Let X be a quasi-Banach space whose dual X* separates the points of X. Then X* is a Banach space under the norm

From X we can construct the Banach envelope Xc of X by defining for x ∊ X, the norm

Then Xc is the completion of (X, ‖ ‖c). Alternatively ‖ ‖c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X → Z into a Banach space extends with preservation of norm to an operator .

A well-known consequence of the König theorem on maximum matchings and minimum covers in bipartite graphs (5) or of the P. Hall theorem on systems of distinct representatives for sets (4) asserts that an n by n (0, 1)-matrix A having precisely p ones in each row and column can be written as a sum of p permutation matrices:

Our main objective is a generalization of (1.1) along the following lines. Let A be an arbitrary m by n (0, 1)-matrix. Call an m by n (0, 1)-matrix P a permutation matrix if PPT = I, where PT is the transpose of P and I is the identity matrix of order m.

Every classical inner function $\varphi$ in the unit disk gives rise to a certain factorization of functions in Hardy spaces. This factorization, which we call the generalized Riesz factorization, coincides with the classical Riesz factorization when $\varphi (z)\,=\,z$. In this paper we prove several results about the generalized Riesz factorization, and we apply this factorization theory to obtain a new description of the commutant of analytic Toeplitz operators with inner symbols on a Hardy space. We also discuss several related issues in the context of the Bergman space.

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers related to the expansion of complete symmetric functions in the Jucys–Murphy elements, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra–Itzykson–Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.

The Newton polygon and the Newton-Puiseux algorithm ([3], p. 370, [8], p. 98), and their generalizations, serve as a powerful tool for analysing the singularities of a given function. Yet experts know how difficult it is to keep track of them when one, or several, blowing-ups are applied. Thus many interesting theorems are stated under the strong, rather undesirable, assumption that the Newton faces are non-degenerate.

In this paper, we introduce a method which is parallel to the classical Newton-Puiseux theory, yet avoids blowing-ups and fractional power series, except in the proofs.

1. Introduction. Under the non-parametric assumption that a set of observations is a sample from an absolutely continuous distribution, the order statistics are known to form a complete sufficient statistic. It is proved in this note that it suffices to have the class of uniform distributions over finite numbers of intervals or the class of uniform distributions over sets of a ring which is a basis for the σ-algebra of Borel sets. This result is derived as a particular case of that of several samples from more general distributions.

The only known simple alternative rings which are not associative are the Cayley algebras. Every such algebra has a scalar extension which is isomorphic over its center F to the algebra where . The elements e11 and e00 are orthogonal idempotents an , for every xij of . The multiplication table of C is then completed by the relations

In this paper we study a codimension 3 form of the 1:2 resonance. It was first noted by Arnold [3] that the study of bifurcations of symmetric vector fields under a rotation of order q yields information about Hopf bifurcation for a fixed point of a planar diffeomorphism F with eigenvalues . The map Fq can be identified to arbitrarily high order with the flow map of a symmetric vector field having a double-zero eigenvalue ([3], [4], [10], [23]). The resonance of order 2 (also called 1:2 resonance) considered here is the case of a pair of eigenvalues —1 with a Jordan block of order 2. The diffeomorphism then has normal form around the origin given by [4]:

In this paper we describe the outlines of the theory of analytic sets from the point of view of recent work on the subject. Our aim is to present the concepts and some of the principal results in a setting useful to workers in analysis, especially those workers not familiar with the field or its current developments. No attempt has been made to include all the results concerning analytic sets—not even in a particular category. There are some excellent monographs (12; 18) as well as chapters in books (2, 8, 9, 15, 17) where the subject is treated extensively. These, however, do not contain the recent results and consider only metric spaces. Our emphasis is on a general topological setting.

In the early days of quantum mechanics, Weyl asked the following question. Let λ be a non-zero real number, ℋa separable Hilbert space. Given certain (unbounded) operators W1,…,Wn,W1+, …, Wn+ on ℋ satisfying

(on a dense subspace D of ℋ) with all other commutators vanishing. Given also a function where ζ ∈ Cn. Let W = (W1 …, Wn) W+ = (W1+ …, Wn+). How does one associate to f an operator f(W, W+)? (Actually, Weyl phrased the question in terms of p = Re ζ, q = Im ζ, P = Re W, Q = Im W+ which represent momentum and position. In this paper, however, we wish to exploit the unitary group on Cn and so prefer complex notation.)

