The α-width ∊A(α) of a (0, 1)-matrix A is the minimal number of columns that can be selected from A in such a way that all row sums of the resulting submatrix of A are at least α. This notion was introduced in (2) and further studied in (3). In these papers the major emphasis was on the minimal α-width sequence for the class of (0, 1)-matrices generated from an arbitrary A by interchanges:

For any radical property Q, a nonzero simple ring (all rings in this paper are assumed to be associative) must make up its mind so to speak and must be either Q radical or Q semi-simple. Every Q thus divides the class of all nonzero simple rings into two disjoint classes. Conversely any partition of the nonzero simple rings into two disjoint classes leads to at least two radicals [1, p. 16].

Let dbe a positive square-free integer and let Od denote the ring of integers in . The groups PSL2(Od) are collectively known as the Bianchi groups and have been widely studied from the viewpoints of group theory, number theory and low-dimensional topology. The interest of the present article is in geometric Fuchsian subgroups of the groups PSL2(Od). Clearly PSL2 is such a subgroup; however results of [18], [19] show that the Bianchi groups are rich in Fuchsian subgroups.

We prove that is quasi-primal, then every algebra in HSPhas a pure embedding into a product of finite algebras. For a general theory of varieties for which every can be purely embedded in an equationally compact algebra , and for all notions not explained here, the reader is referred to [38; 6; or 5]. This theorem was known for Boolean algebras simply as a corollary of the Stone representation theorem and the fact that in the variety of Boolean algebras, all embeddings are pure [2].

Let ${{({{\mathcal{M}}_{i}})}_{i}}{{_{\in }}_{I}},{{({{\mathcal{N}}_{j}})}_{j\in J}}$ be families of von Neumann algebras and $\mathcal{U},\,{\mathcal{U}}'$ be ultrafilters in $I$, $J$, respectively. Let $1\,\le \,p\,<\,\infty $ and $n\,\in \,\mathbb{N}$. Let ${{x}_{1}},...,{{x}_{n}}\,\text{in}\,\prod {{L}_{p}}({{\mathcal{M}}_{i}})$ and ${{y}_{1}},...,{{y}_{n\,}}\text{in }\Pi \,{{L}_{p}}({{\mathcal{N}}_{j}})$ be bounded families. We show the following equality

$$\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}=\,\underset{j,{\mathcal{U}}'}{\mathop{\lim }}\,\,\underset{i,\mathcal{U}}{\mathop{\lim }}\,\,{{\left\| \sum\limits_{k=1}^{n}{{{x}_{k}}(i)\,\,\otimes \,{{y}_{k}}(j)} \right\|}_{{{L}_{p}}({{\mathcal{M}}_{i}}\otimes {{\mathcal{N}}_{j}})}}$$

For $p\,=\,1$ this Fubini type result is related to the local reflexivity of duals of ${{C}^{*}}$ -algebras. This fails for $P=\infty $.

We consider the $k$-osculating varieties ${{O}_{k,\,n.d}}$ to the (Veronese) $d$-uple embeddings of ${{\mathbb{P}}^{n}}$. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes $Y\,\subset \,{{\mathbb{P}}^{n}}$ to $O_{k,n,d}^{s}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

Let I = [0,1] and let be the space of all integrable functions on I, where m denotes Lebesque measure on I. Let ∥ ∥1 be the ℒ-1-norm and let be a measurable, nonsingular transformation on I. Let denote the space of densities. The probability measure μ is invariant under τ if for all measurable sets A, The measure μ is absolutely continuous if there exists an such that for any measurable set A We refer to ƒ* as the invariant density of τ (with respect to m). It is well-known that ƒ * is a fixed point of the Frobenius-Perron operator defined by

A ring A has finite norm properly, abbreviated FNP, if each proper homomorphic image of A is finite. In [3], Chew and Lawn described some of the structural properties of FNP rings with identity, which they called residually finite rings. The twofold aim of this paper is to extend the results of [3] to arbitrary rings with FNP and to give characterizations of FNP rings independent of the results of [3].

If A is a ring, let A+ denote A regarded as an abelian group. In the first section of this paper, we explore the effects of FNP upon the structure of A+. The following theorem is typical of the results in this section.

A lattice ∧ in euclidean n-space, En is a group of vectors under vector addition generated by n independent vectors, X1, X2 … , Xn, called a basis for the lattice. The absolute value of the n × n determinant the rows of which are the co-ordinates of a basis is called the determinant of the lattice and is denoted by d(∧). For any lattice ∧ there is a unique minimal positive number r such that, if spheres of radius r are placed with centres at all points of ∧, the entire space is covered. The density of this covering may be defined as (Jnrn)/(d(∧)) where Jn is the volume of the unit sphere in n-dimensional euclidean space. This density will be denoted by θn(∧). The density of the most efficient lattice covering of n-space by spheres, θn is the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.

Noting that certain restrictions are placed on homeomorphisms of the pseudo-arc, since it is hereditarily indecomposable, in 1955 [4] R. H. Bing asked if the identity is the only stable homeomorphism of the pseudo-arc. In this paper we prove the following theorem.

THEOREM. Let U be an open subset of the pseudo-arc P. Let p and q be distinct points of P such that the subcontinuum M irreducible between p and q does not intersect cl(U). Then there exists a homeomorphism h : P → P with h (p) = q andh│U = 1U.

1. Definitions. A chain C is a collection of open sets C = {C(i))}i≧n such that C(i) ∩ C(j) ≠ ∅ if and only if |i – j| ≦ 1, cl|(C(i)) ∩ cl(C(j)) ≠ ∅ if and only if |i – j| ≦ 1, C(0) – C(l) ≠ ∅, and C(n) – C(n – 1) ≠ ∅. Each C(i) is called a link of C.

