Minimizing geodesic nets and critical points of distance

We establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the n-sphere. This result is used to give conditions under which a minimizing geodesic flower degenerates into a simple closed geodesic.     

Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

In this paper we study quasi-Monte Carlo integration of smooth functions using digital nets. We fold digital nets over ${\mathbb{Z}}_b$ by means of the $b$-adic tent transformation, which has recently been introduced by the authors, and employ such folded digital nets as quadrature points. We first analyze the worst-case error of quasi-Monte Carlo rules using folded digital nets in reproducing kernel Hilbert spaces. Here we need to permit digital nets with “infinite digit expansions”, which are beyond the scope of the classical definition of digital nets. We overcome this issue by considering the infinite product of cyclic groups and the characters on it. We then give an explicit means of constructing good folded digital nets as follows: we use higher order polynomial lattice point sets for digital nets and show that the component-by-component construction can find good folded higher order polynomial lattice rules that achieve the optimal convergence rate of the worst-case error in certain Sobolev spaces of smoothness of arbitrarily high order.     

Visible acyclic differential nets, Part I: Semantics

We give a geometric condition that characterizes the differential nets having a finitary interpretation in finiteness spaces: visible acyclicity. This is based on visible paths, an extension to differential nets of a class of paths we introduced in the framework of linear logic nets. The characterization is then carried out as follows: the differential nets having no visible cycles are exactly those whose interpretation is a finitary relation. Visible acyclicity discloses a new kind of correctness for the promotion rule of linear logic, which goes beyond sequent calculus correctness.     

Bounds for the Quality Parameter of Digital Shift Nets over Z2

Until now, the concept of digital (t,m,s)-nets is the most powerful concept for the construction of low-discrepancy point sets in the s-dimensional unit cube. In this paper we consider a special class of digital nets over $\text{Z}$₂, the so-called shift nets introduced by W. Ch. Schmid, and give bounds for the quality parameter t of such nets.     

Matrix-product constructions of digital nets

We present a new construction of digital nets, and more generally of (d,k,m,s)-systems, over finite fields which is an analog of the matrix-product construction of codes. Examples show that this construction can yield digital nets with better parameters compared to competing constructions.     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Analysis of subdivision schemes for nets of functions by proximity and controllability

In this paper we develop tools for the analysis of net subdivision schemes, schemes which recursively refine nets of bivariate continuous functions defined on grids of lines, and generate denser and denser nets. Sufficient conditions for the convergence of such a sequence of refined nets, and for the smoothness of the limit function, are derived in terms of proximity to a bivariate linear subdivision scheme refining points, under conditions controlling some aspects of the univariate functions of the generated nets. Approximation orders of net subdivision schemes, which are in proximity with positive schemes refining points are also derived. The paper concludes with the construction of a family of blending spline-type net subdivision schemes, and with their analysis by the tools presented in the paper. This family is a new example of net subdivision schemes generating C¹ limits with approximation order 2.     

Neural nets in a group decision process

In recent years there has been some interest in applying Artificial Adaptive Agents (AAA) to the study of complex adaptive systems, especially economic systems. Neural networks are frequently employed as AAA. Artificial neural nets mimic certain aspects of the physical structure and information processing of the human brain and their most attractive characteristic is their ability to learn a pattern from a given set of examples. In this study, we investigated the ability of neural nets to model human behavior in a group decision process. The context was a market entry game with a linear payoff function and binary decisions. The players had to decide, for each trial, whether or not to enter a market whose capacity is public knowledge. Human behavior in this situation has been modeled and empirically validated by the Nash equilibrium for noncooperative n-person games. A simulation of the game was performed with neural nets instead of human subjects. The nets were trained using the results of the games in which they participated. The simulation with groups of neural nets exhibits phenomena very similar to those observed in groups of human players. Received February 2000     

Twelve general position points always form three intersecting tetrahedra

A g cage of valency k is a regular graph of girth g and valency k with the minimum number of vertices consistent with this condition [12]. An embeddable g net of valency k corresponds to a graph of girth g which can be drawn on a surface in such a way that every face is bounded by a circuit of length g. Similarities between g nets and known g cages are investigated and a possible 5 cage of valency 6 is produced. An existence theorem for regular g nets is also given.     

Completeness in quasi-uniform spaces

We introduce a theory of completeness (the π-completeness) for quasi-uniform spaces which extends the theories of bicompleteness and half-completeness and prove that every quasi-uniform space has a π-completion. This theory is based on a new notion of a Cauchy pair of nets which makes use of couples of nets. We call them cuts of nets and our inspiration is due to the construction of the τ-cut on a quasi-uniform space (cf. [1], [20]). This new version of completeness coincides with bicompletion, half-completion and D-completion in extended subclasses of the class of quasi-uniform spaces.     

Combinatorial methods in the construction of point sets with uniformity properties

It is well known that (t, m, s)-nets are useful in numerical analysis. Many of the constructions of such nets arise from number theoretic or algebraic constructions. Less well known is the fact that combinatorial constructions also yield nets with very good and in many cases, optimal parameters. In this paper, we provide a survey of such combinatorial constructions of (t, m, s)-nets.     

