The de Bruijn—van Aardenne Ehrenfest— Smith—Tutte theorem [1] is a theorem which connects the number of Eulerian dicircuits in a directed graph with the number of rooted spanning arborescences. In this paper we obtain a proof of this theorem by considering sequences over a finite alphabet, and we show that the theorem emerges from the generating function for a certain type of sequence. The generating function for the set of sequences is obtained as the solution of a linear system of equations in Section 2. The power series expansion for the solution of this system is obtained by means of the multivariate form of the Lagrange theorem for implicit functions, and is given in Section 3, together with a restatement of the theorem as a matrix identity.

In this paper, we prove results about solutions of the Diophantine equation $x^p+y^p=z^3$ over various number fields using the modular method. First, by assuming some standard modularity conjecture, we prove an asymptotic result for general number fields of narrow class number one satisfying some technical conditions. Second, we show that there is an explicit bound such that the equation $x^p+y^p=z^3$ does not have a particular type of solution over $K=\mathbb {Q}(\sqrt {-d})$, where $d=1,7,19,43,67$ whenever p is bigger than this bound. During the course of the proof, we prove various results about the irreducibility of Galois representations, image of inertia groups, and Bianchi newforms.

Let (X, ) be a topological space equipped with a partial order ≦ and let C (≦) denote the continuous increasing functions mapping X into R (a function f : X → R is increasing provided f(x) ≦ f(y) whenever x ≦ y) Then (X,, ≦) is an N-space (in the terminology of [16], a completely regular order space) provided is the weak topology of C (≦) and if x ≦ y is false, then there is an f ∈ C (≦) such that f(y) < f(x). L. Nachbin's introduction of N-spaces was perspicacious, for these spaces now find application in a wide spectrum of mathematics.

Cet exposé fait suite à des travaux antérieurs et notamment à mon article (4). Il a fait l'objet d'une communication au Colloque de géométrie différentielle et de topologie algébrique de Zürich en juin 1960. Dans une première partie on aborde, par les méthodes cohomologiques, le problème de la restriction du groupe structural G d'un espace fibre à un sous-groupe N, ce qui, lorsque N n'est pas normal, conduit à une "cohomologie à coefficients dans un faisceau d'espaces homogènes." Dans le second on donne une interprétation cohomologique des espaces fibres en théorie simpliciale, une fonction tordante étant interprétée comme un cocycle.

In this paper all rings considered have identity and are commutative, and all modules are finitely generated. We shall make liberal use of the definitions and notation established in [6; 7].

Towber observed in [9] that a local Outer Product ring (OP-ring) must have v-dimension ≦ 2, and so a local OP-ring is either regular of global dimension ≦ 2 or it has infinite global dimension. Since the global dimension of a noetherian ring is the supremum of the global dimensions of its localizations, we immediately obtain the following result.

THEOREM 1.1. The global dimension of a noetherian OP-ring is either∞ or ≦ 2.

α1α2, …, αp are n-dimensional vectors,

,;

they are arranged to form a closed polygon

.

Denote by R(α1, α2, …, αp) the radius of the smallest circumscribed hypersphere with centre at 0 ; by R(α1, α2, …, αp) the minimum of

(α1, α2, …, αp)

for all possible reorderings

of α2, …, αp−1 and by cn the least possible constant such that

for all possible choices of p and α1α2, … , αp.

Let GF(q) denote a finite field of order q = py, p a prime. Let A and C be symmetric matrices of order n, rank m and order s, rank k, respectively, over GF(q). Carlitz [6] has determined the number N(A, C, n, s) of solutions X over GF(q), for p an odd prime, to the matrix equation

1.1

where n = m. Furthermore, Hodges [9] has determined the number N(A, C, n, s, r) of s × n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin [10] has enumerated the s × n matrices of given rank r over GF(q), q = 2y, such that XXT = 0. Finally, the author [3] has determined the number of solutions to (1.1) in case C = 0, where q = 2y.

In this article, we derive a Brunn-Minkowski type theorem for sets bearing some relation to the causal structure on the Minkowski spacetime ${{\mathbb{L}}^{n+1}}$ . We also present an isoperimetric inequality in the Minkowski spacetime ${{\mathbb{L}}^{n+1}}$ as a consequence of this Brunn-Minkowski type theorem.

We introduce a new approach to an enumerative problem closely linked with the geometry of branched coverings, that is, we study the number ${{H}_{\alpha }}({{i}_{2}},{{i}_{3}},...)$ of ways a given permutation (with cycles described by the partition $\alpha $) can be decomposed into a product of exactly ${{i}_{2}}$ 2-cycles, ${{i}_{3}}$ 3-cycles, etc., with certain minimality and transitivity conditions imposed on the factors. The method is to encode such factorizations as planar maps with certain descent structure and apply a new combinatorial decomposition to make their enumeration more manageable. We apply our technique to determine ${{H}_{\alpha }}({{i}_{2}},{{i}_{3}},...)$ when $\alpha $ has one or two parts, extending earlier work of Goulden and Jackson. We also show how these methods are readily modified to count inequivalent factorizations, where equivalence is defined by permitting commutations of adjacent disjoint factors. Our technique permits us to generalize recent work of Goulden, Jackson, and Latour, while allowing for a considerable simplification of their analysis.

