Given a category C and directed partially ordered set J, certain proJ-C on inverse systems in is constructed such that the ordinary pro-category pro-C most special case of singleton J ? {1}. Further, known pro*-category pro*-C becomes proN-C. Moreover, given pro-reflective pair (C, D), J-shape ShJ(C, D) corresponding functor SJ are which, mentioned cases, become well ones. Among several importa...

Two properties must be available in order to construct a fractal set. The first is the selfsimilarity of elements. second real fraction number dimension. In this paper,condensation principle introduced sets. Condensation idea represented threetypes. deduced from rotation –reflection linear transformation. dealt withgroup action. third by graph function.

Brain shapes do not necessarily form a continuum in some descriptor space, but may form clusters related to pre-determined genetical factors or acquired diseases. As a feasibility study for introducing a suitable descriptor space, the use of modal analysis was tested on a large brain database acquired in healthy young subjects. Significant shape differences due to gender were found, and intra-g...

Shape blending has several applications in computer graphics. In this paper, we present a new method for smoothly blending among multiple polygonal shapes. The blended shape is computed as a weighted average of the input shapes. The weight of each input shape is allowed to vary across the shape. This feature increases the flexibility for controlling the local appearance of the blended shapes. O...

T he study of shape has intrigued some of the brightest minds of humanity, from Leonardo Da Vinci and Carl Friedrich Gauss to some of the top scientists of the modern era. The mathematics to analyze shapes are both beautiful and challenging, covering a variety of tools, from topology (1), to metric and differential geometry (2, 3), to statistics (4). The applications are very diverse and potent...

Lines are mysterious. They are drawn by the hand, they are seen by the eye, they appear in the world. Lines are what the hands draw, what the eyes see, and what the page represents. Lines form forms. Simple regular visual/spatial forms like dots, lines, and containers, have meanings that are readily apparent in context. They are used in the service of clear communication, to self and others, no...

We address the question of which convex shapes, when packed as densely as possible under certain restrictions, fill the least space and leave the most empty space. In each different dimension and under each different set of restrictions, this question is expected to have a different answer or perhaps no answer at all. As the problem of identifying global minima in most cases appears to be beyon...

max 0≤≤1 ̄̄̄̄  +  − 2 min ≤≤  ̄̄̄̄ if  =∞ We examined k k earlier [2]; h i is a less familiar random variable but nevertheless important in the study of trees. Note that h i is not a norm since, for any constant , hi = 0 even if  6= 0. Let  be an ordered (strongly) binary tree with = 2+1 vertices. The distance between two vertices of  is the number of edges in the shortest p...

Michael J. Tarr received his PhD from M.I.T. in 1989. He is currently the Fox Professor of Ophthalmology and Visual Sciences and a Professor of Cognitive and Linguistic Sciences at Brown University. He is the recipient of the 1997 APA Distinguished Scientific Award for Early Career Contribution to Psychology in the Area of Cognition/Human Learning and the 2003 Troland Research Award from the Na...

A framework for computing shape statistics in general, and average in particular, for dynamic shapes is introduced in this paper. Given a metric d(·, ·) on the set of static shapes, the empirical mean of N static shapes,C1, . . . , CN , is defined by argminC 1 N ∑N i=1 d(C,Ci) . The purpose of this paper is to extend this shape average work to the case of N dynamic shapes and to give an efficie...