Berezin–Toeplitz quantization for lower energy forms

Let M be an arbitrary complex manifold and let L be a Hermitian holomorphic line bundle over M. We introduce the Berezin–Toeplitz quantization of the open set of M where the curvature on L is nondegenerate. In particular, we quantize any manifold admitting a positive line bundle. The quantum spaces are the spectral spaces corresponding to [0,k^?N], where N>1 is fixed, of the Kodaira Laplace operator acting on forms with values in tensor powers L^k. We establish the asymptotic expansion of associated Toeplitz operators and their composition in the semiclassical limit k→∞ and we define the corresponding star-product. If the Kodaira Laplace operator has a certain spectral gap this method yields quantization by means of harmonic forms. As applications, we obtain the Berezin–Toeplitz quantization for semi-positive and big line bundles.     

An equivalence between varieties of cyclic Post algebras and varieties generated by a finite field

In this paper we give a term equivalence between the simple k-cyclic Post algebra of order p, L _p,k, and the finite field F(p ^k) with constants F(p). By using Lagrange polynomials, we give an explicit procedure to obtain an interpretation Φ₁ of the variety V(L _p,k) generated by L _p,k into the variety V(F(p ^k)) generated by F(p ^k) and an interpretation Φ₂ of V(F(p ^k)) into V(L _p,k) such that Φ₂Φ₁(B) = B for every B ε V(L _p,k) and Φ₁Φ₂(R) = R for every R ε V(F(p ^k)).     

Generalized absolutely monotone functions

The concept of absolutely monotone functions is generalized by replacing the conditionsφ ^(k)(t)≧0,k=0, 1, … by an infinite sequence of differential inequalitiesφ(t)≧0,L _kφ(t)≧0,k=1, 2, …, where theL _k are differential operators of a special type. It is shown that these functions have a valid series expansion in terms of basic functions associated with the operatorsL _k.     

Stochastic anticipative calculus for functionals of a gaussian generalized random field

We give a construction of Skorohod integrals with respect to a Gaussian D'-valued random field W The method is based on the multiple Wiener integral expansion for L ₂-functionals of W We also give a representation of the Malliavin derivative operator of L ²-functionals of W     

A bounded compactness theorem forL 1-embeddability of metric spaces in the plane

LetM=(W, d) be a metric space. LetL ¹ denote theL ¹ metric. AnL ¹-embedding ofM into Cartesiank-space ℝ ^k is a distance-preserving map from (W, d) into (ℝ ^k ,L ¹). Letc(k) be the smallest integer such that for every metric spaceM, M isL ¹-embeddable inR ^k iff everyc(k)-sized subspace ofM isL ¹-embeddable inR ^k. A special case of a theorem of Menger (see p. 94 of [5]) says thatc(1) exists and equals 4. We show thatc(2) exists and satisfies 6≦c(2)≦11. Whether or notc(k) exists for anyk≧3 is an open question. The research of S. M. Malitz was partially supported by NSF Grant CCR-8909953.     

Error estimates for mixed finite element methods for nonlinear parabolic problems

We consider a class of mixed finite element methods for nonlinear parabolic problems over a plane domain. The finite element spaces taken are Raviart-Thomas spaces of index k, k ? 0. We obtain optimal order L²- and almost optimal order L^∞-error estimates for the finite element solution and order optimal L²-error estimates for its gradient. We also derive the error estimates for the time derivatives of the solution. Our results extend those previously obtained by Johnson and Thomée for the corresponding linear problems with k ? 1.     

On kth-order embeddings of K3 surfaces and Enriques surfaces

We give necessary and sufficient conditions for a big and nef line bundle L of any degree on a K3 surface or on an Enriques surface S to be k-very ample and k-spanned. Furthermore, we give necessary and sufficient conditions for a spanned and big line bundle on a K3 surface S to be birationally k-very ample and birationally k-spanned (our definition), and relate these concepts to the Clifford index and gonality of smooth curves in |L| and the existence of a particular type of rank 2 bundles on S. Received: 28 March 2000 / Revised version: 20 October 2000     

The <Emphasis Type="Italic">k</Emphasis>-Fibonacci matrix and the Pascal matrix

We define the k-Fibonacci matrix as an extension of the classical Fibonacci matrix and relationed with the k-Fibonacci numbers. Then we give two factorizations of the Pascal matrix involving the k-Fibonacci matrix and two new matrices, L and R. As a consequence we find some combinatorial formulas involving the k-Fibonacci numbers.     

