrar alguns teoremas da Análise matemática, são eles, O Teorema de Aproximação de Weierstrass, o Teorema de Kakutani-Stone, os Teoremas de Stone-Weierstrass e o Teorema de Nachbin. Para demonstrá-los relembraremos algumas definições e resultados básicos da teoria de Análise e Topologia e abordaremos as demais ferramentas necessárias para suas respectivas demonstrações.; The aim of this dissertation is to present and prove some theorems of mathematical analysis, that are, the Weierstrass Approximation Theorem, the Kakutani-Stone Theorem, the Stone-Weierstrass Theorems and the Nachbin Theorem. To prove them we recall some basic definitions and results of analysis and topology and we discuss other tools that are necessary for their respective proofs.

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## ‣ On the Hausdorff Dimension of Continuous Functions Belonging to Hölder and Besov Spaces on Fractal d-Sets

Fonte: Springer
Publicador: Springer

Tipo: Artigo de Revista Científica

Português

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#Besov spaces#Box counting dimension#Continuous functions#d-Sets#Fractals#Hausdorff dimension#Hölder spaces#Wavelets#Weierstrass function

The Hausdorff dimension of the graphs of the functions in Hölder and Besov spaces (in this case with integrability p≥1) on fractal d-sets is studied. Denoting by s in (0,1] the smoothness parameter, the sharp upper bound min{d+1-s, d/s} is obtained. In particular, when passing from d≥s to d~~
~~

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## ‣ Numerical calculations of Hölder exponents for the Weierstrass functions with (min, +)-wavelets

Fonte: Sociedade Brasileira de Matemática Aplicada e Computacional
Publicador: Sociedade Brasileira de Matemática Aplicada e Computacional

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/12/2014
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One reminds for all function f : n → the so-called (min, +)-wavelets which are lower and upper hulls build from (min, +) analysis [12, 13]. One shows that this analysis can be applied numerically to the Weierstrass and Weierstrass-Mandelbrot functions, and that one recovers their theoretical Hölder exponents and fractal dimensions.

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## ‣ The Weierstrass-Laguerre Transform

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Publicado em /03/1971
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An elegant expression is obtained for the product of the inverse Weierstrass-Laguerre transforms of two functions in terms of their convolution. It is also shown how the main result can be extended to hold for the product of the inverse Weierstrass-Laguerre transforms of several functions.

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## ‣ O Teorema de Stone-Weierstrass e aplicações; The Stone-Weierstrass Theorem and applications

Fonte: Universidade Federal de Uberlândia
Publicador: Universidade Federal de Uberlândia

Tipo: Dissertação

Português

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#Matemática#Aproximação#Funções contínuas#Funções infinitamente diferenciáveis#Espaços compactos#Espaços separáveis#Espaços metrizáveis#Análise funcional#Teoria da aproximação#Approximation#Continuous functions

O objetivo desta dissertação é demonstrar e aplicar o Teorema da Aproximação de Weierstrass, sobre aproximação de funções contínuas em intervalos fechados e limitados da reta por polinômios, e o Teorema de Stone-Weierstrass, sobre aproximação de funções contínuas definidas em espaços topológicos compactos. Como aplicações do Teorema da Aproximação de Weierstrass tratamos o problema dos momentos de uma função contínua e a aproximação de funções contínuas definidas na reta por funções infinitamente diferenciáveis. Como aplicações do Teorema de Stone-Weierstrass provamos que o espaço C(K) das funções contínuas no compacto K é separável se e somente se K é metrizável e também a existência de um compacto K tal que C(K) é isometricamente isomorfo ao espaço `1 das sequências limitadas. __________________________________________________________________________________________ ABSTRACT; The aim of this dissertation is to prove and apply the Weierstrass Approximation Theo- rem, on the approximation of continuous functions on bounded closed intervals by polyno- mials, and the Stone-Weierstrass Theorem, on the approximation of continuous functions on compact topological spaces. As applications of the Weierstrass Approximation Theo- rem we deal with the momentum problem for continuous functions and the approximation of continuous functions on the line by in¯nitely di®erentiable functions. As applications of the Stone-Weierstrass Theorem we prove that the space C(K) of continuous functions on the compact K is separable if and only if K is metrizable and the existence of a com- pact space K such that C(K) is isometrically isomorphic to the space `1 of bounded sequences.; Dissertação (mestrado)-Universidade Federal de Uberlândia...

