In this paper we generalise the concept of a Steinberg cross section to non-connected affine Kac–Moody groups. This Steinberg cross section is a section to the restriction of the adjoint quotient map to a given exterior connected component of the affine Kac–Moody group. (The adjoint quotient is only defined on a certain submonoid of the entire group, however, the intersection of this submonoid with each connected component is non-void.) The image of the Steinberg cross section consists of a “twisted Coxeter cell”, a transversal slice to a twisted Coxeter element. A crucial point in the proof of the main result is that the image of the cross section can be endowed with a ${{\mathbb{C}}^{*}}$-action.

Let $\rho :\,G\,\to \,\text{GL}\left( V \right)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let ${{\sigma }_{1}},\ldots ,{{\sigma }_{n}}$ be a system of generators of the algebra of invariant polynomials $\mathbb{C}{{\left[ V \right]}^{G}}$. We study the problem of lifting mappings $f:\,{{\mathbb{R}}^{q}}\,\supseteq \,U\,\to \,\sigma \left( V \right)\,\subseteq \,{{\mathbb{C}}^{n}}$ over the mapping of invariants $\sigma \,=\,\left( {{\sigma }_{1}},\ldots ,{{\sigma }_{n}} \right):\,V\,\to \,\sigma \left( V \right)$. Note that $\sigma \left( V \right)$ can be identified with the categorical quotient $V//G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $C\subseteq {{C}^{\infty }}$ satisfying some mild closedness properties that guarantee resolution of singularities in $C$, e.g., the real analytic class, then $f$ admits a lift of the same class $C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\text{SB}{{\text{V}}_{\text{loc}}}$, i.e., special functions of bounded variation. If $\rho $ is a real representation of a compact Lie group, we obtain stronger versions.

In this paper we prove that the order of the collineation group of the Veblen-Wedderburn plane of order nine is 311,040. This result was stated by Hall [3] in 1943 and proved by Pierce [9] in 1964. Hall assumed that there were 10 · 8 · 6 · 4 · 2 = 3840 collineations which permute points on the ideal line L and 81 collineations which leave L pointwise fixed. In 1955 André [1] verified this assumption. When it was realized that a harmonic homology with axis L had been overlooked, the number of central collineations with axis L doubled and hence the order of the collineation group became 3840 · 162 = 622,080. This latter figure has been assumed to be correct as recently as 1965 ([6]).

Here it is proved that there are 1920 collineations which move points on L and 162 collineations which leave L pointwise fixed, thus giving the figure 311,040. Pierce's proof of this fact is established from a different viewpoint.

Let $K$ be a knot in ${{S}^{3}}$ . This paper is devoted to Dehn surgeries which create 3-manifolds containing a closed non-orientable surface $\hat{S}$ . We look at the slope $p/q$ of the surgery, the Euler characteristic $\mathcal{X}(\hat{S})$ of the surface and the intersection number $s$ between $\hat{S}$ and the core of the Dehn surgery. We prove that if $\mathcal{X}(\hat{S})\,\ge \,15\,-3q$, then $s\,=\,1$. Furthermore, if $s\,=\,1$ then $q\,\le \,4\,-\,3\,\mathcal{X}(\hat{S})$ or $K$ is cabled and $q\,\le \,8\,-5\mathcal{X}(\hat{S})$ . As consequence, if $K$ is hyperbolic and $\mathcal{X}(\hat{S})\,=\,-1$ , then $q\,\le \,7$.

If x- (i = 1, 2, … , 6) are homogeneous co-ordinates in [5] (the complex projective space of five dimensions), then the equation

1.1

represents the well-known (11) Perazzo cubic primalP43 of order 3 and dimension 4. With it is associated the Segre cubic threefoldS33 (12); specifically, S33 is the section of P43 by a tangent [4].

