Let X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos of quasi-coherent sheaves over X. As it turns out, every topos point of is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.

This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if and are equivalent.

The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ∞, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.

We shall be considering plane closed convex curves Γ (whose interior plus boundary will be denoted by Σ ≡ Σ(Γ)) in which we can inscribe a large set (in various senses to be made precise) of equilateral triangles or e-triangles for short; and in particular an e-triangle of side-length 1 is called an E-triangle or just an E. More formally we define the sets to be the sets of closed convex Γ for which, respectively:

: each point of Γ is the vertex of some inscribed E;

: Γ contains the vertices of some E in each orientation in the plane;

: an E can be moved round continuously, through angle 2π, with all vertices on Γ throughout.

1. Introduction. The historical basis for the work in this paper lies in a remarkable fact discovered by Dean in 1948. He found that time-periodic surface waves in an ideal fluid experience no reflexion when they encounter normally a fixed, submerged, right-circular cylinder. We might reasonably ask if a similar non-reflective property carries over to different geometrical configurations of submerged objects, for example, two or more cylinders. This question motivates the investigation which follows.

Complex variable methods have been applied to isotropic and aelotropic plate problems by several authors. The notation used here is that of Stevenson(14). Dawoud(5) has expressed the continuity conditions between two differently loaded regions in terms of the complex potentials and the particular integrals for the two regions.

The problem of a transverse load at any point of a clamped circular plate was solved by Clebsch(4), Michell(11), Melan(10) and Flügge(6). A series solution for the simply supported circular plate under the same load was given by Foeppel(8). Using Stevenson's tentative method Dawoud(5) applied complex potentials to solve the problem of an eccentric isolated load under certain boundary conditions. Applying Muskhelishvili's method, Washizu(15) obtained the same results for clamped and simply supported boundaries.

It is easy to get solutions for a circular plate concentrically and uniformly loaded. For non-uniform loadings there are the solutions found by Sen (13) for certain distributions of normal thrust over the complete plate or over a concentric circle and the solution of Flügge (7) for a linearly varying load over the simply supported circular plate. The present author and Dawoud(3) obtained the solutions for a circular plate with the load over the complete plate or over a concentric circle, under a general boundary constraint including as special cases the usual clamped and hinged boundaries. Ghose (9) worked out the problem of a clamped circular plate when the load is uniformly distributed between two concentric circles and two radii. Schmidt (12) found the solution for a clamped circular plate uniformly loaded over an eccentric circle. The complex variable method was applied by the author and Dawoud(2) to obtain the solutions for a circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under the general boundary condition mentioned before. The author (1) also found the solution for a linearly varying load over an eccentric circle under the same boundary condition. In this paper the power of the complex variable method is exhibited by rinding the appropriate complex potentials corresponding to the load

over an eccentric circular patch, where R, θ are measured from the centre of the patch and the common diameter of the plate and the patch. Since the two cases n = 0, 1 require special consideration and were dealt with separately (in (2) and (1) respectively), we see that this paper completes the solution of the problem of a circular plate with an eccentric circular patch symmetrically loaded with respect to the common diameter of the plate and patch, the load being in this case expressible in the form .

For a clamped boundary the solution is obtained in finite terms.

F. Riesz's ‘Representation Theorem’ has been proved by methods classical [11, 12], category-theoretic [7], and functional-analytic [2, 9]. (Garling's remarkable proofs [5, 6] owe their brevity to the combined strength of these and other methods.) These proofs often reveal a connection between the Riesz theorem and some unexpected area of mathematics.

It is well known that, if a trigonometric series

converges to zero in 0 ≤ x < 2π then all the coefficients are zero. To generalize this property of the series, sets of uniqueness have been defined. A point-set E in 0 ≤ x < 2π is a set of uniqueness if every series (0·1), converging to zero in [0, 2π) − E, has zero coefficients. Otherwise E is a set of multiplicity. For example, every enumerable set is a set of uniqueness. An account of the theory may be found in Zygmund (2), chapter 11, pp. 267 et seq.

We show that for every $g\geq 2$ there is a compact arithmetic Riemann surface of genus $g$ with at least $4(g-1)$ automorphisms, and that this lower bound is attained by infinitely many genera, the smallest being 24.

Arithmetic and non arithmetic hyperbolic 3-manifolds having infinitely many fibrations over $S^{1}$ of arbitrary high genus are constructed. A concrete example is studied in detail.

If A, B are ideals of a commutative ring R, such that B ⊆ A, and, for some positive integer r, Ar = BAr−1 then B is said to be a reduction of A. (This concept was defined and developed by Northcott and Rees in (1).) In this paper, I shall consider commutative Noetherian rings with the property that no non-zero principal ideal is a reduction of an ideal properly containing it.