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## ‣ The Bolzano-Weierstrass Theorem is the Jump of Weak K\"onig's Lemma

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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We classify the computational content of the Bolzano-Weierstrass Theorem and
variants thereof in the Weihrauch lattice. For this purpose we first introduce
the concept of a derivative or jump in this lattice and we show that it has
some properties similar to the Turing jump. Using this concept we prove that
the derivative of closed choice of a computable metric space is the cluster
point problem of that space. By specialization to sequences with a relatively
compact range we obtain a characterization of the Bolzano-Weierstrass Theorem
as the derivative of compact choice. In particular, this shows that the
Bolzano-Weierstrass Theorem on real numbers is the jump of Weak K\"onig's
Lemma. Likewise, the Bolzano-Weierstrass Theorem on the binary space is the
jump of the lesser limited principle of omniscience LLPO and the
Bolzano-Weierstrass Theorem on natural numbers can be characterized as the jump
of the idempotent closure of LLPO. We also introduce the compositional product
of two Weihrauch degrees f and g as the supremum of the composition of any two
functions below f and g, respectively. We can express the main result such that
the Bolzano-Weierstrass Theorem is the compositional product of Weak K\"onig's
Lemma and the Monotone Convergence Theorem. We also study the class of weakly
limit computable functions...

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## ‣ Some New Addition Formulae for Weierstrass Elliptic Functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Algebraic Geometry#Mathematics - Number Theory#11G05 (Primary) 33E05, 14H45, 14H52, 37K20 (Secondary)

We present new addition formulae for the Weierstrass functions associated
with a general elliptic curve. We prove the structure of the formulae in
n-variables and give the explicit addition formulae for the 2- and 3-variable
cases. These new results were inspired by new addition formulae found in the
case of an equianharmonic curve, which we can now observe as a specialisation
of the results here. The new formulae, and the techniques used to find them,
also follow the recent work for the generalisation of Weierstrass' functions to
curves of higher genus.; Comment: 20 pages

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## ‣ A new construction of Eisenstein's completion of the Weierstrass zeta function

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In the theory of elliptic functions and elliptic curves, the Weierstrass
$zeta$ function (which is essentially an antiderivative of the Weierstrass
$\wp$ function) plays a prominent role. Although it is not an elliptic
function, Eisenstein constructed a simple (non-holomorphic) completion of this
form which is doubly periodic. This theorem has begun to play an important role
in the theory of harmonic Maass forms, and was crucial to work of Guerzhoy as
well as Alfes, Griffin, Ono, and the author. In particular, this simple
completion of $\zeta$ provides a powerful method to construct harmonic Maass
forms of weight zero which serve as canonical lifts under the differential
operator $\xi_{0}$ of weight 2 cusp forms, and this has been shown in to have
deep applications to determining vanishing criteria for central values and
derivatives of twisted Hasse-Weil $L$-functions for elliptic curves.
Here we offer a new and motivated proof of Eisenstein's theorem, relying on
the basic theory of differential operators for Jacobi forms together with a
classical identity for the first quasi-period of a lattice. A quick inspection
of the proof shows that it also allows one to easily construct more general
non-holomorphic elliptic functions.; Comment: 3 pages...

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## ‣ Multi-Dimensional Sigma-Functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/08/2012
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#Mathematical Physics#Nonlinear Sciences - Exactly Solvable and Integrable Systems#Algebraic curves, theta and sigma-functions, completely integrable
equations

In 1997 the present authors published a review (Ref. BEL97 in the present
manuscript) that recapitulated and developed classical theory of Abelian
functions realized in terms of multi-dimensional sigma-functions. This approach
originated by K.Weierstrass and F.Klein was aimed to extend to higher genera
Weierstrass theory of elliptic functions based on the Weierstrass
$\sigma$-functions. Our development was motivated by the recent achievements of
mathematical physics and theory of integrable systems that were based of the
results of classical theory of multi-dimensional theta functions. Both theta
and sigma-functions are integer and quasi-periodic functions, but worth to
remark the fundamental difference between them. While theta-function are
defined in the terms of the Riemann period matrix, the sigma-function can be
constructed by coefficients of polynomial defining the curve. Note that the
relation between periods and coefficients of polynomials defining the curve is
transcendental.
Since the publication of our 1997-review a lot of new results in this area
appeared (see below the list of Recent References), that promoted us to submit
this draft to ArXiv without waiting publication a well-prepared book. We
complemented the review by the list of articles that were published after 1997
year to develop the theory of $\sigma$-functions presented here. Although the
main body of this review is devoted to hyperelliptic functions the method can
be extended to an arbitrary algebraic curve and new material that we added in
the cases when the opposite is not stated does not suppose hyperellipticity of
the curve considered.; Comment: 267 pages...

