C *-algebras generated by orthogonal projections and their applications

We study the following problem: Given a Hilbert spaceH and a set of orthogonal projectionsP, Q ₁, ..., Q_n on it, with the conditionsQ _j ·Q _k = _j,k Q _k , , describe theC ^*-algebraC ^*(P, Q ₁, ..., Q_n) generated by these projections.Applications to Naimark dilation theorems and to Toeplitz operators associated with the Heisenberg group are given.Dedicated to the memory of M. G. Krein.This work was partially supported by CONACYT Project 3114P-E9608, México.     

A superfast algorithm for multi-dimensional Padé systems

For a vector ofk+1 matrix power series, a superfast algorithm is given for the computation of multi-dimensional Padé systems. The algorithm provides a method for obtaining matrix Padé, matrix Hermite Padé and matrix simultaneous Padé approximants. When the matrix power series is normal or perfect, the algorithm is shown to calculate multi-dimensional matrix Padé systems of type (n ₀,...,n _k ) inO(n · log²n) block-matrix operations, where n=n ₀+...+n _k . Whenk=1 and the power series is scalar, this is the same complexity as that of other superfast algorithms for computing Padé systems. Whenk>1, the fastest methods presently compute these matrix Padé approximants with a complexity ofO(n²). The algorithm succeeds also in the non-normal and non-perfect case, but with a possibility of an increase in the cost complexity.Supported in part by NSERC grant No. A8035.Partially supported by NSERC operating grant No. 6194.     

Existence and Construction of H 1-Splines of Class C k on a Three Directional Mesh

Let be the uniform triangulation generated by the usual three directional mesh of the plane and let H ₁ be the regular hexagon formed by the six triangles of surrounding the origin. We study the space of piecewise polynomial functions in C ^k (R ²) with support H ₁ having a sufficiently high degree n, which are invariant with respect to the group of symmetries of H ₁ and whose sum of integer translates is constant. Such splines are called H ₁-splines. We first compute the dimension of this space in function of n and k. Then we prove the existence of a unique H ₁-spline of minimal degree for any fixed k0. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of this spline.     

Sur les formules de quadrature numérique à nombre minimal de noeuds d'intégration

Résumè Cet article a pour objet la recherche, à partir de la théorie des polynômes orthogonaux, de conditions permettant l'obtention de formules de quadrature numérique sur des domaines de ⁿ, avec fonction poids, à nombre minimal de noeuds et exactes sur les espacesQ _k de polynômes de degré k par rapport à chacune de leurn variables. Ces résultats, complétés par des exemples numériques originaux dans ², adaptent à ces espacesQ _k ceux démontréq par H.J. Schmid [14] dans le cadre des espacesP _k de polynômes.

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About Cubature formulas with a minimal number of knots

Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulas on sets of ⁿ, with weight function. which have a minimal number of knots and which are exact on the spaceQ _k of all polynomials of degree k with respect to each variablex _i, 1in.These results, completed by original numerical examples in ², adapt to the spacesQ _k those proved by H.J. Schmid [14] in the case of polynomial spacesP _k.

     

Nonhomogeneous recursions and Generalised Continued Fractions

Consider the (n+1)st order nonhomogeneous recursionX _k+n+1=b _k X _k+n +a _k ⁽ⁿ⁾ X _k+n-1+...+a _k ⁽¹⁾ X _k +X _k .Leth be a particular solution, andf ⁽¹⁾,...,f ⁽ⁿ⁾,g independent solutions of the associated homogeneous equation. It is supposed thatg dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾ andh. If we want to calculate a solutiony which is dominated byg, but dominatesf ⁽¹⁾,...,f ⁽ⁿ⁾, then forward and backward recursion are numerically unstable. A stable algorithm is derived if we use results constituting a link between Generalised Continued Fractions and Recursion Relations.     

On commuting compact self-adjoint operators on a Pontryagin space

Suppose thatA ₁,A ₂, ...,A _n are compact commuting self-adjoint linear maps on a Pontryagin spaceK of indexk and that their joint root subspaceM ₀ at the zero eigenvalue in ⁿ is a nondegenerate subspace. Then there exist joint invariant subspacesH andF inK such thatK=FH,H is a Hilbert space andF is finite-dimensional space withkdimF(n+2)k. We also consider the structure of restrictionsA _j|F in the casek=1.     

Rainbow Numbers for Cycles with Pendant Edges

A subgraph of an edge-colored graph is called rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f (n, H) is the maximum number of colors in an edge-coloring of K _n with no rainbow copy of H. The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of K _n with rb(n, H) number of colors contains a rainbow copy of H. Certainly rb(n, H) = f(n, H) + 1. Anti-Ramsey numbers were introduced by Erdős et al. [4] and studied in numerous papers. We show that for n ≥ k + 1, where C _k ⁺ denotes a cycle C _k with a pendant edge.     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason's theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H⁰(k,(1)) corresponding to a primitive nth root of unity.     

