‎Let $L$ be a Lie algebra‎, ‎$mathrm{Der}(L)$ be the set of all derivations of $L$ and $mathrm{Der}_c(L)$ denote the set of all derivations $alphainmathrm{Der}(L)$ for which $alpha(x)in [x,L]:={[x,y]vert yin L}$ for all $xin L$‎. ‎We obtain an upper bound for dimension of $mathrm{Der}_c(L)$ of the finite dimensional nilpotent Lie algebra $L$ over algebraically closed fields‎. ‎Also‎, ‎we classi...

Numerous studies have detected elevated electroencephalographic (EEG) theta/beta ratio (TBR) or theta power in children with attention deficit/hyperactivity disorder (ADHD) and therefore TBR has been suggested to be a promising biomarker of ADHD. At the same time, recent theoretical models have emphasized the heterogeneity of ADHD and the notion that cognitive deficits in ADHD are not fixed but...

We study a model of proxy voting where the candidates, voters, and proxies are all located on real line, instead directly, each voter delegates its vote to closest proxy. The goal is find set that theta-representative, which entails for any anywhere favorite candidate within distance theta This property guarantees strong form representation as voters not required be fixed in advance, or even fi...

Visual counting, a task that aims to estimate the number of objects from an image/video, is open-set problem by nature as population can vary in $$[0,+\infty )$$ theory. However, collected data are limited reality, which means only closed set observed. Existing methods typically model this through regression, while they prone suffer unseen scenes with counts out scope set. In fact, counting has...

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations ar...

It is shown that if $A \subseteq \mathbb{R}^3$ a Borel set of Hausdorff dimension $\dim A \in (3/2,5/2)$, then for a.e. $\theta [0,2\pi)$ the projection $\pi_{\theta}(A)$ $A$ onto 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos \theta, \sin 1)$ satisfies \pi_{\theta}(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim A+1}{3} \right\}$. This improves bound Oberlin and Ober...

In this article, we introduce a new approach towards the statistical learning problem $\operatorname{argmin}_{\rho(\theta) \in \mathcal P_{\theta}} W_{Q}^2 (\rho_{\star},\rho(\theta))$ to approximate target quantum state $\rho_{\star}$ by set of parametrized states $\rho(\theta)$ in $L^2$-Wasserstein metric. We solve estimation considering Wasserstein natural gradient flows for density operator...

For $N\geq 3$, by the seminal paper of Brezis and V\'eron (Arch. Rational Mech. Anal. 75(1):1--6, 1980/81), no positive solutions $-\Delta u+u^q=0$ in $\mathbb R^N\setminus \{0\}$ exist if $q\geq N/(N-2)$; for $11$ $\theta\in \mathbb R$, we prove that nonlinear ell...