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1.
If { n } is an orthonormal system and {a n} is a sequence of random variables such that n (a n )2=1 a.s. thenf(t)=| n a n n (t)|2 produces a randomly selcted density function. We study the properties off under the assumptions that |a n| is decreasing to zero at a geometric rate and { n } is one of the following four function systems: trigonometric Jacobi, Hermite, or Laguerre. It is shown that, with probability one,f is an analytic function,f has at most a finite number of zeros in any finite interval, and the tail off goes to zero rapidly.  相似文献   

2.
Let n be n-dimensional Lobachevskii space, and {lx:x n} be a family of lines, parallel to a linel 0, 0n (in a given direction). Let {cx:Xn} be a family of circular cones in n of opening with axes lX and vertex X. Then, iff:nn(n>2) is a bijective mapping andf(Cx)=C f(x), it follows thatf is a motion in the space n.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 687–694, May, 1973.  相似文献   

3.
Summary Consider an unconstrained minimization of an uniformly convex functionf(z). Let be an algorithm that, for solving it constructs a sequence {z i} withz i+1 =z i + (i) h i ,h i R n , (i) R = and –h i T f(z i ) > 0. For any algorithm that converges linearly and that uses parabolic or cubic interpolations for the line search, upper bounds on the number of function evaluations needed to approximate the minimum off(z) within a given accuracy, are calculated. The obtained results allow to compare the line search procedure under investigation.  相似文献   

4.
It is proved that for any sequence {R k} k=1 of real numbers satisfyingR kk (k1) andR k=o(k log2 k),k, there exists an orthonormal system {n k(x)} n=1 ,x (0;1), such that none of its subsystems {n k(x)} k=1 withn kRk (k1) is a convergence subsystem.  相似文献   

5.
LetX={x 1,x 2,..., n }I=[–1, 1] and . ForfC 1(I) definef* byfp f =f*, wherep f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT n (x) and obtain x m p x m8eE n–1(x m ), whereE n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE n (f) that omits theassumptionf (n+1)>0 in a similar estimate of Meinardus.  相似文献   

6.
We consider Keller's functions, namely polynomial functionsf:C n C n with detf(x)=1 at allx C n. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh , > 1, defined by power series in a ball with center at the origin, such thath(0)=I andh (f(x))=h (x). So eachh conjugates f to its linear part I in a ball where it is injective.We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s dg(x),g(x)n=0,xC n,sC the conjugationh for f is anentire function.  相似文献   

7.
For the nth order nonlinear differential equation y (n)(t)=f(y(t)), t [0,1], satisfying the multipoint conjugate boundary conditions, y (j)(ai) = 0,1 i k, 0 j n i - 1, 0 =a 1 < a 2 < < a k = 1, and i=1 k n i =n, where f: [0, ) is continuous, growth condtions are imposed on f which yield the existence of at least three solutions that belong to a cone.  相似文献   

8.
LetA be the generator of a cosine functionC t ,t R in a Banach spaceX; we shall connect the existence and uniqueness of aT-periodic mild solution of the equationu = Au + f with the spectral property 1 (C T ) and, in caseX is a Hilbert space, also with spectral properties ofA. This research was supported in part by DAAD, West Germany.  相似文献   

9.
Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx Z, letf |n| denote then'th iterate off. Clearlyf |m|(f |n|(x))=f |m+n|(x) for allm,nZ and allxR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F ||:G} on the real line with the properties (i)F ||(F ||(x))=F |+| (x) for allxR and (ii)F |n|(x)=f |n|(x) for allnZ andxR. That is,f can be embedded in a set ofG-generalized iterates. In particular,f can be embedded in a set of complex generalized iterates.Dedicated to Professor Janos Aczél on his 60th birthday  相似文献   