A survey of orthogonal arrays of strength two

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Sharp Bounds For Some Multicolor Ramsey Numbers

Let H ₁,H ₂, . . .,H _k+1 be a sequence of k+1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H ₁,H ₂,...,H _k+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k+1 colors, there is a monochromatic copy of H _i in color i for some 1ik+1. We describe a general technique that supplies tight lower bounds for several numbers r(H ₁,H ₂,...,H _k+1) when k2, and the last graph H _k+1 is the complete graph K _m on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K ₃,K ₃,K _m ) = (m ³ poly logm), thus solving (in a strong form) a conjecture of Erdos and Sós raised in 1979. Another special case of our result implies that r(C ₄,C ₄,K _m ) = (m ² poly logm) and that r(C ₄,C ₄,C ₄,K _m ) = (m ²/log² m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.* Research supported in part by a State of New Jersey grant, by a USA Israeli BSF grant and by a grant from the Israel Science Foundation. Research supported by NSF grant DMS 9704114.     

Padé approximations to certain power series

Summary An explicit identity involvingQ _n (q ⁱ z) (i = 0, 1,, 4) is shown, whereQ _n (z) is the denominator of thenth Padé approximant to the functionf(z) = _k=0 q ^1/2k(k–1 Z ^k . By using the Padé approximations, irrationality measures for certain values off(z) are also given.

     

Rigidity Theorem for Hypersurfaces in a Unit Sphere

By investigating hypersurfaces M ⁿ in the unit sphere S ⁿ⁺¹(1) with H _k = 0 and with two distinct principal curvatures, we give a characterization of torus the . We extend recent results of Perdomo [9], Wang [10] and Otsuki [8].     

Some families of complete caps of quadrics and symplectic polarities over GF(8)

A cap on a quadric is a set of its points whose pairwise joins are all chords. A cap is complete if it is not part of a larger one. The only field for which all complete quadric caps are known is GF(2). Those caps are small; the biggest for each quadric is of order the dimension of the ambient space. Apart from information about ovoids in dimensions at most 7, little else is known. Here, the evidence is increased by providing caps over GF(2), odd, which, if >1, have size of order the dimension cubed. In particular, complete caps are obtained for the quadrics Q _2m (8), Q ₊ ^8k+7 (8), Q _- ^8k+3 (8), Q ₊ ^8k+1 (8) and Q _- ^8k+5 (8). These caps on Q ₊ ^8k+7 (8) and Q _- ^8k+3 (8) are complete on any Q _n(8) of which their quadrics are sections; so is that that of Q ₄₊₂(8) for any Q _2n (8) of which Q ₄₊₂(8) is a section with the same kernel. From the correspondence with Q _2n (8) complete caps are obtained for symplectic polarities over GF(8).     

Induced operators on symmetry classes of tensors

LetV be ann-dimensional inner product space,T _i ,i=1,...,k, k linear operators onV, H a subgroup ofS _m (the symmetric group of degreem), a character of degree 1 andT a linear operator onV. Denote byK(T) the induced operator ofT onV (H), the symmetry class of tensors associated withH and . This note is concerned with the structure of the setK _{, m} ^H (T₁,...,T_k) consisting of all numbers of the form traceK(T ₁ U ₁...T _k U _k ) whereU _i ,i=1,...k vary over the group of all unitary operators onV. For V=ⁿ or ⁿ, it turns out thatK _{, m} ^H (T₁,...,T_k) is convex whenm is not a multiple ofn. Form=n, there are examples which show that the convexity of _{, m} ^H (T₁,...,T_k) depends onH and .The author wishes to express his thanks to Dr. Yik-Hoi Au-Yeung for his valuable advice and encouragement.     

Cyclic homology and the Macdonald conjectures

Summary LetA+(k) denote the ring [t]/t ^k+1 and letG be a reductive complex Lie algebra with exponentsm ₁, ...,m n. This paper concerns the Lie algebra cohomology ofGA ₊(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA ₊(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m _s +1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m _s +1, one has weight 0 and the others have weights (k+1)m _s +t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A ₊(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA _n ,B _n ,C _n , orD _n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship     

Euler Tours of Maximum Girth in K2 n +1 and K2 n ,2 n

Given an eulerian graph G and an Euler tour T of G, the girth of T, denoted by g(T), is the minimum integer k such that some segment of k+1 consecutive vertices of T is a cycle of length k in G. Let g^E(G)= maxg(T) where the maximum is taken over all Euler tours of G.We prove that g^E(K_2n,2n)=4n–4 and 2n–3g^E(K_2n+1)2n–1 for any n2. We also show that g^E(K₇)=4. We use these results to prove the following:1)The graph K_2n,2n can be decomposed into edge disjoint paths of length k if and only if k4n–1 and the number of edges in K_2n,2n is divisible by k.2)The graph K_2n+1 can be decomposed into edge disjoint paths of length k if and only if k2n and the number edges in K_2n+1 is divisible by k.     

An explicit calculation of the mean of the perimeter of the convex hull of a plane random walk

If (X _n ) _n ⁼¹ is a sequence of i.i.d. random variables in the Euclidean plane such that we compute the mean of the perimeter of theconvex hull ofX ₁++X _k; 0kn}.     

Rational approximants to symmetric formal Laurent series

Rational approximants, in the Padé sense, to a given formal Laurent series,F(z)= _– c _k z ^k , have been considered by several authors (see [3] for a survey about the different kinds of approximants which can be defined). In this paper, we shall be concerned with symmetric series, that is, when the complex coefficients {c _k } _– ⁺ satisfyc _–k=c _k,k=0, 1,....Making use of Brezinski's approach [1], for Padé-type approximation to a formal power series, rational approximants toF(z) with prescribed poles are obtained, and their algebraic properties considered. These results will allow us to give an alternative approach for the Padé-Chebyshev approximants.