The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish a complete theory on the HUM method and give its applications to first order hyperbolic systems. Thus, a deeper relationship between the controllability and the observability can be revealed. Moreover, at the end of the paper, a comparison will be made between these two methods.

The unique continuation on quadratic curves for harmonic functions is discussed in this paper. By using complex extension method, the conditional stability of unique continuation along quadratic curves for harmonic functions is illustrated. The numerical algorithm is provided based on collocation method and Tikhonov regularization. The stability estimates on parabolic and hyperbolic curves for harmonic functions are demonstrated by numerical examples respectively.

The author gives a definition of orbifold Stiefel-Whitney classes of real orbifold vector bundles over special q-CW complexes (i.e., right-angled Coxeter complexes). Similarly to ordinary Stiefel-Whitney classes, orbifold Stiefel-Whitney classes here also satisfy the associated axiomatic properties.

This paper is a continuation of the authors recent work [Beirão da Veiga, H. and Yang, J., On mixed pressure-velocity regularity criteria to the Navier-Stokes equations in Lorentz spaces, Chin. Ann. Math., 42(1), 2021, 1–16], in which mixed pressure-velocity criteria in Lorentz spaces for Leray-Hopf weak solutions of the three-dimensional Navier-Stokes equations, in the whole space ℝ³ and in the periodic torus ${\mathbb{T}^3}$, are established. The purpose of the present work is to extend the result of mentioned above to smooth, bounded domains, under the non-slip boundary condition. Let π denote the fluid pressure and v the fluid velocity. It is shown that if ${\pi \over {{{\left( {1 + \left| v \right|} \right)}^\theta }}} \in {L^p}\left( {0,T;{L^{q,\infty }}\left( \Omega \right)} \right)$;, where 0 ≤ θ ≤ 1, and ${2 \over p} + {3 \over q} = 2 - \theta $ with p ≥ 2, then v is regular on Ω × (0, T].

Let G be a finite group, H be a proper subgroup of G, and S be a unitary subring of C. The kernel of the restriction map S[Irr(G)] → S[Irr(H)] as a ring homomorphism is studied. As a corollary, the main result in [Isaacs, I. M. and Navarro, G., Injective restriction of characters, Arch. Math., 108, 2017, 437–439] is reproved.

Given n samples (viewed as an n-tuple) of a γ-regular discrete distribution π, in this article the authors concern with the weighted and unweighted graphs induced by the n samples. They first prove a series of SLLN results (of Dvoretzky-Erdös’ type). Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1 + γ. They also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distributions with constant exponent 2. This exponent is obviously independent of the parameter γ ∈ (0, 1), which is a surprise to us at first sight.

This paper is concerned with the spreading speeds of time dependent partially degenerate reaction-diffusion systems with monostable nonlinearity. By using the principal Lyapunov exponent theory, the author first proves the existence, uniqueness and stability of spatially homogeneous entire positive solution for time dependent partially degenerate reaction-diffusion system. Then the author shows that such system has a finite spreading speed interval in any direction and there is a spreading speed for the partially degenerate system under certain conditions. The author also applies these results to a time dependent partially degenerate epidemic model.

In this paper, the authors give a comparison version of Pythagorean theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds).

The author proves that the isoperimetric inequality on the graphic curves over circle or hyperplanes over ${\mathbb{S}^{n - 1}}$ is satisfied in the cigar steady soliton and in the Bryant steady soliton. Since both of them are Riemannian manifolds with warped product metric, the author utilize the result of Guan-Li-Wang to get his conclusion. For the sake of the soliton structure, the author believes that the geometric restrictions for manifolds in which the isoperimetric inequality holds are naturally satisfied for steady Ricci solitons.

Suppose that λ₁, ⋯ λ₅ are nonzero real numbers, not all of the same sign, satisfying that ${{{\lambda _1}} \over {{\lambda _2}}}$ is irrational. Then for any given real number η and ε > 0, the inequality$\left| {{\lambda _1}{p_1} + {\lambda _2}p_2^2 + {\lambda _3}p_3^3 + {\lambda _4}p_4^4 + {\lambda _5}p_5^5 + \eta } \right| < {\left( {\mathop {\max }\limits_{1 \le j \le 5} p_j^j} \right)^{ - {{19} \over {756}} + \varepsilon }}$

has infinitely many solutions in prime variables p ₁, ⋯, p ₅. This result constitutes an improvement of the recent results.

In this paper, local unstable metric entropy, local unstable topological entropy and local unstable pressure for partially hyperbolic endomorphisms are introduced and investigated. Specially, two variational principles concerning relationships among the above mentioned numbers are formulated.