A previous note dealt with the summation of the series

and

where n is a fixed positive integer and r = 0, 1, …, n − 1.

The following two theorems are extensions of the Cayley-Hamilton theorem. The first gives the explicit expression for AN as a polynomial in A, where A is an arbitrary square matrix. The second expands the first result to the case of a matrix series.

The congruence x2 ≡ a (mod p), where p is an odd prime and a is any number not divisible by p, sometimes has a solution, but sometimes it has not. Gauss’s symbol (a/p), also known as Legendre’s symbol, is defined as (a/p) = l if there exists at least one x as a solution to this congruence and (a/p) = − l if no such solution exists.

The “Law of Reciprocity”, the famous theorem by Gauss, then states that, if p and q are odd primes,

where

It is remarkable that Cayley, who gave (1895) a purely algebraic treatment of the addition formulae for the Jacobian elliptic functions, never pointed out their essential identity with the ordinary formulae of spherical trigonometry

S. Zylbertrest in his interesting generalisation of the Wallace-line has opened the geometric gate to a new field of research. D. G. Tahta then followed with an algebraic approach and derived a few theorems from his equation of the Zylbertrest-line.

Goodner has studied the equations to conic sections in the elliptic plane by following the methods of Euclidean geometry and using Weierstrassian co-ordinates. In the present note the author has investigated several properties of the elliptic circles from the same analytic standpoint and has shown that some of the well-known properties of the Euclidean circles hold as well for the elliptic circles.

In a recent paper, under the same title, I proved that inversion of certain functional relations is the characteristic property of the Mobius’ μ-function, by establishing the:

THEOREM I′: f(n), g(n) and μ*(n) (possibly depending on f) are three arithmetical functions with

It is a standard textbook exercise to show that by considering the coefficients of xn first in (1 + x)n. (1 + x)n and then in (1 + x)2n It is the purpose of this note to show that formulas for a wide class of sums of products of binomial coefficients can be obtained by using Leibniz’ rule for the differential coefficient of a product; all expressions are derived for three indices, but generalization to any other number is obvious.

Given a positive continuous increasing function f(x), 0 ≤ x < 1, approximations by a convex function g(x) from below and from above are considered. The approximation is measured by the ratio of the integrals

In elementary courses on differential equations the standard method of obtaining particular integrals is probably that of the variation of parameters or, as it is sometimes rather paradoxically called, the variation of constants. In more sophisticated courses the idea of the Green’s function is introduced. This has many advantages, not the least of which is that the method for finding particular integrals based on such a function is much better motivated. Moreover, it is possible to fit in given boundary conditions, homogeneous or otherwise, as part of the process and this avoids artificially breaking the problem up into two parts, in the second of which a plethora of arbitrary constants is related to given boundary or initial conditions. A disadvantage of Green’s function methods is that, if they are to be properly exploited, it is necessary to introduce the concept of the delta function.

Suppose A is an n × n matrix with real elements, and that λ = λ1 is a dominant double real eigenvalue. We will assume for simplicity that none of the order eigenvalues are repeated. Then there are two possibilities

• (a) matrix is non-defective, and we can find two linearly independent eigenvectors corresponding to the double root λ1,

• (b) matrix is defective, and we have only one linearly independent eigenvectors corresponding to the double root λ1.

By a nearly diagonal matrix is meant a matrix in which the off-diagonal elements are small compared with those on the diagonal. Such matrices occur, for instance, in connection with non-orthogonal experimental designs. Inversion may be performed by methods of general applicability such as the Doolittle or Cholesky procedures (direct methods) or relaxation (an iterative method); many such procedures are described in the literature. Those already familiar with any of these methods may well prefer them to the one described below because of their wider applicability, but anyone not so placed who is often concerned with the inversion of nearly diagonal matrices should find the method described here useful.

In Note 2968 a definition of dual numbers a1 + ∊a2, where a1 and a2 are real numbers and ∊2 is put equal to zero whenever it occurs, is given. Similarly, dual vectors are expressions q1 + ∊q2, where q1 and q2 are ordinary three-component vectors (see [1]). The dual vector q1 + ∊q2 is said to be proper if |q1| 0, and normal if |q1| = 1, q1 . q2 ≠ 0.

When Descartes invented and defined exponents about the year 1637, he did not foresee the use of exponents which are not natural numbers. As a result, the standard treatment found in High School Algebra texts developed. While it is correct mathematically, it is very unsatisfactory psychologically. The following treatment is suggested as an improvement.