The ratio $\epsilon'/\epsilon$ measures the size of direct CP violation in $K_L\to\pi\pi$ decays $(\epsilon^\prime)$ relative to indirect one described by $\epsilon$ and is very sensitive new sources violation. As such it played a prominent role particle physics already for 45 years. Due smallness its measurement required heroic efforts 1980s 1990s on both sides Atlantic with final results pres...

We consider the stochastic heat equation $$\partial _{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda )u$$ ? s = 1 2 ? + ( ? V , y ) - ? with a smooth space-time-stationary Gaussian random field V(s, y), in dimensions $$d\ge 3$$ d ? 3 an initial condition $$u(0,x)=u_0(\varepsilon x)$$ 0 x ? and suitably chosen $$\lambda \in {\mathbb {R}}$$ ? R . It is known that, for $$\beta $$ small enough, di...

Given a set P of points in $$\mathbb {R}^2$$ , point burning process is discrete time to burn all the where fires must be initiated at P. Specifically, starts with single burnt from P, and each subsequent step, burns that are within one unit distance currently points, as well other unburnt (if exists). The number smallest steps required If we allow fire anywhere then called an process. One can ...

We study the renormalisation of non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric scalar field theory with interaction $\phi^2(i\phi)^\varepsilon$ using Wilsonian approach and without any expansion in $\varepsilon$. Specifically, we solve Wetterich equation local potential approximation, both ultraviolet regime loop expansion. calculate scale-dependent effective its infrared limit. The is found ...

The purpose of the present paper is to study computation complexity deep ReLU neural networks approximate functions in H\"older-Nikol'skii spaces mixed smoothness $H_\infty^\alpha(\mathbb{I}^d)$ on unit cube $\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\in H_\infty^\alpha(\mathbb{I}^d)$, we explicitly construct nonadaptive and adaptive having an output that approximates $f$ with...