Let $\Sg$ be a nonempty set of prime numbers. In the present paper, we continue the study, initiated in a previous paper by the second author, of the combinatorial anabelian geometry of semi-graphs of anabelioids of pro-$\Sg$ PSC-type, i.e., roughly speaking, semi-graphs of anabelioids associated to pointed stable curves. Our first main result is a partial generalization of one of the main combinatorial anabelian results of this previous paper to the case of nodally nondegenerate outer representations, i.e., roughly speaking, a sort of abstract combinatorial group-theoretic generalization of the scheme-theoretic notion of a family of pointed stable curves over the spectrum of a discrete valuation ring. We then apply this result to obtain a generalization, to the case of proper hyperbolic curves, of a certain injectivity result, obtained in another paper by the second author, concerning outer automorphisms of the pro-$\Sg$ fundamental group of a configuration space associated to a hyperbolic curve, as the dimension of this configuration space is lowered from two to one. This injectivity allows one to generalize a certain well-known injectivity theorem of Matsumoto to the case of proper hyperbolic curves

In this article, we study the infinitesimal isometries on tangent sphere bundles over orientable three-dimensional Riemannian manifolds. Focusing on the vector fields which do not preserve fibers, we show the existence of lifts to the bundles of orthonormal frames. These lifts enable us to analyze the infinitesimal isometries by the symmetry of principal fiber bundles. We prove that the tangent sphere bundle admits a non-fiber-preserving infinitesimal isometry if and only if the base manifold has the same constant sectional curvatures as the fibers have. As an application, we classify the infinitesimal isometries on tangent sphere bundles for the three dimensional case.

We prove that the one dimensional travelling wave solutions corresponding to the Walker wall for the Landau-Lifshitz equation are asymptotically stable for small external magnetic field.

Let $(\mathscr{X},\,d,\,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we give a sufficient condition on the pair of weights $(u,\,v)$ so that the fractional integral operator on spaces of homogeneous type is bounded from $L^p(\mathscr{X},\,v)$ to weak $L^q(\mathscr{X},\,u)$ with $1<p\leq q<\infty$.

In this paper, we prove that there exist entire functions which determines the splitting behavior of polynomials at prime. First, to any monic irreducible polynomial and any prime $p$, we associate a function defined on the set of primes which determines whether the polynomial splits completely at $p$ or not. Then we extend them to entire functions.