Essential closures and supports of multivariate copulas

We generalize the notion of essential closures and use them to formulate a geometric necessary condition for a set to be the support of a multivariate copula. Furthermore, in some special cases, we derive an explicit formula of the support in terms of essential closures and obtain a stronger necessary condition.     

RANS closures for non-neutral microscale CFD simulations sustained with inflow conditions acquired from mesoscale simulations

This study focuses on bridging the gap between the turbulence modelling methodologies of meterological and engineering codes by proposing a novel methodology to define the closure coefficients of Reynolds-Averaged Navier-Stokes turbulence models consistently with the physics of the atmospheric boundary layer. In this framework, different turbulence closures have been developed and tested on different full-scale test cases corresponding to different atmospheric stability conditions by performing microscale simulations with the inflow conditions provided by a numerical weather prediction (NWP) code. Developed turbulence models have been implemented into the open source computational fluid dynamics (CFD) toolbox, OpenFOAM and the inflow conditions have been acquired with another open source code, the Weather Research and Forecasting (WRF) model.     

Rational Hypergeometric Functions

Multivariate hypergeometric functions associated with toric varieties were introduced by Gel'fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function.     

Computing the Conway polynomial of several closures of oriented 3-braids

This paper deals with polynomial invariants of a class of oriented 3-string tangles and the knots (or links) obtained by applying six different closures. In Cabrera-Ibarra (2004) [1], expressions were given to compute the Conway polynomials of four different closures of the composition of two such 3-string tangles. By using the expressions and results from that reference, and using an algorithm developed on the basis of Giller?s calculations for 3-string tangles, we provide new results concerning six closures of 3-braids. Surprisingly, for 3-braids two of the closures turn out to be affine functions of the four previously defined. Among the contributions in this paper one finds computational tools to obtain the Conway polynomial of closures of 3-braids in terms of continuous fractions and their expansions. An interesting feature is that our calculations yield explicit, nonrecursive formulas in the case of 3-braids, thereby considerably lowering the time required to compute them. As a byproduct, explicit expressions are also given to obtain both numerators and denominators of continuous fractions in a nonrecursive way.     

Simple permutations and algebraic generating functions

A simple permutation is one that never maps a nontrivial contiguous set of indices contiguously. Given a set of permutations that is closed under taking subpermutations and contains only finitely many simple permutations, we provide a framework for enumerating subsets that are restricted by properties belonging to a finite “query-complete set.” Such properties include being even, being an alternating permutation, and avoiding a given generalised (blocked or barred) pattern. We show that the generating functions for these subsets are always algebraic, thereby generalising recent results of Albert and Atkinson. We also apply these techniques to the enumeration of involutions and cyclic closures.     

Reorganization of a Sales and Distribution System

The work done by Esso's authorized distributors in a certain area of the U.K. was reviewed in depth, leading to the closure of a number of plants and to a reduction in the number of separate distributor companies operating in the area.The study fell into three phases. The first was a detailed investigation and comparison of the costs and profits of nine distributor companies in 1966. The second phase considered the economics of the physical distribution of our products in the future, and led to the plant closures. The third phase was concerned with changing distributor territories in order to make the best use of the smaller number of plants and larger delivery areas.A special purpose integer-linear-programming code was used to analyse the plant location problem considered in Phase 2. In preparing transport costs for this program, special attention was given to the effect of the significant seasonal variations in demand and to the fact that truck loads are usually divided between several customers.     

Rational smoothness and fixed points of torus actions

We obtain a criterion for rational smoothness of an algebraic variety with a torus action, with applications to orbit closures in flag varieties, and to closures of double classes in regular group completions. Dedicated to the memory of Claude Chevalley     

Profinite completions of some groups acting on trees

We investigate the profinite completions of a certain family of groups acting on trees. It turns out that for some of the groups considered, the completions coincide with the closures of the groups in the full group of tree automorphisms. However, we introduce an infinite series of groups for which that is not so, and describe the kernels of natural homomorphisms of the profinite completions onto the aforementioned closures of respective groups.     

A characterization of the zero-free region of the Riemann zeta function and its applications

We characterize the zero-free regions of a class of functions (including the Riemann zeta function) in half-planes in terms of closures of ranges of the corresponding multiplication operators on Hardy spaces. We give an explicit characterization of these closures. As applications, we obtain a weaker version of the Nyman–Beurling–Báez-Duarte criterion, and provide some investigations on a problem relating to the Riemann hypothesis proposed by Báez-Duarte et al. [Adv. Math. 149 (2000) 130-144].     

A Proof of the Refined PRV Conjecture via the Cyclic Convolution Variety

In this brief note we illustrate the utility of the geometric Satake correspondence by employing the cyclic convolution variety to give a simple proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, along with Kumar’s refinement. The proof involves recognizing certain MV-cycles as orbit closures of a group action, which we make explicit by unique characterization. In an Appendix, joint with P. Belkale, we discuss how this work fits in a more general framework.

     

Moment maps and Riemannian symmetric pairs

We study Hamiltonian actions of a compact Lie group on a symplectic manifold in the presence of an involution on the group and an antisymplectic involution on the manifold. The fixed-point set of the involution on the manifold is a Lagrangian submanifold. We investigate its image under the moment map and conclude that the intersection with the Weyl chamber is an easily described subpolytope of the Kirwan polytope. Of special interest is the integral K?hler case, where much stronger results hold. In particular, we obtain convexity theorems for closures of orbits of the noncompact dual group (in the sense of the theory of symmetric pairs). In the abelian case these results were obtained earlier by Duistermaat. We derive explicit inequalities for polytopes associated with real flag varieties. Received: 8 February 1999 / in revised form: 25 October 1999 / Published online: 8 May 2000     

A note on the serial dictatorship with project closures

In this paper, we consider the problem of assigning agents having preferences to projects with capacities and lower quotas. For this problem, Monte and Tumennasan proposed a strategy-proof and Pareto efficient mechanism, called the serial dictatorship with project closures. In this paper, we show that the serial dictatorship with project closures can be extended to a more general setting.     

