The complete set of minimal obstructions for embedding graphs into the torus is still not determined. In this paper, we present all obstructions for the torus of connectivity 2. Furthermore, we describe the building blocks of obstructions of connectivity 2 for any orientable surface.

This paper is concerned with the Fresnel integrals

1.1

in the complex domain.

Recent research work in different fields of physical and technical applications of mathematics shows that an increasing number of problems require a detailed knowledge of elementary and higher functions for complex values of the argument.

The aim of this paper is the characterization of PSL(2, 31) in terms of its action on a certain polygonal graph. A polygonal graph is a pair (, ) consisting of a graph which is regular, connected and has girth m for some m ≧ 3, and a set of m-gons (circuits of length m) of such that every 2-claw (i.e. path of length 2) of is contained in a unique element of , (See Section 2 for the definitions of the terms used here.) If is the set of all m-gons of H, so that there is in a unique m-gon on every one of its 2-claws, then we write for (, ) and call a strict polygonal graph. If we wish to emphasize the integer m, then we call (, ) an m-gon-graph (respectively, a strict m-gon-graph). For convenience, a strict 5-gon-graph will be called a pentagraph.

In recent work of E. Arthurs and L. A. Shepp on a problem of H. Dym concerning the existence of an ergodic stationary stochastic process with zero entropy (cf. 1), the function dθ(n) was introduced as follows:

For an irrational number θ, let

be the sequence of points {lθ}, 1 ≦ l ≦ n, (where {x} denotes x — [x], the fractional part of x) and define*

dθ(n) = max(ai — ai-1), 1 ≦ in + 1.

We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the 6ϕ5 summation formula of F. H. Jackson and the q-Dixon sum. We also give q-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.

The topic of this paper is the relationship between characters of irreducible supercuspidal representations of the p-adic unramified 3 x 3 unitary group and Fourier transforms of invariant measures on elliptic adjoint orbits in the Lie algebra. We prove that most supercuspidal representations have the property that, on some neighbourhood of zero, the character composed with the exponential map coincides with the formal degree of the representation times the Fourier transform of a measure on one elliptic orbit. For the remainder, a linear combination of the Fourier transforms of measures on two elliptic orbits must be taken. As a consequence of these relations between characters and Fourier transforms, the coefficients in the local character expansions are expressed in terms of values of Shalika germs. By calculating which of the values of the Shalika germs associated to regular nilpotent orbits are nonzero, we determine which irreducible supercuspidal representations have Whittaker models. Finally, the coefficients in the local character expansions of three families of supercuspidal representations are computed.

In (5), the following theorem was proved in a very general setting:

(1) An additive set function is non-negative whenever its lower derivative is non-negative.

For a continuous additive function of intervals, theorem (1) can be improved as follows:

(2) A continuous additive set function is non-negative whenever its lower derivative is non-negative except, perhaps, on a countable set.

Considerable interest has attached to the problem of determining the type of a Riemann surface obtained by performing an identification between the edges of a strip or a half-strip (1, 2, 4, 5, 8). A fairly thorough analysis was made in 1946 by Volkovyskii (6) who gave various sufficient conditions for parabolic and hyperbolic type. The object of the present paper is to show that his principal sufficient condition for hyperbolic type can be substantially improved.

In the article "Moulton Planes" (10), I studied F. R. Moulton's construction over any field containing a multiplicative subgroup of index 2. In "Collineations of Affine Moulton Planes" (11), I determined the collineations between two arbitrary affine Moulton planes.

The purpose now is to describe the collineations between two projective Moulton planes. Since the affine collineations are known from (11), we are concerned with collineations mapping ideal lines onto ordinary lines. Notations and conventions of (10) and (11) are retained.

1. The sextics of the title have five cusps and are particular examples of the curve encountered by Humbert (5). The claim to notoriety of Humbert's curve of genus 5 has been that all its abelian integrals of the first kind are linear combinations of five elliptic integrals; it also has (1) the striking property that its 120 Weierstrassian points are confluent in threes at only 40 distinct points; whereas a general curve of genus 5 has, after Riemann, 12 moduli, a Humbert curve has merely 2. The three curves to be studied now have no free moduli at all, but although this tempts one to construct period matrices, such transcendental topics will not be handled here.

In a paper published in 1936 Burkill (2) proved that, if the trigonometrical series

1.1

is bounded except on a countable set and if the series obtained by integrating series (1.1) once converges everywhere, then the coefficients can be written in Fourier form using the C1P-integral. In §3 of this paper an analogous result is shown to be true when (1.1) is bounded (C, k), k < 0. The proof of this depends on generalizations of theorems by Verblunsky and Zygmund and both of these generalizations are obtained in §2.

