An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties.

Let it be required to arrange v elements into v sets such that each set contains exactly k distinct elements and such that each pair of sets has exactly λ elements in common (0 < λ < k < v). This problem we refer to as the v, k,λ combinatorial problem.

The treatment of semigroups given in a previous paper (3) is based upon representations of a semigroup by means of transformations of a set (cf. also 12). In this paper we try to remove the assumption of the existence of a zero element proposed in (3). In accordance with our general programme explained at the beginning of (3) we utilize certain minimum conditions in order to gain more information on the structure of semigroups.

Our main results are structure theorems on primitive semigroups which have irreducible right ideals generated by idempotents (§§15-17). As we have shown in (5), these theorems permit the explicit construction of primitive semigroups.

A contra variant category-equivalence between categories of right R-modules and left S-modules (all rings have units, all modules are unitary) that contain RR, SS and are closed under submodules and factor modules, is naturally equivalent to a functor Horn (–, U) with a bimodule SUR such that SU, UR are injective cogenerators with S = End UR and R = End SU, and all modules in are [U-reflexive. Conversely, for any SSUR, Hom(–, U) is a contravariant category equivalence between the categories of [U-reflexive modules, and if U has the properties just stated, then these categories are closed under submodules, factor modules, and finite direct sums and contain RR, UR,SS, and SU. Such a functor will be called a (Morita) duality between R and S induced by U (see (5)).

Let R = k[X1, …, Xn] with k a field, and let I ⊂ R be a homogeneous ideal. The algebra R/I is said to have a pure resolution if its homogeneous minimal resolution has the form

Some of the known examples of pure resolutions include the coordinate rings of: the tangent cone of a minimally elliptic singularity or a rational surface singularity [15], a variety defined by generic maximal Pfaffians [2], a variety defined by maximal minors of a generic matrix [3], a variety defined by the submaximal minors of a generic square matrix [6], and certain of the Segre-Veronese varieties [1].

If I is in addition Cohen-Macaulay, then Herzog and Kühl have shown that the betti numbers bi are completely determined by the twists di.

To construct a well behaved localization of a noetherian ring R at a semiprime ideal S, it seems necessary to assume that the set (S) of modulo S regular elements satisfies the Ore condition ; and it is convenient to require the Artin Rees property for the Jacobson radical of the quotient ring Rs in addition: one calls such 5 classical. To determine the classical semiprime ideals is no easy matter; it happens frequently that a prime ideal fails to be classical itself, but is minimal over a suitable classical semiprime ideal.

Let 𝒳 be a complex Banach space, with open unit ball B. We consider the algebra of analytic functions on B that are weakly continuous and that are uniformly continuous with respect to the norm. We show these are precisely the analytic functions on B that extend to be weak-star continuous on the closed unit ball of 𝒳**. If 𝒳* has the approximation property, then any such function is approximable uniformly on B by finite polynomials in elements of 𝒳*. On the other hand, there exist Banach spaces for which these finite-type polynomials fail to approximate. We consider also the approximation of entire functions by finite-type polynomials. Assuming 𝒳* has the approximation property, we show that entire functions are approximable uniformly on bounded sets if and only if the spectrum of the algebra of entire functions coincides (as a point set) with 𝒳**.

If the differential expression l(y) = — y” + g(x)y generates a closed operator L on L2(— ∞, ∞), with domain D consisting of those functions y ∈ L2 with absolutely continuous derivatives and such that l(y) ∈ L2. The case where g(x) is real-valued has been extensively investigated and yields an expansion of any ƒ ∈ L2 in terms of the characteristic functions of L. We shall investigate the case where g is complex-valued.

