In this paper, the authors established a sharp version of the difference analogue of the celebrated Hölder’s theorem concerning the differential independence of the Euler gamma function Γ. More precisely, if P is a polynomial of n + 1 variables in ℂ[X, Y ₀, ⋯, Y _n−1] such that

$P(s,\Gamma (s + {a_0}), \cdots ,\Gamma (s + {a_{n - 1}})) \equiv 0$

for some (a ₀, ⋯, a _n−1) ∈ ℂ ⁿ and a _i − a _j ∉ ℤ for any 0 ≤ i ≤ j ≤ n − 1, then they have

$P \equiv 0.$.

Their result complements a classical result of algebraic differential independence of the Euler gamma function proved by Hölder in 1886, and also a result of algebraic difference independence of the Riemann zeta function proved by Chiang and Feng in 2006.

The higher spin operator of several ℝ⁶ variables is an analogue of the $\overline \partial$ operator in theory of several complex variables. The higher spin representation of ${\mathfrak{s}\mathfrak{o}_6}(\mathbb{C})$ is ⊙ ^kℂ⁴ and the higher spin operator ${{\cal D}_k}$ acts on ⊙ ^kℂ⁴-valued functions. In this paper, the authors establish the Bochner-Martinelli formula for higher spin operator ${{\cal D}_k}$ of several ℝ⁶ variables. The embedding of ℝ⁶ⁿ into the space of complex 4n × 4 matrices allows them to use two-component notation, which makes the spinor calculus on ℝ⁶ⁿ more concrete and explicit. A function annihilated by ${{\cal D}_k}$ is called k-monogenic. They give the Penrose integral formula over ℝ⁶ⁿ and construct many k-monogenic polynomials.

Let (X, G) be a dynamical system (G-system for short), that is, X is a topological space and G is an infinite topological group continuously acting on X. In the paper, the authors introduce the concepts of Hausdorff sensitivity, Hausdorff equicontinuity and topological equicontinuity for G-systems and prove that a minimal G-system (X, G) is either topologically equicontinuous or Hausdorff sensitive under the assumption that X is a T ₃-space and they provide a classification of transitive dynamical systems in terms of equicontinuity pairs. In particular, under the condition that X is a Hausdorff uniform space, they give a dichotomy theorem between Hausdorff sensitivity and Hausdorff equicontinuity for G-systems admitting one transitive point.

Let G be a graph of order n and μ be an adjacency eigenvalue of G with multiplicity k ≥ 1. A star complement H for μ in G is an induced subgraph of G of order n − k with no eigenvalue μ, and the subset X = V(G − H) is called a star set for μ in G. The star complement provides a strong link between graph structure and linear algebra. In this paper, the authors characterize the regular graphs with K _2,2,s (s ≥ 2) as a star complement for all possible eigenvalues, the maximal graphs with K _2,2,s as a star complement for the eigenvalue μ = 1, and propose some questions for further research.

Let M be an open Riemann surface and G: M → ℙ ⁿ(ℂ) be a holomorphic map. Consider the conformal metric on M which is given by ${\rm{d}}{s^2} = ||\tilde G|{|^{2m}}|\omega {|^2}$, where ${\tilde G}$ is a reduced representation of G, ω is a holomorphic 1-form on M and m is a positive integer. Assume that ds ² is complete and G is k-nondegenerate(0 ≤ k ≤ n). If there are q hyperplanes H ₁, H ₂, ⋯, H _q ⊂ ℙ ⁿ(ℂ) located in general position such that G is ramified over H _j with multiplicity at least γ _j(> k) for each j ∈ {1, 2, ⋯, q}, and it holds that $\sum\limits_{j = 1}^q {\left( {1 - {k \over {{\gamma _j}}}} \right) > (2n - k + 1)\left( {{{mk} \over 2} + 1} \right),} $

then M is flat, or equivalently, G is a constant map. Moreover, one further give a curvature estimate on M without assuming the completeness of the surface.

