Quadrature formulas with positive weights

Recent developments in the theory of stability or contractivity of numerical methods for solving ordinary differential equations (see for instance [4], [5], [8]) have renewed the interest for the study of quadrature formulas with positive weights. Nørsett-Wanner [8] and Burrage [2], [3] have given characterisation of such quadrature formulas of order 2m–2 or 2m–3. In this paper we extend these investigations to the case of formulas of order 2m–4 and then to the case where the order is 2m–7. Finally we use these results to characterise the algebraically stable methods out of a 12-parameter family of implicit Runge-Kutta methods of order 2m–4.     

Constructive characterization ofA-stable approximations to exp(z) and its connection with algebraically stable Runge-Kutta methods

Summary All rational approximations to exp(z) of order 2m– (m denotes the maximal degree of nominator and denominator) are given by a closed formula involving real parameters. Using the theory of order stars [9], necessary and sufficient conditions forA-stability (respectivelyI-stability) are given. On the basis of this characterization relations between the concepts ofA-stability and algebraic stability (for implicit Runge-Kutta methods) are investigated. In particular we can partly prove the conjecture that to any irreducibleA-stableR(z) of oderp0 there exist algebraically stable Runge-Kutta methods of the same order withR(z) as stability function.     

A bounded law of the iterated logarithm for Hilbert space valued martingales and its application to U-statistics

Summary A bounded law of the iterated logarithm for martingales with values in a separable Hilbert space H is proved. It is then applied to prove invariance principles for U-statistics for independent identically distributed (-valued) random variables {X _j , j1} and a kernel h: ^m H, m2, which is degenerate for the common distribution function of X _j , j1. This extends to general m results of an earlier paper on this subject and even gives new results in the case H=.     

The limiting behaviour of the last exit time for sequences of independent,identically distributed random variables

Summary Let {X _k , k0} be i.i.d. random variables with EX ⁺< and define t =max{k0: X _k > k} if such a k exists and =0 else, the last exit time of the sequence X _k for fixed >0. We discuss weak limit laws for t as 0; in particular the limit distributions, the stability and the relative stability.This work is partially supported by a grant of the Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung, while the author was at the University of Pittsburgh, USAHerrn Prof. L. Schmetterer zu seinem 60. Geburtstag gewidmet     

Surface order large deviations for Ising,Potts and percolation models

Summary We derive uniform surface order large deviation estimates for the block magnetization in finite volume Ising (or Potts) models with plus or free (or a combination of both) boundary conditions in the phase coexistence regime ford3. The results are valid up to a limit of slab-thresholds, conjectured to agree with the critical temperature. Our arguments are based on the renormalization of the random cluster model withq1 andd3, and on corresponding large deviation estimates for the occurrence in a box of a largest cluster with density close to the percolation probability. The results are new even for the case of independent percolation (q=1). As a byproduct of our methods, we obtain further results in the FK model concerning semicontinuity (inp andq) of the percolation probability, the second largest cluster in a box and the tail of the finite cluster size distribution.     

Updating the singular value decomposition

Summary LetA be anm×n matrix with known singular value decomposition. The computation of the singular value decomposition of a matrixà is considered, whereà is obtained by appending a row or a column toA whenmn or by deleting a row or a column fromA whenm>n. An algorithm is also presented for solving the updated least squares problemà y–b, obtained from the least squares problemAx–b by appending an equation, deleting an equation, appending an unknown, or deleting an unknown.This research was supported by NSF grants MCS 75-06510 and MCS 76-03139     

Monotonie bei den Quadraturverfahren von Gauß und Newton-Cotes

Summary LetQ _n be the quadrature rule of Gauss or Newton-Cotes withn abscissas. It is proven here, thatf ⁽²ⁿ⁾0 impliesQ _n ^G [f]Q _m ^G [f] (for allm>n) andQ _2n–1 ^NC [f]Q _2n ^NC [f]Q _2n+1 ^NC [f]. It follows that the sequenceQ _n[f] (n=1, 2, ...) is monotone, if all derivatives off are positive.

