The following results of Whitehead and Wall are well-known applications of the algebraic K-theoretic functors K0 and K1 to basic homotopy questions in topology.

THEOREM 1 [20]. If f : X → Y is a homotopy equivalence between compact CW complexes, then there is a torsion τ(ƒ) in the algebraically-defined Whitehead group Wh π1(Y) which vanishes if and only if f is a simple homotopy equivalence.

THEOREM 2 [18]. If X is an arbitrary space which is finitely dominated (i.e., homotopically dominated by a compact polyhedron), then there is an obstruction σ(X) in the algebraically-defined reduced projective class group which vanishes if and only if X is homotopy equivalent to some compact polyhedron.

If we direct sum over components, then the above statements make good sense even if the spaces involved are not connected.

Ingham (2) has defined the following summation method. A series ∑ an will be called summable (I) to s if

where as usual [x] is the greatest integer ⩽ x. (An equivalent method was described somewhat earlier by Wintner (7), who called it “an Eratosthenian method“; however, the notation (I) and the name “Ingham summability” introduced by Hardy (1) seem to have become usual.)

Let $G$ be a locally compact group and let $\omega$ be a continuous weight on $G$. We show that for each of the Banach algebras ${{L}^{1}}\left( G,\,\omega \right),\,M\left( G,\,\omega \right),\,LUC{{\left( G,\,{{\omega }^{-1}} \right)}^{*}}$, and ${{L}^{1}}{{\left( G,\,\omega \right)}^{**}}$, the order structure combined with the algebra structure determines the weighted group.

In the literature, considerable attention has been devoted to the study of inversion operators for the Laplace transformation. In particular, much interest attaches to “real” inversion operators, i.e., operators which make use of the values of the generating function arising only from real values of the independent variable. Several of these operators are known (see for example Widder [3, chap. 7, §6; chap. 8, §25], Hirschman [2]).

In the original view-obstruction problem congruent closed, centrally symmetric convex bodies centred at the points of the set are expanded uniformly until they block all rays from the origin into the open positive cone. The central problem is to determine the minimal blocking size and this value is known for balls in dimensions n = 2,3 and for symmetrically placed cubes in dimensions n = 2, 3, 4

In order to explain fully the distinction between rational and irrational rays in the original problem, we extend consideration to the blocking of subspaces of all dimensions. In order to appreciate the special properties of balls and cubes, we give a discussion of the convex body with respect to reflection symmetry, lower dimensional sections, and duality. We introduce topological considerations to help understand when the critical parameter of the theory is an attained maximum and we add substantially to the list of known values of this parameter. In particular, when the dimension is n = 2 our dual body considerations furnish a complete solution to the view-obstruction problem

This paper shows that for the Moore spectrum MG associated with any abelian group G, and for any positive integer n, the order of the Postnikov k-invariant kn+1(MG) is equal to the exponent of the homotopy group πnMG. In the case of the sphere spectrum S, this implies that the exponents of the homotopy groups of S provide a universal estimate for the exponent of the kernel of the stable Hurewicz homomorphism hn: πnX → En(X) for the homology theory E*(—) corresponding to any connective ring spectrum E such that π0E is torsion-free and for any bounded below spectrum X. Moreover, an upper bound for the exponent of the cokernel of the generalized Hurewicz homomorphism hn: En(X) → Hn(X; π0E), induced by the 0-th Postnikov section of E, is obtained for any connective spectrum E. An application of these results enables us to approximate in a universal way both kernel and cokernel of the unstable Hurewicz homomorphism between the algebraic K-theory of any ring and the ordinary integral homology of its linear group.

Suppose that Gis a locally compact group with modular function Δ and that p, q, r are three numbers in the interval (l,∞) satisfying . If cp,q(G) is the smallest constant c such that for all functions f, g ∈ Cc(G) (here the convolution product is with respect to left Haar measure and is the exponent which is conjugate to p) then Young's inequality asserts that cp,q(G) ≤ 1. This paper contains three results about these constants. Firstly, if G contains a compact open subgroup then cp,q(G) = 1 and, as an extension of an earlier result of J. J. F. Fournier, it is shown that there is a constant cp,q < 1 such that if G does not contain a compact open subgroup then c<(G) ≤ c≤. Secondly, Beckner's calculation of is used to obtain the value of cp,q(G) for all simply-connected solvable Lie groups and all nilpotent Lie groups. And thirdly, it is shown that for a nilpotent Lie group the set is not contained in the union of the spaces Ls(G), .

Let and denote infinite dimensional Hilbert spaces and let denote the space of all bounded linear operators from to . For A in and B in , let τAB denote the operator on defined by τAB(X) = AX – XB. The purpose of this note is to characterize the semi-Fredholm domain of τAB (Corollary 3.16). Section 3 also contains formulas for ind(τAB – λ). These results depend in part on a decomposition theorem for Hilbert space operators corresponding to certain “singular points” of the semi-Fredholm domain (Theorem 2.2). Section 4 contains a particularly simple formula for ind(τAB – λ) (in terms of spectral and algebraic invariants of A and B) for the case when τAB – λ is Fredholm (Theorem 4.2). This result is used to prove that (τBA) = –ind(τAB) (Corollary 4.3). We also prove that when A and B are bi-quasi-triangular, then the semi-Fredholm domain of τAB contains no points corresponding to nonzero indices.

