116197572
The cohomology and fundamental group of a hyperplane complement
An electrical game having a board including a representation of an athletic field adjacent to a grid which represents the results of a plurality of athletic plays and which includes a first coordinate representing said plurality of athletic plays and a second coordinate having a plurality of indicator number. A light bar having a plurality of bulbs mounted thereon may be movably oriented or fixed with respect to said grid and may be selectively coupled in circuit relationship with offensive and defensive play result selection switches, the activation of which determines which of the bulbs is illuminated. In the fixed bar embodiment, by first selecting one of said plurality of athletic plays in said first coordinate and by activating one offensive and one defensive play result selection switch, one of the indicator numbers is selected so that the result of the athletic play can be determined by reading the grid at the coodinates as determined by the play selection and the selection of one of said indicator numbers. In the movable bar embodiment the light bar is mounted beneath the grid and may have an indicator which is viewable from the side of the game and can be aligned with the selected play. The bar is mounted on a pair of paralled guide rods. One rod is rotatable and has knobs fixed at its opposite ends. A cord is connected by its opposite ends on opposite sides of the bar and is looped around the knobs so that rotation of either knob will cause the bar to move along the rods.
52206511
On the algebra associated with a geometric lattice
Let L be a geometric lattice. Following P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. math.56 (1980), 167–189, we associate with L a graded commutative algebra A(L). In this paper we introduce a new invariant ψ of the algebra A(L) which suffices to distinguish algebras for which all other known invariants coincide. This result is applied to the study of arrangements of complex hyperplanes, with L being the intersection lattice. In this case A(L) is isomorphic to the cohomology algebra of the associated hyperplane complement. The goal is to find examples of arrangements with non-isomorphic lattices but homotopy equivalent complements. The invariant introduced here effectively narrows the list of candidates. Nevertheless, we exhibit combinatorially inequivalent arrangements for which all known invariants, including ψ, coincide.
92994378
A note on discriminantal arrangements
Let v O be a fixed affine arrangement of n hyperplanes in general position in Kk . Let U(n, k) denote the set of general position arrangements whose elements are parallel translates of the hyperplanes of v o . Then U(n, k) is the complement of a central arrangement R (n, k). These are the well-known discriminantal arrangements introduced by Y. I. Manin and V. V. Schechtman. In this note we give an explicit description of R (n, k) in terms of the original arrangement v o . In terms of the underlying matroids, _q (n, k) realizes an adjoint of the dual of the matroid associated with v o . Using this description we show that, contrary to the conventional wisdom, neither the intersection lattice of S7 (n, k) nor the topology of U(n, k) is independent of the original arrangement v o .
26463172
A geometric duality for order complexes and hyperplane complements
The top cohomology of the complement of an arrangement of complex hyperplanes is canonically isomorphic to the order homology of the associated intersection lattice. We put this result into a geometric framework by constructing a realization of the order complex of the intersection lattice inside the link of the arrangement. The canonical isomorphism is shown to coincide with the restriction of the classical Alexander duality mapping. Thus the isomorphism is a consequence of the linking of ( l − 2)-spheres and l -tori in the (2 l − 1)-sphere.
121498103
Homotopy types of line arrangements
SummaryWe prove that the complement of a real affine line arrangement inC2 is homotopy equivalent to the canonical 2-complex associated with Randell's presentation of the fundamental group. This provides a much smaller model for the homotopy type of the complement of a real affine 2- or central 3-arrangement than the Salvetti complex and its cousins. As an application we prove that these exist (infinitely many) pairs of central arrangements inC3 with different underlying matroids whose complements are homotopy equivalent. We also show that two real 3-arrangements whose oriented matroids are connected by a sequence of flips are homotopy equivalent.
14391043
Arrangements and cohomology
To a matroidM is associated a graded commutative algebraA=A(M), the Orlik-Solomon algebra ofM. Motivated by its role in the construction of generalized hypergeometric functions, we study the cohomologyH* (A, dω) ofA(M) with coboundary mapdω given by multiplication by a fixed element ω ofA1. Using a description of decomposable relations inA, we construct new examples of “resonant” values of ω, and give a precise calculation ofH1 (A, dω) as a function of ω. We describe the setR1(A={ω|H1(A(M,dω)≠}, and use it as a tool in the classification of Orlik-Solomon algebras, with applications to the topology of complex hyperplane complements. We show thatR1(A) is a complete invariant of the quadratic closure ofA, and show under various hypotheses that one can reconstruct the matroidM, or at least its Tutte polynomial, from the varietyR1(A). We demonstrate with several examples thatR1 is easily calculable, may contain nonlocal components, and that combinatorial properties ofR1(A) are often sufficient to distinguish nonisomorphic rank three Orlik-Solomon algebras.
120185810
Asymptotically optimal estimators of general regression functionals
Let X, Y be random vectors with values in d and 1, respectively, and denote by F(Â· x) the conditional distribution function of Y given X = x. It is well known that the kernel estimator of the regression functional [theta](x) := T(F(Â· x)), based on n independent replicates of (X, Y), has optimal asymptotic accuracy in case of T being the mean value and quantile functional. In this paper we discuss conditions under which the kernel estimator has optimal asymptotic accuracy, locally and globally, for a general class of functionals T, containing mean and quantile as particular examples. A weak convergence result for the maximum error over a compact interval completes the paper.
121061906
Weak convergence of the remainder term in the Bahadur representation of extreme quantiles
Weak convergence of the remainder term in the Bahadur representation of the sample q-quantile is established, where q = qn tends to zero or one as the sample size n increases.
120985083
Rates of uniform convergence of extreme order statistics
SummaryBounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution.The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.
123363269
A simple approach to the generation of uniformly distributed random variables with prescribed correlations
In a simulation study we encountered the problem of generating uniformly on (0,1) distributed random variables with a prescribed correlation matrix. This article reports our solution to this problem.We show that the following simple device leads to uniformly on (0,1)distributed random variables, which have a correlation matrix that isfor practical purposes sufficiently close to the prescribed one: Let(X1.., Xm) be a normal vector with theprescribed correlation matrix S = (pij:). Such a vector caneasily be generated. Suppose for simplicity that each Xi is standard normal and denote by Φ the standard normal distribution function. It turns out that the vector(V1:…,Vm:):= (φ(X1),…,φ(Xm)) with uniformly on (0,1)distributed components Vi; then has a correlation matrix S′, whose entries are for practical purposes sufficiently close to that of the target matrix S. This is a consequence of the grade correlation (or Spearman's rho) of normal vectors. Suppose, in addition, that the matrix is positive semidefinite. T...
121496414
On testing the extreme value index via the pot-method
n alternatives, based on the point processes N of the exceedances among Y n i . over a sequence of thresholds t . It turns out that the random number of n . exceedances t n over t is the central sequence for the log-likelihood n . . . ratio d L N rd L N , yielding its local asymptotic normality LAN . b n b n n 0 . This result implies in particular that t n carries asymptotically all the information about the underlying parameter b, which is contained in N. n . We establish sharp bounds for the rate at which t n becomes asymptoti- cally sufficient, which show, however, that this is quite a poor rate. These results remain true if we add an unknown scale parameter. 1 n . A model assumption on the upper tail of the underlying d.f. such as M is, for example, indispensable if one is interested in extreme quantities of the underlying d.f., that is, such quantities of F which are usually outside the range of the given data Y , . . . , Y. A typical example is inference about 1 n y1 . . 4 extreme quantiles F p ( inf t g R: F t G p with p being close to 1. This problem is typically tackled by hydrologists for the prediction of large w