We discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to ${1}/{f}\;$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modiûed to produce inner functions.

The Four Color Theorem says that the faces (or vertices) of a plane graph may be colored with four colors. Vizing’s Theorem says that the edges of a graph with maximum degree $\Delta$ may be colored with $\Delta \,+\,1$ colors. In 1972, Kronk and Mitchem conjectured that the vertices, edges, and faces of a plane graph may be simultaneously colored with $\Delta \,+\,4$ colors. In this article, we give a simple proof that the conjecture is true if $\Delta \,\ge \,6$.

In this paper we prove some properties of the so-called modulus of noncompact convexity. This notion was recently introduced by K. Goebel and T. Sȩkowski [6] and it appears to be an interesting and useful generalization of the classical Clarkson modulus of convexity. We extend the results obtained in [6] showing that the modulus of noncompact convexity is continuous and has some extra properties in reflexive Banach spaces. The properties applicable in the fixed point theory are also stated.

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched coverings of ${{S}^{3}}$ branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in these examples are known as Fibonacci groups which we show not to be left-orderable. Many other examples of non-orderable groups are obtained by taking 3-fold branched covers of ${{S}^{3}}$ branched along various hyperbolic 2-bridge knots. The manifold obtained in such a way from the ${{5}_{2}}$ knot is of special interest as it is conjectured to be the hyperbolic 3-manifold with the smallest volume.

Let $G$ be a hermitian quaternionic group. We determine complementary series for representations of $G$ induced from super-cuspidal representations of a Levi factor of the Siegel maximal parabolic subgroup of $G$.

Problem 7 of Section VI of H. T. Croft's "Research Problems" (August, 1967 edition) is by J. H. Conway:

A is a finite set of integers {ai}. A + A denotes {ai + aj}, A - A denotes {ai - aj}. Prove that A - A always has more members than A + A, unless A is symmetrical about 0.

First note that 'about 0' is redundant and erroneous.

Let be a non-decreasing sequence of non-negative numbers, and let U∘=0. Then we have

Yang [3] proved the following integral inequality:

If y(x) is absolutely continuous on a≤x≤X, with y(a) = 0, then

In this paper it is shown that the maximal spectral type of a general rank one transformation is given by a kind of generalized Riesz product, with possibly some discrete measure.

Several authors have investigated "rings of quotients" of a given ring R. Johnson showed that if R has zero right singular ideal, then the injective hull of RR may be made into a right self injective,- regular (in the sense of von Neumann) ring (see [7] and [12]). In articles by Utumi [10], Findlay and Lambek [6], and Bourbaki [2], various structures which correspond to sub-modules of the injective hull of R are made into rings in a natural manner, in [8], Lambek points out that in each of these cases the rings constructed are subrings of Utumi' s maximal ring of right quotients, which is the maximal rational extension of R in its injective hull. Lambek also shows that Utumi's ring is canonically isomorphic to the bicommutator of the injective hull of RR if R has 1. It thus appears that a "natural" definition of the injective hull of RR as a ring extending module multiplication by R has been carried out only in the case that the injective hull is a rational extension of R. (See [12], [10], or [6] for various definitions of this concept.)

Let a particle move along a straight line a unit distance during every interval of time τ. During the first interval τ it moves to the right with probability ρ1 and to the left with probability ρ2 = 1 - ρ1. Thereafter at the end of each interval τ, the particle with probability p continues its motion in the same direction as in the previous step and with probability q = l - p reverses it.

We prove that generalized free products of finitely generated free-byfinite or nilpotent-by-finite groups amalgamating a cyclic subgroup areconjugacy separable. Applying this result we prove a generalization of a conjecture of Fine and Rosenberger [7] that groups of F-type are conjugacy separable.

We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.

We give a short proof of a result of G. Paouris on the tail behaviour of the Euclidean norm $\left| X \right|$ of an isotropic log-concave random vector $X\,\in \,{{\mathbb{R}}^{n}},$ stating that for every $t\,\ge \,1$,

$$\mathbb{P}\left( \left| X \right|\,\ge \,ct\sqrt{n} \right)\,\le \,\exp (-t\sqrt{n}).$$

More precisely we show that for any log-concave random vector $X$ and any $p\,\ge \,1$,

$${{(\mathbb{E}{{\left| X \right|}^{p}})}^{1/p}}\,\sim \,\mathbb{E}\left| X \right|\,+\,\underset{z\in {{S}^{n-1}}}{\mathop{\sup }}\,\,{{(\mathbb{E}{{\left| \left\langle z,\,X \right\rangle \right|}^{p}})}^{1/p}}.$$

Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write

(1)

if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is written

and means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.

In the present paper, we introduce a modification of the Meyer-König and Zeller $\left( \text{MKZ} \right)$ operators which preserve the test functions ${{f}_{0}}\left( x \right)=1$ and ${{f}_{2}}\left( x \right)={{x}^{2}}$, and we show that this modification provides a better estimation than the classical $\left( \text{MKZ} \right)$ operators on the interval $\left[ \frac{1}{2},1 \right)$ with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the $r$-th order generalization of our operators and study their approximation properties.

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.

Analogues of van der Waerden’s theorem on arithmetic progressions are considered where the family of all arithmetic progressions, $\text{AP}$, is replaced by some subfamily of $\text{AP}$. Specifically, we want to know for which sets $A$, of positive integers, the following statement holds: for all positive integers $r$ and $k$, there exists a positive integer $n={w}'\text{(}k,r)$ such that for every $r$-coloring of $[1,\,n]$ there exists a monochromatic $k$-term arithmetic progression whose common difference belongs to $A$. We will call any subset of the positive integers that has the above property large. A set having this property for a specific fixed $r$ will be called $r$-large. We give some necessary conditions for a set to be large, including the fact that every large set must contain an infinite number of multiples of each positive integer. Also, no large set $\{{{a}_{n}}\,:\,n\,=\,1,\,2,\ldots \}$ can have $\underset{n\to \infty }{\mathop{\lim \,\inf }}\,\,\frac{{{a}_{n+1}}}{{{a}_{n}}}\,>\,1$. Sufficient conditions for a set to be large are also given. We show that any set containing $n$-cubes for arbitrarily large $n$, is a large set. Results involving the connection between the notions of “large” and “2-large” are given. Several open questions and a conjecture are presented.

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.

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.