Two results are obtained in this paper. The first is a generalization of the imprimitivity theorem of Mackey, when the associated projection-valued measure is replaced by a commutative positive operator valued measure. The second is a necessary and sufficient condition for such a system of covariance to possess an overcomplete, covariant family of coherent states.

A canary tree is a tree of cardinality the continuum which has no uncountable branch, but gains a branch whenever a stationary set is destroyed (without adding reals). Canary trees are important in infinitary model theory. The existence of a canary tree is independent of ZFC + GCH.

By applying the Cayley–Dickson process to the division algebra of real octonions, one obtains a 16-dimensional real algebra known as (real) sedenions. We denote this algebra by ${{\text{A}}_{4}}$. It is a flexible quadratic algebra (with unit element 1) but not a division algebra.

We classify the subalgebras of ${{\text{A}}_{4}}$ up to conjugacy (i.e., up to the action of the automorphism group $G$ of ${{\text{A}}_{4}}$) with one exception: we leave aside the more complicated case of classifying the quaternion subalgebras. Any nonzero subalgebra contains 1 and we show that there are no proper subalgebras of dimension 5, 7 or > 8. The proper non-division subalgebras have dimensions 3, 6 and 8. We show that in each of these dimensions there is exactly one conjugacy class of such subalgebras. There are infinitely many conjugacy classes of subalgebras in dimensions 2 and 4, but only 4 conjugacy classes in dimension 8.

The range of the transformation, defined by

is characterized on the spaces Lμ,p defined by the norm

for

Let $U(n)$ be the group of $n\,\times \,n$ unitary matrices. We show that if $\phi $ is a linear transformation sending $U(n)$ into $U(m)$, then $m$ is a multiple of $n$, and $\phi $ has the form

$$A\,\mapsto \,V[(A\,\otimes \,{{I}_{s}})\,\otimes \,({{A}^{t}}\,\otimes \,{{I}_{r}})]W$$

for some $V,\,W\,\in \,U(m)$. From this result, one easily deduces the characterization of linear operators that map $U(n)$ into itself obtained by Marcus. Further generalization of the main theorem is also discussed.

Let there be given two sets of points, P = {P1,…, Pm}1 m and Q={Q1,…, Qn}, such that joining each pair of points (Pi, Qk), for i=1,…, m and k=1…, n, is a line oriented towards one, and only one, of the pair. Such a configuration will be called an m×n bipartite tournament. If the line joining Pi to Qk is oriented towards Qk we may indicate this by pi→Qk, and similarly if the line is oriented in the other sense. The points Pi, Pj, Qk, and Ql. will be said to form a 4-cycle if either Pi→Qk→Pj→Ql→Pi or Pi→Ql→Pj→Qk→Pi. C(m, n), the number of 4-cycles in a given m×n bipartite tournament, provides, in some sense, a measure of the degree of transitivity of the relationship indicated by the orientation of the lines, and the complete configuration may be thought of as representing the outcome of comparing each member of one population with each member of a second population, and making a decision, upon some basis, as to which component of each pair is the preferred one.

In 1927, Jackson [5] obtained a transformation connecting a

where N is any integer, with a

viz.,

1

where | q | > l and |qγ-α-βN| > l.

A ring R with unity is called right (left) self injective if the right (left) R-module R is injective [7]. The purpose of this note is to prove the following: Let R be a prime ring with a maximal annihilator right (left) ideal. If R is right (left) self injective then R is a primitive ring with a minimal one-sided ideal. If R satisfies the maximum condition on annihilator right (left) ideals and R is right (left) self injective then R is a simple ring with the minimum condition on one-sided ideals.

Let p denote a prime and n a positive integer ≥ 2. Let Nn(p) denote the number of polynomials xn + x + a, a = 1, 2,…, p-l, which are irreducible (mod p). Chowla [5] has made the following two conjectures:

Conjecture 1. There is a prime p0(n), depending only on n, such that for all primes p ≥ p0(n)

For an inner function u, we discuss the dual operator for the compressed shift $P_u S|_{{\mathcal {K}}_u}$, where ${\mathcal {K}}_u$ is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.

