L~2(R~d)中MRA小波相位的刻画

设A是d×d实扩展矩阵,ψ是以A为扩展矩阵的小波,f是可测函数.如果对任意以A为扩展矩阵的小波ψ,fψ(其中ψ表示ψ的傅立叶变换)的逆傅立叶变换仍是以A为扩展矩阵的小波,则称f是以A为扩展矩阵的小波乘子.主要刻画了L²(R2(R^d)空间中,以行列式绝对值等于2的整数矩阵为扩展矩阵的MRA小波的线性相位.利用该结果,具体给出了二维情况下,Haar型和Shannon型小波在相似意义下的六类整数扩展矩阵的线性相位的表达形式.最后将具有线性相位的MRA不可分离小波应用到二维图象的边缘检测上.     

带有分布导数的反应扩散方程在Rn中全局吸引子的存在性

该文考虑Rⁿ中一类带有加权项V(x)和分布导数的反应扩散方程解的长时间行为,证明了(L²(Rⁿ),L²(Rⁿ)∩L^p(Rⁿ))-全局吸引子的存在性;而且对任意的δ₁∈[0,+∞),该吸引子能够在L²∩L^p+δ₁-拓扑下吸引L²(Rⁿ)中的有界集.     

Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains

The regularity of random attractors is considered for the non-autonomous fractional stochastic FitzHugh-Nagumo system.We prove that the system has a pullback random attractor that is compact in Hs(Rⁿ)×L²(Rⁿ)and attracts all tempered random sets of L²(Rⁿ)×L²(Rⁿ)in the topology of Hs(Rⁿ)×L²(Rⁿ)with s∈(0,1).By the idea of positive and negative truncations,spectral decomposition in bounded domains,and tail estimates,we achieved the desired results.     

带部分调和势的非齐次非线性Schr?dinger方程的爆破解

该文致力于研究带部分调和势的非齐次非线性Schr?dinger方程的Cauchy问题.该方程是玻色-爱因斯坦凝聚中的一个重要模型.结合非线性椭圆方程基态解的变分特征及质量和能量守恒,首先得到了该问题整体解的存在性,并利用尺度变换技巧证明了该方程在一些特殊初值情形下存在爆破解.其次讨论了爆破解的L²集中现象.最后利用与上述基态解相关的变分结论研究了L²最小质量爆破解的动力学性质,即具有最小质量的爆破解的极限profile、精细质量集中和爆破速率.该文将Zhang^[35]的全局存在性和爆破结果推广到带非齐次非线性项的情形,并将Pan和Zhang^[24]的部分结果改进到空间维数N≥2且非线性项为非齐次的情形.     

Schrödinger方程的连续时空有限元方法

本文讨论Schr?dinger方程的连续时空有限元方法,通过引入相应的时空投影算子,利用实部虚部分离技巧,得到了变量u在时间节点处的L²范数,以及u和u_t的全局L²(H¹)和L²(L²)范数意义下的最优误差估计结果.该文的结论对进一步探索和设计Schr?dinger方程的数值算法是有益的.     

二阶双曲方程的全离散格式下的混合元超收敛分析

通过在空间方向上使用双线性元和最低阶的Nedelec元(即Q₁₁+Q₀₁×Q₁₀)以及在时间方向上使用二阶精度的数值逼近格式,得到了在矩形网格上二阶双曲方程全离散混合元格式下的对原始变量的L^∞(H¹)和流量的L^∞((L²)²)的超逼近和超收敛的误差结果.在分析过程中,巧妙地使用了上述混合单元对在矩形网格上的特有的高精度积分恒等式和精确解的投影和插值之间的在H¹范数意义下的超逼近的估计.最后,给出一些数值结果来验证理论分析的正确性.     

Remarks on Sharp Interface Limit for an Incompressible Navier-Stokes and Allen-Cahn Coupled System*

The authors are concerned with the sharp interface limit for an incompressible Navier-Stokes and Allen-Cahn coupled system in this paper. When the thickness of the diffuse interfacial zone, which is parameterized by ε, goes to zero, they prove that a solution of the incompressible Navier-Stokes and Allen-Cahn coupled system converges to a solution of a sharp interface model in the L^∞(L²) ∩ L²(H¹) sense on a uniform time interval independent of the smal...     

抛物方程的一个新混合元格式的高精度分析

借助双二次元及一阶Raviart-Thomas(R-T)元对抛物方程提出了一种新的协调混合有限元格式,导出了半离散及全离散格式下原始变量在H¹和L1和L²模意义下以及流量(?)在L2模意义下以及流量(?)在L²模意义下的超逼近结果.     

二阶椭圆问题两种新的矩形混合元格式

1引言考虑Poisson方程的第一齐边值问题:（?）其中Ω∈Rⁿ（n=2,3）是有界凸多角形区域.f∈L²（Ω）是已知函数.令（?）=▽u.传统方法是定义:H_l={（?）∈L²（Ω）ⁿ;div（?）∈L²（Ω）},M₁=L²（Ω）,（?）H₁=(（?）+     

BLOW-UP SOLUTIONS OF TWO-COUPLED NONLINEAR SCHR?DINGER EQUATIONS IN THE RADIAL CASE

We consider the blow-up solutions to the following coupled nonlinear Schrodinger equations■.On the basis of the conservation of mass and energy,we establish two sufficient conditions to obtain the existence of a blow-up for radially symmetric solutions.These results improve the blow-up result of Li and Wu [10] by dropping the hypothesis of finite variance((|x|u₀,|x|v₀) ∈L²(R^N) × L²(R^N)).     

