We study certain symplectic quotients of $n$-fold products of complex projective $m$-space by the unitary group acting diagonally. After studying nonemptiness and smoothness of these quotients we construct the action-angle variables, defined on an open dense subset, of an integrable Hamiltonian system. The semiclassical quantization of this system reporduces formulas from the representation theory of the unitary group.

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.

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 $\mathbf{P}_{1,\,2,\,2,\,3,3,\,{{3}^{\centerdot }}}^{5}$.

In this paper we study the fixed point algebra of the automorphism of the rotation algebra , θ = p/q with p, q coprime positive integers, given by u → v-1, v → u. We give a general characterization of the fixed point algebra, determine its K-theory and consider the related crossed-product algebra ⋊Ƭ Z4.

In this paper we study the Lie algebra of derivations of the exterior algebra of a vector space V over a field K of characteristic ≠ 2, and the group A of automorphisms of .

The present investigation is based on two papers: “On the determination of numbers by their sums of a fixed order,” by J. L. Self ridge and E. G. Straus (4), and “On the determination of sets by the sets of sums of a certain order,” by B. Gordon, A. S. Fraenkel, and E. G. Straus (2).

First of all, we explain the terms implicit in the above titles. Throughout these considerations we use the term “set” to mean “a totality having possible multiplicities,” so that two sets will be counted as equal if, and only if, they have the same elements with identical multiplicities. In the most general sense the term “numbers” of (4) can be replaced by “elements of any given torsioniree Abelian group.”

Consider an astronomer and his observation field, i. e., the set of observable (light or radio) signal-emitting loci of the universe. Let the observation field be ordered by attaching a date to each observable locus indicating the time in the history of the universe that the signal was emitted from its source. Whereas both the astronomer and his observation field age with time, the observations of the astronomer may trace a sequence of loci whose time labels proceed forward or backward in time (cf. Appendix).

Consider now a finite set S of events in Ln,n-dimensional space-time (Riemannian n-space having the fundamental form

In an earlier work, we defined and described a new group algebra , which is a von Neumann algebra containing the group G (3). In this paper we continue this study be relating the lattice of normal subgroups of the group G to the lattice of central projections of the group algebra . More precisely, we shall exhibit a one-to-one mapping ϕ of the lattice of closed normal subgroups of G into the lattice of central projections of , having the property that if N1 ⊂ N2, then ϕ(N2) ≤ ϕ(N1).

The present paper is a sequel to that of J. H. Chung (2) and contains a proof of a conjecture made by him, namely, that the number of ordinary (modular) irreducible representations contained in a given p-block of Sn is independent of the p-core. A summary of the results contained herein appeared in the Proceedings of the National Academy of Sciences (9).

While the contents of the author's doctoral thesis (4) have, owing to their lengthy nature, been published only in small part (5, §2; 6; 7), the absence from the literature of graph theory of any characterization of infinite directed graphs with Euler lines seems to constitute a definite gap that prompts the publication in the present paper of some further material from (4). The main results characterizing such directed graphs will be obtained in §§2 and 3. In §4, we shall indicate an alternative (and perhaps better) formulation of one of these results, some extensions obtained in (4), and some comparisons between parallel results for undirected and directed graphs. A familiarity with the definitions and results of (7) will be assumed in §4, but not before.

1. Introduction. In a well-known paper (7) Jacobson has shown how his structure theory for arbitrary rings can be applied to give more precise information about the so-called “algebraic” algebras. This specialization of his general theory is, however, perhaps not completely satisfying in that it deals only with algebras, i.e. rings admitting a field of operators, whereas neither the general structure theory nor the definition of the property of being “algebraic” seems to depend in any essential way on the precise nature of the operators.

About half the following pages are concerned with the “cubic surface group” G, of order 51840, and the geometry cognate to a certain representation thereof. The literature of this group, with its subgroup of order 25920, is already voluminous; the addition of these few pages to it will not, it is hoped, be regarded askance as a perverse and misdirected indulgence in archaism.

We give explicit combinatorial expresssions for the characteristic cycles associated to certain canonical sheaves on Schubert varieties X in the classical Hermitian symmetric spaces: namely the intersection homology sheaves IHX and the constant sheaves ℂX. The three main cases of interest are the Hermitian symmetric spaces for groups of type An (the standard Grassmannian), Cn (the Lagrangian Grassmannian) and Dn. In particular we find that CC(IHX) is irreducible for all Schubert varieties X if and only if the associated Dynkin diagramis simply laced. The result for Schubert varieties in the standard Grassmannian had been established earlier by Bressler, Finkelberg and Lunts, while the computations in the Cn and Dn cases are new.

Our approach is to compute CC(ℂX) by a direct geometric method, then to use the combinatorics of the Kazhdan-Lusztig polynomials (simplified for Hermitian symmetric spaces) to compute CC(IHX). The geometric method is based on the fundamental formula where the Xr ↓ X constitute a family of tubes around the variety X. This formula leads at once to an expression for the coefficients of CC(ℂX) as the degrees of certain singular maps between spheres.