     

Quadrature methods of arbitrary order for solving linear ordinary differential equations

The writer shows that given any positive integern3 there is a one step method for numerical integration of the linear ordinary differential equationY=tAY+B of ordern+1, which employsn evaluations ofA andB. Numerical computations of the method whenn=3 and 4 compare quite favorably with the method of Runge-Kutta in those cases which are considered.     

Rotation numbers for complete tripartite graphs

Arooted graph is a pair (G, x), whereG is a simple undirected graph andx V(G). IfG is rooted atx, then itsrotation number h(G, x) is the minimum number of edges in a graphF of the same order asG such that for allv V(F), we can find a copy ofG inF with the rootx atv. Rotation numbers for all complete bipartite graphs are now known (see [2], [4], [7]). In this paper we calculate rotation numbers for complete tripartite graphs with rootx in the largest vertex class.Funded by the Science and Engineering Research Council.     

Über konvergente Zerlegungen von Matrizen

LetA, M, N ben × n real matrices, letA=M–N, letA andM be nonsingular. LetMy0 implyNy0 (where the prime denotes the transpose). ThenAy0 impliesNy0 if and only if the spectral radius (M ^–1 N) ofM ^–1 N is less than one. This complements a result of Mangasarian, given in [1]. The same conclusions are true ifA, M, andN are replaced byA, M, andN respectively. The proof given here does not make use of the Perron-Frobenius theorem.

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Herrn Professor Dr. Johannes Weissinger zum 60. Geburtstag gewidmet     

On the diophantine equation tan (kπ/m)=k tan π/m

It is proved that the equation tan (k/m)=k tan /m has no solution in integersk andm withk2,m3. This answers a question concerning the problem of approximating a convex disc by polygons.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

Small idempotent clones I

G. Grätzer and A. Kisielewicz devoted one section of their survey paper concerning p _n-sequences and free spectra of algebras to the topic Small idempotent clones (see Section 6 of [18]). Many authors, e.g., [8], [14, 15], [22], [25] and [29, 30] were interested in p _n-sequences of idempotent algebras with small rates of growth. In this paper we continue this topic and characterize all idempotent groupoids (G, ·) with p ₂(G, ·) 2 (see Section 7). Such groupoids appear in many papers see, e.g. [1], [4], [21], [26, 27], [25], [28, 30, 31, 32] and [34].     

Decomposition of operator functions and the multiplication problem for small ideals

Let X be a Banach space, J L(X) a Fréchet ideal and G a region in ^N. If J is non-locally-convex, then it is a problem whether A H(G,L(X)), B H(G,J) implies A·B,B·A H(G,J). A positive answer to this question would sharpen an additive decomposition theorem of B. Gramsch [6] and B. Gramsch-W. Kaballo [8] for resolvents of semi-Fredholm functions. Here it is proved that if J is contained in another ideal J₁ such that the inclusion map i : J J₁ is the product of N exponentially galbed maps in the sense of P. Turpin [18], [19], then A·B,B·A H(G,J₁). An example shows that this is false if i is only a product of N-1 exponentially galbed maps. Thus a sharpening of the decomposition result mentioned above is obtained. Finally, for N=1, a sharper version of a multiplicative decomposition theorem of G.Ph.A. Thijsse [17] for FG-meromorphic functions is proved.     

Higher order methods for a singularly perturbed problem

Summary A finite element method using piecewise polynomials of degree k is used to approximate the problem u+u=f, >0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h ^k+1) are obtained uniformly in ,h the maximum stepsize, fork2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.Work performed while visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y.     

