For the study of mixed type surfaces with singular points, we define a lightcone frame of a surface in the Lorentz-Minkowski 3-space. We give the existence and uniqueness for the basic invariants of the lightcone framed surface. By using the lightcone frame, we investigate differential geometric properties of the mixed type surface. Then we show a method to recognize the type of points on the surface. Moreover, we give the necessary and sufficient conditions for a surface become a lightcone framed surface. We also discuss a relation between lightcone framed surfaces and generalised framed surfaces. As a special case of a lightcone frame, we introduce the lightcone circle frame and call surfaces with such frames lightcone circle framed surfaces. In addition, the conditions for a lightcone framed surface to become a lightcone circle framed surface are also given.

To what extent is the classical maximum principle for scalar-valued functions valid for a vector field on a compact Riemannian manifold? In response, we derive a maximum principle for a smooth vector field with its divergence term on a closed, oriented Riemannian manifold by the conformal geometrical technique that do not widely known. We then use it to obtain some applications to star-shaped hypersurfaces in the Euclidean space and extend the previous spherical result of Hsiung [16] to weak star-shaped hypersurfaces. More precisely, we show that an orientable, connected, compact, weak star-shaped hypersurface in the Euclidean space with constant higher order mean curvature is a sphere.

Let K be a number field with an algebraic closure ${\bar{K}}$ and let S be a finite set of places of K, including all the infinite ones. Let N ≥ 1 be an integer and let D be a nonzero effective cycle of ${\mathbb{G}}{N\\m}({\bar{K}})$. If Supp(D) has at least one point whose coordinates are multiplicatively independent, for example, if D = (α) and ${\alpha}{\in}{\mathbb{G}}{N\\m}({\bar{K}})$ does not lie on any proper torsion subvarieties (= translates of proper subtori by torsion points) of 𝔾^N_m, then we will show that the set $${\bigcup}\;\;_{\array{{Z{\subset}{\mathbb{G}_m^N}\;\text{a torsion hypersurface}}\\{\text{such that}}\;{\bar{Z}}{\cap}{\bar{D}}={\emptyset}\;({\text{i.e.}},\;{{\bar{D}}{\mid}_{\bar{Z}}=0)}}}{\Large Z(\bar{K})}$$, where the bars indicate the respective Zariski closures of Z and D in 𝔾^N_m,𝓞K,S, an 𝓞_K,S-integral model of 𝔾^N_m over K. We will also pose a conjecture, which generalizes this result, in the context of semiabelian varieties.

This paper is devoted to the construction and analysis of an iterative Moser-secant's method. The principal properties that it presents is that is free of both inverses and jacobians. The order of convergence is the same as the classical secant method. A similar Moser-Steffesensen's method with second order is also proposed. Some interesting applications and some numerical experiments, including the approximation of optimization problems, are presented.

Let K be a number field with algebraic closure ${\bar{K}}$, let S be a finite set of places of K containing the archimedean places, and let 𝜑 be a Chebyshev polynomial. In this paper, we prove uniformity results on the number of S-integral preperiodic points relative to a non-preperiodic point 𝛽, as 𝛽 varies over number fields of bounded degree.

We study the Waldschmidt constants of some configurations in the projective plane. In the first part, we show that the Waldschmidt constant of a set 𝕏 of n points where at least n - 3 points among them lie on a line is one of the following: 1, ${\frac{2n-3}{n-1}}$, 2, ${\frac{16}{7}}$, ${\frac{7}{3}}$, ${\frac{17}{7}}$, or ${\frac{5}{2}}$. Together with the Hilbert polynomials, this gives a complete geometric characterization of 𝕏. Next, we study some specific configurations whose Waldschmidt constants are bounded from above by ${\frac{5}{2}}$. Under this condition, we describe all configurations of n points where n - 1 points among them lie on an irreducible conic, and we also study some specific configurations of 9 points.

In this paper, we study the existence and asymptotic behavior of ground state sign-changing solutions for the following Schrödinger-Poisson system with critical growth $$\begin{cases}-{\Delta}_u+V_{\lambda}(x)u+{\phi}u=f(x,u)+{\mu}h(u),&&\text{in}\;{\mathbb{R}}^3,\\-{\Delta}_{\phi}=u^2&&\text{in}\;{\mathbb{R}}^3,\end{cases}$$ where V_λ(x) = λV (x) + 1 with λ > 0, f, h are continuous and 𝜇 > 0. Under some mild assumptions on V , f and h, we first show the existence of ground state sign-changing solutions for each λ > 0 large enough via Nehari manifold methods. Then we prove the ground state sign-changing solution has precisely two nodal domains and characterize the relationship between its energy and the energy of the least energy solution of the above system. Finally, we give the asymptotic behavior of sign-changing solutions as λ → ∞.

We study GIT stability of divisors in products of projective spaces. We characterise all maximal orbits of not stable and strictly semistable pairs, as well as minimal closed orbits of strictly semistable pairs, which we implement via code. This characterisation is applied to classify the GIT quotient of threefolds of bidegree (1,2) and bidegree (4,4) curves in a quadric surface, via singularities, which are in turn used to obtain an explicit description of the K-moduli space of family №2.25 of Fano threefolds, and the K-moduli wall-crossing of log Fano pairs.

Let R = ⊕_γ∈ΓR_γ be a commutative ring graded by a commutative monoid Γ, and let n be a positive integer. This paper introduces the concept of graded n-semiprimary ideals in graded rings, extending the framework of n-semiprimary ideals beyond ungraded structures. We establish foundational properties of these ideals, distinguishing them from their ungraded counterparts and analyzing their connections to graded semiprimary ideals. Additionally, we examine graded n-semiprimary ideals in specific graded rings, such as graded trivial ring extensions and graded amalgamations. Key results include construction techniques for graded n-semiprimary ideals and characterizations in product rings, offering comprehensive insights into their structural properties and behaviors in graded settings.

This article applies an equivalent form of the Carlitz formula, as introduced by the third author, to derive two new q-congruences modulo the square of a cyclotomic polynomial and provides detailed examples of these q-congruences. These developments substantially advance the research previously conducted by Guo et al.

We establish a reciprocity law for degree two special linear local systems on the one-holed torus and their monodromy along simple loops.

We show that every twist of pure supersymmetric Yang-Mills theory with gauge group GL(N) can be realized as an open-string field theory in topological string theory. Our approach reinterprets twists of supersymmetric Yang-Mills theory as generalized Chern-Simons theories, and identifies topological string backgrounds that produce the corresponding Chern-Simons theories.