In the last fifteen years, topology has shown up with an increasing interest in the use of closed subbases. Starting from Frink's internal characterization of complete regularity (Frink [6]), DeGroot and Aarts used closed subbases to obtain Hausdorff compactifications of completely regular spaces, thus giving a characterization of the latter in terms of their subbases [1]. The main tool of that paper is the notion of a linked system, which naturally leads to the notions of supercompactness and superextensions [7]. After 1970, these two topics developed to indepedennt theories, with several deep results available at this moment. Most results up to 1976 are summarized in [12].

It is known that under special conditions, Fourier sine transforms and Fourier cosine transforms behave asymptotically like a power of x, either as x → 0 or as x → ∞ or both. For example (3),

where f(x) = x–αϕ(x), 0 < α < 1, and ϕ(x) is of bounded variation in (0, ∞) and Fc(x) is the Fourier cosine transform of f(x). This suggests that other results connecting the behaviour of a function at infinity with the behaviour of its Fourier or Watson transform near the origin might exist. In this paper wre derive various such results. For example, a special case of these results is

where f(x) is the Fourier sine transform of g(x). It should be noted that the Fourier inversion formula fails to give f(+0) directly in this case. Some applications of these results to show the relationships between various forms of known summation formulae are given.

The study of the modular representation theory of the symmetric group has been greatly facilitated lately by the introduction of the graph (9, III ), the q-graph (5) and the hook-graph (4) of a Young diagram [λ]. In the present paper we seek to coordinate these ideas and relate them to the r-inducing and restricting processes (9, II ).

A Fuchsian group is a discrete subgroup of the hyperbolic group, L.F. (2, R), of linear fractional transformations

each such transformation mapping the complex upper half plane D into itself. If Γ is a Fuchsian group, the orbit space D/Γ has an analytic structure such that the projection map p: D → D/Γ, given by p(z) = Γz, is holomorphic and D/Γ is then a Riemann surface.

If N is a normal subgroup of a Fuchsian group Γ, then N is a Fuchsian group and S = D/N is a Riemann surface. The factor group, G = Γ/N, acts as a group of automorphisms (biholomorphic self-transformations) of S for, if γ ∊ Γ and z ∊ D, then γN ∊ G, Nz ∊ S, and (γN) (Nz) = Nγz. This is easily seen to be independent of the choice of γ in its N-coset and the choice of z in its N-orbit.

Conversely, if S is a compact Riemann surface, of genus at least two, then S can be identified with D/K, where K is a Fuchsian group acting without fixed points in D.

The concept of a mosaic was recently introduced by A. A. Mullin (1). By the fundamental theorem of arithmetic, every integer n > 1 can be uniquely expressed in the form

where the pi are primes satisfying p1 < p2 < . . . < pr. We then express any exponents aj which are greater than unity in the same manner, and continue in this way until the process terminates. The resulting planar configuration of primes is called the mosaic of n.

In this paper, we give a tropical method for computing Gromov–Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds that admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type $\text{A}$ and some moduli space of rank two bundles on a genus two curve.

In this paper we establish a priori${{h}^{1}}$-estimates in a bounded domain for parabolic equations with vanishing $\text{LMO}$ coefficients.

Any real number y leads to a continued fraction of the type

(1)

where ai, bi are integers which satisfy the inequalities

(2)

by means of the algorithm

(3)

the a's being assigned positive integers. The process terminates for rational y; the last denominator bk satisfying bk ≥ ak + 1. For irrational y, the process does not terminate. For a preassigned set of numerators ai ≥ 1, this C.F. development of y is unique; its value being y.

Bankier and Leighton (1) call such fractions (1), which satisfy (2), proper continued fractions. Among other questions, they studied the problem of expanding quadratic surds in periodic continued fractions. They state that “it is well-known that not only does every periodic regular continued fraction represent a quadratic irrational, but the regular continued fraction expansion of a quadratic irrational is periodic.

In [3], C. Davis showed that if a convex polyhedral cone C (the positive span of a finite set of vectors in Euclidean space) contains no nonzero linear subspace, then C is linearly isomorphic to the set V+ of nonnegative points in a linear subspace V of Rn. Moreover n can be taken to be the number of facets (maximal proper faces) of C.

Jacobi proved a curious theorem regarding the solutions of the system of equations

for functions λ=(x1, … , xn) satisfying

This paper introduces a class of domains which we hope to show merits some attention.

Definition. The domain R is said to be a distinguished domain if for any 0 ≠ z ∈ K, the quotient field of R, (1 : z) does not consist entirely of zero divisors modulo (1 : z–l). (Note: Here we use the fact that a zero module has no zero divisors. Thus if z–l ∈ R, so that (1 : z–l) = R, then the condition holds trivially.)

Section 1 of this paper gives numerous examples of distinguished domains, foremost among them being Krull domains and Prufer domains. In fact Prüfer domains are shown to be exactly those distinguished domains whose prime lattice forms a tree. Other distinguished domains can be constructed by the D + M construction. It is shown that distinguished domains are integrally closed but the converse fails.

