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1.
I prove the existence of a weak solution for the Dirichlet problem of a class of elliptic partial differential systems in general Orlicz–Sobolev spaces , where i=1,…,N,α=1,…,n, u:Ω→RN is a vector-valued function, and the summation convention is used throughout with i,j running from 1 to N and α,β running from 1 to n. 相似文献
2.
Let be a family of polynomials such that , i=1,…,r. We say that the family P has the PSZ property if for any set with there exist infinitely many such that E contains a polynomial progression of the form {a,a+p1(n),…,a+pr(n)}. We prove that a polynomial family P={p1,…,pr} has the PSZ property if and only if the polynomials p1,…,pr are jointly intersective, meaning that for any there exists such that the integers p1(n),…,pr(n) are all divisible by k. To obtain this result we give a new ergodic proof of the polynomial Szemerédi theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If are jointly intersective integral polynomials, then for any finite partition of , there exist i{1,…,k} and a,nEi such that {a,a+p1(n),…,a+pr(n)}Ei. 相似文献
3.
Dual generalized Bernstein basis 总被引:1,自引:0,他引:1
The generalized Bernstein basis in the space Πn of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63–78], We give explicitly the dual basis functions for the polynomials , in terms of big q-Jacobi polynomials Pk(x;a,b,ω/q;q), a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula—relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials—is also given. Further, an alternative formula is given, representing the dual polynomial (0jn) as a linear combination of min(j,n-j)+1 big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by , as well as an identity which may be seen as an analogue of the extended Marsden's identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311–346]. 相似文献
4.
By using a fixed point theorem of strict-set-contraction, some new criteria are established for the existence of positive periodic solutions of the following periodic neutral Lotka–Volterra system with state dependent delays where (i,j=1,2,…,n) are ω-periodic functions and (i=1,2,…,n) are ω-periodic functions with respect to their first arguments, respectively. 相似文献
5.
Mihai Stoiciu 《Journal of Approximation Theory》2006,139(1-2):29
The orthogonal polynomials on the unit circle are defined by the recurrence relation where for any k0. If we consider n complex numbers and , we can use the previous recurrence relation to define the monic polynomials Φ0,Φ1,…,Φn. The polynomial Φn(z)=Φn(z;α0,…,αn-2,αn-1) obtained in this way is called the paraorthogonal polynomial associated to the coefficients α0,α1,…,αn-1.We take α0,α1,…,αn-2 i.i.d. random variables distributed uniformly in a disk of radius r<1 and αn-1 another random variable independent of the previous ones and distributed uniformly on the unit circle. For any n we will consider the random paraorthogonal polynomial Φn(z)=Φn(z;α0,…,αn-2,αn-1). The zeros of Φn are n random points on the unit circle.We prove that for any the distribution of the zeros of Φn in intervals of size near eiθ is the same as the distribution of n independent random points uniformly distributed on the unit circle (i.e., Poisson). This means that, for large n, there is no local correlation between the zeros of the considered random paraorthogonal polynomials. 相似文献
6.
Let , and for k=0,1,…, denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c1, depending only on H, α, and β such that Specializing to the case of Chebyshev polynomials, , we apply this theory to obtain a construction of an exponentially localized polynomial basis for the corresponding L2 space. 相似文献
7.
Let LN+1 be a linear differential operator of order N+1 with constant coefficients and real eigenvalues λ1,…,λN+1, let E(ΛN+1) be the space of all C∞-solutions of LN+1 on the real line. We show that for N2 and n=2,…,N, there is a recurrence relation from suitable subspaces to involving real-analytic functions, and with if and only if contiguous eigenvalues are equally spaced. 相似文献
8.
A d-dimensional positive definite correlation matrix R=(ρij) can be parametrized in terms of the correlations ρi,i+1 for i=1,…,d-1, and the partial correlations ρij|i+1,…j-1 for j-i2. These parameters can independently take values in the interval (-1,1). Hence we can generate a random positive definite correlation matrix by choosing independent distributions Fij, 1i<jd, for these parameters. We obtain conditions on the Fij so that the joint density of (ρij) is proportional to a power of det(R) and hence independent of the order of indices defining the sequence of partial correlations. As a special case, we have a simple construction for generating R that is uniform over the space of positive definite correlation matrices. As a byproduct, we determine the volume of the set of correlation matrices in -dimensional space. To prove our results, we obtain a simple remarkable identity which expresses det(R) as a function of ρi,i+1 for i=1,…,d-1, and ρij|i+1,…j-1 for j-i2. 相似文献
9.
Let, for example,where α>0, k1, and expk=exp(exp(…exp())) denotes the kth iterated exponential. Let {An} denote the recurrence coefficients in the recurrence relation
xpn(x)=Anpn+1(x)+An-1pn-1(x)