It is well known that for any real number θ there are infinitely many positive integers n such that

Here ||a|| denotes the distance of a from the nearest integer, taken positively. Indeed, since ||a|| < 1, this implies more generally that if θ1, θ2, . . . , θk are any real numbers, then there are infinitely many positive integers n such that

Let f(x1, … , xn) be a positive definite quadratic form of determinant Δ; let M be its minimum value for integers x1, … , xn not all zero; and let 2s be the number of times this minimum is attained, i.e., the number of solutions of the Diophantine equation

A reverse iterated function system (r.i.f.s.) is defined to be a set of expansive maps $\left\{ {{T}_{1}},...,{{T}_{m}} \right\}$ on a discrete metric space $M$. An invariant set $F$ is defined to be a set satisfying $F\,=\,\bigcup _{j=1}^{m}\,{{T}_{j}}F$, and an invariant measure $\mu $ is defined to be a solution of $\mu \,=\,\sum{_{j=1}^{m}\,{{p}_{j}}\mu {}^\circ T_{j}^{-1}}$ for positive weights ${{p}_{j}}$. The structure and basic properties of such invariant sets and measures is described, and some examples are given. A blowup$\mathcal{F}$ of a self-similar set $F$ in ${{\mathbb{R}}^{n}}$ is defined to be the union of an increasing sequence of sets, each similar to $F$. We give a general construction of blowups, and show that under certain hypotheses a blowup is the sum set of $F$ with an invariant set for a r.i.f.s. Some examples of blowups of familiar fractals are described. If $\mu $ is an invariant measure on ${{\mathbb{Z}}^{+}}$ for a linear r.i.f.s., we describe the behavior of its analytic transform, the power series $\sum{_{n=0}^{\infty }\mu (n){{z}^{n}}}$ on the unit disc.

In [2] by means of a fairly lengthy argument involving Hermitian varieties it was shown that PG(2n, q2) can be partitioned into (q2n++ 1 + 1)/(q + l)-caps. Moreover, these caps were shown to constitute the “large points” of a PG(2n, q) in a natural way. In [3] a similar argument was used to show that once two disjoint (n – l)-subspaces are removed from PG(2n, q2), the remaining points can be partitioned into (q2n – 1)/(q2 – l)-caps.

The purpose of this paper is to give a short proof of the results found in [2], and then use the technique developed to partition PG(2n, q) into (qn + l)-caps for n even and q any prime-power. Moreover, these caps can be treated in a natural way as the “large points” of a PG(n – 1, q).

Ebin [8] gives a thorough study of the space of riemannian metrics on a compact manifold M and of the action of the diffeomorphism group of M on . The purpose of this paper is to study the action of the larger group of conformorphisms, or conformai transformations, on and on . On , the L2-orthogonal decomposition induced by the action of gives a splitting of symmetric tensors into three summands introduced by York [25; 26]. We find submanifolds of tangent to the pieces of this decomposition.

Our present work is divided into three sections. In §2 we study the metrizability of spaces with a Gδ-diagonal (see Definition 2.1). In §3 we study the metrization of topological spaces by means of collections of (not necessarily continuous) real-valued functions on a topological space. Our efforts, in §§2 and 3, are directed toward answering the following question: “Is every normal, metacompact (see Definition 2.4) Moore space a metrizable space?” which still remains unsolved. (However, Theorems 2.12 through 2.15 and Theorem 3.1 may be helpful in answering the preceding question.) In §4 we prove an apparently new necessary and sufficient condition for the metrizability of the Stone-Čech compactification of a metrizable space and hence for the compactness of a metric space.

Suppose that G is a reductive algebraic group defined over a number field F. The trace formula is an identity

of distributions. The terms on the right are parametrized by “cuspidal automorphic data”, and are defined in terms of Eisenstein series. They have been evaluated rather explicitly in [3]. The terms on the left are parametrized by semisimple conjugacy classes and are defined in terms of related G(A) orbits. The object of this paper is to evaluate these terms.

In previous papers we have already evaluated in two special cases. The easiest case occurs when corresponds to a regular semisimple conjugacy class in G(F). We showed in Section 8 of [1] that for such an , could be expressed as a weighted orbital integral over the conjugacy class of σ. (We actually assumed that was “unramified”, which is slightly more general.) The most difficult case is the opposite extreme, in which corresponds to {1}.

