Surface bundles arising from periodic mapping classes may sometimes have non-isomorphic, but profinitely isomorphic fundamental groups. Pairs of this kind have been discovered by Hempel. This paper exhibits examples of nontrivial Hempel pairs where the mapping tori can be distinguished by some Turaev-Viro invariants, and also examples where they cannot be distinguished by any Turaev-Viro invariants.

Let K be a given positive function on a bounded domain Ω of ℝⁿ, n ≥ 3. The authors consider a nonlinear variational problem of the form: $-\Delta u=K\vert u\vert^{4\over{n-2}}u$ in Ω with mixed Dirichlet-Neumann boundary conditions. It is a non-compact variational problem, in the sense that the associated energy functional J fails to satisfy the Palais-Smale condition. This generates concentration and blow-up phenomena. By studying the behaviors of non-precompact flow lines of a decreasing pseudogradient of J, they characterize the points where blow-up phenomena occur, the so-called critical points at infinity. Such a characterization combined with tools of Morse theory, algebraic topology and dynamical system, allow them to prove critical perturbation results under geometrical hypothesis on the boundary part in which the Neumann condition is prescribed.

Let X be a complex smooth quasi-projective variety with a fixed epimorphism ν: π₁(X) ↠ H, where H is a finitely generated abelian group with rank H ≥ 1. In this paper, the authors study the asymptotic behaviour of Betti numbers with all possible field coefficients and the order of the torsion subgroup of singular homology associated to ν, known as the L²-type invariants. When ν is orbifold effective, explicit formulas of these invariants at degree 1 are give. This generalizes the authors’ previous work for H ≌ ℤ.

In the paper, a concept of the essential numerical range W_e(T) of a linear relation T in a Hilbert space is given, other various essential numerical ranges W_ei(T), i = 1, 2, 3, 4, are introduced, and relationships among W_e(T) and W_ei(T) are established. These results generalize relevant results obtained by Bögli et al. in [Bögli, S., Marletta, M. and Tretter, C., The essential numerical range for unbounded linear operators, J. Funct. Anal., 279, 2020, 47–12]. Moreover, several fundamental properties of closed relations related to its operator parts are presented. In addition, singular discrete linear Hamiltonian systems including non-symmetric cases are considered, several properties for the associated minimal relations H₀ are derived, and the above results for abstract linear relations are applied to H₀.

In this article the authors prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if they know presentation of virtual pure braid group V P₄, then they can find presentation of V P_n for arbitrary n > 4. Using this theorem they find the set of generators and defining relations for simplicial group T_* which was defined in [Bardakov, V. G. and Wu, J., On virtual cabling and structure of 4-strand virtual pure braid group, J. Knot Theory and Ram., 29(10), 2020, 1–32]. They find a decomposition of the Artin pure braid group P_n in semi-direct product of free groups in the cabled generators.

In this paper, the authors study the well-posedness and the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term t^−α as t goes to infinity.

In this paper, the authors firstly establish the weak laws of large numbers on the canonical space $(\mathbb{R}^{\mathbb{N}},\cal{B}(\mathbb{R}^{\mathbb{N}}))$ by traditional truncation method and Chebyshev’s inequality as in the classical probability theory. Then they extend them from the canonical space to the general sublinear expectation space. The necessary and sufficient conditions for Peng’s law of large numbers are obtained.

Let $\mathbb{B}(X)$ be the algebra of all bounded linear operators on a Hilbert space X. Consider an operator polynomial

where $A_{i}\in\mathbb{B}(X),i=0,1,\cdots,m$. The numerical range of P(λ) is defined as The main goal of this paper is to respond to an open problem proposed by professor Li, and determine general conditions on connectivity, convexity and spectral inclusion property of W(P(λ)). They also consider the relationship between operator polynomial numerical range and block numerical range.