Acta Mathematica Sinica-English Series最新文献

Recently the quantum toroidal superalgebra ({{cal E}_{m|n}}) associated with ({mathfrak{g}mathfrak{l}_{m|n}}) was introduced by L. Bezerra and E. Mukhin, which is not a quantum Kac–Moody algebra. The quantum toroidal superalgebra ({{cal E}_{m|n}}) exploits infinite sequences of generators and relations of the form, which are called Drinfeld realization. In this paper, we use only finite set of generators and relations to define an associative algebra ({cal E}_{m|n}^prime ) and show that there exists an epimorphism from ({cal E}_{m|n}^prime ) to the quantum toroidal superalgebra ({{cal E}_{m|n}}). In particular, the structure of ({cal E}_{m|n}^prime ) enjoys some properties like Drinfeld–Jimbo realization.

In this article, we provide some sufficient conditions for the dynamical systems with the eventual shadowing property to have positive topological entropy and several equivalent conditions for the dynamical systems with the eventual shadowing property to be mixing.

In this paper, we consider a class of normally degenerate quasi-periodically forced reversible systems, obtained as perturbations of a set of harmonic oscillators,

where 0 ≠ λ ∈ ℝ, l > 1 is an integer and the corresponding involution G is (−θ, x, −y) → (θ, x, y). The existence of response solutions of the above reversible systems has already been proved in [22] if [f₂(ωt, 0, 0)] satisfies some non-zero average conditions (See the condition (H) in [22]), here [ · ] denotes the average of a continuous function on ({mathbb{T}^d}). However, discussing the existence of response solutions for the above systems encounters difficulties when [f₂(ωt, 0, 0)] = 0, due to a degenerate implicit function must be solved. This article will be doing work in this direction. The purpose of this paper is to consider the case where [f₂(ωt, 0, 0)] = 0. More precisely, with 2p < l, if f₂ satisfies ([{f_2}(omega t,0,0)] = [{{partial {f_2}(omega t,0,0)} over {partial x}}] = [{{{partial ^2}{f_2}(omega t,0,0)} over {partial {x^2}}}] = cdots = [{{{partial ^{p - 1}}{f_2}(omega t,0,0)} over {partial {x^{p - 1}}}}] = 0), either ({lambda ^{ - 1}}[{{{partial ^p}{f_2}(omega t,0,0)} over {partial {x^p}}}] < 0) as l − p is even or ({lambda ^{ - 1}}[{{{partial ^p}{f_2}(omega t,0,0)} over {partial {x^p}}}] ne 0) as l − p is odd, we obtain the following results: (1) For (tilde lambda < 0) (see ({tilde lambda }) in (2.2)) and ϵ sufficiently small, response solutions exist for each ω satisfying a weak non-resonant condition; (2) For (tilde lambda < 0) and ϵ_* sufficiently small, there exists a Cantor set ({cal E} in (0,{_ * })) with almost full Lebesgue measure such that response solutions exist for each ( in {cal E}) if ω satisfies a Diophantine condition. In the remaining case where ({lambda ^{ - 1}}[{{{partial ^p}{f_2}(omega t,0,0)} over {partial {x^p}}}] > 0) and l − p is even, we prove the system admits no response solutions in most regions.

In this paper, we construct an ancient solution by using a given shrinking breather and prove a no shrinking breather theorem for noncompact harmonic Ricci flow under the condition that ({rm{Sic}}: = {rm{Ric}} - alpha nabla phi otimes nabla phi ) is bounded from below.

This paper deals with the following prescribed boundary mean curvature problem in ({mathbb{B}^N})

where (tilde K(y) = tilde K(|{y^prime }|,tilde y)) is a bounded nonnegative function with (y = ({y^prime },tilde y) in {mathbb{R}^2} times {mathbb{R}^{N - 3}},,,{2^sharp } = {{2(N - 1)} over {N - 2}}). Combining the finite-dimensional reduction method and local Pohozaev type of identities, we prove that if N ≥ 5 and (tilde K(r,tilde y)) has a stable critical point (r₀, (({r_0},{tilde y_0}))) with r₀ > 0 and (tilde K({r_0},{tilde y_0}) > 0), then the above problem has infinitely many solutions, whose energy can be made arbitrarily large. Here our result fill the gap that the above critical points may include the saddle points of (tilde K(r,tilde y)).

Let λ = (λ₁,…,λ_n) be β-Jacobi ensembles with parameters p₁, p₂, n and β while β varying with n. Set (gamma = {lim _{n to infty }}{n over {{p_1}}}) and (sigma = {lim _{n to infty }}{{{p_1}} over {{p_2}}}). In this paper, supposing ({lim _{n to infty }}{{log n} over {beta n}} = 0), we prove that the empirical measures of different scaled λ converge weakly to a Wachter distribution, a Marchenko–Pastur law and a semicircle law corresponding to σγ > 0, σ = 0 or γ = 0, respectively. We also offer a full large deviation principle with speed βn² and a good rate function to precise the speed of these convergences. As an application, the strong law of large numbers for the extremal eigenvalues of β-Jacobi ensembles is obtained.

A Hom-group is the non-associative generalization of a group whose associativity and unitality are twisted by a compatible bijective map. In this paper, we give some new examples of Hom-groups, and show the first, second and third isomorphism theorems of Hom-groups. We also introduce the notion of Hom-group action, and as an application, we prove the first Sylow theorem for Hom-groups along the line of group actions.