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1.
Summary In this paper we search, from the orthogonal polynomial theory, for conditions which allow to obtain cubature formulae on compacts of n , with weight function, and which are exact on the spaceR( k 1, k2, ..., kn) of all polynomials of degree k i respectively to each variablex i , 1in.  相似文献   

2.
For given data (x i, fi) i=0 n (x 0<x 1<...<x n) we consider the possibility of finding a spline functions of arbitrary degreek (k3) with preassigned smoothnessl, where 1l[(k-1)/2]. The splines should be such thats(x i)=f i (i=0, 1,...,n) ands is convex or nondecreasing and convex on [x 0,x n]. An explicit formula for this function as well as the conditions that guarantee the required properties are established. An algorithm for the determination of the splines and the error bounds is also included.  相似文献   

3.
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.
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.
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4.
We considern-point Lagrange-Hermite extrapolation forf(x), x>1, based uponf(x i ),i=1(1)n, –1x i 1, including non-distinct pointsx i in confluent formulas involving derivatives. The problem is to find the pointsx i that minimize the factor in the remainderP n (x)f (n)()/n, –1<<x subject to the condition|P n (x)|M, –1x1,2n+1M2 n . The solution is significant only when a single set of pointsx i suffices for everyx>1. The problem is here completely solved forn=1(1)4. Forn>4 it may be conjectured that there is a single minimal , 0 rn, whererr(M) is a non-decreasing function ofM, P n (–1)=(–1) n M, and for 0rn–2, thej-th extremumP n (x e, j )=(–1) nj M,j=1(1)n–r–1 (except forM=M r ,r=1(1)n–1, whenj=1(1)n–r).  相似文献   

5.
Summary Let (x i * ) i=1 n denote the decreasing rearrangement of a sequence of real numbers (x i ) i=1 n . Then for everyij, and every 1kn, the 2-nd order partial distributional derivatives satisfy the inequality, . As a consequence we generalize the theorem due to Fernique and Sudakov. A generalization of Slepian's lemma is also a consequence of another differential inequality. We also derive a new proof and generalizations to volume estimates of intersecting spheres and balls in n proved by Gromov.Supported by NSF grant # DMS 8909745, and the USA-Israel Binational Science Foundation grant # 86-00074, and grant for the Promotion of Research at the Technion  相似文献   

6.
In this note maximal order,k step correctors with one nonstep point for the solution ofy=f(x,y),y(x 0)=y 0, introduced by Gragg and Stetter [1] are extended to an arbitrary numbers of nonstep points. These correctors have order 2k + 2s, are proved stable fork8,s2, and unstable for largek.  相似文献   

7.
We characterize generators of sub-Markovian semigroups onL p () by a version of Kato's inequality. This will be used to show (under precise assumptions) that the semigroup generated by a matrix operatorA=(A ij )1i,jn on (L p ()) n is sub-Markovian if and only if the semigroup generated by the sum of each rowA i 1+...+A in (1in), is sub-Markovian. The corresponding result on (C 0(X)) n characterizes dissipative operator matrices.
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8.
Letu be the solution of the differential equationLu(x)=f(x, u(x)) forx(0,1) (with appropriate boundary conditions), whereL is an elliptic differential operator. Letû be the Galerkin approximation tou with polynomial spline trial functions. We obtain error bounds of the form , where 0jm andmk2m+q,p=2 orp=,h is the mesh size andq is a non negative integer depending on the splines being used.This research was supported in part by the Office of Naval Research under Contract N00014-69-A0200-1017.  相似文献   

9.
Summary For a complex polynomial,f:( n+1 ,0) (, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Lê varieties, f (k) , off at 0. The multiplicities of these varieties, f (k) , generalize the Milnor number for an isolated singularity. In particular, we show that ifsn-2, the Milnor, fibre off is obtained fromB 2n by successively attaching f (n – k) k-handles, wheren-skn Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of f (n – 1) circles by successively attaching f (n – k) k-handles, where 2kn.The author is a National Science Foundation, Postdoctoral Research Fellow supported by grant # DMS-8807216  相似文献   

10.
Summary The existence of a joint asymptotic distribution for the windings of a three-dimensional Brownian motion around a finite number of straight lines is obtained. This complements the recent studies, by Pitman- Yor, and the authors, of the joint asymptotic distribution for the windings of planar Brownian motion around a finite number of points.The following principle governs the passage from results in the plane to results in space:Let B be a three-dimensional Brownian motion, and P 1, ..., P k, k planes which intersect two by two. Then, the convergences in distribution concerning the planar Brownian motions B i (1ik), defined respectively as the orthogonal projections of B on P i (1ik), take place jointly, and the corresponding limit variables are independent.  相似文献   

