The dartboard problem is to arrange n numbers on a circle to obtain maximum risk, which is the sum of the q-th power of the absolute differences of adjacent numbers, for q ≥ 1. Curtis showed that the dartboard problem admits a greedy algorithm. We generalize the dartboard problem by considering more circles and the goal is to arrange kn number on k circles to obtain the maximum risk. In this pa...

We study the kernel Kn,s(z) of the remainder term Rn,s( f ) of Gauss–Turán–Kronrod quadrature rules with respect to one of the generalized Chebyshev weight functions for analytic functions. The location on the elliptic contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective L∞-error bounds of Gauss–Turán–Kronrod quadratures. Following Kronrod,...

Herein, rhombohedral self-doped KNbO3 (S-KN) nanorods were fabricated via a one pot, solvothermal method without using any surfactant. The presence of Nb(4+) in S-KN greatly narrows its band gap and thus extends its photoresponse from UV to the visible light region. Moreover, S-KN/Nb4N5 nanorod heterostructures were obtained by nitriding S-KN nanorods for different times, which exhibited signif...

Let the random variable Zn,k denote the number of increasing subsequences of length k in a random permutation from Sn, the symmetric group of permutations of {1, ..., n}. In a recent paper [4] we showed that the weak law of large numbers holds for Zn,kn if kn = o(n 2 5 ); that is, lim n→∞ Zn,kn EZn,kn = 1, in probability. The method of proof employed there used the second moment method and demo...

We study the algebraic properties of the series Kn of semigroups, which is inspired by [Ki] and has origins in convexity theory. In particular, we describe Green’s relations on Kn, prove that there exists a faithful representation of Kn by n × n matrices with non-negative integer coefficients (and even explicitly construct such a representation), and prove that Kn does not admit a faithful repr...