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.
Let Hℓ(N) denote the set of normalized primitive holomorphic cusp forms of even integral weight ℓ for the congruence group Γ0(N). In this paper, our main goal is to derive asymptotic formulas of ∫_1^T S_f^l(x; N) dx, where f ∈ Hℓ(N) and S_f(x; N) = ∑_n ≤ xλ_f(n). Here λf(n) is the n-th Hecke eigenvalue of f.
In this paper we study ℱ -sensitivity for a bounded linear operator T on Banach space and its induced hyperspace dynamics. First, we suggest the necessary and sufficient condition for ℱ -sensitivity for a bounded linear operator T on Banach space. Next, we consider the natural hyperspace extensions T and T̃ of T to the space K(X) of compact subsets of X and C(X) of convex compact subsets of X, respectively. We show that ℱ -sensitivity is equivalent for the maps T, T and T̃ .
Let j ≥ 2 be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = SL(2, ℤ). Denote by λ_sym^jf(n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L(s, symjf). In this paper, we are interested in the average behavior of λ_sym^jf(n) over some certain sequences, which improved the previous results.
Dan Archdeacon raised an open problem: Does every triangulation of a surface by a complete graph have a Grünbaum coloring? In this paper, we partly solve this open problem of Dan Archdeacon by studying the triangular embeddings and Grünbaum coloring of the complete graphs K12s+3, K12s+4 and K12s+7.
Let u(t, x) be the solution to the parabolic Anderson model on ℝ+ × ℝ driven by a space-time white Gaussian noise. In this paper, we prove the central limit theorem and almost sure central limit theorem for product of spatial average of the form ∏_k=1^n S_k as n → ∞ for fixed t > 0, where S_k = ∫_0^k u(t, x)dx .
In 2022, Li, Ye and Yu introduced multivariate sensitivity version of notions of mean m-Sensitivity and m-sensitivity in the mean for m ≥ 2. In this manuscript, we mainly focus on the investigation of multivariate sensitivity in mean forms. First, we prove that for a linear system, the equivalence between mean m-sensitivity and m-sensitivity in the mean holds without any more conditions. Subsequently, we introduce the notion of m-equicontinuity in the mean, and obtain an Auslander–Yorke type dichotomy between m-equicontinuity in the mean and m-sensitivity in the mean for minimal systems. As a consequence, we demonstrate that the equivalence between mean m-sensitivity and m-sensitivity in the mean is valid under the condition of minimality for a general system, thereby affirmatively resolving the conjecture proposed in [Li, J., Ye, X. D., Yu, T.: Equicontinuity and sensitivity in mean forms. J. Dynam. Differential Equations, 34, 133–154 (2022)].
The trace range of products of three projections in a finite factor (ℳ, τ), with their trace equal to λ is studied in this article. When λ≤1/2 , the corresponding boundary curve of the trace range is shown to be a degenerate elliptic curve. For n ≥ 3, when ℳ is of type In and λ=n-1/n , it is shown that the corresponding boundary curve of the trace range is an algebraic curve in the complex plane.
In this work, we prove an L^p_1× L^p_2× L^p_3→ L^p estimate for a trilinear pseudo-differential operators with flag-type symbols taking the form of a(x, ξ1, ξ2, ξ3)b(x, ξ1, ξ2 + ξ3). The study of such operators is motivated by investigations on the corresponding flag Fourier multipliers studied by Muscalu [Rev. Mat. Iberoam., 23, 705–742 (2007); Contemp. Math., 505, 131–151 (2010)], Germain, Masmoudi, Shatah [J. Math. Pures Appl., 97, 505–543 (2012); Ann. of Math., 175, 691–754 (2012)] and Miyachi, Tomita [Math. Z., 282, 577–613 (2016)].
In this paper we study the analytical smoothing effect of Cauchy problem for the incompressible MHD-Boussinesq equations. Precisely, we used the Fourier Gevrey space method to show that the Sobolev H1-solution to the incompressible MHD-Boussinesq equations in periodic domain is analytical for both spatial variable and time variable.
In this paper we introduce a new geometric flow consisting of the Yamabe flow coupled with harmonic heat flow of a function on a closed manifold M, or shortly Yamabe-harmonic flow. It is define by ∂∂ tg = -(R - ϕ)g, ϕ = (n - 1)Δ_g ϕ. To begin with we establish the short-time extsience via conformal transformation for any smooth initial data. Furthermore we derive interior-in-time derivative estimates for the Riemannian curvature and Lapse function and obtain the global existence of Yamabe-harmonic flow. Moreover, compact gradient Yamabe-harmonic solitons are proved to be manifolds with R = ϕ = Constant. Eventually we obtain the classification of singularities by blow-up rate of AC curvature, which is analogous to that of Ricci-harmonic flow and Ricci-connection flow. The relationship between singularity models and solitons shall appear in forthcoming work.
The paper is mainly concerned with the multifractal analysis for smooth dynamical systems in dimension one without uniform hyperbolicity. We characterize the Hausdorff dimension of the saturated set obtained from the empirical measure by the local dimensions of hyperbolic measures for a topologically mixing C2 map modeled by an abstract dynamical system.
For the topological dynamical system with infinite entropy and the specification property, in the context of continuous and affine deformations of empirical measures, the set of divergence points is either empty, or its Bowen and packing metric mean dimensions equal those of the full phase space. We also establish the variational principles of Bowen and packing metric mean dimension for the sup set.
In this paper, we investigate the uniqueness of meromorphic functions sharing four values. We show that if f(z) and g(z) are distinct non-constant meromorphic functions sharing 0, 1, c IM and ∞ CM, then either f(z) and g(z) share 0, 1, c, ∞ CM or 1/13 T(r, f) - 2N̅(r, f) ≤ N_E^(1)(r, 0) + N_E^(1)(r, 1) + N_E^(1)(r, c) + S(r, f) ≤4/3 T(r, f) - 2/3N̅(r, f) holds.
The aim of this article is to introduce a boson-fermionic quantum Weyl algebra W_q(2(m | n)) of type A(m∣n), which is the quantum differential operators (QDO) algebra (also denoted by Diffq(Ωq)) defined on Ωq(m∣n), the quantum Grassmann super-algebra Ωq(m∣n) we defined. Here we develop a constructive approach of Radford–Majid bosonization theory to yield some pointed Hopf algebras via QDO. This addresses a basic problem due to Manin, namely, any QDO approach might help to yield new Hopf algebras (see §2.2. Basic problem, p. 1010 of Manin’s paper in 1992). Remarkably, W_q(2(m | n)) itself is not a Hopf algebra, however, it contains some interesting pointed Hopf algebras as its subquotient objects, like the bosonization A_q(2(m | n)) of the quantum Manin (m∣n)-superspace A ∣ , the bosonization G_q(2(m | n)) of Ωq(m∣n), the multi-rank Taft algebras of 2(m∣n)-type, the homomorphic image of pointed Hopf algebra U_q(gl(m | n)) over Ωq(m∣n) or Ω ! (m∣n) (the quantum dual Grassmann superalgebra), etc.
In this paper, we study Legendrian submanifolds of the Sasakian space form 𝕊^2n+1(c̃) with constant sectional curvature and conformal Maslov form. As main results, we first classify the 2-dimensional Legendrian submanifolds of 𝕊^5(c̃) with constant sectional curvature and conformal Maslov form; then we establish a complete classification of the Legendrian submanifolds of 𝕊^2n+1(c̃) with constant sectional curvature and C-parallel mean curvature vector field.