11.
Some fractal sets determined by stable processes   总被引:2,自引:0,他引:2  
Summary LetY i be independent stable subordinators in (, ,P) with indices 0< i <1 andR i are the ranges ofY i ,i=1, 2. We are able to find the exact Hausdorff measure and packing measure results for the product setsR 1×R 2, and whenever 1 + 2 1/2, we deduce results for the vector sumR 1R 2={x+y:xR 1,yR 2}.  相似文献   

12.
Summary Let {X i , i1} be a random sequence and {u ni ,1in, n1} be an array of boundary values. We consider the asymptotic approximation of the probability P n =P{X i u ni ,1in} by . We give sufficient conditions on X i such that P n–P n * 0 as n. This generalizes the situation considered in extreme-value theory where the boundary is constant in i. The general theory is applied in particular to Gaussian cases.  相似文献   

13.
Zusammenfassung Betrachtet wird eine 3-Pol-Schaltung, die feste Widerstände und endlich viele lineare Potentiometer auf einer gemeinsamen Achse enthält. Wenn diese Schaltung von einer konstanten SpannungE 1 gespeist ist, dann ist die AusgangsspannungE 2 an einem beliebigen Punkt der Schaltung eine rationale Funktion vonx, wobeix den Drehwinkel der Achse bedeutet (0x1).In dieser Arbeit wird folgender Satz bewiesen: «Die Nullstellen vonE 2 sind entweder reell ausserhalb des Intervalls 0<x<1, oder konjugiert-komplex. Die Pole vonE 2 sind einfach, reell, negativ oder >1.»  相似文献   

14.
Summary Ann×n complex matrixB is calledparacontracting if B21 and 0x[N(I-B)]Bx2<x2. We show that a productB=B k B k–1 ...B 1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofB in terms of the subdominant singular values of theB i 's and in terms of the angles between certain subspaces. Our results here extend earlier results due to Halperin and due to Smith, Solomon and Wagner. We also determine necessary and sufficient conditions forn numbers in the interval [0, 1] to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product. In the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the SOR iteration matrix associated with ann×n hermitian positive semidefinite matrixA none of whose diagonal entries vanish.The work of this author was supported in part by NSF Research Grant No. MCS-8400879  相似文献   

15.
Summary The problem is considered of fitting a linear manifold of dimensions with 1sn–1 to a given set of points in n such that the sum of orthogonal squared distances attains a minimum.  相似文献   

16.
In this paper, we study the Hodge decompositions ofK-theory and cyclic homology induced by the operations k and k , and in particular the decomposition of the Loday symbols x,y, ...z. Except in special cases, these Loday symbols do not have pure Hodge index. InK n (A) they can project into every componentK n (i) for 2in, and the projection of the Loday symbol x,y, ...,z intoK n (n) is a multiple of the generalized Dennis-Stein symbol x,y, ...,z. Our calculations disprove conjectures of Beilinson and Soulé inK-theory, and of Gerstenhaber and Schack in Hochschild homology.Partially supported by National Security Agency grant MDA904-90-H-4019.Partially supported by National Science Foundation grant DMS-8803497.  相似文献   

17.
Let(n) be the least integer such thatn may be represented in the formn=x 1 2 +x 2 3 +...+x (n) (n)+1 wherex 1,x 2, ...,x (n) are natural numbers. We computed(n) forn 250 000 and found that(n) 5 for all thesen exceptn=56, 160 for which(n)=6. Also(n) 4 for 41542<n<=250 000.  相似文献   

18.
Summary A quantum diffusion (A, A, j) comprises of unital *-algebras A and A and a family of identity preserving *-homomorphisms j=(j t : t0) from A into A. Also j satisfies a system of quantum stochastic differential equations dj t (x 0=j t( j i (x 0))dM i i , j 0(x 0)=x 0I for all x 0A where j i , 1i, jN are maps from A to itself and are known as the structure maps. In this paper an existence proof is given for a class of quantum diffusions, for which the structure maps are bounded in the operator norm sense. A solution to the system of quantum stochastic differential equations is first produced using a variation of the Picard iteration method. Another application of this method shows that the solution is a quantum diffusion.  相似文献   

19.
For the numerical solution of the initial value problemy=f(x,y), –1x1;y(–1)=y 0 a global integration method is derived and studied. The method goes as follows.At first the system of nonlinear equations is solved. The matrix (A i,k (n) ) of quadrature coefficients is nearly lower left triangular and the pointsx k,n ,k=1,2,...,n are the zeros ofP n P n–2, whereP n is the Legendre polynomial of degreen. It is showed that the errors From the valuesf(x i,n ,y i,n ),i=1,2,...,n an approximation polynomial is constructed. The approximation is Chebyshevlike and the error at the end of the interval of integration is particularly small.  相似文献   

20.
Denote byV æ (f,x) the number of solutionsnx of the equationf(n)=æ·n. Then there are multiplicative functions, such that the lower estimate holds for infinitely many pairs (æ,x),x.  相似文献   

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