Sample path behaviour in connection with generalized arcsine laws

Summary LetG=(G(t),t0) be the process of last passage times at some fixed point of a Markov process. The Dynkin-Lamperti theorem provides a necessary and sufficient condition forG(t)/t to converge in law ast to some non-degenerate limit (which is then a generalized arcsine law). Under this condition, we give a simple integral test that characterizes the lower-functions ofG. We obtain a similar result forA ⁺=(A ⁺ (t),t0), the time spent in [0, ) by a real-valued diffusion process, in connection with Watanabe's recent extension of Lévy's second arcsine law.     

Some dimension results for super-Brownian motion

Summary The Dawson-Watanabe super-Brownian motion has been intensively studied in the last few years. In particular, there has been much work concerning the Hausdorff dimension of certain remarkable sets related to super-Brownian motion. We contribute to this study in the following way. Let (Y _t)_t0 be a super-Brownian motion on ^d (d2) andH be a Borel subset of ^d . We determine the Hausdorff Dimension of {t0; SuppY _tHØ}, improving and generalizing a result of Krone. We also obtain a new proof of a result of Tribe which gives, whend4, the Hausdorff dimension of SuppY _t as a function of the dimension ofB.     

Zonoide und verwandte Klassen konvexer Körper

Forn3 and every j{2,...,n-1} the existence of ann-dimensional convex bodyK is shown which is not a zonoid, although all itsj-dimensional projections are zonoids. Possible characterizations of such bodies with zonoidal projections are discussed. Further, a characterization of the generating measures of generalized zonoids is given, which extends a result ofLindquist.     

Convergence of linear multistep methods for differential equations with discontinuities

Summary A new stability functional is introduced for analyzing the stability and consistency of linear multistep methods. Using it and the general theory of [1] we prove that a linear multistep method of design orderqp1 which satisfies the weak stability root condition, applied to the differential equationy (t)=f (t, y (t)) wheref is Lipschitz continuous in its second argument, will exhibit actual convergence of ordero(h ^p–1) ify has a (p–1)th derivativey ^(p–1) that is a Riemann integral and ordero(h ^p) ify ^(p–1) is the integral of a function of bounded variation. This result applies for a functiony taking on values in any real vector space, finite or infinite dimensional.This work was supported by Grant GJ-938 from the National Science Foundation     

The algebraic radical of a normed ideal inL 1(G)

LetA be a proper normed ideal (in the sense ofCigler) insideL ¹ (G), whereG is a non-discrete LCA group. This is proved: For each integern1 there existsfL ¹ (G) such thatf, f ² ,..., f ^nA whilef ⁿ⁺¹ A.     

On abelian automorphism group of a surface of general type

Summary LetS be a minimal surface of general type over,K the canonical divisor ofS. LetG be an abelian automorphism group ofS. IfK ²140, then the order ofG is at most 52K ²+32. Examples are also provided with an abelian automorphism group of order 12K ²+96.The automorphism groups for a complex algebraic curve of genusg2 have been thoroughly studied by many authors, including many recent ones. In particular, various bounds have been established for the order of such groups: for example, the order of the total automorphism group is 84(g–1) [Hu], that of an abelian subgroup is 4g+4 [N], while the order of any automorphism is 4g+2 ([W], see also [Ha]).It is an intriguing problem to generalise these bounds to higher dimensions. For example, for surfaces of general type, it is well known that the automorphism groups are finite, and the bound of the orders of these groups depends only on the Chern numbers of the surface [A].In the attempts to such generalisations, the order of abelian subgroups has a special importance. Due to Jordan's theorem on group representations (and its followers), a bound on the order of abelian subgroups induces a bound on that of the whole automorphism group, although bounds thus obtained are generally far from satisfactory. In [H-S], it is shown that for surfaces of general type, the order of such an abelian subgroup is bounded by the square of the Chern numbers times a constant.The purpose of this article is to give a further analysis to the abelian case for surfaces of general type, in proving that the order is bounded linearly by the Chern numbers of the surface, in good analogy with the case of curves. More precisely, our main result is the following.Oblatum 11-IX-1989 & 29-I-1990