For any group G and integer n, let Pn:G→G be the function defined by Pn(g) = gn for all g ϵ G. If G is abelian then Pn is a homomorphism for all n. Conversely, it is well known (and easy to show) that if P2 or P-1 is a homomorphism then G is abelian. As the groups Gn described below show, for every n other than 2 and -1 there exist non-abelian groups for which Pn is a homomorphism.

The following problem is studied. Generalized Fermat's problem: in an n-dimensional Hadamard manifold M, locate a point whose distances from the given k vertices of M have the smallest possible sum.

How few three-term arithmetic progressions can a subset $S\,\subseteq \,{{\mathbb{Z}}_{N}}\,:=\,\mathbb{Z}\,/\,N\mathbb{Z}$ have if $|S|\,\ge \,\upsilon N$ (that is, $S$ has density at least $\upsilon$)? Varnavides showed that this number of arithmetic progressions is at least $c(v)\,{{N}^{2}}$ for sufficiently large integers $N$. It is well known that determining good lower bounds for $c\left( \upsilon \right)\,>\,0$ is at the same level of depth as Erdös's famous conjecture about whether a subset $T$ of the naturals where $\sum{_{n\in T}\,1/n}$ diverges, has a $k$-term arithmetic progression for $k\,=\,3$ (that is, a three-term arithmetic progression).

We answer a question posed by B. Green about how this minimial number of progressions oscillates for a fixed density $\upsilon$ as $N$ runs through the primes, and as $N$ runs through the odd positive integers.

Let $\mathcal {B}(\mathcal {H})$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathcal {H}$ . In this paper, we first establish several sharp improved and refined versions of the Bohr’s inequality for the functions in the class $H^{\infty }(\mathbb {D},\mathcal {B}(\mathcal {H}))$ of bounded analytic functions from the unit disk $\mathbb {D}:=\{z \in \mathbb {C}:|z|<1\}$ into $\mathcal {B}(\mathcal {H})$ . For the complete circular domain $Q \subset \mathbb {C}^{n}$ , we prove the multidimensional analogues of the operator valued Bohr-type inequality which can be viewed as a special case of the result by G. Popescu [Adv. Math. 347 (2019), 1002–1053] for free holomorphic functions on polyballs. Finally, we establish the multidimensional analogues of several improved Bohr’s inequalities for operator valued functions in Q.

It is shown that there exists a sequence of integers a1 < a2 <... such that each positive integer is a difference of elements of the sequence in exactly one way, and such that ak does not exceed a constant times k3. In fact we construct such a sequence with each ak in [C(k -1)3, Ck3), where C is an absolute constant.

Let V(G) and E(G) denote the vertex set and the edge set of a graph G; let Kn denote the complete graph with n vertices and let Kn, m denote the complete bipartite graph on n and m vertices. A Hamiltonian cycle (Hamiltonian path, respectively) in a graph G is a cycle (path, respectively) in G that contains all the vertices of G.

The notion of nuclear object in an autonomous category is studied. It is shown that the full subcategory determined by the nuclear objects is also autonomous.

We prove that, in ordered plane geometries endowed with a very weak notion of orthogonality, one can always triangulate any triangle into seven acute triangles, and, in case the given triangle is not acute, into no fewer than seven.

This article introduces the concept of a condensed domain, that is, an integral domain R for which IJ = {ij: i ∊ I, j ∊ J} for all ideals I and J of R. This concept is used to characterize Bézout domains (resp., principal ideal domains; resp., valuation domains) in suitably larger classes of integral domains. The main technical results state that a condensed domain has trivial Picard group and, if quasilocal, has depth at most 1. Special attention is paid to the Noetherian case and related examples.

The paper describes polyhedral realizations for Felix Klein's map {3, 7}8 of genus 3, for Gordan's map {4, 5}6 of genus 4, and for two maps of genus 5, the Klein-Fricke map of type {3, 8} and Sherk's map of type {4, 6}. The polyhedra have self-intersections but high symmetry and thus are close analogues to the Kepler-Poinsot-polyhedra.