The authors investigate H-surfaces into static Lorentzian manifolds and show the Hölder continuity of weak solutions.

In this paper, the authors develop the theory of

-graded modules to study some kinds of graded structures on reducing subspaces. They introduce some concepts and find an effective way to study

{\mathcal{S}}_T

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{S}}_{T}$$\end{document}

-graded modules, which they name the kernel method. For its application, they define standard models for some multiplication operators and special Toeplitz operators, and show that these standard models can be solved by the kernel method. The authors also generalize the kernel method to arbitrary Hilbert spaces without any

{\mathcal{S}}_T

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{S}}_{T}$$\end{document}

-graded condition. Finally, for any bounded operator T, they can find a reducing subspace that is

{\mathcal{S}}_T

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\cal{S}}_{T}$$\end{document}

-graded.

This paper is concerned with the existence and optimal boundary behavior of large solutions to the Monge-Ampère type equations det D²u(x) = λuⁿ(x)+b(x)g(∣∇u(x)∣), x ∈ Ω, where Ω is a uniformly convex, bounded smooth domain in ℝⁿ with n ≥ 2, b ∈ C^∞(Ω) is positive in Ω, g ∈ C[0, ∞) ∩ C¹ (0, ∞), g(0) = 0 and g is increasing on [0, ∞). The author finds new structure conditions on g which play a crucial role in boundary behavior of such solutions.

In this paper, the authors study the homogenization of the third boundary value problem for semilinear parabolic PDEs with rapidly oscillating periodic coefficients in the weak sense. Their method is entirely probabilistic, and builds upon the work of Tanaka (2020) and Buckdahn (1999). Backward stochastic differential equations with singular coefficients play an important role in this approach.

In this paper, the authors study gradient Ricci-Harmonic solitons on warped product manifolds. First, they prove triviality results for the potential and warping functions that reach a maximum or a minimum. In order to provide nontrivial examples, they consider the base and the fiber conformal to a semi-Euclidean space, which is invariant under the action of a translation group of co-dimension one. This approach allows them to produce infinitely many examples of geodesically complete semi-Riemannian Ricci-Harmonic solitons not present in the literature.

In this paper, the authors prove a big Picard type theorem concerning derivative: Let f(z) be meromorphic in D = {z: 0 < ∣z − z₀∣ < δ} for each δ > 0, if z₀ is an essential singularity of f(z), then either f(z) assumes every finite value infinitely often or f′(z) assumes every finite value except possibly zero infinitely often. As an application of this result, they extend Nevo, Pang and Zalcman’s quasinormal criterion: Let {f_n(z)} be a sequence of meromorphic functions on the plane domain D, all of whose zeros are multiple such that f′_n(z) − 1 has zeros with multiplicity at least n for all n on D, then {f_n(z)} is quasinormal of order 1 on D. Then they obtain a corresponding result in value distribution theory: Let f(z) be a meromorphic function on ℂ, all but finitely many of whose zeros are multiple such that

then there exist a positive integer M and R₀ > 0 such that for each r > R₀, there exists z₀ ∈ ℂ satisfying ∣z₀∣ > r such that z₀ is a zero of f′(z) − 1 with multiplicity at most M.

Let H be a Hilbert space of dimension greater than 2 and B(H) be the algebra of all bounded linear operators on H. In this paper, the authors show that G ∈ B(H) is a Lie all-derivable point of B(H) if the range of G is not dense in H.

In this paper, the authors consider the equation

, for distinct odd prime exponents p, q, and show that, for p > 3, it has no solutions under the condition that q does not divide h_p⁻, the minus part of the class number of the p-th cyclotomic field.

For an integer m ≥ 2, let ℤ/mℤ be the set of all residue classes mod m. For S ⊆ ℤ/mℤ and

is defined as the number of solutions to the equation

\overline{n}=\overline{s}+\overline{s^{\mathrm{\prime }}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\bar{n}=\bar{s}+\bar{s^{\prime}}$$\end{document}

with an unordered pair

(\overline{s},\overline{s^{\mathrm{\prime }}})\in S^2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\bar{s},\bar{s^{\prime}})\in S^{2}$$\end{document}

and

\overline{s}\ne \overline{s^{\mathrm{\prime }}}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\bar{s}\ne\bar{s^{\prime}}$$\end{document}

. In this paper, the author determines the structures of sets A and B such that A ⋃ B = ℤ/mℤ,

A\cap B=\overline{k}\mathbb{Z}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$A\;\cap\;B=\bar{k}\mathbb{Z}$$\end{document}

and

R_A(\overline{n})=R_B(\overline{n})

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_{A}(\bar{n})=R_{B}(\bar{n})$$\end{document}

for all

\overline{n}\in \mathbb{Z}/m\mathbb{Z}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\bar{n}\in\mathbb{Z}/m\mathbb{Z}$$\end{document}

, where k is an integer.

In this paper, the authors firstly give a simple proof and strengthened version of a uniqueness theorem of meromorphic functions which partially share 0, ∞ CM and 1 IM with their difference operators. In addition, they partially solve a conjecture given by Chen-Yi (2013) and generalize some previous theorems by Chen (2018) and Chen-Xu (2022). Furthermore, the authors obtain a uniqueness result of the k-th derivative of meromorphic functions with its shift, which is a generalization of some previous theorems by Chen-Xu (2022), Qi-Li-Yang (2018) and Qi-Yang (2020).