If we define the weight b of a Young diagram containing n nodes to be the number of removable p-hooks where n = a + bp, then three fundamental theorems stand out in the modular representation theory of the symmetric group Sn.

Introduction. Certain aspects of the theory of subspaces of a Finsler space had been treated by the present author in earlier papers (7). These developments were based on an approach essentially different from the classical theory of Cartan (2) and subsequent writers, whose use of the element of support enables one to introduce the so-called “euclidean connection,” which effects the vanishing of the covariant derivative of the metric tensor.

For G a group, S a subset of G which generates G, the length problem in G with respect to S is to find, for g ∈ G, the least integer r such that g can be written as the product of r elements of S. For G an orthogonal group Of(F) (here F is a field, and the elements of Of(F) preserve the quadratic form f) and S the set of reflections in Of(F) the length problem has been studied by E. Cartan [2], J. Dieudonné [4, 5], E. Ellers [7], P. Scherk [8], and others. In all of these investigations, however, the problem posed by requiring that S be a single conjugacy class of reflections in Of(F) has been ignored. And it is generally the case that the reflections in Of(F) fall into several conjugacy classes.

This paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

A new approach to prove weak convergence of random polytopes on the space of compact convex sets is presented. This is used to show that the profile of the rescaled Schläfli random cone of a random conical tessellation, generated by n independent and uniformly distributed random linear hyperplanes in $\mathbb {R}^{d+1}$ , weakly converges to the typical cell of a stationary and isotropic Poisson hyperplane tessellation in $\mathbb {R}^d$ , as $n\to \infty $ .

An intrinsic quadric is a normal projective variety with a Cox ring defined by a single quadratic relation. We provide explicit descriptions of these varieties in the smooth case for small Picard numbers. As applications, we figure out in this setting the Fano examples and (affirmatively) test Fujita’s freeness conjecture.

Introduction. Let K be an algebraically closed field, G = SL(3, K) the group of 3 × 3 matrices over K of determinant 1. Let denote the monoid of all 3 × 3 matrices over K. If e is an idempotent in , then

are opposite parabolic subgroups of G in the usual sense [1], [28]. However the map

does not exhaust all pairs of opposite parabolic subgroups of G. Now consider the representation ϕ:G → SL(6, K) given by

In [2], we have briefly described, as examples of the general theory developed there, a scale of symmetric Cesaro-Perron integrals, namely SCnP-integral for n = 1, 2, 3, … . The purpose of this paper is to consider the integrals in a greater detail.

As a preliminary, we prove some lemmas, which are also interesting for their own sake, concerning the de la Vallée Poussin derivatives in Section 1, and we also state two deep theorems concerning the n-convex functions in Section 2.

The spaces P(ω), 1 ⩽ p ⩽ ∞, ω real are defined to consist of those analytic functions ƒ(s), regular for Re s > ω and for which μP(ƒ;x) is bounded for x > ω where

(1.1), 1 ⩽ p ⩽ ∞and

(1.2)

Let denote a rooted tree with n nodes. (For definitions not given here, see, e.g. [4]). For any node v of , let B(v) denote the subtree of determined by v and all nodes u such that v is between u and the root of ; node v serves as the root of B(v). The branches of are the subtrees B(v) such that node v is joined to the root of . A branch B with i nodes is a primary branch of if n/2 ≦ i ≦ n – 1; if has a primary branch B with i nodes, then a branch C with j nodes is a secondary branch if (n – i)/2 ≦ j ≦ n – 1 ≦ i; if has a primary branch B with i nodes and a secondary branch C with j nodes, then a branch D with h nodes is a tertiary branch if

Let $\Gamma $ be a rank-one arithmetic subgroup of a semisimple Lie group $G$. For fixed $K$-Type, the spectral side of the Selberg trace formula defines a distribution on the space of infinitesimal characters of $G$, whose discrete part encodes the dimensions of the spaces of square-integrable $\Gamma $-automorphic forms. It is shown that this distribution converges to the Plancherel measure of $G$ when $\Gamma $ shrinks to the trivial group in a certain restricted way. The analogous assertion for cocompact lattices $\Gamma $ follows from results of DeGeorge-Wallach and Delorme.

Following (2) we say that a measure μ on a ring is semifinite if

Clearly every σ-finite measure is semifinite, but the converse fails.

In § 1 we present several reformulations of semifiniteness (Theorem 2), and characterize those semifinite measures μ on a ring that possess unique extensions to the σ-ring generated by (Theorem 3). Theorem 3 extends a classical result for σ-finite measures (3, 13.A). Then, in § 2, we apply the results of § 1 to the study of product measures; in the process, we compare the “semifinite product measure” (1; 2, pp. 127ff.) with the product measure described in (4, pp. 229ff.), finding necessary and sufficient conditions for their equality; see Theorem 6 and, in relation to it, Theorem 7.