Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equation

so that there corresponds the matrix B to the bilinear form

4.1

on the linear space

4.2

and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.

The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

This is the first of (presumably) three articles on the subject mentioned in the title. The exposition is based on a course of fifteen lectures which formed part of the Edmonton (1957) Seminar of the Canadian Mathematical Congress. Limitations of space (and, originally, of time) compel us to be selective in two directions. First, while we shall refer to other branches of logic in passing, we shall be concerned principally with the two fundamental calculi - of propositions and of predicates (of the first order). Thus, except for a number of modern developments which are included here, our exposition will be similar in scope, though not in detail, to the first and third chapters of the well-known "Principles of Mathematical Logic" by D. Hilbert and W. Ackermann (English translation, Chelsea, New York, 1950) and this was in fact the recommended text for the Edmonton course. However, there exists a growing number of other good introductions to the field and some of these will be listed later.

For the past four years we have given, at the University of Alberta, a course entitled "The Nature of Mathematics". This course is open to first and second year students in arts and science and in education, and is designed primarily for those who will be taking only a single course in mathematics at the university. The only prerequisite for the course is high school mathematics and the course is not prerequisite for any other course.

1. Explain briefly what is meant by each of the following:

• (i) Euler path

• (ii) fixed point theorem

• (iii) four color problem

• (iv) unit braid of order n

• (v) MBbius band.

The sixth mathematics seminar was held on the University of Alberta campus in Edmonton, Alberta, in conjunction with the first seminar of the Canadian Association of Physicists (Theoretical Physics Division). The following courses were given.

Immediately following the seminar in Edmonton the fourth quadrennial Mathematical Congress was held in Banff, Alberta. The Canadian Association of Physicists (Theoretical Physics Division) participated in this, as they did in the seminar.

Finance. Grants to support the Journal were received as follows

In addition, 35,530 was received from subscriptions, advertising, reprints, etc. Total cost of printing (at the University of Totonto Press) was about 312,500, leaving, after other expenses, a balance of $5,700.

The Institute was in session from May 15 to August 15, 1957. There were 32 Fellows and visitors from the staffs of 19 Canadian universities and from the University of London and Rutgers University. Fellowship holders.

The new Bulletin has started off most successfully and congratulations to Peter Scherk, Jim Lambek, John Peck and all the others who have worked together to launch it so successfully. The reports on the Seminar at Edmonton and the Congress at Banff, which appeared in the first number of the Bulletin, happily make it unnecessary for me to do more than to say that these seemed to be as successful meetings as any we have held. The settings both at Edmonton and Banff were ideal, but without our hosts1 careful planning and continued attention to detail, these would have been of no avail.

The Montreal Mathematical Colloquium is an organization of mathematicians in the Montreal area formed for the purpose of holding regular mathematical meetings and for inviting outside speakers. This year (1957-1958) the following talks were given by the invited speakers.

University of Alberta: Professor J.G. Kemeny spent four days here, March 12 to 15, as a visiting lecturer for the Mathematical Association of America. He gave two colloquium talks on "Markov chains in probability theory" and public lectures on "New developments in the teaching of mathematics" and on "Mathematics in the Social Sciences". He was guest speaker at a dinner meeting of the Mathematics and Physics club, at which he gave a delightful popular talk on the way group theory might have originated, and he also addressed several groups of mathematics students. His visit was greatly appreciated.