Generalised Mertens and Brauer-Siegel Theorems

In this article, we prove a generalisation of the Mertens theorem for prime numbers to number fields and algebraic varieties over finite fields, paying attention to the genus of the field (or the Betti numbers of the variety), in order to make it tend to infinity and thus to point out the link between it and the famous Brauer-Siegel theorem. Using this we deduce an explicit version of the generalised Brauer-Siegel theorem under GRH, and a unified proof of this theorem for asymptotically exact families of almost normal number fields

202 Helvetica Physica Acta, Vol. 51 (1978), Birkhliuser Verlag, Basel Stochastic processes II:

Abstract. Linear and nonlinear response theory are developed for stationary Markov systems describing systems in equilibrium and nonequilibrium. Generalized fluctuation theorems are derived which relate the response function to a correlation of nonlinear fluctuations of the unperturbed stationary process. The necessary and sufficient stochastic operator condition for the response tensor,;�:(t), of classical nonlinear stochastic processes to be linearly related to the two-time correlations of the fluctuations in the stationary state (fluctuation theorems) is given. Several classes of stochastic processes obeying a fluctuation theorem are presented. For example, the fluctuation theorem in equilibrium is recovered when the system is described in terms of a mesoscopic master equation. We also investigate generalizations of the Onsager relations for non-equilibrium systems and derive sum rules. Further, an exact nonlinear integral equation for the total response is derived. An efficient recursive scheme for the calculation of general correlation functions in terms of continued fraction expansions is given. The purpose of this work on stochastic Markov processes is to develop a general scheme for the calculation of transport coefficients for systems whose unperturbed time-dependence is described by a master equation of a stationary Markov process

Supplementary Material for “Combinatorial Partial Monitoring Game with Linear Feedback and Its Application”.

If the reader will recall, we have the following problem-specific constants in the main text: the size of the global observer set |σ|, parameter L> 0 from the continuity assumption, error bound βσ = max ‖(M ⊺ σ Mσ) −1 ∑ |σ| i=1 M ⊺ xiMxi (νi − ν0)‖2, where the max is taken from ν0, ν1, · · · , ν |σ | ∈ [0, 1] n, and the maximum difference in the expected reward Rmax = maxx1,x2∈X,ν∈[0,1] n |r(x1, ν) − r(x2, ν)|. For technical reasons, we also defined φ(ν) = max(min(ν, ⃗1), ⃗0) to adjust ν to the nearest vector in [0, 1] n, and r(x, ν) = r(x, φ(ν)), ∀ν ∈ R n \ [0, 1] n to preserve the Lipschitz continuity throughout R n. To make our proof clearer, we define v(t) as the state of any variable v by the end of time step t. Our analysis is based on the snapshot of all variables just before the statement t ← t + 1 (Line 14 and 30). One batch processing in exploration phase is called one round, and then nσ is increased by 1. Denote ˆν (j) as the estimated mean of outcomes after j rounds of exploration. For example, at time t, the estimated mean of outcomes is ˆν(t) and the exploration counter is nσ(t), so we have ˆν (nσ(t)) = ˆν(t). And for time step t + 1, the player will use the previous knowledge of ˆν(t) to get ˆx(t + 1) = argmax x∈X r(x, ˆν(t)) and ˆx − (t + 1) = argmax x∈X \{ˆx(t+1)} r(x, ˆν(t)). In the following analysis, the frequency function is set to fX (t) = ln t + 2 ln |X |. Note that by using fX (t), we can construct the confidence interva