1. Mikusiński ((2), Problem 84) asks: can we express the surface of a sphere (in Euclidean 3-space) as the union of non-overlapping congruent great-circle arcs? We give a partial answer to this question.

In this article we exhibit a method complementary to the method presented in [4], that allows us, at least in some important cases, to obtain exact expressions for the orders of ideal classes of cyclotomic fields in terms of properties of the units of the field. We consider only the particular case in which the classes belong to the p-Sylow subgroup (A)p of the ideal class group of a real p-cyclotomic field, but it appears that the results can be generalized.

Our primary objective is to give necessary and sufficient conditions for a triangular subalgebra of a groupoid C-algebra to be semisimple, i.e. to have vanishing Jacobson radical. If, in addition, the subalgebra is the analytic subalgebra determined by a real-valued cocycle on the groupoid, then we can give an explicit description of the radical in terms of the cocycle. As a consequence of this analysis, we are able to determine when certain analytic crossed products are semisimple.

Let g denote the moduli space of compact Riemann surfaces of genus g > 3. It is known that g is a non-complete quasi-projective variety that contains many complete curves. This is because the Satake compactification g of g is projective and the boundary \ has co-dimension 2; thus by intersecting with hypersurfaces in sufficiently general position one obtains a complete curve in g passing through any given set of points [8].

The original approach of Blasius to the solution of the differential equation now associated with his name was to develop the unknown function as a power series. Unfortunately, this series has a limited radius of convergence, so that such a representation is not valid over the whole range of interest. It is shown here that, if we work instead with a particular inverse function, this can be expanded as a power series which converges for all relevant values of the independent variable. Moreover, the number associated with the solution which is of principal physical interest can be expressed in terms of the asymptotic properties of the coefficients of this series. Exploiting this relationship, we find upper and lower bounds for this number in terms of the zeros of two particular families of polynomials.

1.Introduction. 1.1. Let f(t) be a periodic function with period 2π and integrable in the Lebesgue sense over ( -π,π). We assume as we may without loss of generality, that the Fourier series of f(t) is .

Let G be an Abelian group, the symmetric algebra of G and the associated graded ring of the integral group ring ZG, where (AG denotes the augmentation ideal of ZG). Then there is a natural epimorphism (4)

which is given on the nth component by

In general θ is not an isomorphism. In fact Bachmann and Grünenfelder(1) have shown that for finite Abelian G, θ is an isomorphism if and only if G is cyclic. Thus it is of interest to investigate ker θn for finite Abelian groups. In view of proposition 3.25 of (3) it is enough to consider finite Abelian p-groups.

A table of totally complex quartic fields, with small discriminants, which possess quadratic subfields was given by Delone and Faddeev(1); the purpose of the present paper is to give a table of such fields which have no subfield. The method used is based on an idea used recently (2) in the case of totally real quartic fields and depends on a special case of the following theorem, which is stated in full generality.

1. Definitions and notations. Let be a given infinite series with the sequence of its partial sums {sn}. Let {pn} be a sequence of constants, real or complex, and let us write

In a recent paper (1) expressions were found for the statistical distributions of the velocities of the zero-crossings and of the maxima and minima of a random moving wave-form. In the present note it will be shown that these distributions are to some extent arbitrary, depending on how the points of observation are selected.

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

This paper improves a method of Godwin (4) for finding a pair of fundamental units in a totally real cubic field. The determination of such a unit pair is a well known computational problem. There is an old algorithm (circa 1896) of Voronoi which solves this problem, but the algorithm is quite complicated (an account of it is given in the book of Delone and Faddeev ((3), chapter IV, part A)). The method of Godwin is, in principle, much simpler. However, this method also has its drawbacks (more is said about this in Section 4 below). Indeed, when Godwin's student Angell produced his large table (see (1)) of totally real cubic fields some 15 years after (4) appeared, Voronoi's algorithm was used to compute the pairs of fundamental units.

We give a general method that may be effectively applied to the question of whether two components of a function space map(X, Y) have the same homotopy type. We describe certain group-like actions on map(X, Y). Our basic results assert that if maps f, g: X → Y are in the same orbit under such an action, then the components of map(X, Y) that contain f and g have the same homotopy type.

