인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

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Orthogonal Polynomials of Several Variables

1. (i) Suppose K is a conjugacy class of Sn contained in An; then K is called split if K is a union of two conjugacy classes of An. Show that the number of split conjugacy classes contained in An is equal to the number of characters χ ∈ Irr(Sn) such that χAn is not irreducible. (Hint. Consider the vector space of class functions on An which are invariant under conjugation by the transposition (12).)

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Character Theory of Finite Groups

When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This paper explains the mathematical connections between mesh geometry, interpolation errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these two purposes do not agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation. Several of the upper and lower bounds on interpolation errors and element stiffness matrix conditioning given here are tighter than those that have appeared in the literature before, so the quality measures are likely to be unusually precise indicators of element fitness.

What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures.

Coming to the end of Part 1, we hope you have a different picture of a frame in your mind from a "picture frame." While necessarily colored by our personal bias, we intended this tutorial as a basic introduction to frames, geared primarily toward engineering students and those without extensive mathematical training. Frames are here to stay; as wavelet bases before them, they are becoming a standard tool in the signal processing toolbox, spurred by a host of recent applications requiring some level of redundancy.

Life Beyond Bases: The Advent of Frames (Part II)

An Introduction to Finite Tight Frames

The aim of this paper is to investigate the symmetry properties of tight frames, with a view to constructing tight frames of orthogonal polynomials in several variables which share the symmetries of the weight function, and other similar applications. This is achieved by using representation theory to give methods for constructing tight frames as orbits of groups of unitary transformations acting on a given finite-dimensional Hilbert space. Along the way, we show that a tight frame is determined by its Gram matrix and discuss how the symmetries of a tight frame are related to its Gram matrix. We also give a complete classification of those tight frames which arise as orbits of an abelian group of symmetries.

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Tight Frames and Their Symmetries

This article shows what are called the Welch (1974) bound equality (WBE) sequences by the signal processing community are precisely the isometric/equal norm-normalized/uniform tight frames which are currently being investigated for a number of applications, and in the real case are the spherical 2-designs of combinatorics. Recent applications include wavelet expansions, Grassmannian frames, frames robust to erasures, and quantum measurements. This is done by giving an elementary proof of a generalization of Welch's inequality to vectors which need not have equal energy, and then showing that equality occurs in this exactly when the vectors form a tight frame.

Shayne Waldron

Papers

An Introduction to Finite Tight Frames

Tight Frames and Their Symmetries

Generalized Welch bound equality sequences are tight frames

On the construction of equiangular frames from graphs

The Eigenstructure of the Bernstein Operator