An example of a non-Borel locally-connected finite-dimensional topological group

Answering a question posed by S.Maillot in MathOverFlow, for every $n\in\mathbb N$ we construct a locally connected subgroup $G\subset\mathbb R^{n+1}$ of dimension $dim(G)=n$, which is not locally compact.Comment: 2 page

Criteria of efficiency for conformal prediction

We study optimal conformity measures for various criteria of efficiency of classification in an idealised setting. This leads to an important class of criteria of efficiency that we call probabilistic; it turns out that the most standard criteria of efficiency used in literature on conformal prediction are not probabilistic unless the problem of classification is binary. We consider both unconditional and label-conditional conformal prediction.Comment: 31 page

Matrix Metalloproteinase-9 and Inflammation in Different Types of Multiple Sclerosis

Different clinical courses of multiple sclerosis, heterogeneity of its clinical implications, different effect of immunomodulatory therapy for the same clinical forms implies various pathogenetic mechanisms of central nervous system damage at this disease. Applicability of immunological and biochemical markers for the estimation of immunocorrecting and anti-inflammatory therapy efficacy is important. This research aims at improvement of pathological process stages diagnostics at multiple sclerosis and further therapy optimization depending on the activity of the inflammatory process. In the article matrix metalloproteinase-9 rate was assessed in 135 patients with multiple sclerosis of different course types and at different activity stages of the pathological process. The highest matrix metalloproteinase-9 rates were in patients with relapsing-remitting type at the stage of exacerbation, with the lowest rate being in patients with primary-progressive multiple sclerosis. Determination of matrix metalloproteinase-9 rate allows to assess the degree of inflammatory process expression and to monitor the efficacy of multiple sclerosis treatment

Rough paths in idealized financial markets

This paper considers possible price paths of a financial security in an idealized market. Its main result is that the variation index of typical price paths is at most 2, in this sense, typical price paths are not rougher than typical paths of Brownian motion. We do not make any stochastic assumptions and only assume that the price path is positive and right-continuous. The qualification "typical" means that there is a trading strategy (constructed explicitly in the proof) that risks only one monetary unit but brings infinite capital when the variation index of the realized price path exceeds 2. The paper also reviews some known results for continuous price paths and lists several open problems.Comment: 21 pages, this version adds (in Appendix C) a reference to new results in the foundations of game-theoretic probability based on Hardin and Taylor's work on hat puzzle

Extreme State Aggregation Beyond MDPs

We consider a Reinforcement Learning setup where an agent interacts with an environment in observation-reward-action cycles without any (esp.\ MDP) assumptions on the environment. State aggregation and more generally feature reinforcement learning is concerned with mapping histories/raw-states to reduced/aggregated states. The idea behind both is that the resulting reduced process (approximately) forms a small stationary finite-state MDP, which can then be efficiently solved or learnt. We considerably generalize existing aggregation results by showing that even if the reduced process is not an MDP, the (q-)value functions and (optimal) policies of an associated MDP with same state-space size solve the original problem, as long as the solution can approximately be represented as a function of the reduced states. This implies an upper bound on the required state space size that holds uniformly for all RL problems. It may also explain why RL algorithms designed for MDPs sometimes perform well beyond MDPs.Comment: 28 LaTeX pages. 8 Theorem