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## ‣ Fractional Weierstrass function by application of Jumarie fractional trigonometric functions and its analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/08/2015
Português

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The classical example of no-where differentiable but everywhere continuous
function is Weierstrass function. In this paper we define the fractional order
Weierstrass function in terms of Jumarie fractional trigonometric functions.
The Holder exponent and Box dimension of this function are calculated here. It
is established that the Holder exponent and Box dimension of this fractional
order Weierstrass function are the same as in the original Weierstrass
function, independent of incorporating the fractional trigonometric function.
This is new development in generalizing the classical Weierstrass function by
usage of fractional trigonometric function and obtain its character and also of
fractional derivative of fractional Weierstrass function by Jumarie fractional
derivative, and establishing that roughness index are invariant to this
generalization.; Comment: 17 pages, 2 figures, submitted to Physics Letters A

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## ‣ Some addition formulae for Abelian functions for elliptic and hyperelliptic curves of cyclotomic type

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We discuss a family of multi-term addition formulae for Weierstrass functions
on specialized curves of genus one and two with many automorphisms. In the
genus one case we find new addition formulae for the equianharmonic and
lemniscate cases, and in genus two we find some new addition formulae for a
number of curves, including the Burnside curve.; Comment: 19 pages. We have extended the Introduction, corrected some typos and
tidied up some proofs, and inserted extra material on genus 3 curves

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## ‣ The Application of Weierstrass elliptic functions to Schwarzschild Null Geodesics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In this paper we focus on analytical calculations involving null geodesics in
some spherically symmetric spacetimes. We use Weierstrass elliptic functions to
fully describe null geodesics in Schwarzschild spacetime and to derive
analytical formulae connecting the values of radial distance at different
points along the geodesic. We then study the properties of light triangles in
Schwarzschild spacetime and give the expansion of the deflection angle to the
second order in both $M/r_0$ and $M/b$ where $M$ is the mass of the black hole,
$r_0$ the distance of closest approach of the light ray and $b$ the impact
parameter. We also use the Weierstrass function formalism to analyze other more
exotic cases such as Reissner-Nordstr\om null geodesics and Schwarzschild null
geodesics in 4 and 6 spatial dimensions. Finally we apply Weierstrass functions
to describe the null geodesics in the Ellis wormhole spacetime and give an
analytic expansion of the deflection angle in $M/b$.; Comment: Latex file, 19 pages 4 figures references and two comments added

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## ‣ Boundary values of harmonic gradients and differentiability of Zygmund and Weierstrass functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/02/2012
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We study differentiability properties of Zygmund functions and series of
Weierstrass type in higher dimensions. While such functions may be nowhere
differentiable, we show that, under appropriate assumptions, the set of points
where the incremental quotients are bounded has maximal Hausdorff dimension.

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## ‣ Hausdorff dimension of the graphs of the classical Weierstrass functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/05/2015
Português

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We obtain the explicit value of the Hausdorff dimension of the graphs of the
classical Weierstrass functions, by proving absolute continuity of the SRB
measures of the associated solenoidal attractors.; Comment: 42 pages

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## ‣ An entropy formula for a non-self-affine measure with application to Weierstrass-type functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Let $ \tau : [0,1] \rightarrow [0,1] $ be a piecewise expanding map with full
branches. Given $ \lambda : [0,1] \rightarrow (0,1) $ and $ g : [0,1]
\rightarrow \mathbb{R} $ satisfying $ \tau ' \lambda > 1 $, we study the
Weierstrass-type function \[ \sum _{n=0} ^\infty \lambda ^n (x) \, g (\tau ^n
(x)), \] where $ \lambda ^n (x) := \lambda(x) \lambda (\tau (x)) \cdots \lambda
(\tau ^{n-1} (x)) $. Under certain conditions, Bedford proved that the box
counting dimension of its graph is given as the unique zero of the topological
pressure function \[ s \mapsto P ((1-s) \log \tau ' + \log \lambda) . \] We
give a sufficient condition under which the Hausdorff dimension also coincides
with this value. We adopt a dynamical system theoretic approach which was
originally used to investigate special cases including the classical
Weierstrass functions. For this purpose we prove a new Ledrappier-Young entropy
formula, which is a conditional version of Pesin's formula, for non-invertible
dynamical systems. Our formula holds for all lifted Gibbs measures on the graph
of the above function, which are generally not self-affine.

