First, we prove the function g is an injection (i.e., g is one-to-one). We need to prove that if g(m) = g(n) --(1), where m and n are two integers, then m = n --(2). Since g(m) (and also g(n)) can be defined by two different formulas depending whether the argument m (and n) is even or odd, we consider the following 3 cases: (Case 1) Both m and n are even. In this case, (1) implies 1 – m = 1 – n...

Our research program at the Brookhaven National Laboratory’s Alternating Gradient Synchrotron (AGS) focuses on rare kaon decays. Only at the AGS intense low-momentum neutral and charged kaon beams are still available today. Though Experiment BNL E-865 finished data-taking in 1998 the analysis of its high statistics data sample on semi-leptonic and leptonic charged kaon decays still supplies nov...

Lemmas 1, 2, 3 and 4, and Corollary 1, were originally derived by Toulis and Airoldi (2014). These intermediate results (and Theorem 1) provide the necessary foundation to derive Lemma 5 (only in this supplement) and Theorem 2 on the asymptotic optimality of θ̄n, which is the key result of the main paper. We fully state these intermediate results here for convenience but we point the reader to t...

We show that unconverged stochastic gradient descent can be interpreted as a procedure that samples from a nonparametric approximate posterior distribution. This distribution is implicitly defined by the transformation of an initial distribution by a sequence of optimization steps. By tracking the change in entropy over these distributions during optimization, we form a scalable, unbiased estim...

This paper proposes an efficient probabilistic method that computes combinatorial gradient fields for two dimensional image data. In contrast to existing algorithms, this approach yields a geometric Morse-Smale complex that converges almost surely to its continuous counterpart when the image resolution is increased. This approach is motivated using basic ideas from probability theory and builds...

We verify strong rates of convergence for a time-implicit, finite-element based space-time discretization the backward stochastic heat equation, and forward-backward equation from optimal control. The fully discrete version is then used within gradient descent algorithm to approximately solve linear-quadratic control problem driven by additive noise. This work thus giving theoretical foundation...

Topology optimization methods for inverse design of nano-photonic systems have recently become extremely popular and are presented in various forms under names. Approaches comprise gradient non-gradient based algorithms combined with more or less systematic ways to improve convergence, discreteness solutions satisfaction manufacturing constraints. We here provide a tutorial the efficient struct...

The introduction of Eurocode in Kazakhstan allows for the application modern technological innovations and elimination technical barriers realization international projects. It is significant to study standards design requirements provided Eurocode. This presents a comparative analysis Kazakhstani European approaches geotechnical foundations provides methods considered codes practice. Three dif...