with kn ≥ 0, rn ≥ 0, kn → 1, rn → 0 n → ∞ , for each x, y ∈ C, n 0, 1, 2, . . . . If kn 1 and rn 0, 1.1 reduces to nonexpansive mapping; if rn 0, 1.1 reduces to asymptotically nonexpansive mapping; if kn 1, 1.1 reduces to asymptotically nonexpansive-type mapping. So, a generalized asymptotically nonexpansive mapping is much more general than many other mappings. Browder 1 introduced the followi...

In a recent work, the combinatorial interpretation of the polynomial α(n; k1, k2, . . . , kn) counting the number of Monotone Triangles with bottom row k1 < k2 < · · · < kn was extended to weakly decreasing sequences k1 ≥ k2 ≥ · · · ≥ kn. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Genera...

In this paper, we present an involution on some kind of colored k-ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers Ck,γ(n) = γ kn+γ ( kn+γ n ) . From the combinatorial sum, we refine the formula for k-ary trees and obtain an implicit formula for the generating function of the generalized Catalan numbers which obviously implies a Van...

The three-parameter Indian buffet process is generalized. The possibly different role played by customers is taken into account by suitable (random) weights. Various limit theorems are also proved for such generalized Indian buffet process. Let Ln be the number of dishes experimented by the first n customers, and let Kn = (1/n) ∑n i=1 Ki where Ki is the number of dishes tried by customer i. The...

In general, the optimal computational complexity of Arnoldi iteration is O(kN) for solving a generalized eigenvalue problem, with k being the number of dominant eigenvalues and N the matrix size. In this work, we reduce the computational complexity of the Arnoldi iteration from O(kN) to O(N), thus paving the way for full-wave extraction of very large-scale onchip interconnects, the k of which i...

A weighted embedding for the generalized pontogram {Kn(t): 06t61} corresponding pointwise to a renewal process {N (s): 06s¡∞} via Kn(t)=n−1=2(N (nt)− tN (n)) is studied in this paper. After proper normalization, weak convergence results for the processes {Kn(t): 06t61} are derived both in sup-norm as well as in Lp-norm. These results are suggested to serve as asymptotic testing devices for dete...

It has been shown [Balister, 2001] that if n is odd and m1, . . . , mt are integers with mi ≥ 3 and ∑t i=1 mi = |E(Kn)| then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1, . . . , mt. This result was later generalized [Balister, to appear] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider the corresponding questions for direc...

It is well known that fluctuations of marginals high-dimensional random vectors satisfy a certain concentration estimate called the thin shell condition are approximately Gaussian. In this article we identify general on sequence under which one can exponential decay rate large deviation probabilities corresponding marginals. More precisely, consider projection an n-dimensional vector onto kn-di...