A map wil be called k-chromatic or said to have chromatic number k if k is the least positive integer having the property that the countries of the map can be divided into k mutually disjoint (colour) classes in such a way that no two countries which have a common frontier line are in the same (colour) class.Heawood [4] proved that for h > 1 the chromatic number of a map on a surface of connectivity h is at most nh, where

The theory developed for the study of flows in networks (2; 3; 4; 5; 6; 7) sometimes provides a useful tool for dealing with certain kinds of combinatorial problems, as has been previously indicated in (3; 4; 6; 7). In particular, Hall-type theorems for the existence of systems of distinct representatives which contain a prescribed set of marginal elements (10; 11), or, more generally, whose intersection with each member of a given partition of the fundamental set has a cardinality between prescribed lower and upper bounds (9), can be obtained in this way (7).

A generalized Riccati transformation will be utilized to derive a Riccati-type inequality (3) associated with a semilinear elliptic inequality yL(y; x) ≦ 0 possessing a positive solution y in an exterior domain in Euclidean n-space. On the basis of (3), general sufficient conditions for the elliptic inequality to be oscillatory are developed in § 3. The matrix of coefficients of the second derivative terms in L(y;x) (i.e. (Aij) in (1)) is not restricted in any way beyond the usual ellipticity hypothesis (iv) below, and thereby one of the difficulties mentioned in [9] and inherent in the method there is resolved. Furthermore, the nonlinear term B﹛x, y) in (1) is not required to be one-signed.

Let $X$ be a separated finite type scheme over a noetherian base ring $\mathbb{K}$. There is a complex ${{\hat{C}}^{\cdot }}\left( X \right)$ of topological ${{\mathcal{O}}_{X}}$-modules, called the complete Hochschild chain complex of $X$. To any ${{\mathcal{O}}_{X}}$-module $\mathcal{M}$—not necessarily quasi-coherent—we assign the complex $Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,\mathcal{M} \right)$ of continuous Hochschild cochains with values in $\mathcal{M}$. Our first main result is that when $X$ is smooth over $\mathbb{K}$ there is a functorial isomorphism

$$Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{C}^{\cdot }}\left( X \right),\,M \right)\,\cong \,\text{R}\,Hom_{{{\mathcal{O}}_{X}}^{2}}^{{}}\,\left( {{\mathcal{O}}_{X}},\,M \right)$$

in the derived category $\text{D}\left( \text{Mod}\,{{\mathcal{O}}_{{{X}^{2}}}} \right)$, where ${{X}^{2}}\,:=\,X\,{{\times }_{\mathbb{K}}}\,X$.

The second main result is that if $X$ is smooth of relative dimension $n$ and $n!$ is invertible in $\mathbb{K}$, then the standard maps $\text{ }\!\!\pi\!\!\text{ }\,\text{:}\,{{\hat{C}}^{-q}}\left( X \right)\,\to \,\Omega _{X/\mathbb{K}}^{q}$ induce a quasi-isomorphism

$$Hom_{{{\mathcal{O}}_{X}}}^{{}}\,\left( \underset{q}{\mathop \oplus }\,\,\Omega _{X/\mathbb{K}}^{q}\,\left[ q \right],\,M \right)\,\,\to \,Hom_{{{\mathcal{O}}_{X}}}^{\text{cont}}\left( {{{\hat{C}}}^{\cdot }}\left( X \right),\,M \right).$$

When $M\,=\,{{\mathcal{O}}_{X}}$ this is the quasi-isomorphism underlying the Kontsevich Formality Theorem.

Combining the two results above we deduce a decomposition of the global Hochschild cohomology

$$\text{Ext}_{{{\mathcal{O}}_{{{X}^{2}}}}}^{i}\,\left( {{\mathcal{O}}_{X}}\,,\,M \right)\,\cong \,\underset{q}{\mathop \oplus }\,\,\,{{\text{H}}^{i-q}}\,\left( X,\,\left( \underset{{{\mathcal{O}}_{X}}}{\overset{q}{\mathop \Lambda }}\,\,{{T}_{X/\mathbb{K}}} \right)\,{{\otimes }_{{{\mathcal{O}}_{X}}}}\,M \right),$$

where ${{T}_{X/\mathbb{K}}}$ is the relative tangent sheaf.

In this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.