Ideal convergence and divergence of nets in (ℓ)-groups

In this paper we introduce the I- and I*-convergence and divergence of nets in (?)-groups. We prove some theorems relating different types of convergence/divergence for nets in (?)-group setting, in relation with ideals. We consider both order and (D)-convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that I*-convergence/divergence implies I-convergence/divergence for every ideal, admissible for the set of indexes with respect to which the net involved is directed, and we investigate a class of ideals for which the converse implication holds. Finally we pose some open problems.     

Mathematical programming approach to the Petri nets reachability problem

This paper focuses on the resolution of the reachability problem in Petri nets, using the mathematical programming paradigm. The proposed approach is based on an implicit traversal of the Petri net reachability graph. This is done by constructing a unique sequence of Steps that represents exactly the total behaviour of the net. We propose several formulations based on integer and/or binary linear programming, and the corresponding sets of adjustments to the particular class of problem considered. Our models are validated on a set of benchmarks and compared with standard approaches from IA and Petri nets community.     

On the Fell topology

Let 2 ^X denote the closed subsets of a Hausdorff topological space <X, {gt}>. The Fell topology τ_F on 2 ^X has as a subbase all sets of the form {A ∈ 2 ^X :A ∩V ≠ 0}, whereV is an open subset ofX, plus all sets of the form {A ∈ 2 ^X :A ?W}, whereW has compact complement. The purpose of this article is two-fold. First, we characterize first and second countability for τ_F in terms of topological properties for τ. Second, we show that convergence of nets of closed sets with respect to the Fell topology parallels Attouch-Wets convergence for nets of closed subsets in a metric space. This approach to set convergence is highly tractable and is well-suited for applications. In particular, we characterize Fell convergence of nets of lower semicontinuous functions as identified with their epigraphs in terms of the convergence of sublevel sets.     

Quantitative approximation of certain stochastic integrals

We approximate certain stochastic integrals, typically appearing in Stochastic Finance, by stochastic integrals over integrands, which are path-wise constant within deterministic, but not necessarily equidistant, time intervals. We ask for rates of convergence if the approximation error is considered in L 2 . In particular, we show that by using non-equidistant time nets, in contrast to equidistant time nets, approximation rates can be improved considerably.     

Generalized Dini theorems for nets of functions on arbitrary sets

We characterize the uniform convergence of pointwise monotonic nets of bounded real functions defined on arbitrary sets, without any particular structure. The resulting condition trivially holds for the classical Dini theorem. Our vector-valued Dini-type theorem characterizes the uniform convergence of pointwise monotonic nets of functions with relatively compact range in Hausdorff topological ordered vector spaces. As a consequence, for such nets of continuous functions on a compact space, we get the equivalence between the pointwise and the uniform convergence. When the codomain is locally convex, we also get the equivalence between the uniform convergence and the weak-pointwise convergence; this also merges the Dini-Weston theorem on the convergence of monotonic nets from Hausdorff locally convex ordered spaces. Most of our results are free of any structural requirements on the common domain and put compactness in the right place: the range of the functions.     

Extensions and new observations of Tychonoff,Stone-Weierstrass theorems,compactifications and the realcompactification

An extension of the Tychonoff theorem is obtained in characterizing a compact space by the nets and the images induced by any family of continuous functions on it. The idea of this extension is applied to get a new process and new observations of compactifications and the realcompactification. Finally, a sufficient and necessary condition of a vector sublattice or a subalgebra of C¹(X) to be dense in (C¹(X),∥·∥) is provided in terms of the nets in X induced by C¹(X), where C¹(X) is the space of all bounded real continuous functions on a topological space X with pointwise ordering, and ∥·∥ is the supremum norm.     

Local conformal nets arising from framed vertex operator algebras

We apply an idea of framed vertex operator algebras to a construction of local conformal nets of (injective type III₁) factors on the circle corresponding to various lattice vertex operator algebras and their twisted orbifolds. In particular, we give a local conformal net corresponding to the moonshine vertex operator algebras of Frenkel-Lepowsky-Meurman. Its central charge is 24, it has a trivial representation theory in the sense that the vacuum sector is the only irreducible DHR sector, its vacuum character is the modular invariant J-function and its automorphism group (the gauge group) is the Monster group. We use our previous tools such as α-induction and complete rationality to study extensions of local conformal nets.     

On I-Cauchy nets and completeness

In this paper we extend the idea of usual Cauchy condition of nets to I-Cauchy condition by using the concept of ideals. This Cauchy condition arises naturally from the notion of I-convergence of nets introduced by Lahiri and Das (2008). As the underlying structure for the whole study we take a uniform space so that our notion and results extend the idea of statistical Cauchy sequences very recently introduced in uniform spaces by Di Maio and Ko?inac (2008). In particular we try to give partial answers to an open problem posed by Di Maio and Ko?inac and examine the relationship between this new Cauchy condition and usual completeness of a uniform space.