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## ‣ Canonical Weierstrass Representation of Minimal and Maximal Surfaces in the Three-dimensional Minkowski Space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/02/2008
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We prove that any minimal (maximal) strongly regular surface in the
three-dimensional Minkowski space locally admits canonical principal
parameters. Using this result, we find a canonical representation of minimal
strongly regular time-like surfaces, which makes more precise the Weierstrass
representation and shows more precisely the correspondence between these
surfaces and holomorphic functions (in the Gauss plane). We also find a
canonical representation of maximal strongly regular space-like surfaces, which
makes more precise the Weierstrass representation and shows more precisely the
correspondence between these surfaces and holomorphic functions (in the Lorentz
plane). This allows us to describe locally the solutions of the corresponding
natural partial differential equations.; Comment: 15 pages

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## ‣ Cubic Algebraic Equations in Gravity Theory, Parametrization with the Weierstrass Function and Non-Arithmetic Theory of Algebraic Equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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A cubic algebraic equation for the effective parametrizations of the standard
gravitational Lagrangian has been obtained without applying any variational
principle.It was suggested that such an equation may find application in
gravity theory, brane, string and Rundall-Sundrum theories. The obtained
algebraic equation was brought by means of a linear-fractional transformation
to a parametrizable form, expressed through the elliptic Weierstrass function,
which was proved to satisfy the standard parametrizable form, but with $g_{2}$
and $g_{3}$ functions of a complex variable instead of the definite complex
numbers (known from the usual arithmetic theory of elliptic functions and
curves). The generally divergent (two) infinite sums of the inverse first and
second powers of the poles in the complex plane were shown to be convergent in
the investigated particular case, and the case of the infinite point of the
linear-fractional transformation was investigated. Some relations were found,
which ensure the parametrization of the cubic equation in its general form with
the Weierstrass function.; Comment: v.2; submitted to Journ.Math.Phys.(October 2001); Latex
(Sci.Word,amsmath style), 77 pages, no figures, 4 appendixes; Sect.III
rewritten for more clear derivation of the cubic algebraic equation;
clarifying comments in Sect.VI and in the Introduction; new Sect.VII added;2
references corrected; acknowledgments added

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## ‣ The Commutativity of Integrals of Motion for Quantum Spin Chains and Elliptic Functions Identities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/11/2007
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We prove the commutativity of the first two nontrivial integrals of motion
for quantum spin chains with elliptic form of the exchange interaction. We also
show thair linear independence for the numbers of spins larger than 4. As a
byproduct, we obtained several identities between elliptic Weierstrass
functions of three and four arguments.; Comment: 13 pages

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## ‣ A primer on elliptic functions with applications in classical mechanics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/11/2007
Português

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The Jacobi and Weierstrass elliptic functions used to be part of the standard
mathematical arsenal of physics students. They appear as solutions of many
important problems in classical mechanics: the motion of a planar pendulum
(Jacobi), the motion of a force-free asymmetric top (Jacobi), the motion of a
spherical pendulum (Weierstrass), and the motion of a heavy symmetric top with
one fixed point (Weierstrass). The problem of the planar pendulum, in fact, can
be used to construct the general connection between the Jacobi and Weierstrass
elliptic functions. The easy access to mathematical software by physics
students suggests that they might reappear as useful tools in the undergraduate
curriculum.; Comment: 17 pages, 20 figures

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## ‣ Weierstrass mock modular forms and elliptic curves

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Mock modular forms, which give the theoretical framework for Ramanujan's
enigmatic mock theta functions, play many roles in mathematics. We study their
role in the context of modular parameterizations of elliptic curves
$E/\mathbb{Q}$. We show that mock modular forms which arise from Weierstrass
$\zeta$-functions encode the central $L$-values and $L$-derivatives which occur
in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a
kernel recently studied by H\"ovel, we obtain canonical weight 1/2 harmonic
Maass forms whose Fourier coefficients encode the vanishing of these values for
the quadratic twists of $E$. We employ results of Bruinier and the third
author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger,
and Zagier. We also obtain $p$-adic formulas for the corresponding weight 2
newform using the action of the Hecke algebra on the Weierstrass mock modular
form.; Comment: To appear in Research in Number Theory

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