In this paper we study the Chen–Ruan cohomology ring of weighted projective spaces. Given a weighted projective space ${{\mathbf{P}}^{n}}_{{{q}_{0}},\ldots ,{{q}_{n}}}$, we determine all of its twisted sectors and the corresponding degree shifting numbers. The main result of this paper is that the obstruction bundle over any 3-multisector is a direct sum of line bundles which we use to compute the orbifold cup product. Finally we compute the Chen–Ruan cohomology ring of weighted projective space $\mathbf{P}_{1,\,2,\,2,\,3,3,\,{{3}^{\centerdot }}}^{5}$.

We let , where the xi are positive integers and i ∈ N, the set of all integers. We define , where where . We let = [0; x1, x2, … ] where

We let

and define

The range of L(ξ) is known as the Lagrange spectrum and the range of M(ξ) as the Markov spectrum. It is known that both are closed and that the Markov spectrum includes the Lagrange spectrum.

This paper presents a list of all finite groups having exactly six and seven conjugate classes and an outline of the background necessary for the proof, and gives, in particular, two results which may be of independent interest. In 1903 E. Landau (8) proved, by induction, that for each the equation

*

has only finitely many solutions over the positive integers.

A variety (primitive class) is a class of abstract algebras which is closed under the formation of subalgebras, homomorphic images, and products. For a given variety we shall call a function μ*, which assigns to each algebra a natural number or ∞, denoted by μ*(A), a variety invariant if for every natural number n the class of all with μ*(A) ≦ n is again a variety. In this paper, a general method of finding variety invariants for the variety of all modular lattices will be developed. This method will be based on the concept of a quotient tree of a modular lattice. As examples of variety invariants we shall define, using the general result, the primitive length and the primitive width of modular lattices.

f(n, k) sera l'entier maximum t tel qu'il existe un graphe G ayant les propriétés suivantes:

• (a) G possède n sommets;

• (b) le nombre chromatique de G est égal à k;

• (c) Gest minimal par rapport à la propriété (b); c'est-à-dire, la suppression ede n'importe quelle arête rend G(k – 1)-colorable;

• (d) il existe t sommets indépendants de G, c'est-à-dire dont nulle paire ne se joigne par une arête.

Un graphe sera k-minimal s'il possède les propriétés (b) et (c). Puisque les graphes 3-minimaux sont tous des polygones impairs, il s'ensuit que f(n, 3) = [n/2] (n = 3, 5, 7, …), Il y a quelque temps T. Gallai a posé la conjecture:

1.1

M. Simonovits a réfuté l'inégalité stricte dans (1.1) en prouvant que

1.2

By a submethod of a regular matrix method A we mean a method (see 1 or 3) whose matrix is obtained by deleting a set of rows from the matrix A. We establish a one-one correspondence between the submethods of A and the points of the interval 0 < ξ ≤We designate the submethod which corresponds to 𝞷 by A (𝞷) and are accordingly able to speak of sets of submethods of measure 0, of the first category, etc.

Let Δ be the open unit disk in the complex plane and let be the group of automorphisms of Δ onto Δ, define by

The Banach spaces Hp = Hp(Δ), 1 ≦ p < ∞, are the Hardy spaces of functions analytic in Δ with their integral p means bounded,

The summability fields of generalized Nörlund means (N,p*α,p), α ∈ Ν, are increasing with a and are contained in that of the corresponding power series method (P,p). Particular cases are the Cesàro- and Euler-means with corresponding power series methods of Abel and Borel. In this paper we generalize a convexity theorem, which is well-known for the Cesàro means and which was recently shown for the Euler means to a large class of generalized Nörlund means.

An interesting identity is obtained for ideals A and B in a Noetherian ring :

A = (A + Bn) ∩ (A : Bn)

for sufficiently large n. This identity is applied to obtain Fuchs' quasi-primary decomposition of A in an improved form, and to obtain Krull's theorem on the intersection of the powers of A, both developments making no use of the Noetherian primary decomposition of A. Finally, the identity is used to obtain the primary decomposition without reference to irreducible ideals, in a largely constructive manner which yields the decomposition in an illuminating, automatically normal form, and which, subject to certain simple conditions, is unique.

1. Introduction. There are many problems, particularly in circuit theory, where the inverse of a Laplace transform is required but only for large values of the independent variable. For example (2, p. 66) the Laplace transform

of the current function for a semi-infinite cable under certain conditions turns out to be

.

