J. N. Boyd

Representation theory of finite abelian groups applied to a linear diatomic crystal

After a brief review of matrix representations of finite abellan groups, projection operators are defined and used to compute symmetry coordinates for systems of coupled harmonic oscillators. The Lagrangian for such systems is discussed in the event that the displacements along the symmetry coordinates are complex. Lastly, the natural frequencies of a linear, dlatomic crystal are deter- mined through application of the Born cyclic condition and the determination of the symmetry coordinates.

The Gergonne point generalized through convex coordinates

The Gergonne point of a triangle is the point at which the three cevians to the points of tangency between the incircle and the sides of the triangle are concurrent. In this paper, we follow Koneĉný [7] in generalizing the idea of the Gergonne point and find the convex coordinates of the generalized Gergonne point. We relate these convex coordinates to the convex coordinates of several other special points of the triangle. We also give an example of relevant computations.

PROJECTION OPERATOR TECHNIQUES IN LAGRANGIAN MECHANICS: SYMMETRICALLY COUPLED OSCILLATORS

We apply projection operator techniques to the computation of the natural frequencies of oscillation for three symmetrically coupled mechanical systems. In each case, the rotation subgroup of the full symmetry group is used to determine the projection operators with the result that the Lagrangian must be expressed in terms of complex-valued coordinates. In the coordinate system obtained from the action of the projection operators upon the original coordinates, the Lagrangian yields equations of motion which are separated to the maximum extent made possible by symmetry considerations.

A geometrical approach to maximizing a variance

Abstract The matrix form for the variance of n independently chosen real numbers, x 1 , x 2 ,…, x n , is observed to be the same as that for the potential energy matrix for n coupled oscillators symmetrically arranged on a circle. Unitary matrices that have previously been developed from the irreducible representations of the circular symmetries to diagonalize the potential energy matrices are given. These transformations are applied to the variance. A geometric setting is provided for the transformations, and the maximum variance is found for real numbers x i subject to linear constraints on the choices of the numbers.

A GROUP THEORETIC APPROACH TO GENERALIZED HARMONIC VIBRATIONS IN A ONE DIMENSIONAL LATTICE

Beginning with a group theoretical simplification of the equations of motion for harmonically coupled point masses moving on a fixed circle, we obtain the natural frequencies of motion for the array. By taking the number of vibrating point masses to be very large, we obtain the natural frequencies of vibration for any arbitrary, but symmetric, harmonic coupling of the masses in a one dimensional lattice. The result is a cosine series for the square of the frequency, s f2 i

Computations for a vibrating system diagonalize the variance

The transformations to diagonalize potential energy matrices for coupled harmonic oscillators will also diagonalize the variance when written in matrix form. After a brief review of a geometrical interpretation of the variance, the transformations are described and an example is given.

TWO DIMENSIONAL LATTICE VIBRATIONS FROM DIRECT PRODUCT REPRESENTATIONS OF SYMMETRY GROUPS

Arrangements of point masses and ideal harmonic springs are used to model two dimensional crystals. First, the Born cyclic condition is applied to a double chain composed of coupled linear lattices to obtain a cylindrical arrangement. Then the quadratic Lagrangian function for the system is written in matrix notation. The Lagrangian is diagonalized to yield the natural frequencies of the system. The trans- formation to achieve the diagonalization was obtained from group theorectic considera- tions. Next, the techniques developed for the double chain are applied to a square lattice. The square lattice is transformed into the toroidal Ising model. The direct product nature of the symmetry group of the torus reveals the transformation to diag- onalize the Lagrangian for the Ising model, and the natural frequencies for the prin- cipal directions in the model are obtained in closed form.

Random processes with convex coordinates on triangular graphs

Probabilities for reaching specified destinations and expectation values for lengths for random walks on triangular arrays of points and edges are computed. Probabilities and expectation values are given as functions of the convex (barycentric) coordinates of the starting point.