A-posteriori error estimates for optimal control problems with state and control constraints

We discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints. We provide the first reliable a-posteriori error estimator that contains only computable quantities for this class of problems. Moreover, we show, that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem. The theory is illustrated by numerical tests.     

On summability of nonlinear mappings: a new approach

The main goal of this paper is to characterize arbitrary nonlinear (non-multilinear) mappings ${f:X_{1}\times\cdots\times X_{n}\rightarrow Y}$ between Banach spaces that satisfy a quite natural Pietsch Domination-type theorem around a given point ${(a_{1},\ldots,a_{n})\in X_{1}\times\cdots\times X_{n}}$ . As a consequence of our approach a notion of weighted summability arises naturally, which may be an interesting topic for further investigation.     

Level 17 Ramanujan–Sato series

The Ramanujan Journal - Two level 17 modular functions $$\begin{aligned} r = q^2 \prod _{n=1}^{\infty } (1-q^{n})^{\left( \frac{n}{17} \right) },\qquad s = q^{2} \prod _{n=1}^{\infty } \frac{(1 -...     

Optimality of the Expert Rule Under Partial Information

We study the uncertain dichotomous choice model. In this model a group of decision makers is required to select one of two alternatives. The applications of this model are relevant to a wide variety of areas, such as medicine, management and banking. The decision rule may be the simple majority rule; however, it is also possible to assign more weight to the opinion of members known to be more qualified. The extreme example of such a rule is the expert decision rule. We are concerned with the probability of the expert rule to be optimal. Our purpose is to investigate the behaviour of this probability as a function of the group size for several rather general types of distributions. One such family of distributions is that where the density function of the correctness probability is a polynomial (on the interval [1/2,1]). Our main result is an explicit formula for the probability in question. This contains formerly known results as very special cases.     

Worst Traffic Passing Virtual Frame Regulation: Analysis with Dynamic Programming

This paper studies a discrete-time queueing system in which the traffic regulation places an upper bound on the number of arrivals in each of the non-overlapping time windows (virtual frames) of a fixed length. The worst traffic passing through this traffic regulation is characterized. As a criterion for traffic comparison, this paper employs the weighted sum of queueing delays with the weighting factors decaying geometrically over the time horizon. This paper uses dynamic programming techniques and convexity argument to characterize the worst traffic.     

The $${\bar\mu}$$-invariant of Seifert fibered homology spheres and the Dirac operator

We derive a formula for the [`(m)]{\bar\mu}-invariant of a Seifert fibered homology sphere in terms of the η-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres.     

Nevanlinna counting function and Carleson function of analytic maps

We show that the maximal Nevanlinna counting function and the Carleson function of analytic self-maps of the unit disk are equivalent, up to constants.     

Regge?s Einstein-Hilbert functional on the double tetrahedron

The double tetrahedron is the triangulation of the three-sphere gotten by gluing together two congruent tetrahedra along their boundaries. As a piecewise flat manifold, its geometry is determined by its six edge lengths, giving a notion of a metric on the double tetrahedron. We study notions of Einstein metrics, constant scalar curvature metrics, and the Yamabe problem on the double tetrahedron, with some reference to the possibilities on a general piecewise flat manifold. The main tool is analysis of Regge?s Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds. We study the Einstein-Hilbert-Regge functional on the space of metrics and on discrete conformal classes of metrics.     

Cotype and absolutely summing linear operators

New applications of cotype to the theory of absolutely summing linear operators between Banach spaces are proved in this paper. Among other consequences we extend/complement some classical results of Bennett (Duke Math J 44:603?C639, 1977) on the existence of non-absolutely summing operators between ? _p spaces and of Davis and Johnson (Stud Math 51:81?C85, 1974) on the existence of compact non-absolutely summing linear operators. We also point out that some of our results are sharp.