Let G be a group of finite order g, A a splitting field of G of characteristic p (which may be 0) and R = AG the group algebra of G over A. In [2], the author studied some of the properties of the Grothendieck ring K(R) of the category of all finitely generated R-modules, and derived a number of consequences. This paper continues the study carried out in [2]. The study is concerned with the structure and irreducible representations of K(R). The ring K(R) is proved to be semisimple and the primitive idempotents of K(R) are explicitly constructed. If the ring K(R) is identified with the ‘algebra of representations', then Robinson's idempotent [3; 4; 5] follow from our description as a special case.

We study the moderate growth generalized Whittaker functions, associated to a unitary character $\psi $ of a unipotent subgroup, for the non-tempered cohomological representation of $G\,=\,\text{Sp}\left( 2,\,\mathbb{R} \right)$. Through an explicit calculation of a holonomic system which characterizes these functions we observe that their existence is determined by the including relation between the real nilpotent coadjoint $G$-orbit of $\psi $ in $\mathfrak{g}_{\mathbb{R}}^{*}$ and the asymptotic support of the cohomological representation.

In the study of direct differential geometry, families of oriented arcs and curves have been employed extensively to define the differentiability of an arc at a point in various kinds of planes; cf. [2]. In [6], P. Scherk used lines in the projective plane; in [3] and [4], N. D. Lane and P. Scherk used circles in the conformai plane; conic-sections in the projective plane were employed in [5] and [7] by N. D. Lane and K. D. Singh; in [1], M. Gupta and N. D. Lane used the graphs of polynomials of degree at most n in the affine plane.

1. Introduction. In 1931, Newman [9] showed that a connected manifold cannot accept arbitrarily small period-n homeomorphisms, for any n > 1. In this paper we are concerned with the existence of chainable continua with arbitrarily small homeomorphisms.

For a long time the only known periodic homeomorphisms of chainable continua had periods 1, 2 or 4 [4]. Recently, Wayne Lewis [8] showed that the pseudo-arc admits periodic homeomorphisms of every order, as well as p-adic cantor group actions. We will see that such homeomorphisms can be made arbitrarily small.

In Section 4, a different chainable indecomposable continuum accepting arbitrarily small period-2 homeomorphisms is constructed.

The “k-dense” subgraphs of a connected graph G are connected and contain neighbours of all but at most k-1 points. We consider necessary and sufficient conditions that a point be in Γk, the union of the minimal k-dense subgraphs. It is shown that Γk contains all the [m, k]-isthmuses” and [m, k]-articulators“— minimal subgraphs which disconnect the graph into at least k + 1 disjoint graphs—and that an [m, k]-isthmus or [m, k]-articulator of Γk disconnects G. We define “central points,” “degree” of a point, and “chromatic number” and examine the relationship of these concepts to connectivity. Many theorems contain theorems previously proved (1) as special cases.

A remarkable theorem of Herstein [1, Theorem 2] of which we have made several uses states: If R is a semiprime ring of characteristic different from 2 and if U is both a Lie ideal and a subring of R then either U ⊂ Z (the centre of R) or U contains a nonzero ideal of R. In a recent paper [3] Herstein extends the above mentioned result significantly and has proved that if R is a semiprime ring of characteristic different from 2 and V is an additive subgroup of R such that [V, U] ⊂ V, where U is a Lie ideal of R, then either [V, U] = 0 or V ⊃ [M, R] ≠ 0 where M is an ideal of R. In this paper our object is to prove the following.

A Banach space B is called an isometric predual, or simply a predual, of a Banach space X if the dual B* of B is isometrically isomorphic to X. A Banach space X is said to have a unique (isometric) predual if X has a predual and all preduals are mutually isometrically isomorphic. In general a Banach space does not have a unique predual even if it has a predual. A simple example of this is the space l1, because c0 and c are isometric preduals of l1 but not isometrically isomorphic. A. Grothendieck [3] first noticed that L∞-spaces have unique preduals, and then S. Sakai generalized this to von Neumann algebras (see p. 30 of [9]).