On Frobenius splitting of orbit closures of spherical subgroups in flag varieties

Let H denote a spherical subgroup within a semisimple algebraic group G. In this paper we study the closures of the finitely many H-orbits in the flag variety of G. Using the language of Frobenius splitting we provide a criterion for these closures to have nice geometric and cohomological properties. We then show how the criterion applies to the spherical subgroups of minimal rank studied by N. Ressayre. Finally, we also provide applications of the criterion to orbit closures which are not multiplicity-free in the sense defined by M. Brion.     

A general Galois theory for operations and relations in arbitrary categories

In this paper, we generalize the notions of polymorphisms and invariant relations to arbitrary categories. This leads us to a Galois connection that coincides with the classical case from universal algebra if the underlying category is the category of sets, but remains applicable no matter how the category is changed. In analogy to the situation in the classical case, we characterize the Galois closed classes by local closures of clones of operations and local closures of what we will introduce as clones of (generalized) relations. Since the approach is built on purely category-theoretic properties, we will also discuss the dualization of our notions.     

Weakening Additivity in Adjoining Closures

In this paper, we weaken the conditions for the existence of adjoint closure operators, going beyond the standard requirement of additivity/co-additivity. We consider the notion of join-uniform (lower) closure operators, introduced in computer science, in order to model perfect lossless compression in transformations acting on complete lattices. Starting from Janowitz’s characterization of residuated closure operators, we show that join-uniformity perfectly weakens additivity in the construction of adjoint closures, and this is indeed the weakest property for this to hold. We conclude by characterizing the set of all join-uniform lower closure operators as fix-points of a function defined on the set of all lower closures of a complete lattice.     

Program Explanation and Higher-Order Properties

Our aim in this paper is to evaluate Frank Jackson and Philip Pettit’s ‘program explanation’ framework as an account of the autonomy of the special sciences. We argue that this framework can only explain the autonomy of a limited range of special science explanations. The reason for this limitation is that the framework overlooks a distinction between two kinds of properties, which we refer to as ‘higher-level’ and ‘higher-order’ properties. The program explanation framework can account for the autonomy of special science explanations that appeal to higher-level properties but it does not account for the autonomy of most of those explanations that appeal to higher-order properties.     

Special Points on the Product of Two Modular Curves

We prove, assuming the generalized Riemann hypothesis for imaginary quadratic fields, the following special case of a conjecture of Oort, concerning Zarsiski closures of sets of CM points in Shimura varieties. Let X be an irreducible algebraic curve in C², containing infinitely many points of which both coordinates are j-invariants of CM elliptic curves. Suppose that both projections from X to C are not constant. Then there is an integer m 1such that X is the image, under the usual map, of the modular curve Y₂0(m). The proof uses some number theory and some topological arguments.     

Verallgemeinerte topogene Ordnungen

Topogenous orders in the sense of Császár are a common generalization of proximity and topology. ech closures are a generalization of the topological closure operators in the sense of Kuratowski. We show that the topogenous orders as well as the ech closures are special cases of the so called compressed operators. Moreover, the now defined categoryCOM (in germanBAL) of compress spaces and compress faithful maps is a properly fibred topological category in the sense of Herrlich which is weakly cartesian closed, that means the product map of two quotient maps inCOM is a quotient map inCOM. Therefore by results of L. D. Nel it is possible to construct a cartesian closed properly fibred topological category in whichCOM can be nicely embedded. Further it turns out that the compressed operators be in a natural connexion with the uniform convergence structures in the sense of Cook and Fischer and in addition with the limit structures in the sense of Fischer. For principal ideal uniform convergence structures we prove that they are precompact and complete iff the properly constructed compressed operator is compact.     

On some properties of symplectic Grothendieck polynomials

Grothendieck polynomials, introduced by Lascoux and Schützenberger, are certain K-theory representatives for Schubert varieties. Symplectic Grothendieck polynomials, described more recently by Wyser and Yong, represent the K-theory classes of orbit closures for the complex symplectic group acting on the complete flag variety. We prove a transition formula for symplectic Grothendieck polynomials and study their stable limits. We show that each of the K-theoretic Schur P-functions of Ikeda and Naruse arises from a limiting procedure applied to symplectic Grothendieck polynomials representing certain “Grassmannian” orbit closures.     

Asymptotic gauges: Generalization of Colombeau type algebras

We use the general notion of set of indices to construct algebras of nonlinear generalized functions of Colombeau type. They are formally defined in the same way as the special Colombeau algebra, but based on more general “growth condition” formalized by the notion of asymptotic gauge. This generalization includes the special, full and nonstandard analysis based Colombeau type algebras in a unique framework. We compare Colombeau algebras generated by asymptotic gauges with other analogous construction, and we study systematically their properties, with particular attention to the existence and definition of embeddings of distributions. We finally prove that, in our framework, for every linear homogeneous ODE with generalized coefficients there exists a minimal Colombeau algebra generated by asymptotic gauges in which the ODE can be uniquely solved. This marks a main difference with the Colombeau special algebra, where only linear homogeneous ODEs satisfying some restrictions on the coefficients can be solved.