If X is a topological space and Y is any set, then we call a family $\mathcal {F}$ of maps from X to Y nowhere constant if for every non-empty open set U in X there is $f \in \mathcal {F}$ with $|f[U]|> 1$, i.e., f is not constant on U. We prove the following result that improves several earlier results in the literature.

If X is a topological space for which $C(X)$, the family of all continuous maps of X to $\mathbb {R}$, is nowhere constant and X has a $\pi $-base consisting of connected sets then X is $\mathfrak {c}$-resolvable.

We introduce a new type of differentiability, called cofinite Fréchet differentiability. We show that the convex point-of-continuity property of Banach spaces is dual to the cofinite Fréchet differentiability of all equivalent norms. A corresponding result for dual spaces with the weak* convex point-of-continuity property is also established.

It is well known that in the Euclidean plane there are seven distinct frieze patterns, i.e. seven ways to generate an infinite design bounded by two parallel lines. In the hyperbolic plane, this can be generalized to two types of frieze patterns, those bounded by concentric horocycles and those bounded by concentric equidistant curves. There are nine such frieze patterns; as in the Euclidean case, their symmetry groups are and

In what follows small latin letters denote rational integers (whole numbers) and we write

Let us consider the system of k equations

1

that is

These equations are obviously satisfied if the b are a permutation of the a; such a solution we call trivial.

We classify real hypersurfaces in complex projective space whose structure Jacobi operator satisfies two conditions at the same time.

Let K be a compact subset of the real Euclidean space En. We say that K has constant width if the distance between each pair of distinct parallel hyperplanes which support K is constant. The collection of all compact convex subsets of En which have constant width is denoted .

Raghavarao and Chandrasekhararao [3] introduced a family of PBIB designs having three associate classes known as cubic designs. In this paper a detailed analysis of the case of PBIB designs having four associate classes, which are called quartic designs, is given. Results are obtained pertaining to construction and existence of quartic designs. Moreover, using methods similar to those used by Shah [5], [6], [7], the block structure of certain quartic designs is studied.

It is shown that the operator of integration Fy(x) = ∫x0y(t) dt defined on the space C(—∞, ∞) of all continuous real valued functions on (—∞, ∞) is a contraction relative to a certain family of seminorms generating the topology of uniform convergence on compacta. However, as a contrast to this it is proved that F is not contractive with respect to any metric on C(—∞, ∞) inducing the above topology on C(—∞, ∞).

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

In this paper we study the Hilbert transformations over ${{L}^{2}}(\mathbb{R})$ and ${{L}^{2}}(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over ${{L}^{2}}(\mathbb{R})$ commutative with the $ax\,+\,b$ group, we show that the operator is of the form $\lambda I+\eta H$, where $I$ and $H$ are the identity operator and Hilbert transformation, respectively, and $\lambda ,\eta $ are complex numbers. In the related literature this result was proved by first invoking the boundedness result of the operator using some machinery. In our setting the boundedness is a consequence of the boundedness of the Hilbert transformation. The methodology that we use is the Gelfand–Naimark representation of the $ax\,+\,b$ group. Furthermore, we prove a similar result on the unit circle. Although there does not exist a group like the $ax\,+\,b$ group on the unit circle, we construct a semigroup that plays the same symmetry role for the Hilbert transformations over the circle ${{L}^{2}}(\mathbb{T})$.

Under certain conditions the reduced product space JX of a space X has the same homotopy type as ΩΣX, the loop-space on the suspension of X. Several proofs can be found in the literature. The original proof [6] made unnecessarily strong assumptions. Later, in the last chapter of [3], torn Dieck, Kamps and Puppe gave a proof under much weaker conditions and showed that they could not be further weakened. The question then arose as to whether the reduced product construction could be generalized to provide combinatorial models not only of ΩΣX but also of Ω2Σ2X, Ω3Σ3X and so on. This was answered in the affirmative by May [10], using ideas of Boardman and Vogt [1], and the construction was further developed by Segal [11] and others.

The present note, however, is not concerned with these generalizations. Its purpose is to give a simple proof of the original result, without striving for maximum generality, and to show that the same method can be used to prove an equivariant version of the reduced product theorem and hence a fibrewise version. Fibrewise versions of the reduced product theorem have previously been given by Eggar [5] and, more recently, by one of us [8] but it may be useful to have a relatively simple treatment which is adequate for the majority of applications.

The extended completeness theorem of the predicate calculas of the first order. In section 12, we developed a deductive theory of the first order predicate calculus, while in section II we dealt with the semantic theory of that calculus. We now have to consider the connection between these two theories. We recall that a sentence X can be satisfied by a structure M only if X is defined in M. Given a sentence X (a set of sentences K) we shall say that the structure M is a model of X (of K) if X is (all the sentences of K are) satisfied by M.

The material sketched here is mostly well known and concerns the geometrical inter-relations of vectors in a Hilbert (that is, complete inner product) space. The discussion and references are (obviously) not exhaustive but I hope the reader will find here some interesting problems.

We shall permit the scalars to be real, complex or quaternionic numbers. The inner product of vectors x and y will be denoted (x|y).

The cover product of disjoint graphs $G$ and $H$ with fixed vertex covers $C\left( G \right)$ and $C\left( H \right)$, is the graph $G\circledast H$ with vertex set $V\left( G \right)\cup V\left( H \right)$ and edge set

$$E\left( G \right)\,\cup \,E\left( H \right)\,\cup \,\left\{ \left\{ i,\,j \right\}\,:\,i\,\in \,C\left( G \right),\,j\,\in \,C\left( H \right) \right\}.$$

We describe the graded Betti numbers of $G\circledast H$ in terms of those of $G$ and $H$. As applications we obtain: (i) For any positive integer k there exists a connected bipartite graph $G$ such that $\text{reg}\,R/I\left( G \right)\,=\,{{\mu }_{s}}\left( G \right)\,+\,k$, where, $I\left( G \right)$ denotes the edge ideal of $G$, $\text{reg}\,\text{R/I}\left( G \right)$ is the Castelnuovo–Mumford regularity of $\text{R/I}\left( G \right)$ and ${{\mu }_{s}}\left( G \right)$ is the induced or strong matching number of $G$; (ii)The graded Betti numbers of the complement of a tree depends only upon its number of vertices; (iii)The $h$-vector of $R/I\left( G\circledast H \right)$ is described in terms of the $h$-vectors of $\text{R/I}\left( G \right)$ and $R/I\left( H \right)$. Furthermore, in a different direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.

The number of cyclic cubic fields with a given conductor and a given index is determined.