This paper is devoted to a global existence theorem of meromorphic solutions of the form Z(z)=Zo(z)+R(z) of a nonlinear Riemann–Hilbert problem (RHP) for multiply connected domains Gq(q[ges ]1), where Zo(z) is the singular part of the solution, R(z) is the regular part which is a holomorphic solution of some appropriate nonlinear RHP for Gq(q[ges ]1). Under appropriate conditions on the characteristics of both the singular part Zo(z) (number of poles) and regular part (winding number) we prove the existence of meromorphic solutions Z(z) of the form Z(z)=Zo(z)+R(z). The proof is based on a special construction of the singular part Zo(z) and an adequate formulation of Newton's method for the regular part R(z).

Spherical diagrams were introduced by Lyndon and Schupp[14] in order to study asphericity in group presentations and in 2-complexes. They have since been studied by several authors [2, 3, 5]. In particular, some technical loopholes in the original approach were closed in [5]. For many purposes the dual notion of pictures, introduced by Rourke[17], is more useful. These arise naturally through transversality. Pictures have also been studied and applied in [2, 5, 6, 12, 19].

In this paper we study the question of characterizing those positive Borel measures μ on the unit disc Δ for which the differentiation operator D defined by Df = f′ maps the Hardy space Hp continuously into the Bergman space Bp(dμ) of all functions f analytic in Δ which belong to Lp (Δ, dμ).

If I and J are ideals in a Noetherian ring R, then I and J are projectively equivalent in case (Ii)a = (Jj)a for some positive integers i, j (where Ka denotes the integral closure in R of the ideal K) and the form ring F(R, I) of R with respect to I is the graded ring R/I ⊕ I/I2 ⊕ I2/I3 ⊕ …. These two concepts have played an important role in many research problems in commutative algebra, so they have been deeply studied and many of their properties have been discovered. In a recent paper [13] they were combined to show that a semi-local ring R is unmixed if and only if for every ideal J in R there exists a projectively equivalent ideal J in R such that every prime divisor of zero in F(R, J) has the same depth. It seems to us that results similar to this are interesting and potentially quite useful, so in this paper we prove several additional such theorems. Namely, it is shown that all ideals in all local rings have a projectively equivalent ideal whose form ring is fairly nice. Also, a characterization similar to the just mentioned result in [13] is given for the class of local rings whose completions have no embedded prime divisors of zero, and several analogous new characterizations are given for locally unmixed Noetherian rings. In particular, it is shown that if I is an ideal in an unmixed local ring R such that height(I) = l(I) (where l(I) denotes the analytic spread of I), then there exists a projectively equivalent ideal J in R such that Ass (F(R, J)) has exactly m elements, all minimal, where m is the number of minimal prime divisors of I (so if I is open, then F(R, J) has exactly one prime divisor of zero and is a locally unmixed Noetherian ring).

Suppose an amenable group G is acting freely on a simply connected simplicial complex $\~{X}$ with compact quotient X. Fix n ≥ 1, assume $H_n(\~{X}, \mathbb{Z}) = 0$ and let (Hi) be a Farber chain in G. We prove that the torsion of the integral homology in dimension n of $\~{X}/H_i$ grows subexponentially in [G : Hi]. This fails if X is not compact. We provide the first examples of amenable groups for which torsion in homology grows faster than any given function. These examples include some solvable groups of derived length 3 which is the minimal possible.

In this paper an approximate formula is developed for the phase shifts required in the calculation of the scattering of particles according to wave mechanics. It is shown to be more accurate than the current approximations and it is not difficult to handle. The same method can also be used to compute any quantity which is sensitive to the form of the wave function near the centre of force (e.g. a matrix element).

A classical link in 3-space can be represented by a Gauss paragraph encoding a link diagram in a combinatorial way. A Gauss paragraph may code not a classical link diagram, but a diagram with virtual crossings. We present a criterion and a linear algorithm detecting whether a Gauss paragraph encodes a classical link. We describe Wirtinger presentations realizable by virtual link groups.