Let π be a projective or an affine plane ; a configuration C of π is a subset of points and a subset of lines in π such that a point P of C is incident with a line I of C if and only if P is incident with I in π. A configuration of a projective plane π which is a projective plane itself is called a projective subplane of π, and a configuration of an affine plane π’ which is an affine plane with the improper line of π‘ is an affine subplane of π‘.

Let π be a finite projective (respectively, an affine) plane of order n and π0 a projective (respectively, an affine) subplane of π of order n0 different from π; then n0 ≦ . If n0 = , then π0 is called a Baer subplane of π. Thus, Baer subplanes are the “biggest” possible proper subplanes of finite planes.

The purpose of this note is to give a new proof of a theorem of L. Carlitz [2] and R. McConnel [5]. The theorem is as follows:

THEOREM 1. Let F = GF(pn) be the finite field of order q = pn and let K — {x ∈ F|xd = 1} for some proper divisor d of q — 1.

Although the problem of finding the minimum number of planar graphs into which the complete graph can be decomposed remains partially unsolved, the corresponding problem can be solved for certain other surfaces. For three, the torus, the double-torus, and the projective plane, a single proof will be given to provide the solutions. The same questions will also be answered for bicomplete graphs.

Let g denote an odd prime, and h = h(g) the class number of the cyclotomic field R(), where is a primitive gth root of unity. It is known that we can write

We provide an explicit thick and thin decomposition for oriented hyperbolic manifolds $M$ of dimension 5. The result implies improved universal lower bounds for the volume $\text{vo}{{\text{l}}_{\text{5}}}\left( M \right)$ and, for $M$ compact, new estimates relating the injectivity radius and the diameter of $M$ with $\text{vo}{{\text{l}}_{\text{5}}}\left( M \right)$. The quantification of the thin part is based upon the identification of the isometry group of the universal space by the matrix group $\text{P}{{\text{S}}_{\Delta }}\text{L}\left( 2,\,\mathbb{H} \right)$ of quaternionic $2\,\times \,2$-matrices with Dieudonné determinant $\Delta$ equal to 1 and isolation properties of $\text{P}{{\text{S}}_{\Delta }}\text{L}\left( 2,\,\mathbb{H} \right)$.

We study the case of an Axiom $\text{A}$ holomorphic non-degenerate (hence non-invertible) map $f:{{\mathbb{P}}^{2}}\mathbb{C}\to {{\mathbb{P}}^{2}}\mathbb{C}$, where ${{\mathbb{P}}^{2}}\mathbb{C}$ stands for the complex projective space of dimension 2. Let $\Lambda $ denote a basic set for $f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda $; we denote by ${{\delta }^{s}}(x)$ the Hausdorff dimension of $\text{W}_{r}^{s}(x) \cap \,\Lambda$, where $r$ is some fixed positive number and $\text{W}_{r}^{s}(x)$ is the local stable manifold at $x$ of size $r;{{\delta }^{s}}(x)$ is called the stable dimension at$x$. Mihailescu and Urbański introduced a notion of inverse topological pressure, denoted by ${{P}^{-}}$, which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates of $f$, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on $\Lambda $. When each point $x$ from $\Lambda $ has the same number ${d}'$ of preimages in $\Lambda $, then we show that ${{\delta }^{s}}(x)$ is independent of $x$; in fact ${{\delta }^{s}}(x)$ is shown to be equal in this case with the unique zero of the map $t\to P(t{{\phi }^{s}}-\log {d}')$. We also prove the Lipschitz continuity of the stable vector spaces over $\Lambda $; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.

The purpose of this paper is to characterize weak compactness in Orlicz spaces. Though an Orlicz space is a Banach space, it will be viewed from the standpoint of the theory of Köthe spaces. Considering that a norm-bounded subset is not weakly compact in general, we shall give some criteria for weak compactness in terms of the functional defining an Orlicz space. Because weak compactness is closely connected with the continuity of the semi-norms on the conjugate space, at the same time some properties of continuous semi-norms on Orlicz spaces will be brought to light.

The first characterization (Theorem 1) is concerned with degree of smoothness of the functional at the origin. In Theorem 2 a connection between weak compactness and boundedness (by another functional) is obtained. In Theorem 3 the result in Theorem 2 is stated as a proposition about continuous seminorms.

A system of openly disjoint discs in the plane is said to form a 6-neighboured circle packing if every is tangent to at least 6 other elements of (It is evident that such a system consists of infinitely many discs.) The simplest example is the regular circle packing all of whose circles are of the same size and have exactly 6 neighbours. L. Fejes Tóth conjectured that the regular circle packing has the interesting extremal property that, if we slightly “perturb” it, then there will necessarily occur either arbitrarily small or arbitrarily large circles. More precisely, he asked whether or not the following “zero or one law” (cf. [3], [6]) is valid: If is a 6-neighboured circle packing, then

where r(C) denotes the radius of circle C, inf and sup are taken over all C ∊

Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,\,p,\,n)$ were constructed recently in the work of Ariki and Koike [AK], Broué and Malle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of $G(r,\,p,\,n)$ given by Ariki and Koike [AK] and Ariki [Ari].

1. Introduction. Let Q be a matrix of order v, all of whose entries are 0's and l's. Let the total number of l's in Q be t, and let the absolute value of the determinant of Q be denoted by |det Q|. In this paper we study the problem of determining the maximum of |det Q| for fixed t and v. It turns out that this problem is closely related to the v, k, λ problem, which has been extensively studied of late.