Le radical d'un anneau, dans le sens général de N. Jacobson (1; 2), peut être caractérisé de plusieurs manières. On peut, par exemple, le définir comme l'intersection de tous les idéaux primitifs de l'anneau envisagé. Dans ce travail, nous considérons une classe particulière d'idéaux primitifs, la classe des idéaux corpoïdaux: un idéal est dit corpoïdal si et seulement si l'anneauquotient correspondant est un corps. L'objet principal de ce travail est de donner quelques caractérisations de l'intersection C de tous les idéaux corpoïdaux d'un anneau A, intersection appelée le radical corpoïdal de l'anneau A. Si C est distinct de A, l'anneau-quotient A/C est isomorphe à une somme sous-directe de corps Dans le cas d'un anneau commutatif, tout idéal primitif est corpoïdal, et par conséquent son radical corpoïdal coïncide avec son radical.

Let R be an associative ring with identity, and let denote the category of unital left R-modules. The Walkers [6] raised the question of characterizing the maximal torsion radicals of , and showed that if R is commutative and Noetherian, then there is a one-to-one correspondence between maximal torsion radicals and minimal prime ideals of R [6, Theorem 1.29]. Popescu announced [5, Theorem 2.5] that the result remains valid for commutative rings with Gabriel dimension (in the terminology of [2]). Theorem 4.6 below shows that the result holds for rings (not necessarily commutative) with Krull dimension on either the left or right, extending the previous theorem for right Noetherian rings which appeared in [1].

A Hopf surface is the quotient of the complex surface ${{\mathbb{C}}^{2}}\,\backslash \,\left\{ 0 \right\}$ by an infinite cyclic group of dilations of ${{\mathbb{C}}^{2}}$. In this paper, we study the moduli spaces ${{\mathcal{M}}^{n}}$ of stable $\text{SL}\left( 2,\,\mathbb{C} \right)$-bundles on a Hopf surface $\mathcal{H}$, from the point of view of symplectic geometry. An important point is that the surface $\mathcal{H}$ is an elliptic fibration, which implies that a vector bundle on $\mathcal{H}$ can be considered as a family of vector bundles over an elliptic curve. We define a map $G:\,{{\mathcal{M}}^{n}}\,\to \,{{\mathbb{P}}^{2n+1}}$ that associates to every bundle on $\mathcal{H}$ a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map $G$ is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on ${{\mathcal{M}}^{n}}$. We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

We characterize the class of RFD $C^{\ast }$-algebras as those containing a dense subset of elements that attain their norm under a finite-dimensional representation. We show further that this subset is the whole space precisely when every irreducible representation of the $C^{\ast }$-algebra is finite-dimensional, which is equivalent to the $C^{\ast }$-algebra having no simple infinite-dimensional AF subquotient. We apply techniques from this proof to show the existence of elements in more general classes of $C^{\ast }$-algebras whose norms in finite-dimensional representations fit certain prescribed properties.

Since Tutte initiated the systematic enumeration of planar maps [11], most of the literature on the subject has dealt with rooted maps (i.e., with maps whose automophism group has been trivialized by distinguishing a doubly-oriented edge). In particular, Tutte proved [11] that the number B′(n) of rooted planar 2-connected (i.e., non-separable) maps with n ≧ 1 edges is expressed by the formula

Recently one of the authors developed a general technique for enumerating unrooted planar maps considered up to orientationpreserving isomorphisms (see [6] and [8]). This technique, which is based on combinatorial map theory, Burnside’s lemma [3, p. 181] and the concept of a quotient map (see Section 1.4), was used to find, with little algebraic manipulation, simple counting formulae for the numbers of non-isomorphic planar maps of several types [7].

This paper considers the rational system ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})\,\,:=\,\left\{ \frac{P(x)}{\Pi _{k=1}^{n}(x-{{a}_{k}})},\,P\,\in \,{{P}_{n}} \right\}$ with nonreal elements in $\left\{ {{a}_{k}} \right\}_{k=1}^{n}\,\subset \,\mathbb{C}\,\backslash \,[-1,\,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$. The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$ are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results

All rings considered in this note have an identity element, and all R-modules are unitary.

Bass (2) defined a left perfect ring as a ring R satisfying the minimum condition on principal right ideals. A commutative ring R is perfect if and only if R is a direct sum of finitely many local rings whose radicals are T-nilpotent. Therefore, the commutative perfect rings with finite global projective dimension are just the direct sums of finitely many commutative fields, and hence they trivially satisfy the minimum condition for all ideals. However, in the non-commutative case, even hereditary perfect rings are not necessarily right or left artinian (cf. Example 3.4).

Each left perfect ring R has only finitely many idempotent (two-sided) ideals (Corollary 2.3), where the ideal X of R is called idempotent, if X = X2. Hence, it makes sense to consider minimal idempotent idealsof the left perfect ring R, i.e., ideals of R which are minimal in the set of all idemponent ideals of R.