The authors prove that a 3-dimensional small cover M is a Haken manifold if and only if M is aspherical or equivalently the underlying simple polytope is a flag polytope. In addition, they find that M being Haken is also equivalent to the existence of a Riemannian metric with non-positive sectional curvature on M.

for stationary hypersonic-limit Euler flows passing a solid body in three-dimensional space, the shock-front coincides with the upwind surface of the body, hence there is an infinite-thin layer of concentrated mass, in which all particles hitting the body move along its upwind surface. By proposing a concept of Radon measure solutions of boundary value problems of the multi-dimensional compressible Euler equations, which incorporates the large-scale of three-dimensional distributions of upcoming hypersonic flows and the small-scale of particles moving on two-dimensional surfaces, the authors derive the compressible Euler equations for flows in concentration layers, which is a stationary pressureless compressible Euler system with source terms and independent variables on curved surface. As a by-product, they obtain a formula for pressure distribution on surfaces of general obstacles in hypersonic flows, which is a generalization of the classical Newton-Busemann law for drag/lift in hypersonic aerodynamics.

The authors show that if Θ = (θ _jk) is a 3 × 3 totally irrational real skew-symmetric matrix, where θ _jk ∈ [0, 1) for j, k = 1, 2, 3, then for any ε > 0, there exists δ > 0 satisfying the following: For any unital C*-algebra A with the cancellation property, strict comparison and nonempty tracial state space, any four unitaries u ₁, u ₂, u ₃,w ∈ A such that (1) $||{u_k}{u_j} - {{\rm{e}}^{2\pi {\rm{i}}{\theta _{jk}}}}{u_j}{u_k}||\, < \delta $ wu _j w ⁻¹ = u _j ⁻¹, w ² = 1 _A for j, k = 1, 2, 3; (2) τ(aw) = 0 and $\tau ({({u_k}{u_j}u_k^ * u_j^ * )^n}) = {{\rm{e}}^{^{2\pi {\rm{i}}n{\theta _{jk}}}}}$ for all n ∈ ℕ, all a ∈ C*(u ₁, u ₂, u ₃), j, k = 1, 2, 3 and all tracial states τ on A, where C*(u ₁, u ₂, u ₃) is the C*-subalgebra generated by u ₁, u ₂ and u ₃, there exists a 4-tuple of unitaries ${\tilde u_1},{\tilde u_2},{\tilde u_3},\tilde w$ in A such that ${\tilde u_k}{\tilde u_j} = {{\rm{e}}^{2\pi {\rm{i}}{\theta _{jk}}}}{\tilde u_j}{\tilde u_k},\,\,\,\,\,\,\,\tilde w{\tilde u_j}{\tilde w^{ - 1}} = \tilde u_j^{ - 1},\,\,\,\,\,{\tilde w^2} = {1_A}\,\,\,\,$ and $\Vert{u_j} - {\tilde u_j}\Vert < \varepsilon ,\,\,\,\,\Vert w - \tilde w\Vert < \varepsilon $ for j, k = 1, 2, 3. The above conclusion is also called that the rotation relations of three unitaries with the flip action is stable under the above conditions.

Zhang (2021), Luo and Yin (2022) initiated the study of Lagrangian submanifolds satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ ⁿ or ℂℙ ⁿ, where $T = {\nabla ^ * }\tilde h$ and ${\tilde h}$ is the Lagrangian trace-free second fundamental form. They proved several rigidity theorems for Lagrangian surfaces satisfying ∇*T = 0 or ∇*∇*T = 0 in ℂ² under proper small energy assumption and gave new characterization of the Whitney spheres in ℂ². In this paper, the authors extend these results to Lagrangian submanifolds in ℂ ⁿ of dimension n ≥ 3 and to Lagrangian submanifolds in ℂℙ ⁿ.

Let V be a finite set. Let ${\cal K}$ be a simplicial complex with its vertices in V. In this paper, the author discusses some differential calculus on V. He constructs some constrained homology groups of ${\cal K}$ by using the differential calculus on V. Moreover, he defines an independence hypergraph to be the complement of a simplicial complex in the complete hypergraph on V. Let ${\cal L}$ be an independence hypergraph with its vertices in V. He constructs some constrained cohomology groups of ${\cal L}$ by using the differential calculus on V.