Let φ and ψ be two bounded harmonic functions on the open unit disk, 𝕋_φ and ℍ_ψ be the Toeplitz and small Hankel operators on the harmonic Bergman space, respectively. In this paper, we establish a number of necessary conditions and sufficient conditions for 𝕋_φ to commute with ℍ_ψ . Moreover, we obtain equivalent characterizations for such commutativity problem in two cases: (i) φ is analytic and ψ is harmonic; (ii) φ is harmonic and the coefficients of the Taylor series of the harmonic function ψ are all real.
In this paper, we obtain a Banach–Tarski type paradox preserving the self-similar measure: Let K be a self-similar set on a complete metric space and ν a self-similar measure on K, then there are bijections gi: K → K preserving ν-measure such that K = _i=1^n A_i = _i=1^k g_i(A_i) = _i=k+1^n g_i(A_i) with k < n, where means the disjoint union.
In this paper, we investigate the large-time behavior of solutions to an initial-boundary value problem for the compressible non-isentropic Navier–Stokes/Allen–Cahn system in a half line ℝ+:= (0,+∞). We focus our attention on the inflow problem and give a rigorous proof of the asymptotic stability of the composite wave which is composed of the boundary layer solution, the contact wave and the rarefaction wave under some smallness conditions. Meanwhile, we obtain the global existence of solutions based on the basic energy method by taking into account the complexity of composite wave.
We consider the random graph model G(w) for a given expected degree sequence w = (w1, w2,…, wn), where the probability pij of an edge between vertices vi and vj is defined as wiwjρ with ρ = 1/∑_i=1^n w_i . In this paper, we apply a matrix concentration inequality to derive an upper bound on the largest eigenvalue of the adjacency matrix of the random graph with given expected degrees. We further analyze the expectation of the largest eigenvalue of the adjacency matrix and establish its concentration properties. Additionally, by utilizing an extension of the matrix Chernoff inequality that incorporates an intrinsic dimension parameter, we investigate the expectation and tail behavior of the largest eigenvalue of the Laplacian matrix for the random graph model G(w).
In this paper, we study the extinction profiles of solutions to weighted fast diffusion equation u_t = | x|^γ∇· (| x|^-β∇ u^m) , with 0 < m < 1, posed in ℝN × [0, ∞). The range of parameters for weights ∣x∣γ and ∣x∣−β satisfies γ < N and γ - 2 < β≤γ(N-2)/N , which is optimal for the validity of a class of Caffarelli–Kohn–Nirenberg inequalities. When the initial data belongs to the Marcinkiewicz space and m be in the very fast diffusion range, we obtain the extinction profiles of solutions to the weighted fast diffusion equation.
