The Structural Nature of the Nerve Functor for n-Groupoids

Given a left exact category B, the construction of the nerve functor _n for n-groupoids in B is related to a certain property of the category S-S i m p l _{n – 1} B of the split (n – 1)-truncated simplicial objects in B, which allows us to define the split n-truncated simplicial objects in B completely internally to S-S i m p l _{n – 1} B and thus to construct intrisincally from it the category S-S i m p l _n B.     

Orthogonal polynomials and Gaussian quadrature for refinable weight functions

We show how to compute the modified moments of a refinable weight function directly from its mask in O(N²n) rational operations, where N is the desired number of moments and n the length of the mask. Three immediate applications of such moments are:

• the expansion of a refinable weight function as a Legendre series;

• the generation of the polynomials orthogonal with respect to a refinable weight function;

• the calculation of Gaussian quadrature formulas for refinable weight functions.

In the first two cases, all operations are rational and can in principle be performed exactly.

Beste Quadraturformeln für Integrale mit einer Gewichtsfunktion

Let be a fixed matrix with elements that are 0 or 1 and letX be a fixed set ofm+1 different knots. The problem is to find necessary and sufficient conditions for (E, X) to guarantee the existence of a quadrature formula with a remainder term of type for any choice of a weight functionw(t) and satisfyingR(f)=0 forf a polynomial of degree at mostn–1. The result generalizes the corresponding result ofI. J. Schoenberg for the special case of quasi-Lagrange-matricesE. —in case of the existence ofR it is possible to calculate the best quadrature formulaR ^* in the sense ofSard by integrating splines of degree 2n–1. But ifE contains onlyn ones it is sufficient to integrate polynomials of degreen–1.     

Shape-Preserving Smoothing of 3-Convex Splines of Degree 4

For every 3-convex piecewise-polynomial function s of degree ≤4 with n equidistant knots on [0, 1] we construct a 3-convex spline s ₁ (s ₁ ∈ C ⁽³⁾) of degree ≤4 with the same knots that satisfies the inequality

Behavior of alternation points in best rational approximation

The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [–1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [–1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [–1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when n (0 < 1) poles stay away from [–1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.The research of this author was supported, in part, by NSF grant DMS 920-3659.     

Nonexistence of certain symmetric spherical codes

A spherical 1-codeW is any finite subset of the unit sphere inn dimensionsS ^n–1, for whichd(u, v)1 for everyu, v fromW, uv. A spherical 1-code is symmetric ifuW implies –uW. The best upper bounds in the size of symmetric spherical codes onS ^n–1 were obtained in [1]. Here we obtain the same bounds by a similar method and improve these bounds forn=5, 10, 14 and 22.     

Skew-symmetric association schemes with two classes and strongly regular graphs of type L 2n−1(4n−1)

A construction of a pair of strongly regular graphs ⁿ and ⁿ of type L _2n–1(4n–1) from a pair of skew-symmetric association schemes W, W of order 4n–1 is presented. Examples of graphs with the same parameters as ⁿ and ⁿ, i.e., of type L _2n–1(4n–1), were known only if 4n–1=p ³, where p is a prime. The first new graph appearing in the series has parameters (v, k, )=(225, 98, 45). A 4-vertex condition for relations of a skew-symmetric association scheme (very similar to one for the strongly regular graphs) is introduced and is proved to hold in any case. This has allowed us to check the 4-vertex condition for ⁿ and ⁿ, thus to prove that ⁿ and ⁿ are not rank three graphs if n>2.     

Room designs and one-factorizations

The existence of a Room square of order 2n is known to be equivalent to the existence of two orthogonal one-factorizations of the complete graph on 2n vertices, where orthogonal means any two one-factors involved have at most one edge in common. DefineR(n) to be the maximal number of pairwise orthogonal one-factorizations of the complete graph onn vertices.The main results of this paper are bounds on the functionR. If there is a strong starter of order 2n–1 thenR(2n) 3. If 4n–1 is a prime power, it is shown thatR(4n) 2n–1. Also, the recursive construction for Room squares, to obtain, a Room design of sidev(u – w) +w from a Room design of sidev and a Room design of sideu with a subdesign of sidew, is generalized to sets ofk pairwise orthogonal factorizations. It is further shown thatR(2n) 2n–3.     

Adaptive choice of trimming proportions

We consider Jaeckel's (1971,Ann. Math. Statist.,42, 1540–1552) proposal for choosing the trimming proportion of the trimmed mean in the more general context of choosing a trimming proportion for a trimmedL-estimator of location. We obtain higher order expansions which enable us to evaluate the effect of the estimated trimming proportion on the adaptive estimator. We find thatL-estimators with smooth weight functions are to be preferred to those with discontinuous weight functions (such as the trimmed mean) because the effect of the estimated trimming proportion on the estimator is of ordern ^–1 rather thann ^–3/4. In particular, we find that valid inferences can be based on a particular smooth trimmed mean with its asymptotic standard error and the Studentt distribution with degrees of freedom given by the Tukey and McLaughlin (1963,Sankhy Ser. A,25, 331–352) proposal.     

Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

The real spectrum of an ideal and KO-theory exact sequences

A construction in abstract real algebra is used to define invariants S _n(A) of commutative rings, with or without identity. If A=C(X) is the ring of continuous real functions on a compact space, then S _n(A) = k0^–n(X), and, for any A, S _n(A) Z[1/2]-W _n(A) Z[1/2], where the W _n(A) are the Witt groups of A. In addition, a short exact sequence of rings yields a long exact sequence of the groups S _n. The functors S _n(A) thus provide a solution of a problem proposed by Karoubi. This paper primarily deals with the exact sequences involving a ring A and an ideal I A. Work supported in part by NSF Grant DMS85-06816.     

On the graph of a function over a prime field whose small powers have bounded degree

Let f be a function from a finite field with a prime number p of elements, to . In this article we consider those functions f(X) for which there is a positive integer with the property that f(X)ⁱ, when considered as an element of , has degree at most p−2−n+i, for all i=1,…,n. We prove that every line is incident with at most t−1 points of the graph of f, or at least n+4−t points, where t is a positive integer satisfying n>(p−1)/t+t−3 if n is even and n>(p−3)/t+t−2 if n is odd. With the additional hypothesis that there are t−1 lines that are incident with at least t points of the graph of f, we prove that the graph of f is contained in these t−1 lines. We conjecture that the graph of f is contained in an algebraic curve of degree t−1 and prove the conjecture for t=2 and t=3. These results apply to functions that determine less than directions. In particular, the proof of the conjecture for t=2 and t=3 gives new proofs of the result of Lovász and Schrijver [L. Lovász, A. Schrijver, Remarks on a theorem of Rédei, Studia Sci. Math. Hungar. 16 (1981) 449–454] and the result in [A. Gács, On a generalization of Rédei’s theorem, Combinatorica 23 (2003) 585–598] respectively, which classify all functions which determine at most 2(p−1)/3 directions.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

A bound on the plurigenera of projective varieties

We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space P ^r , and classify the varieties attaining the bound, when n2, r2n+1, d>>r and i>>r. When n=2 and r=5, or n=3 and r=7, we give a complete classification, i.e. for any i1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n+1 and minimal degree in P ^r , with r>(n ² +3n)/(n–1), has not maximal i-th plurigenus, for i>>r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef. Mathematics Subject Classification (2000):Primary 14J99; Secondary 14N99     

Notes on Free Monadic Boolean Algebras

If F B(2 ⁿ – 1) denotes the Boolean algebra with 2 ⁿ – 1 free generators and P(2 ⁿ ) is the Cartesian product of 2 ⁿ Boolean algebras all equal to F B(2 ⁿ – 1), we define on P(2 ⁿ ) an existential quantifier by means of a relatively complete Boolean subalgebra of P(2 ⁿ ) and we prove that (P(2ⁿ),) is the monadic Boolean algebra with n free generators. Every element of P(2 ⁿ ) is a 2 ⁿ -tuple whose coordinates are in F B(2 ⁿ – 1); in particular, so are the n generators of P(2 ⁿ ). We indicate in this work the coordinates of the n generators of P(2 ⁿ ).     

A Lower Bound for the Height of a Rational Function at S-unit Points

Let a,b be given, multiplicatively independent positive integers and let >0. In a recent paper jointly with Y. Bugeaud we proved the upper bound exp(n) for g.c.d.(aⁿ–1, bⁿ–1); shortly afterwards we generalized this to the estimate g.c.d.(u–1,v–1)<>u,v) for multiplicatively independent S-units u,vZ. In a subsequent analysis of those results it turned out that a perhaps better formulation of them may be obtained in terms of the language of heights of algebraic numbers. In fact, the purposes of the present paper are: to generalize the upper bound for the g.c.d. to pairs of rational functions other than {u–1,v–1} and to extend the results to the realm of algebraic numbers, giving at the same time a new formulation of the bounds in terms of height functions and algebraic subgroups of G_m².     

An upper bound on the size of the snake-in-the-box

A snake in a graph is a simple cycle without chords. We give an upper bound on the size of a snake S in then-dimensional cube of the form |S|2 ^n–1(1–n ^1/2/89+O(1/n)).     

Lefschetz Complex Conditions for Complex Manifolds

For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L _1,0: H ^1,0(M) H ^{{n, n–1}(M) and L _{0, 1}: H ^{0, 1}(M) H ^{n – 1, n} (M) given by the cup product with [] ^{n – 1}, n being the complex dimension of M andH ^{*, *}(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L _{1, 0} (resp.L _{0, 1}) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L _{1, 0}characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom. 13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).