Let G be a connected locally finite simplicial graph with rk(π1(G))[ges ]2 and let T be the universal cover of G. Consider a π1(G)-invariant conformal density μ of dimension d on ∂T. The total mass function ϕμ of μ is defined on the set of vertices of G. Let |ϕμ| be its l2-norm. Let Ω be the geodesic flow space of G and mμ the invariant measure on Ω associated to μ. The main results of this paper are the following:

(i) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [les ]k. Then, if d>½log(k−1) and mμ(Ω)<∞, we have ϕμ∈l2(V) and

|ϕμ|2[les ]Cmμ(Ω),

with C=(e2d−1)/(e2d−k+1).

(ii) Assume that there exists an integer k[ges ]3 such that the degree at each vertex of G is [ges ]k. Then, if ϕμ∈l2(V), we have d>½log(k−1) and

Cmμ(Ω)[les ]|ϕμ|2.

In the geometry of the cubic curve Γ in space, we have to study the quadrics Q which stand in certain special relations to the curve. We shall find it convenient to refer to these relations as A, B, B′, C, C′; these are defined below.

It is familiar that a “double plane”, i.e. an algebraic surface conceived as consisting of each point of a plane counted twice, is unambiguously determined when its “branch curve”, or locus of points in the plane which count as two coincident points of the surface (the general point of the plane counting as two distinct points), is assigned; the only condition which must be satisfied by this curve being that it is of even order. In fact, if f(x, y) = 0 is the Cartesian equation of the curve, the double plane can be regarded as the limit of the surface

where A is a large constant, or as the projection of this surface, for finite A, from the point at infinity on the z-axis. If, however, f = 0 is of odd order, it is easily seen that the line at infinity also forms part of the branch curve of the double plane. A repeated portion of the branch curve is ineffective, for if φ (x, y) = 0 is any other curve, the surface

is birationally identical with the former merely by putting

Introduction. This paper is a continuation of (1), where definitions may be found. All manifolds, maps, bundles, etc., are to be piecewise linear. Recall that cells are always closed.

It is the purpose of this paper to prove a general compactness theorem and deduce from it, as corollaries, the Tychonoff theorem on the product of compact spaces and Ascoli-type theorems giving sufficient conditions for function spaces with the topology of uniform convergence on compact sets to be compact.

1. The problems of stress distributions in an infinite plate containing a circular hole were solved in a general manner by Bickley(1) for isotropic materials. An alternative method, using complex stress functions, was later given by Green(5) and extended so as to apply to aeolotropic materials. In the present paper Green's method is employed to determine the stress distributions that arise when the stresses are produced by isolated forces acting at points on, or near to, the edge of the circular hole, and though some of the solutions are cumbersome they are all obtained in finite form.

Let $p$ be an odd prime and $G$ the finite cyclic group of order $p$. We use the Casson–Walker–Lescop invariant to find a necessary condition for a three-manifold to have an action of $G$ with a circle as the set of fixed points and an integral homology sphere as the quotient.

This paper is a sequel to one recently published, in which the regularity of surfaces was investigated by means of certain criteria suggested by Comessatti. We suppose throughout that the surface F under consideration is of general type in Sr, and that it has curve sections C of order n and genus π. It may be recalled that in the previous work the two most useful criteria were the following theorems:

If π < ½ (n + 2), then F is regular or referable. (I)

If πr (n) ≤ π ≤ n, where πr (n) is the maximum genus of a Cn in Sr, then F is regular or referable. (II)

Functional analytic properties of the Banach Algebras $A_p^r(G)$ such as Arens regularity, amenability, weak sequential completeness, sets of synthesis and strict containment for fixed $p$ and increasing $r$, are investigated.

It is found that $A_p^r(G)$, $1\leq r<\infty$, behave very differently to the Figa–Talamanca–Herz algebras $A_p(G)$.

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the property

a partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

In an earlier paper (Helliwell and Mackie(3)) it was shown that steady two-dimensional flow patterns of a compressible inviscid fluid at high subsonic speed past a finite wedge could be determined quite simply when sonic velocity is attained at the shoulder of the wedge and thereafter the flow breaks away from the shoulder with a free streamline. In a subsequent paper (Helliwell (4)) a similar method of analysis has been applied to determine a flow pattern of the same general type past a finite wedge symmetrically placed in a channel, from which the case of the wedge in the free stream may be deduced as a special case. However, in a general investigation into transonic flow past wedges (Mackie and Pack (5)) it was argued that when the wedge angle or the free stream (subsonic) velocity is too small no supersonic region would develop on the wedge side, and the flow would break away from the wedge shoulder with some higher subsonic velocity, giving a free stream line. The present note examines the flow pattern which develops under these conditions for a wedge symmetrically placed in a channel with parallel walls.