Search results for " Geometry"
showing 10 items of 2294 documents
First-Order Calculus on Metric Measure Spaces
2020
In this chapter we develop a first-order differential structure on general metric measure spaces. First of all, the key notion of cotangent module is obtained by combining the Sobolev calculus (discussed in Chap. 2) with the theory of normed modules (described in Chap. 3). The elements of the cotangent module L2(T∗X), which are defined and studied in Sect. 4.1, provide a convenient abstraction of the concept of ‘1-form on a Riemannian manifold’.
Spaces of weighted symbols and weighted sobolev spaces on manifolds
1987
This paper gives an approach to pseudodifferential operators on noncompact manifolds using a suitable class of weighted symbols and Sobolev spaces introduced by H.O. Cordes on ℙ. Here, these spaces are shown to be invariant under certain changes of coordinates. It is therefore possible to transfer them to manifolds with a compatible structure.
Luigi Cremona’s Years in Bologna: From Research to Social Commitment
2011
Luigi Cremona (1830–1903), unanimously considered to be the man who laid the foundations of the prestigious Italian school of Algebraic Geometry, was active at the University of Bologna from October 1860, when assigned by the Minister Terenzio Mamiani (1799–1885) to cover the Chair of Higher Geometry, until September 1867 when Francesco Brioschi (1824–1897) called him to the Politecnico di Milano. The “Bolognese years” were Cremona’s richest and most significant in terms of scientific production, and, at the same time, were the years when he puts the basis for its most important interventions in the social and political life of the “newborn” kingdom of Italy. In this article we present thes…
Learning Dynamic Geometry: Implementing Rotations
1995
This paper presents research in which we observed students of various ages and ability levels solving problems by using several pieces of software for solid geometry with different user interfaces. We evaluated the influence of the software on students’ learning, ways of reasoning, and the kind of mental images generated.
Coxeter on People and Polytopes
2004
H. S. M. Coxeter, known to his friends as Donald, was not only a remarkable mathematician. He also enriched our historical understanding of how classical geometry helped inspire what has sometimes been called the nineteenth-century’s non-Euclidean revolution (Fig. 35.1). Coxeter was no revolutionary, and the non-Euclidean revolution was already part of history by the time he arrived on the scene. What he did experience was the dramatic aftershock in physics. Countless popular and semi-popular books were written during the early 1920s expounding the new theory of space and time propounded in Einstein’s general theory of relativity. General relativity and subsequent efforts to unite gravitati…
30 years of finite-gap integration theory
2007
The method of finite-gap integration was created to solve the periodic KdV initial problem. Its development during last 30 years, combining the spectral theory of differential and difference operators with periodic coefficients, the algebraic geometry of compact Riemann surfaces and their Jacobians, the Riemann theta functions and inverse problems, had a strong impact on the evolution of modern mathematics and theoretical physics. This article explains some of the principal historical points in the creation of this method during the period 1973–1976, and briefly comments on its evolution during the last 30 years.
CliffoSor: A Parallel Embedded Architecture for Geometric Algebra and Computer Graphics
2006
Geometric object representation and their transformations are the two key aspects in computer graphics applications. Traditionally, compute-intensive matrix calculations are involved to model and render 3D scenery. Geometric algebra (a.k.a. Clifford algebra) is gaining growing attention for its natural way to model geometric facts coupled with its being a powerful analytical tool for symbolic calculations. In this paper, the architecture of CliffoSor (Clifford Processor) is introduced. ClifforSor is an embedded parallel coprocessing core that offers direct hardware support to Clifford algebra operators. A prototype implementation on an FPGA board is detailed. Initial test results show more …
Design of the CGAL Spherical Kernel and application to arrangements of circles on a sphere
2009
International audience; This paper presents a CGAL kernel for algorithms manipulating 3D spheres, circles, and circular arcs. The paper makes three contributions. First, the mathematics underlying two non trivial predicates are presented. Second, the design of the kernel concept is developed, and the connexion between the mathematics and this design is established. In particular, we show how two different frameworks can be combined: one for the general setting, and one dedicated to the case where all the objects handled lie on a reference sphere. Finally, an assessment about the efficacy of the \sk\ is made through the calculation of the exact arrangement of circles on a sphere. On average …
A mathematical model of mass transfer in spherical geometry: plum (Prunus domestica) drying
2003
In this paper the analytical solution of a mathematical model of mass transfer in spherical geometry is presented for boundary conditions useful for simulating drying processes of fruit with near spherical stones. This model is applied to analyse the efficiency of a new pre-treatment for a prune drying processes. The new proposed physical abrasion pre-treatment increases the plum drying rate at 60 degreesC. The mathematical model here presented allows a complete comparison of the experimental results obtained with this pre-treatment and with traditional ones. In particular the greater efficiency of the new physical pre-treatment appears to be due to the enhancement of the water diffusivity …
Shell structure and level spacing distribution in metallic clusters
1993
The lattice gas Monte Carlo and tight binding method is used to study the electronic shell structure in large metallic clusters. The average density of states of a large ensemble of deformed clusters shows the same shell structure as the most spherical geometry. The level spacing distribution at the Fermi level is a Wigner distribution.