We prove that any ergodic endomorphism on an n-torus admits a sequence of periodic orbits uniformly distributed in the metric sense. As a corollary, an endomorphism on the torus is ergodic if and only if the Haar measure can be approximated by periodic measures.

We prove Liouville theorem for the equation Δ_mv + v^p + M∣∇v∣^q = 0 in a domain Ω ⊂ ℝⁿ, with M ∈ ℝ in the critical and subcritical case. As a natural extension of our recent work [2023, arXiv:2311.04641], the proof is based on an integral identity and Young’s inequality.

Whether or not classical solutions to the hyperbolic Navier-Stokes equations (NSE) can develop finite-time singularities remains a challenging open problem. For general data without smallness condition, even the L²-norm of solutions is not known to be globally bounded in time. This paper presents a systematic approach to the global existence and stability problem by examining the difference between a general hyperbolic NSE and its corresponding Navier-Stokes counterpart. We make use of the integral representations. The functional setting is taken to be critical Sobolev spaces for the NSE. As a special consequence, any d-dimensional (d ≥ 2) hyperbolic NSE with general fractional dissipation is shown to possess a unique global solution if the coefficient of the double-time derivative and the initial data obey a suitable constraint.

Let $1 < k < {7 \over 6},{\lambda _1},{\lambda _2},{\lambda _3}$ and λ₄ be non-zero real numbers, not all of the same sign such that ${{{\lambda _1}} \over {{\lambda _2}}}$ is irrational and let ω be a real number. We prove that the inequality $\left| {{\lambda _1}p_1^2 + {\lambda _2}p_2^2 + {\lambda _3}p_3^2 + {\lambda _4}p_4^k - \omega} \right| \le {\left({{{\max}_j}{p_j}} \right)^{- {{7 - 6k} \over {14k}} + \varepsilon}}$ has infinitely many solutions in prime variables p₁, p₂, p₃, p₄ for any ε > 0.

Estimates are obtained for the number of natural numbers n in certain residue classes that do not have a representation of the form

The coupled q-oscillator algebra ${\cal A}$ is a smash product of the polynomial algebra in one variable with the Hopf algebra $U_{q}({\mathfrak s}{\mathfrak l}_{2})$. We show that the centre of the algebra ${\cal A}$ is trivial. We find a distinguished normal element of ${\cal A}$ that plays a role in studying its structures and representations. We give explicit descriptions of the prime, completely prime, primitive and maximal ideals of the algebra ${\cal A}$. As a result, we show that ${\cal A}$ cannot have a Hopf algebra structure, and ${\cal A}$ has no finite-dimensional representations. The group of automorphisms of ${\cal A}$ is explicitly described. A classification of all simple weight modules over the algebra ${\cal A}$ is obtained. Using the classification of primitive ideals of ${\cal A}$, we determine the annihilator of every simple weight module.

Over an algebraically closed field of characteristic p > 2, this paper gives a sufficient and necessary condition for a map from the Cartesian product of a finite-dimensional Lie algebra with itself two times to its any nontrivial and simple module to be a symmetric biderivation, and determines all biderivations of the 3-dimensional simple Lie algebra ${\mathfrak s}{\mathfrak l}(2)$ on its any finite-dimensional simple module.

Consider the invariance principle for a random walk with a random environment (denoted by μ) in time on ℝ in a weak quenched sense. We show that a sequence of random probability measures on ℝ generated by μ and a bounded Lipschitz functional f will converge in distribution to another random probability measure, which can be represented by f and two independent Brownian motions. The upper bound of the convergence rate has been obtained. We also explain that in general, this convergence can not be strengthened to the almost surely sense.

The aim of this work is to approximate the random periodic solutions of neutral type stochastic differential equations (SDEs) with non-uniform dissipativity via the discretization method. The non-uniform dissipativity here means the drift satisfying dissipativity on average rather than uniform dissipativity concerning the time variables. On one hand, we show the existence and uniqueness of random periodic solutions for neutral type SDEs via the synchronous coupling approach when the starting time tends to −∞. On the other hand, using the Euler–Maruyama scheme on an infinite time horizon we study the existence and uniqueness of the numerical approximation of random periodic solution. During this procedure, the difficulties, which arose from the time-discretization of both the neutral term and the functional solutions, have to be dealt with.

In this paper we study the extension of holomorphic canonical forms on complete d-bounded Kähler manifolds by using L² analytic methods and L² Hodge theory, which generalizes some classical results to noncompact cases.

Let (X, ω) be an n-dimensional compact Hermitian manifold with ω a pluri-closed Hermitian metric, i.e., dd^cω = 0. Let $\left\{\alpha \right\},\left\{\beta \right\} \in H_{BC}^{1,1}\left({X,\mathbb{R}} \right)$ be two nef classes, such that αⁿ − nαⁿ⁻¹ · β > 0. In this short note, we prove that if there is a bounded quasi-plurisubharmonic potential ρ, such that α + dd^cρ ≥ 0 in the weak sense of currents, then the class {α − β} contains a Kähler current. This gives a partial solution of Demailly’s transcendental Morse inequalities conjecture.