A proper subgraph of a connected linear graph is said to disconnect the graph if removing it leaves a disconnected graph. In this paper we characterize, in the following sense, the disconnecting subgraphs of a fixed connected graph. We define two distinct types of disconnecting subgraphs (isthmuses and articulators) which are minimal in the sense that no proper subgraph of either type can disconnect the graph. We then show that any disconnecting subgraph must contain either an isthmus or an articulator. We also define a set of subgraphs (called dense) which form a lattice. We show that the union of the minimal dense subgraphs contains all isthmuses and articulators. In terms of these subgraphs we investigate some of the consequences of assuming that a disconnecting subgraph must contain at least m points.

Let A be a separable C*-algebra. Two representations π and π1 of A on the Hilbert spaces H and H1, respectively are said to be quasi-equivalent (denoted by π ~ π1) if projections of H ⊕ H1 on the invariant subspaces H and H1 of (π ⊕ π1)(A) have the same central support in the commutant (π ⊕ π1) (A)′ of (π ⊕ π1) (A), or equivalently, if there is an isomorphism ϕ of π(A)″ onto π1(A)″ such that ϕ(π(x)) = π(x) for all x ∊ A (cf. [5, § 5]). A representation π of A is said to be a factor representation if the center of π(A)″ consists of scalar multiples of the identity.

Let Λ be a ring and σ a kernel functor (left exact preradical) on the category of left Λ-modules. A left Λ-module M is called σ-semi-simple if whenever N is a submodule of M with M/Nσ-torsion, N is a direct summand of M. In Section 1 we consider alternative characterizations and properties of σ-semi-simplicity for modules. In Section 2 conditions equivalent to the σ-semi-simplicity of the ring are obtained. Section 3 is devoted to the condition, which frequently arises in Section 2, that every σ-torsion module be semisimple.

A locally convex Hausdorff topological vector space is said to be quasicomplete if closed bounded subsets of the space are complete, and von Neumann complete if closed totally bounded subsets are complete (equivalently, compact). Clearly quasi-completeness implies von Neumann completeness, and the converse holds in, for example, metrizable locally convex spaces. In this note we obtain a class of locally convex spaces for which the converse fails. Specifically, let X be a completely regular Hausdorff space, and let CC(X) denote the space of continuous real-valued functions on X, endowed with the compact-open topology.

In [1] Atiyah described how to use the complex representations of the symmetric group, Sn, to define and investigate operations in complex topological K-theory. In this paper operations for more general Grothendieck groups are described in terms of the integral representations of Sn using the representations directly without passing to the dual as Atiyah did. The principal tool, which is proved in the first section, is the theorem that the direct sum of the Grothendieck groups of finite integral representations of Sn form a bialgebra isomorphic to a polynomial ring with a sequence of divided powers. A consequence of this theorem is that the only operations that can be constructed from the symmetric groups will be polynomials in the symmetric powers.

In connection with the study of perturbation methods for differential eigenvalue problems, Aronszajn put forth a theory of systems (X, Y; A, B) consisting of a pair of linear transformations A, B:X → Y (see [1]; cf. also [2]). Here X and Y are complex vector spaces, possibly of infinite dimension. The algebraic aspects of this theory, where no restrictions of topological nature are imposed, where developed in [3] and [5]. We hasten to point out that the category of C2-systems (definition in § 1) in which this algebraic investigation takes place is equivalent to the category of all right modules over the ring of matrices of the form

In this paper, we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to enhance the quandle coloring invariants. The notion of indexed quandle is introduced, which generalizes the quandle idea. Some applications of this new invariant is discussed. We also study how to define a generalized chord index via a fixed finite biquandle. Finally, the chord index and its applications in twisted knot theory are discussed.

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

Matrices whose entries belong to certain rings of algebraic integers can be associated with discrete groups of transformations of inversive $n$-space or hyperbolic $(n+1)-\text{space}\,{{\text{H}}^{n+1}}$. For small $n$, these may be Coxeter groups, generated by reflections, or certain subgroups whose generators include direct isometries of ${{\text{H}}^{n+1}}$. We show how linear fractional transformations over rings of rational and (real or imaginary) quadratic integers are related to the symmetry groups of regular tilings of the hyperbolic plane or 3-space. New light is shed on the properties of the rational modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z})$, the Gaussian modular (Picard) group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}[i])$, and the Eisenstein modular group $\text{PS}{{\text{L}}_{2}}(\mathbb{Z}\left[ \omega \right])$.