Let $v \ne 0$ be a vector in ${\mathbb {R}}^n$ . Consider the Laplacian on ${\mathbb {R}}^n$ with drift $\Delta _{v} = \Delta + 2v\cdot \nabla $ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , with respect to which $\Delta _{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood–Paley–Stein functions associated with the heat and the Poisson semigroups.

The major portion of this paper is devoted to an investigation of the conditions which imply that a semigroup (no identity or commutativity assumed) with a bounded invariant measure is a group. We find in §3 that a weakened form of “shearing” is sufficient and a counter-example (§5) shows that “shearing” may not be dispensed with entirely. In §4 we discuss topological measures in locally compact semigroups and find that shearing may be dropped without affecting the results of the earlier sections (Theorem 2). The next two theorems show that under certain circumstances (shearing or commutativity) the topology of the semigroup (already known to be a group by virtue of earlier results) can be weakened so that the structure becomes a separated compact topological group. The last section treats the problem of extending an invariant measure on a commutative semigroup to an invariant measure on its quotient structure.

In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.

An intrinsic quadric is a normal projective variety with a Cox ring defined by a single quadratic relation. We provide explicit descriptions of these varieties in the smooth case for small Picard numbers. As applications, we figure out in this setting the Fano examples and (affirmatively) test Fujita’s freeness conjecture.

Un anneau R est dit primitif (3) s'il existe un R-module irréductible et fidèle M. L'objet de cet article est d'étudier une généralisation naturelle de la notion d'anneau primitif en remplaçant "irréductible" par "sous-directement irréductible" dans la définition ci-dessus. D'une façon plus précise, un anneau R sera dit métaprimitif s'il existe un R-module sous-directement irréductible et fidèle M, avec SR ≠ {0}, S étant le sous-module minimal de M. Nous établissons pour les anneaux métaprimitifs un théorème de structure (théorème 2.3) faisant apparaître certaines analogies avec le théorème de structure correspondant pour les anneaux primitifs.

In 1974 [2] we used the n + 2 zeros of (1 - x2)Pn(α, β)(x), α, β > — 1, where Pn(α, β)(x) denotes Jacobi polynomials, to construct a sequence of linear operators {An(α, β)(f, x)} which has the following properties:

(i) An(α, β)(f, x) is a linear polynomial operator mapping C[— 1, 1] into polynomials of degree ≦ n(1 + c), (c > 0 fixed but arbitrary)

Let $F$ be a p-adic field of odd residual characteristic. Let $\overline{GS{{p}_{2n}}(F)}$ and $\overline{S{{p}_{2n}}(F)}$ be the metaplectic double covers of the general symplectic group and the symplectic group attached to the $2n$ dimensional symplectic space over $F$, respectively. Let $\sigma$ be a genuine, possibly reducible, unramified principal series representation of $\overline{GS{{p}_{2n}}(F)}$. In these notes we give an explicit formula for a spanning set for the space of Spherical Whittaker functions attached to $\sigma$. For odd $n$, and generically for even $n$, this spanning set is a basis. The significant property of this set is that each of its elements is unchanged under the action of the Weyl group of $\overline{S{{p}_{2n}}(F)}$. If $n$ is odd, then each element in the set has an equivariant property that generalizes a uniqueness result proved by Gelbart, Howe, and Piatetski-Shapiro.Using this symmetric set, we construct a family of reducible genuine unramified principal series representations that have more then one generic constituent. This family contains all the reducible genuine unramified principal series representations induced from a unitary data and exists only for $n$ even.

This paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonal

of a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

Let A be an integral domain and K its quotient field. A is called a Krull domain if there is a set {Vα} of rank one discrete valuation rings such that A = ∩αVα and such that each non-zero element of A is a non-unit in only finitely many of the Vα. The structure of these rings was first investigated by Krull, who called them endliche discrete Hauptordungen (4 or 5, p. 104).

Let $G$ be a locally compact group and let $\omega$ be a continuous weight on $G$. We show that for each of the Banach algebras ${{L}^{1}}\left( G,\,\omega \right),\,M\left( G,\,\omega \right),\,LUC{{\left( G,\,{{\omega }^{-1}} \right)}^{*}}$, and ${{L}^{1}}{{\left( G,\,\omega \right)}^{**}}$, the order structure combined with the algebra structure determines the weighted group.