We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds.
The paper deals with applying the strong aggregating algo- rithm to games with asymmetric loss function. A particular example of such games is the problem of time series forecasting where specific losses from under-forecasting and over-forecasting may vary considerably. We use the aggregating algorithm for building compositions of adaptive fore- casting algorithms. The paper specifies sufficient conditions under which a composition based on the aggregating algorithm performs as well as the best of experts. As a result, we find a theoretical bound for the loss process of a given composition under asymmetric loss function. Finally we compare the composition based on the aggregating algorithm to other well-known compositions in an experiment with real data.
Market graph is built on the basis of some similarity measure for financial asset returns. The paper considers two similarity measures: classic Pearson correlation and sign correlation. We study the associated market graphs and compare the conditional risk of the market graph construction for these two measures of similarity. Our main finding is that the conditional risk for the sign correlation is much better than for the Pearson correlation for larger values of threshold for several probabilistic models. In addition, we show that for some model the conditional risk for sign correlation dominates over the conditional risk for Pearson correlation for all values of threshold. These properties make sign correlation a more appropriate measure for the maximum clique analysis.
The problem of comparison of several branches efficiency is formulated as a multiple decision problem. The main difficulty to handle this problem lies in the compatibility condition. Solution of this difficulty, based on a method of combination of testing compatible generating hypotheses is given. The additivity condition of the loss function is investigated. This condition is used to construct the optimal multiple decision statistical procedures for comparison of network branches efficiency. Some examples are given.
This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. The appendix is an expanded version of my notes on open problems posted on the site of this workshop. It contains formulations of some open problems and the relevant comments.
We present some results recently obtained by the authors for the so-called constrained stochastic and parabolic equations, including Navier–Stokes Equations.