Cheeger's energy on the Harmonic Sierpinski Gasket

Koskela and Zhou have proven that, on the harmonic Sierpinski gasket with Kusuoka's measure, the "natural" Dirichlet form coincides with Cheeger's energy. We give a different proof of this result, which uses the properties of the Lyapounov exponent of the gasket

Young measures, Cartesian maps, and polyconvexity

We consider the variational problem consisting of minimizing a polyconvex integrand for maps between manifolds. We offer a simple and direct proof of the existence of a minimizing map. The proof is based on Young measures

Harmonic embeddings of the Stretched Sierpinski gasket.

P. Alonso-Ruiz, U. Freiberg and J. Kigami have defined a large family of resistance forms on the Stretched Sierpinski Gasket $G$. In the present paper we introduce a system of coordinates on $G$ (technically, an embedding of $G$ into $\R^2$) such that \noindent$\bullet$) these forms are defined on $C^1(\R^2,\R)$ and \noindent$\bullet$) all affine functions are harmonic for them. We do this adapting a standard method from the Harmonic Sierpinski Gasket: we start finding a sequence $G_l$ of pre-fractals such that all affine functions are harmonic on $G_l$. After showing that this property is inherited by the stretched harmonic gasket $G$, we use the formula for the Laplacian of a composition to prove that, for a natural measure $\mu$ on $G$, C^2(\R^2,\R)\subset\dc(\Delta) and Teplyaev's formula for the Laplacian of $C^2$ functions holds. Lastly, we use the expression for $\Delta u$ to show that the form we have found is closable in $L^2(G,\mu)$

Another point of view on Kusuoka's measure

Kusuoka's measure on fractals is a Gibbs measure of a very special kind, because its potential is discontinuous, while the standard theory of Gibbs measures requires continuous (actuallly, H\"older) potentials. In this paper, we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measures completely within the scope of the standard theory; there are naturally some minor modifications, but they are only due to the fact that we are dealing with matrix-valued functions and measures. We shall use these matrix-valued Gibbs measures to build self-similar Dirichlet forms on fractals. Moreover, we shall see that Kusuoka's measure can be recovered in a simple way from the matrix-valued Gibbs measure

Another point of view on Kusuoka's measure.

Kusuoka's measure on fractals is a Gibbs measure of a very special kind, since its potential is discontinuous while the standard theory of Gibbs measures requires continuous (in its simplest version, H\"older) potentials. In this paper we shall see that for many fractals it is possible to build a class of matrix-valued Gibbs measures completely within the scope of the standard theory; there are naturally some minor modifications, but they are only due to the fact that we are dealing with matrix-valued functions and measures. We shall use these matrix-valued Gibbs measures to build self-similar bilinear forms on fractals. Moreover, we shall see that Kusuoka's measure and bilinear form can be recovered in a simple way from the matrix-valued Gibbs measure

Cristoforo Buondelmonti: Greek Antiquities in Florentine Humanism

No abstrac

Cristoforo Buondelmonti: Greek Antiquities in Florentine Humanism

No abstrac

Laterality in artistic gymnastics

Worldwide trainers ask if there is a rotation scheme, which facilitate the learning of the elements with longitudinal rotations. Although there are some research on it, they did not attempt to verify a total scheme, but merely to see the relationship between two elements or four elements. In this study we analyse the appreciation of experts N = 161 coaches (age: 34.9 ± 10.9) from different levels of expertise and from different countries (ARG, BOL, BRA, CHI, ECU, ELS, GER, GUA, HON, MEX, PAN, PER, URU, VEN) with 12 ± 8.8 years of experience regardinghow gymnasts should execute 27 different elements in 5 male apparatus. We chose these elements, because we wanted to have movements with rotation in upright stance, upside down and in combination with transversal rotation. Through a questionnaire for coaches, we tried to verify if there are differences, coincidences or even immovable rules in the rotation scheme that gymnasts use (or should use). The answers were typologized with three categories of rotational preference: unilateral consistent twister, bilateral consistent twister and inconsistent twister. The study aimed to answer several questions: Do coaches agree on how the rotation scheme should be in gymnastics? How do coaches (former gymnasts) determined which way to turn? Do the handedness or the footedness influence on the direction of rotation? Does the personal rotation scheme influence on the concept of appropriate rotation scheme? Do the national practices influence the rotation scheme? Are there differences in appreciation between coaches at different levels? Are unambiguous rules among the elements?Técnicos do mundo todo questionam se há um esquema de rotação, que facilita a aprendizagem dos elementos nas rotações longitudinais. As pesquisas sobre o assunto verificam a relação entre dois a quatro elementos daquelas, então, neste estudo, analisamos a apreciação de especialistas n = 161 treinadores (idade: 34,9 ± 10,9) de diferentes níveis de especialização e de diferentes países (ARG, BOL, BRA, CHI, ECU, ELS, GER, GUA, HON, MEX, PAN , PER, URU, VEN) com 12 ± 8,8 anos de experiência a respeito de como um ginasta deve executar 27 diferentes elementos em cinco aparelhos masculinos. Escolhemos esses elementos, pois desejávamos investigar movimentos com rotação na posição em pé, de cabeça para baixo e em combinação com a rotação transversal. Por meio de um questionário para técnicos, averiguamos se existem diferenças, coincidências ou regras ainda imóveis no esquema de rotação que ginastas devem usar (ou deveriam usar). As respostas foram conceituadas em três categorias de preferência de rotação: rotação consistente unilateral, rotação consistente bilateral e rotação inconsistente. O estudo teve como objetivo responder a várias perguntas: Os técnicos concordam em relação a qual esquema de rotação deve estar na ginástica? Como técnicos (ex-ginastas) determinaram que caminho tomar? O posicionamento das mãos e dos pés influenciam na rotação? O modelo individual de rotação influência no conceito apropriado de rotação? As práticas nacionalizadas influênciam em um modelo de rotação? Existem diferenças na apreciação entre técnicos de diferentes níveis? Há regras ambíguas entre os elementos