Correlation Wavelet and its Applications

§１．　ＤａｕｂｅｃｈｉｅｓＷａｖｅｌｅｔｓａｎｄＣｏｒｒｅｌａｔｉｏｎＷａｖｅｌｅｔｓＡｓｗｅｋｎｏｗ,Ｄａｕｂｅｃｈｉｅｓｓｃａｌｉｎｇｆｕｎｃｔｉｏｎφ（ｘ）ａｎｄｗａｖｅｌｅｔｆｕｎｃｔｉｏｎψ（ｘ）ｈａｖｅｔｈｅｆｏｌ－ｌｏｗｉｎｇｐｒｏｐｅｒｔｉｅｓ：１．φ（ｘ）ｉｓａｇｅｎｅｒａｔｉｎｇｅｌｅｍｅｎｔｆｏｒａｍｕｌｔｉｒｅｓｏｌｕｔｉｏｎａｎａｌｙｓｉｓ（ＭＲＡ）,ｓｕｐｐφ（ｘ）＝［０,２Ｎ－１］,ｏｒｔｈｏｇｏｎａｌｗｉｔｈｉｎｔｅｇｅｒｔｒａｎｓｌａｔｉｏｎｓ．Ｇｅｎｅｒａｌ…     

Frame Wavelets with Compact Supports for L^2（R^n）

The construction of frame wavelets with compact supports is a meaningful problem in wavelet analysis. In particular, it is a hard work to construct the frame wavelets with explicit analytic forms. For a given n × n real expansive matrix A, the frame-sets with respect to A are a family of sets in R^n. Based on the frame-sets, a class of high-dimensional frame wavelets with analytic forms are constructed, which can be non-bandlimited, or even compactly supported. As an application, the construction is illustrated by several examples, in which some new frame wavelets with compact supports are constructed. Moreover, since the main result of this paper is about general dilation matrices, in the examples we present a family of frame wavelets associated with some non-integer dilation matrices that is meaningful in computational geometry.     

基于高斯型窗函数的基小波构造

阐述了基于高斯型窗函数的可容基小波构造,讨论了若干类基小波.首先引入若干经典基小波如墨西哥草帽小波、莫莱小波、DOG犬小波和盖博解析小波,作者发现它们具有统一的结构,即均由高斯窗函数生成;进而在犬小波结构的启示下,构造了由高斯窗函数的差形成的犬小波族,对之验证了可容性条件;并且将它推广为有限个高斯窗函数的线性组合形成的小波,确定了带通条件.     

二元正交小波的构造

高维小波是处理多维信息的工具。本文给出的构造紧支撑不可分二元正交小波函数的算法，当尺度函数和符号中所含因子[(1 z1/2)(1 z2/2)]^2的幂指数r越高时，尺度函数越光滑。     

区间上的双正交小波的一种构造方法

小波分析是近十几年来十分热门的课题,早期的小波都是定义在无穷区间上的．而实际问题常常是有限区间,常用的方法是将有限区间上的数据向区间外延拓,但这样做容易产生边界误差,如何构造区间上的具有良好性质的小波,如光滑性、对称性等是非常有意义的．在文献［２］中,Cohen,Darbechies与Vial在总结了构造区间周期小波,折叠小波等方法的基础上,提出了一种新的构造方法,并把无穷区间上的Daubechies小波改造成区间上的Daubechies小波．但是,Daubechies小波没有对称性,光滑性也差．本文用类似的方法把无穷区间上的双正交小波改造…     

REAL-VALUED PERIODIC WAVELETS: CONSTRUCTION ANDRELATION WITH FOURIER SERIES

1.IntroductionWaveletshaverecentlyreceivedagreatdealofattentioninsuchareasassignalprocessingandimageprocessing([12],[8]).Variousmethodstoconstructwaveletshavebeengiven([14],[13],[9],[7]).Itiswellknownthatinmathematicsandmathematicphysicsmanyperiodicp...     

紧支撑二元正交小波滤波器的构造

高维小波是处理多维信号的有力工具，张量积小波有其自身的缺点．本文给出矩形域上二元正交小波滤波器的一种参数化构造算法，二元小波滤波器的这种构造方法使我们能更方便地研究非张量积的二元正交小波．最后给出算例．     

二元实值正交周期Box样条小波

In this paper, we construct a kind of bivariate real-valued orthogonal periodic box-spline wavelets. There are only 4 terms in the two-scale dilation equations. This implies that the corresponding decomposition and reconstruction algorithms involve only 4 terms respectively which are simple in practical computation. The relation between the periodic wavelets and Fourier series is also discussed.     

Numerical solution of high-order differential equations by using periodized Shannon wavelets

In this paper, periodized Shannon wavelets are applied as basis functions in solution of the high-order ordinary differential equations and eigenvalue problem. The first periodized Shannon wavelets are defined. The second the connection coefficients of periodized Shannon wavelets are related by a simple variable transformation to the Cattani connection coefficients. Finally, collocation method is used for solving the high-order ordinary differential equations and eigenvalue problem. Some equations are solved in order to find out advantage of such choice of the basis functions.     

Pointwise convergence of a class of non-orthogonal wavelet expansions

Non-orthogonal wavelet expansions associated with a class of mother wavelets is considered. This class of wavelets comprises mother wavelets that are not necessarily integrable over the whole real line, such as Shannon's wavelet. The pointwise convergence of these wavelet expansions is investigated. It is shown that, unlike other wavelet expansions, the ones under consideration do not necessarily converge pointwise to the functions at points of continuity, unless a more stringent condition, such as bounded variation, is imposed.