One of the more popular problems today in the area of orthogonal polynomials is the classification of all orthogonal polynomial solutions to the second order differential equation:

In this paper, we show that the Laguerre type and Jacobi type polynomials satisfy such a second order equation.

In this paper we prove Chen inequalities for submanifolds of real space forms endowed with a semi-symmetric non-metric connection, i.e., relations between the mean curvature associated with a semi-symmetric non-metric connection, scalar and sectional curvatures, Ricci curvatures and the sectional curvature of the ambient space. The equality cases are considered.

Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.

In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed.

At the Edmonton Meeting of the Canadian Mathematical Congress E. Wigner asked me whether one knew something about the distribution of the characteristic roots of the linear transformations that leave invariant the quadratic form t2+x2-y2-z2, just as one knows that a Lorentz transformation has two complex conjugate characteristic roots and two real characteristic roots that are either inverse to one another or the numbers 1 and -1.

In this paper an answer to E. Wigner’s question will be obtained.

We are concerned with the pairs of matrices (X,A) with coefficients in a field of reference F such that the condition

0.1

is satisfied, where XT = (ξki) is the transpose of the matrix X = (ξki). It follows that both matrices are quadratic of the same degree d.

A graph G is defined as a set X = {x1, …, xn} of elements xi called vertices, and a collection Γ of (not necessarily distinct) unordered pairs of distinct vertices, called edges. An edge (xi, xj) is said to be incident to xi and xj which are its end-vertices.

In 1879 Kempe [5] presented what has become the most famous of all incorrect proofs of the Four Colour Conjecture, but even though his proof was erroneous his method has become quite useful. In 1890 Heawood [4] was able to modify Kempe's method to establish the Five Colour Theorem for planar graphs. In this article we show that other modifications of Kempe's method can be made which enable one to establish more results about planar graphs. By this process we obtain upper bounds for several parameters which involve partitioning the point set of a graph. In particular, we show that the point set of any planar graph can be partitioned into four or less subsets such that the subgraph induced by each subset is either disconnected or trivial (consists of a single point). We also show that the point set of any planar graph can be partitioned into three or less subsets such that the subgraph induced by each subset contains no cycles.

We consider a piecewise monotonie and piecewise continuous map T on the interval. If T has a derivative of bounded variation, we show for an ergodic invariant measure μ with positive Ljapunov exponent λμ that the Hausdorff dimension of μ equals hμ / λμ.

In [1] G. Birkhoff stated an algorithm for expressing a doubly stochastic matrix as an average of permutation matrices. In this note we prove two graphical lemmas and use these to find an upper bound for the number of permutation matrices which the Birkhoff algorithm may use.

A doubly stochastic matrix is a matrix of non-negative elements with row and column sums equal to unity and is there - fore a square matrix. A permutation matrix is an n × n doubly stochastic matrix which has n2-n zeros and consequently has n ones, one in each row and one in each column. It has been shown by Birkhoff [1],Hoffman and Wielandt [5] and von Neumann [7] that the set of all doubly stochastic matrices, considered as a set of points in a space of n2 dimensions constitute the convex hull of permutation matrices.

The relationships between synchronous and asynchronous reversible Markov systems are investigated. It is shown that the invariant measure of such systems is a second order Markov random field. The conditions under which the invariant measure is a first order Markov random field are obtained.

A partial homomorphism ϕ from B into A will be called a fractional homomorphism if dom ϕ ≤ B (A). Every extension of a fractional homomorphism is again fractional, and each fractional homomorphism has a unique irreducible extension.

Let $C$ be a curve over a complete valued field having an infinite residue field and whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on $C$, confirming a conjecture of Cools, Draisma, Robeva, and the third author.

The eigenvalues of a second order self-adjoint elliptic differential operator on Riemannian n-space R will be considered. Our purpose is to obtain asymptotic variational formulae for the eigenvalues under the topological deformations of (i) removing an ɛ -cell (and adjoining an additional boundary condition on the boundary component thereby introduced); and (ii) attaching an ɛ -handle, valid on a half-open interval 0 < ɛ ≤ ɛo. In particular the formulae will exhibit the non-analytic nature of the variation. Similar variational problems for singular ordinary differential operators have been considered by the writer in [3] and [4].

A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer ${{C}_{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $S$ of the Schrödinger (Lie) algebra. The simple ${{C}_{S}}(H)$-modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $S$ (over the centre). It is proved that some (prime) factor algebras of $S$ and ${{C}_{S}}(H)$ are tensor homological$/$Krull minimal.

The following question has been asked by J. Dénes [2]: If n - 1 elements of the diagonal of an n × n array are prescribed, is it possible to complete the array to form an n × n latin square?" It is known that if n diagonal elements are given such a completion is not always possible.

That the answer to Dénes' question is yes follows directly from a theorem of M. Hall Jr. [1].

In this note we consider some problems related to the following question: What is the smallest integer m(n) for which there exists a family Fn of sets A1, A2,…, Am(n) with the following properties, (i) each member of Fn has n elements and (ii) if S is a set which meets each member of Fn, then S contains at least one member of Fn?

Erdős and Hajnal [1] observed that

and that m(l) = 1, m(2) = 3, m(3) = 7.

The action by multiplication of the class of transpositions of the symmetric group on the other conjugacy classes defines a graded poset as described by Birkhoff ([2]). In this paper, the edges of this poset are given a weight and the structure obtained is called the poset of conjugacy classes of the symmetric group. We use weights of chains in the posets to obtain new formulas for the decomposition of products of conjugacy classes of the symmetric group in its group algebra as linear combinations of conjugacy classes and we derive a new identity involving partitions of n.

If S is an ordered set we write tp S to denote the order type of S and |5| for the cardinal of S. We also write [S]k for the set {X:X ⊂ S, |X|=k}. The partition symbol

(1)

Let G(x,s) be the Green's function for the boundary value problem y(n) = 0, Ty = 0, where Ty = 0 represents boundary conditions at two points. The signs of G(x,s) and certain of its partial derivatives with respect to x are determined for two classes of boundary value problems. The results are also carried over to analogous classes of boundary value problems for difference equations.

In answer to a question raised by Leo Moser, A. Meir proved some years ago that every plane arc of unit length lies in some closed semidisk of radius ½. His elegant, unpublished argument is reproduced here with his kind permission.