Let $X$ be a set and $\mathcal{T}_X$ the full transformation semigroup on $X$. For partition $\mathcal{P}$ of $X$, we consider semigroups $T(X, \mathcal{P}) = \{f\in \mathcal{T}_X| (\forall X_i\in (\exists X_j \in \mathcal{P})\;X_i f \subseteq X_j\}$, $\Sigma(X, T(X, \mathcal{P})|(\forall X_i \mathcal{P})\; Xf \cap \neq \emptyset\}$, $\Gamma(X, \mathcal{T}_X|(\forall \mathcal{P})(\exists X_j\in...

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...

This article concerns the existence and multiplicity of radially symmetricnodal solutions to nonlinear equation $$\displaylines{ -\mathcal{M}_\mathcal{C}^{\pm}(D^2u)=\mu f(u) \quad \text{in } \mathcal{B},\cru=0 \text{on \partial\mathcal{B}, }$$ where \(\mathcal{M}_\mathcal{C}^{\pm}\) are general Hamilton-Jacobi-Bellman operators, (\mu\) is a real parameter \(\mathcal{B}\) unit ball. By using bi...

In this article we study localisation of spectral sums $\{S_R\}_{R > 0}$ associated to the sub-Laplacian $\mathcal{L}$ on Heisenberg Group $\mathbb{H}^d$ where $S_R f := \int_0^R dE_{\lambda }f$, with $\mathcal{L} = \int_0^{\infty} \lambda \, dE_{\lambda}$ being resolution $\mathcal{L}.$ We prove that for any compactly supported function $f \in L^2(\mathbb{H}^d)$, and $\gamma < \frac{1}{2}$, $R...

In this paper, we study almost nef regular foliations. We give a structure theorem of smooth projective variety $X$ with an foliation $\mathcal{F}$: admits morphism $f: X \rightarrow Y$ rationally connected fibers such that $\mathcal{F}$ is pullback numerically flat on $Y$. Moreover, $f$ characterized as relative MRC fibration algebraic part $\mathcal{F}$. As corollary, tangent bundle generical...

We present the coupling of torsion scalar T and trace energy-momentum tensor $\mathcal {T}$ , which produces new modified f ( ) $f(T,\mathcal {T})$ gravity. Moreover, we consider functional form = α + β 2 {T}) =\alpha \mathcal {T}+\beta T^2$ where are free parameters. As an alternative to a cosmological constant, theory may offer theoretical explanation late-time acceleration. The recent observ...