We analyze algorithms for approximating a function $f(x) = \Phi x$ mapping $\Re^d$ to $\Re^d$ using deep linear neural networks, i.e. that learn a function $h$ parameterized by matrices $\Theta_1,...,\Theta_L$ and defined by $h(x) = \Theta_L \Theta_{L-1} ... \Theta_1 x$. We focus on algorithms that learn through gradient descent on the population quadratic loss in the case that the distribution...

‎Let $\Omega$ be a bounded domain in $R^{n}$ with smooth boundary‎ ‎$\partial\Omega$‎. ‎In this article‎, ‎we will investigate the spectral‎ ‎properties of non-self adjoint elliptic differential operator\\‎ ‎$(Au)(x)=-\sum^{n}_{i,j=1}\left(\omega^{2\alpha}(x)a_{ij}(x)‎ ‎\mu(x)u'_{x_{i}}(x)\right)'_{x_{j}}$‎, ‎acting Hilbert space ‎$H=L^{2}{(\Omega)}$. Dirichlet-type boundary conditions‎. ‎Here‎...

Double blind $T$-private information retrieval (DB-TPIR) enables two users, each of whom specifies an index ($\theta_1, \theta_2$, resp.), to efficiently retrieve a message $W(\theta_1,\theta_2)$ labeled by the indices, from set $N$ servers that store all messages $W(k_1,k_2), k_1\in\{1,2,\cdots,K_1\}, k_2\in\{1,2,\cdots,K_2\}$, such users' indices are kept private any up $T_1,T_2$ colluding se...

We investigate the localization transition for a simple model of interface which interacts with an inhomonegeous defect plane. The is modeled by graph function $\phi: \mathbb Z^2 \to Z$,and disorder given fixed realization field IID centered random variables$(\omega_x)_{x\in Z^2}$. Hamiltonian system depends on three parameters $\alpha,\beta>0$ and $h\in R$ determine respectively intensity near...

We implement the cellular automata model proposed by Stauffer and Weisbuch in 1992 to describe the response of the immune system to antigens in the presence of medications. The model contains two thresholds, $\theta_1$ and $\theta_2$, suggested by de Boer, Segel, and Perelson to present the minimum field needed to stimulate the proliferation of the receptors and to suppress it, respectively. Th...

We consider the problem of simultaneous approximation real numbers $\theta_1, …,\theta_n$ by rationals and dual approximating zero values linear form $x_0+\theta_1x_1+…+\theta_nx_n$ at integer points. In this setting we analyse two transference inequalities obtained Schmidt Summerer. present a rather simple geometric observation which proves their result. also derive several previously unknown ...

We present chemical abundances and velocities of five stars between 0.3 kpc to 1.1 from the center Tucana II ultra-faint dwarf galaxy (UFD) high-resolution Magellan/MIKE spectroscopy. find that every star is deficient in metals (-3.6 < [Fe/H] -1.9) neutron-capture elements as characteristic UFD stars, unambiguously confirming their association with II. Other (e.g., C, iron-peak) largely follow ...

We investigate strong field gravitational lensing by rotating Simpson-Visser black hole, which has an additional parameter ($0\leq l/2M \leq1$), apart from mass ($M$) and rotation ($a$). A metric correspond to (i) a Schwarzschild for $l/2M=a/2M=0$ $ M \neq 0 $, (ii) Kerr $l/2M=0$, $|a/2M|< 0.5$ (iii) regular hole 0.5$, $l/2M$ in the range $0<l/2M<0.5 + \sqrt{\left(0.5\right)^2-(a/2M)^2}$, (iv) ...