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1.
A nonnegative, infinitely differentiable function ø defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and 0 1 ø(t)dt=1. In this article the following problem is considered. Determine k =inf 0 1(k)(t)dt, k=1,..., where ø(k) denotes thekth derivative of ø and the infimum is taken over the set of all mollifier functions. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. In this article, the structure of the problem of determining k is analyzed, and it is shown that the problem is reducible to a nonlinear programming problem involving the minimization of a strictly convex function of [(k–1)/2] variables, subject to a simple ordering restriction on the variables. An optimization problem on the functions of bounded variation, which is equivalent to the nonlinear programming problem, is also developed. The results of this article and those from approximation of functions theory are applied elsewhere to derive numerical values of various mathematical quantities involved in this article, e.g., k =k~22k–1 for allk=1, 2, ..., and to establish certain inequalities of independent interest. This article concentrates on problem reduction and equivalence, and not numerical value.This research was supported in part by the National Science Foundation under Grant No. GK-32712.  相似文献   

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Summary This paper deals with polynomial approximations ø(x) to the exponential function exp(x) related to numerical procedures for solving initial value problems. Motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument, denoted by –R(ø), at which ø is absolutely monotonic. For given integersp1,m1 we determine the maximum ofR(ø) when ø varies over the class of all polynomials of a degree m with (forx0).  相似文献   

4.
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  相似文献   

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Summary This paper deals with rational functions ø(z) approximating the exponential function exp(z) related to numerical procedures for solving initial value problems. Motivated by positivity and contractivity requirements imposed on these numerical procedures we study the greatest nonnegative numberR, denoted byR(ø), such that ø is absolutely monotonic on (–R, 0]. An algorithm for the computation ofR(ø) is presented. Application of this algorithm yields the valueR(ø) for the well-known Padé approximations to exp(z). For some specific values ofm, n andp we determine the maximum ofR(ø) when ø varies over the class of all rational functions ø with degree of the numerator m, degree of the denominator n and ø(z)=exp(z)+(z p+1 ) (forz0).  相似文献   

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Summary In this paper, it is shown that all expected lifetimes ofh-processes inD are finite if and only if the area ofD is finite ifD={(x,y):ø_(x)>y+(x), – <x<}, where ø_(x)<ø+ are two Lipschitz functions. We show that if is a bounded convex region in the plane, there is anh-process in with expected lifetime at leastc area (). We also give an example of a planar domainD of infinite area such that the expected lifetime of eachh-process inD is finite.  相似文献   

7.
A semilatticeS isrepresentable by subspaces of R k if, to eachx S we can assign a subspace so thatx y=z inS if and only if . Every height-2 semilattice is representable inR 2. We show that for everyk there is a height-3 semilattice which is not representable by subspaces ofR k.Presented by J. Berman.Research supported in part by the National Science Foundation.Research supported in part by the Office of Naval Research.  相似文献   

8.
We study toric varieties over a field k that split in a Galois extension using Galois cohomology with coefficients in the toric automorphism group. Part of this Galois cohomology fits into an exact sequence induced by the presentation of the class group of the toric variety. This perspective helps to compute the Galois cohomology, particularly for cyclic Galois groups. We use Galois cohomology to classify k‐forms of projective spaces when is cyclic, and we also study k‐forms of surfaces.  相似文献   

9.
We consider k‐dimensional random simplicial complexes generated from the binomial random (k + 1)‐uniform hypergraph by taking the downward‐closure. For 1 ≤ jk ? 1, we determine when all cohomology groups with coefficients in from dimension one up to j vanish and the zero‐th cohomology group is isomorphic to . This property is not deterministically monotone for this model, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the j‐th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced by Linial and Meshulam, previously only known for dimension two.  相似文献   

10.
Problems of distance geometry and convex properties of quadratic maps   总被引:1,自引:0,他引:1  
A weighted graph is calledd-realizable if its vertices can be chosen ind-dimensional Euclidean space so that the Euclidean distance between every pair of adjacent vertices is equal to the prescribed weight. We prove that if a weighted graph withk edges isd-realizable for somed, then it isd-realizable for (this bound is sharp in the worst case). We prove that for a graphG withn vertices andk edges and for a dimensiond the image of the so-called rigidity map ℝ dn →ℝ k is a convex set in ℝ k provided . These results are obtained as corollaries of a general convexity theorem for quadratic maps which also extends the Toeplitz-Hausdorff theorem. The main ingredients of the proof are the duality for linear programming in the space of quadratic forms and the “corank formula” for the strata of singular quadratic forms. This research was supported by the United States Army Research Office through the Army Center of Excellence for Symbolic Methods in Algorithmic Mathematics (ACSyAM), Mathematical Sciences Institute of Cornell University, Contract DAAL03-91-C0027.  相似文献   

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