5 research outputs found
McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions
Evidence is presented to suggest that, in three dimensions, spherical
6-designs with N points exist for N=24, 26, >= 28; 7-designs for N=24, 30, 32,
34, >= 36; 8-designs for N=36, 40, 42, >= 44; 9-designs for N=48, 50, 52, >=
54; 10-designs for N=60, 62, >= 64; 11-designs for N=70, 72, >= 74; and
12-designs for N=84, >= 86. The existence of some of these designs is
established analytically, while others are given by very accurate numerical
coordinates. The 24-point 7-design was first found by McLaren in 1963, and --
although not identified as such by McLaren -- consists of the vertices of an
"improved" snub cube, obtained from Archimedes' regular snub cube (which is
only a 3-design) by slightly shrinking each square face and expanding each
triangular face. 5-designs with 23 and 25 points are presented which, taken
together with earlier work of Reznick, show that 5-designs exist for N=12, 16,
18, 20, >= 22. It is conjectured, albeit with decreasing confidence for t >= 9,
that these lists of t-designs are complete and that no others exist. One of the
constructions gives a sequence of putative spherical t-designs with N= 12m
points (m >= 2) where N = t^2/2 (1+o(1)) as t -> infinity.Comment: 16 pages, 1 figur
Cubature formulas, geometrical designs, reproducing kernels, and Markov operators
Cubature formulas and geometrical designs are described in terms of
reproducing kernels for Hilbert spaces of functions on the one hand, and Markov
operators associated to orthogonal group representations on the other hand. In
this way, several known results for spheres in Euclidean spaces, involving
cubature formulas for polynomial functions and spherical designs, are shown to
generalize to large classes of finite measure spaces and
appropriate spaces of functions inside . The last section
points out how spherical designs are related to a class of reflection groups
which are (in general dense) subgroups of orthogonal groups
Balanced Frames: A Useful Tool in Signal Processing with Good Properties
So far there has not been paid attention to frames that are balanced, i.e. those frames which sum is zero. In this paper we consider balanced frames, and in particular balanced unit norm tight frames, in finite dimensional Hilbert spaces. Here we discover various advantages of balanced unit norm tight frames in signal processing. They give an exact reconstruction in the presence of systematic errors in the transmitted coefficients, and are optimal when these coefficients are corrupted with noises that can have non-zero mean. Moreover, using balanced frames we can know that the transmitted coefficients were perturbed, and we also have an indication of the source of the error. We analyze several properties of these types of frames. We define an equivalence relation in the set of the dual frames of a balanced frame, and use it to show that we can obtain all the duals from the balanced ones. We study the problem of finding the nearest balanced frame to a given frame, characterizing completely its existence and giving its expression. We introduce and study a concept of complement for balanced frames. Finally, we present many examples and methods for constructing balanced unit norm tight frames.Fil: Heineken, Sigrid Bettina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Morillas, Patricia Mariela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; ArgentinaFil: Tarazaga, Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; Argentin