We propose the notion of nondecreasing Lyapunov functions which can be used to prove stability or other properties of the system in question. This notion is in particular useful in studying switched or hybrid systems. We illustrate the concept by a general construction of such a nondecreasing Lyapunov function for a class of planar hybrid systems. It is noted that this class encompasses switche...

The Monotone Upper Bound Problem asks for the maximal number M(d, n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d, n) ≤ Mubt(d, n) provided by McMullen’s (1970) Upper Bound Theorem is tight, where Mubt(d, n) is the number of vertices of a dual-to-cyclic d-polytope with n facets. It was recently shown ...

A class of α-potentials represented as the sum of modified Green potential and modified Poisson integral are proved to have the growth estimates Rα,l,l x o x β n|x| h |x| −1 at infinity in the upper-half space of the n-dimensional Euclidean space, where the function h |x| is a positive nondecreasing function on the interval 0,∞ satisfying certain conditions. This result generalizes the growth p...

Interval-censored data occur naturally in many fields and the main feature is that the failure time of interest is not observed exactly, but is known to fall within some interval. In this paper, we propose a semiparametric probit model for analyzing case 2 interval-censored data as an alternative to the existing semiparametric models in the literature. Specifically, we propose to approximate th...

The Robinson–Schensted–Knuth bijection, denoted by RSK, is a bijection between sequences π of real numbers of length n ≥ 1 and ordered pairs 〈P,Q〉 where P = P (π) is a semi–standard tableau with entries π1, . . . , πn, Q = Q(π) is a standard Young tableau with entries one through n, and P and Q both have shape λ = λ(π) where |λ| = n. Schensted’s theorem, which has been generalized by Greene, st...

We design a class of submodular functions meant for document summarization tasks. These functions each combine two terms, one which encourages the summary to be representative of the corpus, and the other which positively rewards diversity. Critically, our functions are monotone nondecreasing and submodular, which means that an efficient scalable greedy optimization scheme has a constant factor...

A production function f is called quasi-sum if there are continuous strict monotone functions F, h1, . . . , hn with F > 0 such that f(x) = F (h1(x1) + · · · + hn(xn)) (cf. [1]). A quasi-sum production function is called quasi-linear if at most one of F, h1, . . . , hn is a nonlinear function. For a production function f , the graph of f is called the production hypersurface of f . In this pape...

This paper investigates stability conditions and positivity of the solutions of a coupled set of nonlinear difference equations under very generic conditions of the nonlinear real functions which are assumed to be bounded from below and nondecreasing. Furthermore, they are assumed to be linearly upper bounded for sufficiently large values of their arguments. These hypotheses have been stated in...

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we add...

TheMonotone Upper Bound Problem (Klee, 1965) asks if the numberM(d, n) of vertices in a monotone path along edges of a d-dimensional polytope with n facets can be as large as conceivably possible: Is M(d, n) = Mubt(d, n), the maximal number of vertices that a d-polytope with n facets can have according to the Upper Bound Theorem? We show that in dimension d = 4, the answer is “yes”, despite the...