In this paper we make an exhaustive study of the fourth order linear operator u((4)) + M u coupled with the clamped beam conditions u(0) = u(1) = u'(0) = u'(1) = 0. We obtain the exact values on the real parameter M for which this operator satisfies an anti-maximum principle. Such a property is equivalent to the fact that the related Green's function is nonnegative in [0, 1] x [0, 1]. When M < 0 we obtain the best estimate by means of the spectral theory and for M > 0 we attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation u((4)) + M u = 0. By using the method of lower and upper solutions we deduce the existence of solutions for nonlinear problems coupled with this boundary conditions. (C)

We consider the boundary value problem for the nonlinear Poisson equation with a nonlocal term −Δu = f (u, U g(u)), u| ∂U = 0. We prove the existence of a positive radial solution when f grows linearly in u, using Krasnoselskii's fixed point theorem together with eigenvalue theory. In presence of upper and lower solutions, we consider monotone approximation to solutions.

In this paper we prove the existence of bounded solutions in the real line for the equationü + sign(u) = p(t), where p is a function with average. Some useful density results for the space of functions with zero average are also obtained.

We start by studying the existence of positive solutions for the differential equation, with u ''(0) = u(+infinity) = 0, where a is a positive function, and g is a power or a bounded function. In other words, we are concerned with even positive homoclinics of the differential equation. The main motivation is to check that some well-known results concerning the existence of homoclinics for the autonomous case (where a is constant) are also true for the non-autonomous equation. This also motivates us to study the analogous fourth-order boundary value problem {u( (4)u(+infinity) = u'(+infinity) = 0 for which we also find nontrivial (and, in some instances, positive) solutions. (C)

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We prove general existence results forwhere f and g need not be continuous or monotone. Moreover neither f nor g need be bounded around, respectively, x0 and x1, thus allowing singularities in the equation. Several other basic topics such as uniqueness, continuation, extremality and periodicity are studied in our general framework.

In this paper, we model mass running urban races, taking into consideration several conditioning factors. The main goal is to find ideal configurations of the start of the race, splitting it into several waves, reducing the density of athletes and the overall time lost, when comparing the normal race results with a race without density constraints. This model takes into account distinct realistic runners’ profiles, changes in slope and width on the race course and its influence on the running pace. Moreover, density levels, dynamics of the start of the race and time between the departure of waves are also considered.

We consider a time series with real data from a water lift station, equipped with three water pumps which are activated and deactivated depending on certain starting and halting thresholds. Given the water level and the number of active pumps, both read every 5 min, we aim to infer when each pump was activated or deactivated. To do so, we build an algorithm that sets a hierarchy of criteria based on the past and future of a given interval to identify which thresholds have been crossed during that interval. We then fill the gaps between the 5 min time steps, modeling the water level continuously with a piecewise linear function. This filling takes into account not only every water level reading and every previously identified change of status, but also the fact that activation and deactivation of a pump has no immediate effect on water level. This allows for the fulfillment of the ultimate objective of the problem in its real context, which is to provide the water management company an estimate of how long each pump has been working. Additionally, our estimates correct the errors contained in the time series regarding the number of active pumps.