Soit M un facteur de type IIIλ 0 ≦ λ< 1, agissant sur un espace hilbertien à base dénombrable. Le propos de ce travail est l'étude des états normaux d'un tel facteur à équivalence unitaire près. L'espace quotient n'est pas séparé; l'espace séparé associé est obtenu comme quotient de l'espace des états normaux par la relation d'équivalence R dont les classes sont les fermetures des orbites précédentes.

Pour étudier cet espace, on définit un calcul fonctionnel sur M+* (espace des formes linéaires positives normales sur M) à valeurs dans M. Ce calcul fonctionnel est déterminé par la donnée d'une C* algèbre d'applications de M+ * dans M “continues” en un certain sens, notée C*(M).

In [10], Witt laid the foundation for the study of quadratic forms over fields. Suppose Q is a quadratic form defined on a finite dimensional vector space V over a field of characteristic not equal to 2. Witt showed that non-zero vectors x and y in V satisfying Q(x) = Q(y) can be mapped into each other via an isometry of the vector space V. More generally, if τ : W ⟶ W’ is an isometry between subspaces of V, then τ extends to an isometry ϕ of V.

This paper is concerned with the description of principal congruence relations. Given elements a and b of a universal algebra , let θ(a, b) denote the smallest congruence relation on containing the pair 〈a, b〉. One of the earliest characterizations of θ(a, b) is Mal'cev's well-known result [5, Theorem 1.10.3], which says that c ≡ d(θ(a, b)) if and only if there exists a sequence z0, z1, …, zn of elements of and a sequence f1, f2, …, fn of unary algebraic functions such that c = z0, d = zn, and for each i = 1, …, n,

Although this describes θ(a, b) in terms of a set of unary algebraic functions, it is not possible to predict the number or complexity of the unary functions used independently of the choice of a, b, c and d. Several recent papers ([1], [2], [3], [4], [6]) investigate classes of algebras in which principal congruences are simpler.

Hyperbolic differential operators with constant coefficients introduced and studied systematically by Gårding (4), were characterized by the existence of the fundamental solution with some cone condition, according to Hörmander (6). Recently Ehrenpreis, extending the notion of hyperbolicity due to Gårding, has defined hyperbolic operators for distributions with compact support in the convolution sense. Under the hypothesis that the operator is invertible as a distribution, he has established a theorem analogous to the theorem of Hörmander mentioned above (3).

In this paper we discuss a generalization of the split exceptional Jordan algebra M38() of the 3 X 3 hermitian matrices with elements in the split Cayley-Dickson algebra (1). The generalization consists of replacing by the non-commutative Jordan algebra ≡ (A, ƒ, s, t) discussed in (2; 3) and forming the set of 3 X 3 hermitian matrices M3m() ≡ M with elements in the m-dimensional algebra . With the usual definition of multiplication X · Y = ½(XY + YX), M becomes a commutative algebra and we have the following theorem, which shows how the structure of M is reflected by that of .

For k ≧ 2 denote by Vk the class of functions f regular in and having the representation

(1.1)

where μ is a real-valued function of bounded variation on [0, 2π] with

(1.2)

Vk is the class of functions with boundary rotation at most kπ.

Let S be a spiral contained in D = {|z| < 1} such that S tends to C = {|z| = 1}. For the sake of brevity, by “f is bounded on S” we shall mean that f is holomorphic in D, unbounded, and bounded on S. The existence of such functions was first discussed by Valiron (9; 10); see also (1; 3; 8). Valiron also proved that any function that is “bounded on a spiral” must have the asymptotic value ∞ (10, p. 432). Functions that are bounded on a spiral may also have finite asymptotic values (1, p. 1254). In view of the above, Seidel has raised the question (oral communication): “Does there exist a function bounded on a spiral that has only the asymptotic value ∞?”. The following theorem answers this question affirmatively.

One of the nicest results in elementary number theory is the following, giving the relation between quadratic residues and sums of squares.