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F D Murnaghan

Publications and source records attributed to F D Murnaghan.

At least 19 recordsLinked to original sources
PubMed1972

Airey's Converging Factor.

Asmptotic series for the calculation of functions, for values of the argument numerically >1, start off with terms whose numerical values decrease but, at a certain stage, the terms begin to increase in numerical value and must be ignored. At this stage, there may be two adjacent terms of equal numerical value; when the least term of the asymptotic series is spoken of, it is in reference to the first of these two terms. The sum of the initial terms of the asymptotic series up to, and including, the least term often furnishes a fair approximation to the desired value of the function being evaluated. It was early observed by computers that if the terms of the asymptotic series alternate in sign, this approximation was often improved by replacing the least term by its half. The factor by which the least term of the asymptotic series must be multiplied so that the true value of the function being evaluated is obtained by addition of this modified least term to the remaining initial terms of the asymptotic series is known as the converging factor for the asymptotic series. The converging factor for the asymptotic series involved in the calculation of the exponential integral, for large negative values of the argument, was given as a power series in the reciprocal of the argument by Airey; the first term of this series is (1/2). A method for the determination of the coefficients of this series is given.

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PubMed1972

Powers of representations of the rotation group (their symmetric, alternating, and other parts).

It has been known for more than three quarters of a century that the square of the irreducible representation D(j), of dimension 2j + 1, of the 3-dimensional rotation group R(3) contains the representations D(2j),D(2j - 1),...,D(1),D(0) of R(3), each occurring once. I give in this paper a method, involving no more than the expansion of (1 + x +... + x(2j))(n-1), of analyzing (D(j))(n), the nth power of D(j), n = 3, 4, 5,..., into the sum of the irreducible representations D(nj), D(nj - 1),..., D(1), D(0) of R(3), each occurring a stated number of times. When n = 2, the part D(2j) + D(2j - 2) +... + D(2) + D(0) of (D(j))(2) is associated with the partition (2) of 2 and is termed the symmetric part of (D(j))(2), while the remaining part, D(2j - 1) + D(2j - 3) +... + D(3) + D(1), of (D(j))(2) is associated with the partition (1(2)) of 2 and is termed the alternating part of (D(j))(2). For any value of n, (D(j))(n) contains various parts, each associated with a k-part partition of n, where k </= 2j + 1, the part associated with the l-part partition (n) of n being termed the symmetric part of (D(j))(n) and the part associated with the n-part partition (1(n)) of n, where n </= 2j + 1, being termed the alternating part of (D(j))(n). If n > 2j + 1, (D(j))(n) has no alternating part. I show how to determine these various parts giving full details when j = 1 and n is arbitary, and when j is arbitrary and n = 3 or 4. I also show that when n = 2m + 1 is odd the alternating part of (D(j))(n) is, when it exists, i.e., when m </= j, the same as the symmetric part of (D(j - m))(n) and that the alternating part of (D(j))(n) is the same as the alternating part of (D(j))(n'), where n + n' = 2j + 1. This implies that, when n is even and <2j + 1, the alternating part of (D(j))(n) is the same as the symmetric part of (D(n/2))(2j+1-n).

Journal Article↗
PubMed1971

The characters of the symmetric group.

A short and simple derivation of the formula of Frobenius, which gives the dimensions of the irreducible representations of S(n), the symmetric group on any number, n, of symbols, is given. These dimensions are the characters of the identity element of the group, i.e., of the element all of whose cycles are unary. It is shown how a slight modification of Frobenius' formula yields, when n = 2p is even, the characters of an element of S(n) all of whose cycles are binary and, when n = 3p is a multiple of 3, the characters of an element of S(n) all of whose cycles are ternary and, generally, when n = kp is a multiple of any positive integer k, the characters of an element of S(n) all of whose cycles are of length k. It is noteworthy that the calculations become simpler, rather than more complicated, as k increases. Finally, this paper shows how to derive from Frobenius' formula the characters of an element of S(n) which has at least one unary cycle and, from the present modifications of Frobenius' formula, the characters of an element of S(n) which has at least one cycle of length k, k = 2, 3,..., n.

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