An upper dominating set is a minimal in graph. In the Upper Dominating Set problem, goal to find an of maximum size. We study complexity parameterized algorithms for Set, as well its sub-exponential approximation. First, we prove that, under ETH, k -Upper cannot be solved time \(O(n^{o(k)})\) (improving on \(O(n^{o(\sqrt{k})})\)), and same show assumption that any constant ratio r \(\varepsilon...

We determine the exact minimax rate of a Gaussian sequence model under bounded convex constraints, purely in terms local geometry given constraint set $K$ . Our main result shows that risk notation="LaTeX">$\vphantom {_{\int _{\int }}...

We present efficient differentially private algorithms for learning unions of polygons in the plane (which are not necessarily convex). Our $(\alpha,\beta)$--probably approximately correct and $(\varepsilon,\delta)$--differentially using a sample size $\tilde{O}\left(\frac{1}{\alpha\varepsilon}k\log d\right)$, where domain is $[d]\times[d]$ $k$ number edges union polygons. obtained by designing...

For each given $n\ge 2$, we construct a family of entire solutions $u\_\varepsilon (z,t)$, $\varepsilon > 0$, with helical symmetry to the three-dimensional complex-valued Ginzburg–Landau equation $$ \Delta u+(1-|{u}|^2)u=0, \quad (z,t) \in \mathbb{R}^2\times \mathbb{R} \simeq \mathbb{R}^3. These are $2\pi/\varepsilon$-periodic in $t$ and have $n$ helix-vortex curves, asymptotic behavior, as $\...

For a prime number $$p \ge 5$$ , we explicitly construct family of imaginary quadratic fields K with ideal class groups $$Cl_{K}$$ having p-rank $${\mathrm{{rk}}_{p}(Cl_{K})}$$ at least 2. We also quantitatively prove, under the abc-conjecture that for sufficiently large positive real numbers X and any $$\varepsilon $$ $$0< \varepsilon < \frac{1}{p - 1}$$ absolute value discriminant $$d_{K}\le ...

We present a systemic theoretical study of the electronic properties of the quantum dots inserted in quantum dot infrared photodetectors (QDIPs). The strain distribution of three different shaped quantum dots (QDs) with a same ratio of the base to the vertical aspect is calculated by using the short-range valence-force-field (VFF) approach. The calculated results show that the hydrostatic strai...

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, varepsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lattices, and then an extensive set of initial-value calculations, the phenomenology of solutions over the parameter plane is broadly classified. The lattice possesses many...

Erdős, Hajnal and Szemerédi proved that any subset $G$ of vertices a shift graph $\text{Sh}_{n}^{k}$ has the property independence number subgraph induced by satisfies $\alpha(\text{Sh}_{n}^{k}[G])\geq \left(\frac{1}{2}-\varepsilon\right)|G|$, where $\varepsilon\to 0$ as $k\to \infty$. In this note we prove for $k=2$ $n \to \infty$ there are graphs $G\subseteq \binom{[n]}{2}$ with $\alpha(\text...

In this paper, we study a class of quasilinear Schrödinger equation the form \begin{document}$ -\varepsilon^2\Delta u+V(x)u-\varepsilon^2(\Delta(|u|^{2}))u = K(x)|u|^{q-2}u,\quad x\in{\mathbb{R}^N}, $\end{document} where $ V $, K are smooth functions and may vanish at infinity, 2&lt;q&lt;2(2^*) $. We prove existence positive ground state solution which possesses unique local maximum decays expo...