In this paper we study the initial boundary value problem for two-dimensional semilinear wave equations with small data, in asymptotically Euclidean exterior domains. We prove that if 1<p≤pc(2), the ...problem admits almost the same upper bound of the lifespan as that of the corresponding Cauchy problem, only with a small loss for 1<p≤2. It is interesting to see that the logarithmic increase of the harmonic function in 2-D has no influence to the estimate of the upper bound of the lifespan for 2<p≤pc(2). One of the novelties is that we can deal with the problem with flat metric and general obstacles (bounded and simply connected), and it will be reduced to the corresponding problem with compact perturbation of the flat metric outside a ball.
It has been known for some time that the Green's function of a planar domain can be defined in terms of the exit time of Brownian motion, and this definition has been extended to stopping times more ...general than exit times. In this paper, we extend the notion of conformal invariance of Green's function to analytic functions which are not injective, and use this extension to calculate the Green's function for a stopping time defined by the winding of Brownian motion. These considerations lead to a new proof of the Riemann mapping theorem. We also show how this invariance can be used to deduce several identities, including the standard infinite product representations of several trigonometric functions.
Issue Title: Special Issue: Computer Vision and Pattern Recognition-CVPR 2004 The study of 2D shapes and their similarities is a central problem in the field of vision. It arises in particular from ...the task of classifying and recognizing objects from their observed silhouette. Defining natural distances between 2D shapes creates a metric space of shapes, whose mathematical structure is inherently relevant to the classification task. One intriguing metric space comes from using conformal mappings of 2D shapes into each other, via the theory of Teichmüller spaces. In this space every simple closed curve in the plane (a "shape") is represented by a 'fingerprint' which is a diffeomorphism of the unit circle to itself (a differentiable and invertible, periodic function). More precisely, every shape defines to a unique equivalence class of such diffeomorphisms up to right multiplication by a Möbius map. The fingerprint does not change if the shape is varied by translations and scaling and any such equivalence class comes from some shape. This coset space, equipped with the infinitesimal Weil-Petersson (WP) Riemannian norm is a metric space. In this space, the shortest path between each two shapes is unique, and is given by a geodesic connecting them. Their distance from each other is given by integrating the WP-norm along that geodesic. In this paper we concentrate on solving the "welding" problem of "sewing" together conformally the interior and exterior of the unit circle, glued on the unit circle by a given diffeomorphism, to obtain the unique 2D shape associated with this diffeomorphism. This will allow us to go back and forth between 2D shapes and their representing diffeomorphisms in this "space of shapes". We then present an efficient method for computing the unique shortest path, the geodesic of shape morphing between each two end-point shapes. The group of diffeomorphisms of S^sup 1^ acts as a group of isometries on the space of shapes and we show how this can be used to define shape transformations, like for instance 'adding a protruding limb' to any shape.PUBLICATION ABSTRACT
One of the substantial topics in the forward extrusion process is using theoretical methods to predict the exit profile distortion due to the non-symmetry of sections or the off-centricity of die ...cavity. In this paper, a new approach based on the Riemann mapping theorem and upper bound method is developed to obtain the velocity field and the strain distribution. Then, the strain distribution is determined for the die with off-centered square sections. Afterwards, the exit profile curvature is calculated using an approach based on elastic–plastic bending of beams. Theoretical results are compared to the results yielded through experimental and finite element simulation and validated. Subsequently, the effect of various parameters such as relative die length and friction factor is examined on the amount of the exit profile distortion. The results show that the obtained theory can be used to predict the strain distribution and the exit profile distortion in addition to the process pressure.
Locations and patterns of functional brain activity in humans are difficult to compare across subjects because of differences in cortical folding and functional foci are often buried within cortical ...sulci. Unfolding a cortical surface via flat mapping has become a key method for facilitating the recognition of new structural and functional relationships. Mathematical and other issues involved in flat mapping are the subject of this paper. It is mathematically
impossible to flatten curved surfaces without metric and area distortion. Nevertheless, “metric” flattening has flourished based on a variety of computational methods that minimize distortion. However, it is mathematically possible to flatten without any
angular distortion — a fact known for 150 years. Computational methods for this “conformal” flattening have only recently emerged. Conformal maps are particularly versatile and are backed by a uniquely rich mathematical theory. This paper presents a tutorial level introduction to the mathematics of conformal mapping and provides both conceptual and practical arguments for its use. Discrete conformal mapping computed via circle packing is a method that has provided the first practical realization of the Riemann Mapping Theorem (RMT). Maps can be displayed in three geometries, manipulated with Möbius transformations to zoom and focus on particular regions of interest, they respect canonical coordinates useful for intersubject registration and are locally Euclidean. The versatility and practical advantages of the circle packing approach are shown by producing conformal flat maps using MRI data of a human cerebral cortex, cerebellum and a specific region of interest (ROI).
Cortical flattening algorithms are becoming more widely used to assist in visualizing the convoluted cortical gray matter sheet of the brain. Metric-based approaches are the most common but suffer ...from high distortions. Conformal, or angle-based algorithms, are supported by a comprehensive mathematical theory. The conformal approach that uses circle packings is versatile in the manipulation and display of results. In addition, it offers some new and interesting metrics that may be useful in neuroscientific analysis and are not available through numerical partial differential equation conformal methods.
In this paper, we begin with a brief description of cortical “flat” mapping, from data acquisition to map displays, including a brief review of past flat mapping approaches. We then describe the mathematics of conformal geometry and key elements of conformal mapping. We introduce the mechanics of circle packing and discuss its connections with conformal geometry. Using a triangulated surface representing a cortical hemisphere, we illustrate several manipulations available using circle packing methods and describe the associated “ensemble conformal features” (ECFs). We conclude by discussing current and potential uses of conformal methods in neuroscience and computational anatomy.
We introduce a new method to compute conformal parameterizations using a recent definition of discrete conformity, and establish a discrete version of the Riemann mapping theorem. Our algorithm can ...parameterize triangular, quadrangular and digital meshes. It can also be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section.
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