Acta Mathematica Sinica-English Series最新文献

We construct a new class of subspace lattices ({cal L}) on an infinite-dimensional Hilbert space ({cal K}). We show that the bounded cohomology groups (H^{n}({rm Alg} , {cal L},,{cal B}({cal K}))) of the corresponding lattice algebras Alg ({cal L}) with coefficients in ({cal B}({cal K})) are trivial for all n ≥ 1, and every derivation ϕ from Alg ({cal L}) into Alg ({cal L}) is an inner derivation under some conditions. We also prove that every Lie derivation δ from Alg ({cal L}) into ({cal B}({cal K})) is standard.

Polya–Carlson theorem asserts that if a power series with integer coefficients and convergence radius 1 can be extended holomorphically out of the unit disc, it must represent a rational function. In this note, we give a generalization of this result to multivariate case and give an application to rationality theorem about D-finite power series.

We construct an explicit example of a smooth isotopy {ξ_t}_t∈[0,1] of volume- and orientation-preserving diffeomorphisms on [0, 1]ⁿ (n ≥ 3) that has infinite total kinetic energy. This isotopy has no self-cancellation and is supported on countably many disjoint tubular neighbourhoods of homothetic copies of the isometrically embedded image of (M, g), a “topologically complicated” Riemannian manifold-with-boundary. However, there exists another smooth isotopy that coincides with {ξ_t} at t = 0 and t = 1 but of finite total kinetic energy.

DeBiasio and Krueger showed the following result: For all 0 ≤ δ ≤ 1 and ϵ > 0, there exists n₀ such that if G is a balanced bipartite graph on 2n ≥ 2n₀ vertices with δ(G) = δn, then in every 2-coloring of G, there exists a monochromatic cycle of order at least (f(δ) − ϵ)n, where

Zhang and Peng (2023) extended the above result to off-diagonal cases when ({delta} > {3 over 4}). In this paper, we relax the condition ({delta} > {3 over 4}) to ({delta} > {2 over 3}). We show the following result: For every η > 0, there exists a positive integer N₀ such that for every integer N > N₀ the following holds. Let ({2 over 3} < {delta} leq {3 over 4}), and let ({alpha_1} geq {{deltaalpha}_{2} over {3delta - 2}} > 0) such that α₁ + α₂ = 1. Let G[X, Y] be a balanced bipartite graph on 2N vertices with δ(G) = (δ + 3η)N. Then for each red-blue-edge-coloring of G, either there exist red even cycles of each length in {4, 6, 8, …, 2(2δ − 1)(2 − 3η²)α₁N}, or there exist blue even cycles of each length in {4, 6, 8, …, 2(2δ − 1)(2 − 3δ²)α₂N}. There are constructions of colorings showing that the length of a longest monochromatic cycle is asymptotically tight and the condition ({alpha_1} geq {{deltaalpha}_{2} over {3delta - 2}}) cannot be removed.

In this paper, we consider the recovery of third-order differential operators from two spectra, as well as fourth-order or fifth-order differential operators from three spectra, where these differential operators are endowed with complex-valued distributional coefficients. For the case of multiple spectra, we first establish the relationship between spectra and the Weyl–Yurko matrix. Secondly, we prove the uniqueness theorem for the solution of the inverse problems. Our approach allows us to obtain results for the general case of complex-valued distributional coefficients.

We study the L^p boundedness of singular integral operators along surfaces of revolution on product spaces. The L^p boundedness for these operators are obtained under very weak conditions on kernels. Our results are new and they improve several previously known results. Furthermore, they are natural extensions of many known results on singular integrals in the one-parameter setting and they subsume many other corresponding results on the product space setting.

Given a finite tensor category ({cal C}), an exact indecomposable ({cal C})-module category ({cal M}), and a tensor subcategory ({cal D} subseteq {cal C}_{cal M}^{ast}), we describe a way to produce exact commutative algebras in the center (Z({cal C})), measuring this inclusion. The construction of such algebras is done in an analogous way as presented by Shimizu [20], but using instead the relative (co)end, a categorical tool developed in [1] in the realm of representations of tensor categories. We provide some explicit computations.

Let V ⋃_S W be a Heegaard splitting of M with distance n ≥ 2 and F a boundary component of ∂_V. A simple closed curve J in F is called distance degenerating if the distance of M_J = V_J ⋃_S W is less than n, where M_J is obtained by attaching a 2-handle to M along J. In this paper, by considering the distance between J and the image of the essential disks of W under the projection map, we obtain a sufficient condition for the diameter of the set of distance degenerating curves in F to be bounded in C(F). Moreover, for F = ∂M, an upper bound of the diameter of the set of the boundary reducible curves in F is given under some circumstance.

Combing the weak KAM method for contact Hamiltonian systems and the theory of viscosity solutions for Hamilton–Jacobi equations, we study the Lyapunov stability and instability of viscosity solutions for evolutionary contact Hamilton–Jacobi equation in the first part. In the second part, we study the existence and multiplicity of time-periodic solutions.

In this paper, we give a slight improvement of El Soufi–Ilias–Ros’s upper bound of the first Laplace eigenvalue on a torus in a fixed conformal class. We also optimize Montiel–Ros’s argument to obtain a better upper bound of the conformal area for certain rectangular tori.