The Complexity of Linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces

We study linear problems defined on tensor products of Hilbert spaces with an additional (anti-) symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the singular values of the univariate problem. Moreover, we show that this algorithm is optimal with respect to a wide class of algorithms and investigate its complexity. We clarify the influence of different (anti-) symmetry conditions on the complexity, compared to the classical unrestricted problem. In particular, for symmetric problems we give characterizations for polynomial tractability and strong polynomial tractability in terms of the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schr\"odinger equation.Comment: Extended version (53 pages); corrected typos, added journal referenc

Tractability of multivariate problems for standard and linear information in the worst case setting: part II

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: 1) the minimal errors for the univariate case decay polynomially fast to zero, 2) the largest singular value for the univariate case is simple and 3) the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces

Film screening and discussion: Citizen Vaclav Havel Goes on Vacation

This is the archive of a film screening and discussion given by Jan Novak, author and filmmaker

Shortest movie: Bose-Einstein correlation functions in e+e- annihilations

Bose-Einstein correlations of identical charged-pion pairs produced in hadronic Z decays are analyzed in terms of various parametrizations. A good description is achieved using Levy stable distributions. The source function is reconstructed with the help of the tau-model.Comment: 6 pages, 3 figures, presented at the 5th Budapest Winter School on Heavy Ion Physic

Quantum Complexity of Integration

It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hoelder classes with d variables. The optimal orders for the complexity of deterministic and (general) randomized methods are known. We obtain the respective optimal orders for quantum algorithms and also for restricted Monte Carlo (only coin tossing instead of general random numbers). To summarize the results one can say that (1) there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if the smoothness is small; (2) there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical methods, if the smoothness is small.Comment: 13 pages, some minor correction