For a smooth strongly convex Minkowski norm $F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$ , we study isometries of the Hessian metric corresponding to the function $E=\tfrac 12F^2$ . Under the additional assumption that F is invariant with respect to the standard action of $SO(k)\times SO(n-k)$ , we prove a conjecture of Laugwitz stated in 1965. Furthermore, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$ -symmetry.

1. Introduction. If m and n are two integers chosen at random, the probability that they are relatively prime (2, p. 267) is 6π-2. This result may still hold when m and n are functionally related. Thus, Watson (3) recently proved that for α irrational, the positive integers n for which (n, [αn]) = 1, have density 6π-2. A different proof of a slightly more general result was given by Estermann (1).

Let $(Y,\,T)$ be a minimal suspension flow built over a dynamical system $(X,\,S)$ and with (strictly positive, continuous) ceiling function $f:\,X\,\to \,\mathbb{R}$. We show that the eigenvalues of $(Y,\,T)$ are contained in the range of a trace on the ${{K}_{0}}$-group of $(X,\,S)$. Moreover, a trace gives an order isomorphism of a subgroup of ${{K}_{0}}\left( C(X)\,{{\rtimes }_{S}}\,\mathbb{Z} \right)$ with the group of eigenvalues of $(Y,\,T)$. Using this result, we relate the values of $t$ for which the time-$t$ map on the minimal suspension flow is minimal with the $K$-theory of the base of this suspension.

In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

We study the family of elliptic curves Em X3 + Y3 = m where m is a cubefree integer.

The elliptic curves Em with even analytic rank and those with odd analytic rank are proved to be equally distributed. It is proved that the number of cubefree integers m ≤ X such that the analytic rank of Em is even and ≥ 2 is at least CX2/3-ε, where ε is arbitrarily small and C is a positive constant, for X large enough. Therefore, if we assume the Birch and Swinnerton-Dyer conjecture, the number of all cubefree integers m ≤ X such that the equation X3 + Y3 = m have at least two independent rational solutions is at least CX2/3-ε.

Petrie polygons, especially as they arise in the study of regular polytopes and Coxeter groups, have been studied by geometers and group theorists since the early part of the twentieth century. An open question is the determination of which polyhedra possess Petrie polygons that are simple closed curves. The current work explores combinatorial structures in abstract polytopes, called Petrie schemes, that generalize the notion of a Petrie polygon. It is established that all of the regular convex polytopes and honeycombs in Euclidean spaces, as well as all of the Grünbaum–Dress polyhedra, possess Petrie schemes that are not self-intersecting and thus have Petrie polygons that are simple closed curves. Partial results are obtained for several other classes of less symmetric polytopes.

In an earlier paper [6] we considered the application of Lagrangian methods to the enumeration of plane rooted trees with given colour partition. We obtained an expression which generalised Tutte’s result [9], and a correspondence, which, when specialised, gives the de Bruijn-van Aardenne Ehrenfest-Smith-Tutte Theorem [1]. A corollary of these results is a one-to-one correspondence [4], between trees and generalised derangements, for which no combinatorial description has yet been found.

In this paper we extend these methods to the enumeration of rooted labelled trees to demonstrate how another pair of well-known and apparently unrelated theorems may be obtained as the result of a single enumerative approach. In particular, we show that a generalisation of Good's result [3], also considered by Knuth [7], and the matrix tree theorem [8] have a common origin in a single system of functional equations, and that they correspond to different coefficients in the power series solution.

Kantor pairs arise naturally in the study of 5-graded Lie algebras. In this article, we introduce and study Kantor pairs with short Peirce gradings and relate themto Lie algebras graded by the root system of type $\text{B}{{\text{C}}_{2}}$. This relationship allows us to define so-called Weyl images of short Peirce graded Kantor pairs. We use Weyl images to construct new examples of Kantor pairs, including a class of infinite dimensional central simple Kantor pairs over a field of characteristic $\ne$ 2 or 3, as well as a family of forms of a split Kantor pair of type ${{\text{E}}_{6}}$.

The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on “square roots” of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.

We show that the assignment of the (left) completely bounded multiplier algebra $M_{cb}^{l}({{L}^{1}}(\mathbb{G}))$ to a locally compact quantum group $\mathbb{G}$, and the assignment of the intrinsic group, form functors between appropriate categories. Morphisms of locally compact quantum groups can be described by Hopf $*$-homomorphisms between universal ${{C}^{*}}$-algebras, by bicharacters, or by special sorts of coactions. We show that the whole theory of completely bounded multipliers can be lifted to the universal ${{C}^{*}}$-algebra level, and that the different pictures of both multipliers (reduced, universal, and as centralisers) and morphisms then interact in extremely natural ways. The intrinsic group of a quantum group can be realised as a class of multipliers, and so our techniques immediately apply. We also show how to think of the intrinsic group using the universal ${{C}^{*}}$-algebra picture, and then, again, how the differing views on the intrinsic group interact naturally with morphisms. We show that the intrinsic group is the “maximal classical” quantum subgroup of a locally compact quantum group, that it is even closed in the strong Vaes sense, and that the intrinsic group functor is an adjoint to the inclusion functor from locally compact groups to quantum groups.

Let Σ represent the class of analytic functions

(1)

which are regular, except for a simple pole at infinity, and univalent in |z| > 1 and map |z| > 1 onto a domain whose complement is starlike with respect to the origin. Further let Σ- 1 be the class of inverse functions of Σ which at w = ∞ have the expansion

(2).

In this paper we develop variational formulas for functions of the classes Σ and Σ- 1 and obtain certain properties of functions that extremalize some rather general functionals pertaining to these classes. In particular, we obtain precise upper bounds for |b2| and |b3|. Precise upper bounds for |b1|, |b2| and |b3| are given by Springer (8) for the general univalent case, provided b0 =0.