Edgeworth Expansion of Densities of Order Statistics with Fixed Rank

We give an Edgeworth expansion for densities of order statistics with fixed rank k.The Edgeworth expansion for densities of extreme values is then obtained as a special case k=1.     

Orthonormal expansions of invariant densities for expanding maps

We give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding C^ω Markov maps. This is a classical problem, with one of the standard approaches being Ulam's method [Problems in Modern Mathematics, Interscience, New York, 1960] of phase space discretisation.Our method hinges instead on the expansion of the density function with respect to an L² orthonormal basis, and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map. The efficiency of the method, and its extension to C^k expanding maps, are also discussed.     

Improved algorithms for path partition and related problems

We study the L_∞ path partition problem: given a path of n weighted vertices and an integer k, remove k−1 edges from the path so that the maximum absolute deviation of the weights of the resulting k sub-paths from their mean is minimized. Previously, the best algorithm solves this problem in O(nklogk) time. We present an O(nk) time algorithm. We also give improved solutions for two related problems: the L_d path partition problem and the web proxies placement problem.     

Isomorphic classification of the spaces $H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R)$

We give a representation of the spaces H^k_loc(\Bbb R)?L²(\Bbb R)H^k_{\rm loc}(\Bbb R)\cap L^2(\Bbb R) as Köthe sequence spaces and use some recent topological invariants to investigate their isomorphic classification.     

Commutativity of kth‐order slant Toeplitz operators

In this paper, commutativity of k^th‐order slant Toeplitz operators are discussed. We show that commutativity and essential commutativity of two slant Toeplitz operators are the same. Also, we study k^th‐order slant Toeplitz operators on the Bergman space L²_a(D) and give some commuting properties, algebraic and spectral properties of k^th‐order slant Toeplitz operators on the Bergman space (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dense Packing of Patterns in a Permutation

We study the length L _k of the shortest permutation containing all patterns of length k. We establish the bounds e ⁻² k ² < L _k ≤ (2/3 + o(1))k ². We also prove that as k → ∞, there are permutations of length (1/4 + o(1))k ² containing almost all patterns of length k. Received January 2, 2007     

On the Expected Number of Shadow Vertices of the Convex Hull of Random Points

Leta₁, . . . ,a_mbe independent random points in ⁿthat are independent and identically distributed spherically symmetrical in ⁿ. Moreover, letXbe the random polytope generated as the convex hull ofa₁, . . . ,a_mand letL_kbe an arbitraryk-dimensional subspace of ⁿwith 2 ≤k≤n− 1. LetX_kbe the orthogonal projection image ofXinL_k. We call those vertices ofXwhose projection images inL_kare vertices ofX_kshadow vertices ofXwith respect to the subspaceL_k. We derive a distribution independent sharp upper bound for the expected number of shadow vertices ofXinL_k.     

Removability of singularities for a class of fully non-linear elliptic equations

In this paper we address the problem of understanding the singularities of the fully non-linear elliptic equation σ _k (v) = 1. These σ _k curvature are defined as the symmetric functions of the eigenvalues of the Schouten tensor of a Riemannian metric and appear naturally in conformal geometry, in fact, σ₁ is just the scalar curvature.Here we deal with the local behavior of isolated singularities. We give a sufficient condition for the solution to be bounded near the singularity. The same result follows for a more general singular set Λ as soon as we impose some capacity conditions. The main ingredient is an estimate of the L ^∞ norm in terms of a suitable L ^p norm. Mathematics Subject Classification (2000) 35J60, 53A30     

Weighted estimates for the <Emphasis Type="Italic">k</Emphasis>-plane transform of radial functions on euclidean spaces

We discuss the L ^p − L ^q mapping property of k-plane transforms acting on radial functions in certain weighted L ^p spaces with power weight. We show that for all admissible power weights it is not always possible to get strong (p, q) boundedness of the k-plane transform. However, we prove the best possible estimates with respect to the Lorentz norms.