Let be a finite family of finite sets with the property that |A ∩ B| ≦ 1 whenever A and B are distinct members of . It is a conjecture of Erdös, Faber, and Lovász ([1] a 50 pound problem and [2] a 100 dollar problem) that there is an n-coloring of (i.e., a function such that A ∩ B = ∅ whenever A and B are distinct members of with f(A) = f(B). They actually state the conjecture in a different form. They actually state the conjecture in a different form. Namely, if n is a positive integer and is a family of n sets satisfying (1) |Bp| = n for each p and (2) |Bp ∩ Bq| ≦ 1 when p ≠ q, then there is an n-coloring of the elements of so that each set Bp gets all n colors. The equivalence of the two forms is easily seen by interchanging the roles of elements and sets.

This paper is primarily concerned with the existence of solutions of the fourth-order self-adjoint differential equation

(1)

(where r(x) > 0, q(x) ≥ 0, p(x) ≥ 0 and all three coefficients are continuous on [a, ∞)) and one of the two-point boundary conditions:

(2)

or

(3)

the subscript notation for any solution y(x) denoting:

(4)

Definition. An orthogonal design in order n and of type (u1, … , us) on the commuting variables x1, . . . , xs is an n X n matrix, X, with entries from the set ﹛0, ±x1, … , ±xs﹜ such that

A Sturm-Liouville function is simply a non-trivial solution of the Sturm-Liouville differential equation

(1.1)

considered, together with everything else in this study, in the real domain. The associated quantities whose higher monotonicity properties are determined here are defined, for fixed λ > –1, to be

(1.2)

where y(x) is an arbitrary (non-trivial) solution of (1.1) and x1, x2, … is any finite or infinite sequence of consecutive zeros of any non-trivial solution z(x) of (1.1) which may or may not be linearly independent of y(x). The condition λ > –1 is required to assure convergence of the integral defining Mk, and the function W(x) is taken subject to the same restriction.

The notion of a quasi-metric was introduced by Wilson (7) and has been studied by himself, Albert (1), and Ribeiro (6) among others. In this paper, we extend and unify some of their work, and connect it with results of Csaszar (2) and Pervin (4, 5) on quasi-uniformities.

This paper develops some techniques for the study of derivation algebras and cohomology groups of Lie algebras. We are especially concerned with solvable algebras over arbitrary fields with structural properties like those of the Borel subalgebras of complex semi-simple Lie algebras. In particular, these algebras are semi-direct sums of nilpotent ideals and abelian subalgebras which act on the ideals in a semi-simple fashion. We make strong use, in our discussion, of a cohomology theorem of Hochschild-Serre. This result is stated herein (§ 2) in a modified form which allows us to omit the original hypothesis that the base field have characteristic 0.

In many cases Lattice Theory has proven itself to be useful in the study of the totality of mathematical systems of a given type. In this paper we shall continue one of such studies by investigating further the lattice of all topologies on a given set S. A considerable amount of research has been done in this field (1; 2; 3; 5; 6). This research, besides satisfying the intrinsic interest in the lattice theoretic properties of this lattice, has aided the study of interconnections of different properties of point set topologies.

Let be a complete Heyting algebra (CHA). An -valued set is a pair (X, δ) where X is a set and δ is a function from X × X to such that

for all x, y z in X. -valued sets form a category as follows: a morphism from (X, δ) to (Y, δ) is a function f from X × Y to such that

• (i) f(x) ∧ δ(x, x′) ≦ f(x′, y), f(x, y) ∧ δ(y, y′) ≧ f(x, y′),

• (ii) f(x, y) ∧ f(x, y′) ≦ δ(y, y′), and

• (iii)

Let Mn(C) be the vector space of all w-square complex matrices. Denote by (• , •) the standard inner product in the space Cn of complex n-tuples. For a matrix A ∈ Mn(C) and an n-tuple c = (c1,… , cn) ∈ Cn, define the c-numerical range of A to be the set

(1)

Let ∑ be a projective space PG(3, q) of dimension 3 and finite order q. Then ∑ contains (q + 1)(q2 + 1) points and an equal number of planes, and (q2 + 1) (q2 + q + 1) lines. It will be convenient to consider lines and planes as sets of points and to treat the incidence relation as set inclusion. Each plane contains q2 + q + 1 points and an equal number of lines. Each line contains q + 1 points and is contained in an equal number of planes. Each point is contained in q2 + q + 1 planes and an equal number of lines.

A spread of lines of ∑ is a set of q2 + 1 lines of ∑ which are pairwise disjoint, or skew; it can also be defined as a set of lines such that each point (or each plane) is incident with exactly one of the lines.