10.
Let (n) be a system, close to the orthonormal complete system (x n). An estimate is obtained for the deviation of the system {fn}, obtained from {n} by Schmidt's method, from the system {xn}. This estimate is used to show that, in any LP(–1,1), withp (1,4/3] [4,), and for any >e¦4 = i,13..., there exists an orthogonal algebraic system (P n (x)) n=0 , forming a basis in LP and such that n = degP n (x) n for n>no(p,).Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 223–230, February, 1978.  相似文献   

11.
Let w be an element of the Weyl group of sl n + 1. We prove that for a certain class of elements w (which includes the longest element w0 of the Weyl group), there exist a lattice polytope R l(w) , for each fundamental weight i of sl n + 1, such that for any dominant weight = i = 1 n a i i , the number of lattice points in the Minkowski sum w = i = 1 n a i i w is equal to the dimension of the Demazure module E w (). We also define a linear map A w : R l(w) P Z R where P denotes the weight lattice, such that char E w () = e eA(x) where the sum runs through the lattice points x of w .  相似文献   

12.
Let R be a subring of the rationals with 1/2, 1/3R; let S R n denote the R-local n-sphere and define R n :=S R n for n odd, R n :=S R n for n>0 even. An H-space (resp. a 1-conn. co-H-space) is decomposable over R, if it is homotopy equivalent to a weak product of spaces R n (resp. to a wedge of R-local spheres). We prove that, if E is grouplike decomposable of finite type over R, the functor [-,E] is determined on finite dim. complexes by the Hopf algebra M*(E;R); here M* denotes the unstable cohomotopy functor of H.J. Baues. If C is cogrouplike decomposable over R, the functor [C,-] is determined on 1-conn. R-local spaces by *(C) as a cogroup in the category of M-Lie algebras. For R = the functor [-,E] is also determined by the Lie algebra *(E) and [C,-] by the Berstein coalgebra associated to the comultiplication of C.  相似文献   

13.
We obtained that for any n N, C = 1 is the smallest constant for which the inequality ||B n (f) - f|| C 2(f, 1/n) holds on the class of continuous functions f, as well as on the class of bounded functions f, where B n is the Bernstein operators of degree n, 2 is the second order modulus and || || is the sup-norm.  相似文献   

14.
Consider a classical cusp eigenform f= n=1 a n (f)q n of weight k2 for 0(N) with a Dirichlet character mod N, and let L f (s,)= n=1 (n)a n (f)n -s denote the L-function of f twisted with an arbitrary Dirichlet character . For a prime number p5, consider a family of cusp eigenforms f (k) of weight k , k {f (k)= n=1 a n (f (k))q n } containing f=f (k), such that the Fourier coefficients a n (f (k)) are given by certain p-adic analytic functions k a n (f (k)). The purpose of this paper is to construct a two variable p-adic L function attached to Colemans family {f (k)} of cusp eigenforms of a fixed positive slope =v p ( p )>0 where p = p (k ) is an eigenvalue (which depends on k ) of the Atkin operator U=U p . Our p-adic L-function interpolates the special values L f(k)(s,) at points (s,k ) with s=1,2,...,k -1. We give a construction using the Rankin-Selberg method and the theory of p-adic integration on a profinite group Y with values in an affinoid K-algebra A, where K is a fixed finite extension of Q p . Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. In their turn, such measures come from Eisenstein distributions with values in certain Banach A-modules M =M (N;A) of families of overconvergent forms over A. To Robert Alexander Rankin in memoriam  相似文献   

15.
Training Multilayer Perceptrons Via Minimization of Sum of Ridge Functions   总被引:5,自引:0,他引:5  
Motivated by the problem of training multilayer perceptrons in neural networks, we consider the problem of minimizing E(x)= i=1 n f i ( i x), where i R s , 1in, and each f i ( i x) is a ridge function. We show that when n is small the problem of minimizing E can be treated as one of minimizing univariate functions, and we use the gradient algorithms for minimizing E when n is moderately large. For large n, we present the online gradient algorithms and especially show the monotonicity and weak convergence of the algorithms.  相似文献   

