We consider the question introduced by [16] of identifying all $$\varepsilon $$ -optimal arms in a finite stochastic multi-armed bandit with Gaussian rewards. give two lower bounds on sample complexity any algorithm solving problem confidence at least $$1-\delta . The first, unimprovable asymptotic regime, motivates design Track-and-Stop strategy whose average is asymptotically optimal when ris...

In this paper, we shall study the convergence of Taylor approximations for backward Loewner differential equation (driven by Brownian motion) near origin. More concretely, whenever initial condition (which lies in upper half plane) is small and has form $Z_{0} = \varepsilon i$, show these exhibit an $O(\varepsilon)$ error provided time horizon $\varepsilon^{2+\delta}$ $\delta > 0$. Statements t...

The response of a semi-infinite ocean to slowly travelling atmospheric perturbation crossing the coast provides simple example breakdown nearly geostrophic balance induced by boundary. We examine this in linear shallow-water model at small Rossby number $\varepsilon \ll 1$. Using matched asymptotics we show that long Kelvin wave, with $O(\varepsilon^{-1})$ length scale and $O(\varepsilon)$ ampl...

We present the best known separation between tree-like and general resolution, improving on the recent $exp(n^{\varepsilon}$ separation of [BEGJ98]. This is done by constructing a natural family of contradictions, of size $n$, that have $O(n)$-size resolution refutations, but only $exp(\Omega(n / log n))$size tree-like refutations. This result implies that the most commonly used automated theor...

In this paper, we study the line bundle mean curvature flow defined by Jacob and Yau. The is a kind of parabolic flows to obtain deformed Hermitian Yang-Mills metrics on given K\"ahler manifold. goal paper give an $\varepsilon$-regularity theorem for flow. To establish theorem, provide scale invariant monotone quantity. As critical point quantity, define self-shrinker solution Liouville type se...

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network consisting two symmetrically linked star subnetworks identical oscillators with shear Kuramoto--Sakaguchi coupling. show that states may be metastable or asymptotically stable. If intra-star coupling strength is order $\varepsilon$, persist on time scales at least $1/\vare...