If h is a nondecreasing real valued function and 0 ≤ q ≤ 2, we analyse the boundary behaviour of the gradient of any solution u of −∆u+ h(u) + |∇u|q = f in a smooth N-dimensional domain Ω with the condition that u tends to infinity when x tends to ∂Ω. We give precise expressions of the blow-up which, in particular, point out the fact that the phenomenon occurs essentially in the normal directio...

The minimum cost multiple-source unsplittable flow problem is studied in this paper. A simple necessary condition to get a solution is proposed. It deals with capacities and demands and can be seen as a generalization of the well-known semi-metric condition for continuous multicommdity flows. A cutting plane algorithm is derived using a superadditive approach. The inequalities considered here a...

Sufficient conditions are found for the applicability of generalized law iterated logarithm sums dependent random variables in case where sequence normalizing constants is not necessarily nondecreasing.

In this article, we investigate the existence of a solution arising from the following fractional q-difference boundary value problem by using the p-Laplacian operator: Dq (φp(Dqy(t))) + f (t, y(t)) = 0 (0 < t < 1; 0 < γ < 1; 3 < δ < 4), y(0) = (Dqy)(0) = (D 2 qy)(0) = 0, a1(Dqy)(1) + a2(Dqy)(1) = 0, a1 + |a2| = 0, D0+y(t)|t=0 = 0. We make use of such a fractional q-difference boundary value pr...

We give a representation of the solution for a stochastic linear equation of the form Xt = Yt+ ∫ (0,t] Xs−dZs where Z is a càdlàg semimartingale and Y is a càdlàg adapted process with bounded variation on finite intervals. As an application we study the case where Y and −Z are nondecreasing, jointly have stationary increments and the jumps of −Z are bounded by 1. Special cases of this process a...

We find necessary conditions for every solution of the neutral delay difference equation Δ(rnΔ(yn − pnyn−m)) + qnG(yn−k)= fn to oscillate or to tend to zero as n→∞, where Δ is the forward difference operator Δxn = xn+1 − xn, and pn, qn, rn are sequences of real numbers with qn ≥ 0, rn > 0. Different ranges of {pn}, including pn =±1, are considered in this paper. We do not assume that G is Lipsc...

(ii) {g(n)} is a nondecreasing sequence, and limn→∞ g(n)=∞; (iii) f ∈ (R,R), x f (x) > 0, and f ′(x)≥ 0 for x = 0; (iv) αi, i= 1,2, are quotients of positive odd integers. The domain (L3) of L3 is defined to be the set of all sequences {x(n)}, n ≥ n0 ≥ 0 such that {Ljx(n)}, 0≤ j ≤ 3 exist for n≥ n0. A nontrivial solution {x(n)} of (1.1;δ) is called nonoscillatory if it is either eventually posi...

We study countably piecewise continuous, piecewise monotone interval maps. We establish a necessary and sufficient criterion for the existence of a nondecreasing semiconjugacy to a map of constant slope in terms of the existence of an eigenvector of an operator acting on a space of measures. Then we give sufficient conditions under which this criterion is not satisfied. Finally, we give example...

We show that it is impossible to define big-O notation for functions on more than one variable in a way that implies the properties commonly used in algorithm analysis. We also demonstrate that common definitions do not imply these properties even if the functions within the big-O notation are restricted to being strictly nondecreasing. We then propose an alternative definition that does imply ...

It is well-known that the generating function for tableaux of a given skew shape with r rows where the parts in the i'th row are bounded by some nondecreasing upper and lower bounds which depend on i can be written in form of a determinant of size r. We show that the generating function for tableaux of a given skew shape with r rows and c columns where the parts in the i'th row are bounded by n...