Let $\Gamma$ be a group and $\mathscr{C}$ class of groups endowed with bi-invariant metrics. We say that is $\mathscr{C}$-stable if every $\varepsilon$-homomorphism $\Gamma \rightarrow G$, $(G,d) \in \mathscr{C}$, $\delta_\varepsilon$-close to homomorphism, $\delta_\varepsilon\to 0$ when $\varepsilon\to 0$. If $\delta_\varepsilon < C \varepsilon$ for some $C$ we $ \mathscr{C} $-stable linear ra...

We consider a fractal refinement of Carleson's problem for pointwise convergence solutions to the periodic Schr\"odinger equation their initial datum. For $\alpha \in (0,d]$ and \[ s < \frac{d}{2(d+1)} (d + 1 - \alpha), \] we find function in $H^s(\mathbb{T}^d)$ whose corresponding solution diverges limit $t \to 0$ on set with strictly positive $\alpha$-Hausdorff measure. conjecture this regula...

let $r$ be a ring and $m$ a right $r$-module with $s=end_r(m)$. a module $m$ is called semi-projective if for any epimorphism $f:mrightarrow n$, where $n$ is a submodule of $m$, and for any homomorphism $g: mrightarrow n$, there exists $h:mrightarrow m$ such that $fh=g$. in this paper, we study sgq-projective and$pi$-semi-projective modules as two generalizations of semi-projective modules. a m...

Trigraph list homomorphism problems (also known as list matrix partition problems) have generated recent interest, partly because there are concrete problems that are not known to be polynomial time solvable or NP -complete. Thus while digraph list homomorphism problems enjoy dichotomy (each problem is NP -complete or polynomial time solvable), such dichotomy is not necessarily expected for tri...

We define the Homomorphism Extension (HomExt) problem: given a group G, a subgroup M ≤ G and a homomorphism φ : M → H , decide whether or not there exists a homomorphism φ̃ : G → H extending φ, i.e., φ̃|M = φ. This problem arose in the context of list-decoding homomorphism codes but is also of independent interest, both as a problem in computational group theory and as a new and natural problem i...

It has been proposed that extracellular ATP may be involved in visceral mechanosensory transduction by activating ligand-gated ion channels (P2X receptors). In this study, we have investigated the effects of the P2X(3) agonist alpha,beta-methylene ATP (alpha,beta-meATP) and antagonist 2',3'-O-trinitrophenyl-ATP (TNP-ATP) on pelvic afferents innervating the urinary bladder using an in vitro mous...

There is a deep astrophysical interest in the structure of $^{19}\mathrm{F}$ states close to $\ensuremath{\alpha}$-decay threshold. The nuclear these important for understanding development $\ensuremath{\alpha}$ clustering $^{20}\mathrm{Ne}$ region. Emergence clustered states, and generally that favor coupling reaction channels near corresponding decay thresholds, currently special theoretical ...