A K-coloured rooted tree t is said to have colour partitionL if L is a K × ∞ matrix with elements lij equal to the number of non-root vertices of colour i and degree j. If adjacent vertices are of different colours then t is called a chromatic tree and L a chromatic partition. The tree has edge partitionD where D is a K × K matrix with elements dij equal to the number of edges, directed away from the root, from a vertex of colour i to a vertex of colour j.

In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions.

Ramanujan famously found congruences like $p(5n\,+\,4)\,\equiv \,0$ mod 5 for the partition function. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on ${{\Gamma }_{1}}(4)$ that is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored $F$-partitions.

One of the main purposes of the present paper is to provide a proof for the following statement:

Theorem A. Let M be a complex manifold of a complex dimension n. Let be a fixed point in M such that there exists a square integrable holomorphic n-form a(z) on M with .

Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

Let G be a group and F a field of arbitrary characteristic. In [4] Kaplansky asks under what conditions is F[G] primitive, where F[G] is the group algebra of G over F. We give some necessary conditions on G that F[G] be primitive and propose a conjecture.

Definition. A ring R is primitive if it has a faithful irreducible right module.

The above should really be considered as a definition of right primitive. One can analogously define left primitive and the two properties are not equivalent. For our purposes, the two concepts are equivalent, for the group algebra possesses a nice involution.

If we assume that F[G] is primitive, there are some immediate restrictions on G. First of all G cannot be Abelian since the only primitive commutative rings are fields. (I exclude of course the case when G consists of one element.)

1. Introduction. In (1) we have introduced the notions of “construction” and “ generic acyclicity” in order to determine a homology theory for any class of multiplicative systems defined by identities. Among these classes the most interesting one is the class of associative and commutative systems II with a unit element (containing the class of abelian groups).

Darmon cycles are a higher weight analogue of Stark–Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on ${{\Gamma }_{0}}\left( N \right)$ of even weight ${{k}_{0}}\,\ge \,2$. They are conjectured to be the restriction of global cohomology classes in the Bloch–Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross–Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur– Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$.

We compute some Hodge and Betti numbers of the moduli space of stable rank $r$, degree $d$ vector bundles on a smooth projective curve. We do not assume $r$ and $d$ are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank $r$, degree $d$ vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of $\text{S}{{\text{L}}_{n}}$ is one.

Virasoro-toroidal algebras, ${{\tilde{J}}_{[n]}}$, are semi-direct products of toroidal algebras ${{J}_{[n]}}$ and the Virasoro algebra. The toroidal algebras are, in turn, multi-loop versions of affine Kac-Moody algebras. Let $\Gamma $ be an extension of a simply laced lattice $\dot{Q}$ by a hyperbolic lattice of rank two. There is a Fock space $V\left( \Gamma \right)$ corresponding to $\Gamma $ with a decomposition as a complex vector space: $V\left( \Gamma \right)=\coprod{_{m\in z}K\left( m \right)}$ . Fabbri and Moody have shown that when $m\ne 0,\,K\left( m \right)$ is an irreducible representation of ${{\tilde{J}}_{[2]}}$ . In this paper we produce a filtration of ${{\tilde{J}}_{[2]}}$-submodules of $K\left( 0 \right)$. When $L$ is an arbitrary geometric lattice and $n$ is a positive integer, we construct a Virasoro-Heisenberg algebra $\tilde{H}\left( L,n \right)$ . Let $Q$ be an extension of $\dot{Q}$ by a degenerate rank one lattice. We determine the components of $V\left( \Gamma \right)$ that are irreducible $\tilde{H}\left( Q,1 \right)$ -modules and we show that the reducible components have a filtration of $\tilde{H}\left( Q,1 \right)$-submodules with completely reducible quotients. Analogous results are obtained for $\tilde{H}\left( \dot{Q},2 \right)$ . These results complement and extend results of Fabbri and Moody.

In § 1, we begin with a basic theorem which describes a convenient embedding of a nonsingular left R-module into a complete direct product of copies of the left injective hull of R (Theorem 2). Several applications follow immediately. Notably, the injective hull of a finitely generated nonsingular left R-module is isomorphic to a direct sum of injective hulls of closed left ideals of R (Corollary 4). In particular, when R is left self-injective, every finitely generated nonsingular left R-module is isomorphic to a finite direct sum of injective left ideals (Corollary 6).

In § 2, where it is assumed for the first time that rings have identity elements, we investigate more generally the class of left R-modules which are embeddable in direct products of copies of the left injective hull Q of R. Such modules are called torsion free, and can also be characterized by the property that no nonzero element is annihilated by a dense left ideal of R (Proposition 12).