In this paper we introduce a new approximation operator of the Arato-Renyi type and study its properties. Special cases of our operator are the power-series, of W. Meyer-König and K. Zeller (see [9]) and the generalized Berenstein power series introduced by A. Jakimovski and D. Leviatan in [5] and analyzed by them in [6].

The concept of extending to a topological space X a continuous pseudometric defined on a subspace S of X has been shown to be very useful. This problem was first studied by Hausdorff for the metric case in 1930 [9]. Hausdorff showed that a continuous metric on a closed subset of a metric space can be extended to a continuous metric on the whole space. Bing [4] and Arens [3] rediscovered this result independently. Recently, Shapiro [15] and Alo and Shapiro [1] studied various embeddings. It has been shown that extending pseudometrics can be characterized in terms of extending refinements of various types of open covers. In this paper we continue our study of extending pseudometrics. First we show that extending pseudometrics can be characterized in terms of σ-locally finite and σ-discrete covers. We then investigate when can certain types of covers be extended.

In this article we are concerned with how to compute the cohomology ring of a symplectic quotient by a circle action using the information we have about the cohomology of the original manifold and some data at the fixed point set of the action. Our method is based on the Tolman-Weitsman theorem which gives a characterization of the kernel of the Kirwan map. First we compute a generating set for the kernel of the Kirwan map for the case of product of compact connected manifolds such that the cohomology ring of each of them is generated by a degree two class. We assume the fixed point set is isolated; however the circle action only needs to be “formally Hamiltonian”. By identifying the kernel, we obtain the cohomology ring of the symplectic quotient. Next we apply this result to some special cases and in particular to the case of products of two dimensional spheres. We show that the results of Kalkman and Hausmann-Knutson are special cases of our result.

Let X be a space. We are interested in the homotopy-commutativity of the loop-space ΩX and the suspension ΣX, that is, in the question whether or not nil X ≦ 1, conil X ≦ 1, respectively. Let c: ΩX× ΩX ⟶ ΩX, c': ΣX ⟶ ΣX V ΣX be the commutator and co-commutator maps, respectively.

Let (Q, +, ·) be a finite quasifield of dimension d over its kernel K = GF(q), where q = pk with p a prime and k ≧ 1. (See p. 18-22 and p. 74 of [7] or Section 5 of [9] for the definition of quasifield.) For the remainder of this article we will follow standard conventions and omit, whenever possible, the binary operations + and · in discussing a quasifield. For example, the notation Q will be used in place of the triple (Q, +, ·) and Q* will be used to represent the multiplicative loop (Q − {0}, ·).

Let m be a non-zero element of the quasifield Q; the right multiplicative mapping ρm:Q → Q is defined by

1

For graph theoretic terms, see Tutte [1], A rank 2 tactical configuration of girth 2g and order (s, t) may be regarded as a (1 + s, 1 + t)-regular bipartite graph of girth 2g. We assume s ≦ t. Using a technique of Friedman [2] we show (i) and (ii).

If ℭ and ℭ′ are classes of finite abelian groups, we write ℭ + ℭ′ for the smallest class containing the groups of ℭ and of ℭ′. For any positive number r, ℭ < r is the smallest class of abelian groups which contains the groups Zp for all primes p less than r.

Our aim in this paper is to prove the following theorem.

THEOREM. Iƒ ℭ is a class of finite abelian groups and

(i) πi(Y) ∈ℭ for i < n,

(ii) H*(X; Z) is finitely generated,

(iii) Hi(X;Z)∈ ℭ for i > n + k,

Then

This statement contains many of the classical results of homotopy theory: the Hurewicz and Hopf theorems, Serre's (mod ℭ) version of these theorems, and Eilenberg's classification theorem. In fact, these are all contained in the case k = 0.

We extend a theorem of L. E. Dubins on “purely finitely additive disintegrations” of measures (cf. [4]) and apply this result to the disintegrations of extremal Gibbs states with respect to the asymptotic algebra enlarging another result of L. E. Dubins on the symmetric coin tossing game.

We recall the following definition of L. E. Dubins (cf. [3], [4]): Let (X , , μ) be a measure space, a sub σ-algebra of . A real function σx (A), is called a measurable-disintegration of μ if:

• (i) ∀x ∊ X , σx(.) is a finitely additive measure .

• (ii) ∀A ∊ , σ. (A) is constant on each -atom.

• (iii) For each A ∊ , σ. (A) is measurable with respect to the completion of by μ and

• (iv)σx(B) = 1 if x ∊ B ∊ .

In recent years, a considerable amount of research has been devoted to questions involving set porosity, particularly as it relates to differentiation theory. We may express the type of question in which we are interested by using the language of path derivatives and sequential derivatives. A path derivative of a function/is defined by writing

where at each point x a set Ex is given. A special case of the path derivative is the sequential derivative, defined by writing

where hn is a fixed sequence of nonzero numbers converging to zero.