Suppose $G$ is a countable, Abelian group with an element of infinite order and let $\text{ }\!\!\chi\!\!\text{ }$ be a mixing rank one action of $G$ on a probability space. Suppose further that the Følner sequence $\{{{F}_{n}}\}$ indexing the towers of $\text{ }\!\!\chi\!\!\text{ }$ satisfies a “bounded intersection property”: there is a constant $p$ such that each $\{{{F}_{n}}\}$ can intersect no more than $p$ disjoint translates of $\{{{F}_{n}}\}$. Then $\text{ }\!\!\chi\!\!\text{ }$ is mixing of all orders. When $G\,=\,\mathbf{Z}$, this extends the results of Kalikow and Ryzhikov to a large class of “funny” rank one transformations. We follow Ryzhikov’s joining technique in our proof: the main theorem follows from showing that any pairwise independent joining of $k$ copies of $\text{ }\!\!\chi\!\!\text{ }$ is necessarily product measure. This method generalizes Ryzhikov’s technique.

In [7] Richardson constructed a Stone-Čech type compactification R(E) of a Hausdorff convergence space E. Two questions arise in this regard. First, when is R(E) homeomorphic to β(E), β(E) the topological Stone-Čech compactification of E, for a Tychonoff topological space E? Second, if E is a regular convergence space, when is R(E) regular? The last question is motivated by the study of regular compactifications in [6]. In section 2 it will be shown that a necessary and sufficient condition in answer to both questions, is that α = cl(α) for each nonconvergent ultrafilter α on E.

Let $X$ be a complex Banach space and let ${{B}_{p}}\left( X \right)$ denote the vector-valued Bergman space on the unit disc for $1\,\le \,p\,<\,\infty $. A sequence ${{\left( {{T}_{n}} \right)}_{n}}$ of bounded operators between two Banach spaces $X$ and $Y$ defines a multiplier between ${{B}_{p}}\left( X \right)$ and ${{B}_{q}}\left( Y \right)$ (resp. ${{B}_{p}}\left( X \right)$ and ${{l}_{q}}\left( Y \right)$) if for any function $f\left( z \right)\,=\,\sum _{n=0}^{\infty }\,{{x}_{n}}{{z}^{n}}$ in ${{B}_{p}}\left( X \right)$ we have that $g\left( z \right)\,=\,\sum _{n=0}^{\infty }\,{{T}_{n}}\left( {{x}_{n}} \right){{z}^{n}}$ belongs to ${{B}_{q}}\left( Y \right)$ (resp. ${{\left( {{T}_{n}}\left( {{x}_{n}} \right) \right)}_{n}}\,\in \,{{\ell }_{q}}\left( Y \right)$). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces $X$ and $Y$. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in ${{B}_{p}}\left( X \right)$ are introduced.

Throughout this paper “space” means “T1 topological space.“

1. The concept of an ortho-base was introduced by W. F. Lindgren and P. J. Nyikos.

Definition 1. A base of a space X is called an ortho-base provided that for each subcollection either is open or is a local base of a point x ∊ X [17].

Ortho-bases are related to interior-preserving collections which have been known for some time.

Definition 2. A collection of open sets of a space X is called interior-preserving provided that the intersection of any subcollection is open. A space X is called orthocompact provided that each open cover has an open interior-preserving refinement.

It was proved in [17], in particular, that each space with an ortho-base is orthocompact, and each orthocompact developable space (which is the same as a non-archimedean quasi-metrizable developable space [4]) has an ortho-base.

An appropriate definition of equivariant formality for spaces equipped with the action of a finite group G, and for equivariant maps between such spaces, is given. Kahler manifolds with holomorphic G-actions, and equivariant holomorphic maps between such Kàhler manifolds, are proven to be equivariantly formal, generalizing results of Deligne, Griffiths, Morgan, and Sullivan

In [2] Sandler studied generalizations of projective planes called pseudo planes. These structures gave rise to ternary operations for which addition and multiplication are loop operations. Our aim in this paper is to investigate the pseudo planes for which the operations of addition and multiplication give rise to nearfields. The pseudo planes for which this is true will be called π-systems in this paper, and a list of postulates for such systems is given in § 2. In that section some results on collineations of π-systems are given which are stronger than analogous results for general pseudo planes.

We give a complete classification of mixed Tsirelson spaces $T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$ for finitely many pairs of given compact and hereditary families ${{F}_{i}}$ of finite sets of integers and $0<{{\theta }_{i}}<1$ in terms of the Cantor–Bendixson indices of the families ${{F}_{i}}$, and ${{\theta }_{i}}(1\le i\le r)$. We prove that there are unique countable ordinal $\alpha $ and $0<\theta <1$ such that every block sequence of $T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$ has a subsequence equivalent to a subsequence of the natural basis of the $T({{S}_{{{\omega }^{\alpha }}}},\theta )$. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.

Main results. Let G be a group generated by two subnormal subgroups H and K. Denoting the class of nilpotent groups by 𝔑, and the limit of the lower central series by G𝔑, Wielandt showed in [14], for groups with a composition series that

(*)

The letter k denotes an infinite cardinal. A space is a compact Hausdorff space unless otherwise indicated. A space is called extremally disconnected (k-extremally disconnected) if it is the Stone space for a complete (k-complete) Boolean algebra. A map is a continuous function from one space into another. A map f:X —> Y is called minimal if f is onto, but f(M) is properly contained in Y for each closed proper subset M of X. A space F is called free if F has a dense subset X such that every space-valued function on X extends to a map on all of F or, equivalently, if F is the Stone-Cech compactification of some discrete topological space X.