数学季刊

李姣龙, 秦玉明

2017, 32(3): 221-237. doi:10.13371/j.cnki.chin.q.j.m.2017.03.001

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In this paper, we first establish the global existence and asymptotic behavior of solutions by multiplicative techniques, then further prove the existence of a uniform attractor for a thermoelastic system by using the method of uniform contractive functions. We finally investigate an alternative result of solutions for the semilinear thermoelastic systems by virtue of the semigroup method.

不含四圈的k圈图的无符号拉普拉斯谱半径

孔琪, 王力工

2017, 32(3): 238-245. doi:10.13371/j.cnki.chin.q.j.m.2017.03.002

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A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C₄-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C₄-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the（combinatorial） Laplacian

非自治Timoshenko系统解的整体存在性和渐进性

郑秀珍, 秦玉明

2017, 32(3): 246-254. doi:10.13371/j.cnki.chin.q.j.m.2017.03.003

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Considering Timoshenko systems under the Cattaneo thermal law, the purpose of the article is to obtain the global existence of linear Timoshenko system and semilinear Timoshenko system for the solutions of non-autonomous situations. The main method is converting the systems into abstract ODE, constructing proper spaces and proving the global existence by using semigroup methods. For asymptotic behavior, there will be a remark to describe the results.

具有有限整体Gorenstein维数的环

任伟, 张瑜

2017, 32(3): 255-260. doi:10.13371/j.cnki.chin.q.j.m.2017.03.004

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As a proper setting to study Gorenstein projective and injective dimensions of modules via vanishing of Gorenstein Ext-functors, a notion of a generalized Gorenstein ring is introduced, which is a non-trivial generalization of Gorenstein rings. Moreover, a new proof for Bennis and Mahdou’s equality of global Gorenstein dimension is given.

应用（G^’/G²）-展开法求解非线性分数阶微分方程

康周正

2017, 32(3): 261-270. doi:10.13371/j.cnki.chin.q.j.m.2017.03.005

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In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie’s modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.

关于特定同余式方程解数的余项

张义锋, 史三英

2017, 32(3): 271-276. doi:10.13371/j.cnki.chin.q.j.m.2017.03.006

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Let f（x） be an irreducible polynomial of degree m ≥ 2 with integer coefficients,and let r（n） denote the number of solutions x of the congruence f（x） ≡ 0（mod n） satisfying0 ≤ x < n. Define ?（x） =Σ 1≤n≤x r（n）-αx, where α is the residue of the Dedekind zeta function ζ（s, K） at its simple pole s = 1. In this paper it is shown that ∫1^X?2（x）dx?

ε{X^3-6/m+3+εif m ≥ 3,X^2+ε if m = 2,for any non-Abelian polynomial f（x） and any ε > 0. This result constitutes an improvement upon that of Lü for the error terms on average.

二维区域上椭圆均匀化问题的收敛率

赵杰, 李红, 王娟

2017, 32(3): 277-293. doi:10.13371/j.cnki.chin.q.j.m.2017.03.007

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In this paper, we study the convergence rates of solutions for second order elliptic equations with rapidly oscillating periodic coefficients in two-dimensional domain. We use an extension of the "mixed formulation" approach to obtain the representation formula satisfied by the oscillatory solution and homogenized solution by means of the particularity of solutions for equations in two-dimensional case. Then we utilize this formula in combination with the asymptotic estimates of Green or Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in Lp for solutions as well as gradient error estimates for Dirichlet or Neumann problems respectively.

l~0({X_i})型赋准范空间的共轭空间的表示问题

王见勇

2017, 32(3): 294-304. doi:10.13371/j.cnki.chin.q.j.m.2017.03.008

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Extending the results of an article published in(Acta Mathematica Sinica（2016,59（4）） by the author, for a sequence of normed spaces {X_i}, the representation problem of conjugate spaces of some l⁰（{X_i}） type F-normed spaces are studied in this paper. The algebraic representation continued equalities l⁰（{X_i}） * A=c₀₀⁰（{X_i}） * A= c₀₀（{X_i*}）,（l⁰（X））* A=（c⁰（X））* A=（c₀⁰（X））* A=(c₀₀⁰（X）)* A= c₀₀（X*）are obtained in the first part. Under weak-star topology, the topological representation c₀₀⁰（{X_i}） *, w* = c₀₀⁰（{X_i*}） is obtained in the second part. For the sequence of inner product spaces and number fields with the usual topology, the concrete forms of the basic representation theorems are obtained at last.

关于衰减Timoshenko系统的解的整体存在性、渐近性及一致吸引子

胡文松, 秦玉明

2017, 32(3): 305-321. doi:10.13371/j.cnki.chin.q.j.m.2017.03.009

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In this paper, we consider the Timoshenko system as an initial-boundary value problem in a one-dimensional bounded domain. And then we establish the global existence and asymptotic behavior of solution by using the semigroup method and multiplicative techniques, then further prove the existence of a uniform attractor by using the method of uniform contractive function. The main advantage of this method is that we need only to verify compactness condition with the same type of energy estimate as that for establishing absorbing sets.

参外激励下强非线性广义Van der Pol方程的亚谐共振解

许东亮

2017, 32(3): 322-330. doi:10.13371/j.cnki.chin.q.j.m.2017.03.010

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In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.

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