Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the hig...

Descriptive set theory was originally developed on Polish spaces. It was later extended to $\omega$-continuous domains [Selivanov 2004] and recently to quasi-Polish spaces [de Brecht 2013]. All these spaces are countably-based. Extending descriptive set theory and its effective counterpart to general represented spaces, including non-countably-based spaces has been started in [Pauly, de Brecht ...

We present a general framework for balancing expressions (terms) in form of so called tree straight-line programs. The latter can be seen as circuits over the free term algebra extended by contexts (terms with a hole) and the operations which insert terms/contexts into contexts. It is shown that for every term one can compute in DLOGTIME-uniform TC$^0$ a tree straight-line program of logarithmi...

$P_c$ resonances are studied in the approach of quark model and group theory. It is found that there totally 17 possible pentaquark states with contents $q^3Q \bar Q$ ($q$ $u$ $d$ quarks; $Q$ $c$ quark) compact picture, where hidden heavy may take color singlet-singlet ($[111]_{{qqq}}\otimes [111]_{{c c}}$) octet-octet ($[21]_{{qqq}}\otimes [21]_{{c configurations. The partial decay widths calc...

A graph class $\mathcal{F}$ has the induced Erd\H{o}s-P\'osa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $\mathcal{F}$, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $\mathcal{F}$. Kim and Kwon (SODA'18) showed that for a cy...

Stimulated by the meson-baryon molecular interpretations of $P_c$ states ($P_c(4312)/P_c(4440)/P_c(4457)$), we systematically study interactions between an $S-$wave charm-strange baryon $\Xi_c^{(\prime,*)}$ and anti-charmed meson $\bar{D}^{(*)}$ in a coupled channel analysis. Effective potentials for $\Xi_c^{(\prime,*)}\bar{D}^{(*)}$ one-boson-exchange model can be related to those $\Sigma_c^{(...

Distributed optimization algorithms are essential for training machine learning models on very large-scale datasets. However, they often suffer from communication bottlenecks. Confronting this issue, a communication-efficient primal-dual coordinate ascent framework (CoCoA) and its improved variant CoCoA+ have been proposed, achieving a convergence rate of $\mathcal{O}(1/t)$ for solving empirica...

Hierarchical classification is supervised multi-class classification problem over the set of class labels organized according to a hierarchy. In this report, we study the work by Ramaswamy et. al. on hierarchical classification over symmetric tree distance loss. We extend the consistency of hierarchical classification algorithm over asymmetric tree distance loss. We design a $\mathcal{O}(nk\log...