New Class Of Examples Research Articles

On the base locus of the generalized theta divisor

In response to a question of Beauville, we give a new class of examples of base points for the linear system ¦Θ¦on the moduli space SU X ( r) of semistable rank r vector bundles of trivial determinant on a curve X and we prove that for sufficiently large r the base locus is positive dimensional.

Symmetric functions and exact Lyapunov exponents

Newton's method applied to the elementary symmetric functions of polynomials generates a class of dynamical systems. These systems have invariant lines on which the Lyapunov exponents can be found analytically, thus predicting the exact parameter values for which these structures are chaotic attractors (in the sense of Milnor) and precisely when bifurcations, such as blowout, destroy their stability. Often, blowout bifurcations lead to a behavior called on-off intermittency. We also present evidence that the bursting mechanism of on-off intermittency frequently occurs in neighborhoods focal points, a common feature of maps that have singularities. Finally, because of an embedding property, this new class of examples can be extended to construct systems having this bifurcation in higher dimensions.

HIGH DIMENSIONAL KNOT GROUPS AND HNN EXTENSIONS OF THE FIBONACCI GROUPS

We shall present a new class of examples of high dimensional knot groups. All of them are HNN extensions of the Fibonacci groups. We give also some characterization of these groups.

D-Modules and Darboux Transformations

A method of G. Wilson for generating commutative algebras of ordinary differential operators is extended to higher dimensions. Our construction, based on the theory of D-modules, leads to a new class of examples of commutative rings of partial differential operators with rational spectral varieties. As an application, we briefly discuss their link to the bispectral problem and to the theory of lacunas.

(4,0) and (4,4) sigma models with a tri-holomorphic Killing vector

We analyze (4,0) supersymmetric σ-models on a four dimensional target space which possess one tri-holomorphic Killing vector which is also assumed to leave invariant the torsion. The problem is reduced to two stages: first finding “special” three dimensional Einstein-Weyl spaces, and second solving a monopole-like equation on the special Einstein-Weyl space. A new class of examples is constructed using as Einstein-Weyl geometry the Berger sphere which includes the round three sphere as a particular case. When the Einstein-Weyl geometry is taken to be the round three sphere we show that the corresponding (4,0) geometries can be lifted to (4,4) geometries with two sets of non-commuting hyper-complex structures.

On the Completeness of Wave Operators under Loss of Local Compactness

We present a general criterion for the existence and completeness of the wave operators in obstacle (or potential) scattering which does not require local compactness near the boundary of the obstacle. The scope of our basic theorem is illustrated by new classes of examples, namely, exterior "Rooms and Passages"-domains, where completeness holds, and exterior "Jelly Roll"-domains, where it fails.

New perspectives on the BRST-algebraic structure of string theory

Motivated by the descent equation in string theory, we give a new interpretation for the action of the symmetry charges on the BRST cohomology in terms of what we callthe Gerstenhaber bracket. This bracket is compatible with the graded commutative product in cohomology, and hence gives rise to a new class of examples of what mathematicians call aGerstenhaber algebra. The latter structure was first discussed in the context of Hochschild cohomology theory [11]. Off-shell in the (chiral) BRST complex, all the identities of a Gerstenhaber algebra hold up to homotopy. Applying our theory to thec=1 model, we give a precise conceptual description of the BRST-Gerstenhaber algebra of this model. We are led to a direct connection between the bracket structure here and the anti-bracket formalism in BV theory [29]. We then discuss the bracket in string backgrounds with both the left and the right movers. We suggest that the homotopy Lie algebra arising from our Gerstenhaber bracket is closely related to the HLA recently constructed by Witten-Zwiebach. Finally, we show that our constructions generalize to any topological conformal field theory.

Translation divisible designs

An (s,k,λ)-translation divisible design (s,k,λ)-TDD) is a (group) divisible design without repeated blocks and with an automorphism group T (translation group) which acts regularly on all points and such that each T-orbit on the blocks is a partition of the point set. We construct a new class of examples of regular (s,k,λ)-TDD's, with non-abelian p-group as translation group. We study the translation group for regular TDD's and for some transversal TDD's. Finally we characterize regular TDD's by balanced incomplete block designs endowed of a suitable automorphism group

Minimax‐variance L‐ and R‐estimators of location

AbstractWe consider the problem of minimax‐variance, robust estimation of a location parameter, through the use of L‐ and R‐estimators. We derive an easily checked necessary condition for L‐estimation to be minimax, and a related sufficient condition for R‐estimation to be minimax. Those cases in the literature in which L‐estimation is known not to be minimax, and those in which R‐estimation is minimax, are derived as consequences of these conditions. New classes of examples are given in each case. As well, we answer a question of Scholz (1974), who showed essentially that the asymptotic variance of an R‐estimator never exceeds that of an L‐estimator, if both are efficient at the same strongly unimodal distribution. Scholz raised the question of whether or not the assumption of strong unimodality could be dropped. We answer this question in the negative, theoretically and by examples. In the examples, the minimax property fails both for L‐estimation and for R‐estimation, but the variance of the L‐estimator, as the distribution of the observation varies over the given neighbourhood, remains unbounded. That of the R‐estimator is unbounded.

Matched pairs of lie groups and Hopf algebra bicrossproducts

We construct a new class of examples of non-commutative and non-cocommutative Hopf algebras. There is essentially one for every Lie bialgebra, and several other examples including one for every simple real Lie algebra. The latter are obtained from a canonical solution of the Classical Yang-Baxter Equations. Physically, these Hopf algebras arise as the algebra of observables of quantum mechanics on homogeneous spacetimes.