16.
Givenf: R n R n* with some conditions, our aim is to compute a fixed pointx f(x) off; hereR n isn-dimensional Euclidean space andR n* is the collection of nonempty subsets ofR n . A typical application of the algorithm can be motivated as follows: Beginning with the constant mapf 0:R n {0} R n and its fixed pointx 0 = 0, we deformf t ast tof f and follow the pathx t of fixed points off t . Cluster points of thex t 's ast are fixed points off. This research was supported in part by Army Research Office-Durham Contract DAHC-04-71-C-0041 and by National Science Foundation Grant GK-5695.  相似文献   

17.
The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR + n ofR n intoR n can be written as the system of equationsF(x, y) = 0 and(x, y) R + 2n , whereF denotes the mapping from the nonnegative orthantR + 2n ofR 2n intoR + n × Rn defined byF(x, y) = (x 1y1,,xnyn, f1(x) – y1,, fn(x) – yn) for every(x, y) R + 2n . Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR + 2n ontoR + n × Rn. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x 0, y0) and(x, y) R + 2n from an arbitrary initial point(x 0, y0) R + 2n witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.  相似文献   

18.
Summary Let ( k ) k =– be a stationary sequence of random variables, and, forA, let where n is an affine transformation of (has the forma n·+b n,a n>0,b n). ThenM n is a random sup measure, that is, for arbitrary collections of open setsG . We show that the possible limiting random sup measures for such sequences (M n) are those which are stationary (M(·+b)= d M forb) and self-similar (M(a·)= d loga (M) fora>0, where is an affine transformation of ). By applying simple transformations, we need only study stationaryM such thatM(a·)= d aM fora>0. We show that these processes retain some but not all of the properties of the classical case. In particular, we display a nontrivial example such thattM (0,t] is continuous wp1. The classical planar point process representation of extremal processes is a special case of the present approach, but is not adequate for describing all possible limits.  相似文献   

19.
Summary A random timeT is a future independent time for a Markov chain (X n ) 0 ifT is independent of (X T+n ) n / =0 and if (X T+n ) n / =0 is a Markov chain with initial distribution and the same transition probabilities as (X n ) 0 . This concept is used (with the conditional stationary measure) to give a new and short proof of the basic limit theorem of Markov chains, improving somewhat the result in the null-recurrent case.This work was supported by the Swedish Natural Science Research Council and done while the author was visiting the Department of Statistics, Stanford University  相似文献   

20.
Summary For a frictionless, isentropic, unsteady flow of an ideal gas in a tube of constant cross section there is a possibility to calculate the pressures and velocities directly at the ends of the tube, if the boundary conditions are known. The approximative method described is quicker than the ordinary method of characteristics which requires a step by step calculation in the entire physical plane. For Mach numbers up to 0.5, the accuracy is generally sufficient for practical problems.
Symbolliste a Schallgeschwindigkeit,A=a/a 0 - c Wellengeschwindigkeit,C=c/a 0 - d h hydraulischer Durchmesser des Kanals - Isentropenexponent,k=(3–)/(1+) - l Kanallänge - N, Q rechts-und linkslaufende Charakteristik (2)N a ,Q a für Verfahren benötigte AnfangscharakteristikenN G ,Q G Durchschnitt der Gegencharakteristiken (5)N *,Q * verzerrte Charakteristiken (9)N **, Q** für Stossberechnung verzerrte Charakteristiken (12) - p Druck,P=p/p 0 - R spez. Gaskonstate - Dichte, - s Entropie - t Zeit,T=t a 0/l - T Q , T N Laufzeit derN-bzw.Q-Charakteristik durch Kanal - T NQa , T QNa hypothetische Laufzeit (10) - Temperatur - W Gasgeschwindigkeit,W=w/a 0 - X laufende Kanallänge,X=x/l - Y Ausdruck in Grundformel (11) - konstant Indices 0 Bezugszustand - l linkes Kanalende - r rechtes Kanalende - K im Kanal - B im Behälter ausserhalb des Kanals  相似文献   

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