Superposition operator in Sobolev spaces on domains

For an arbitrary open set we characterize all functions on the real line such that for all . New element in the proof is based on Maz'ya's capacitary criterion for the imbedding .     

态|ψ3(3)〉q广义电场分量的等幂次N次方H压缩

构造了由多模复共轭相干态|{iZ_j^*}〉_q、多模复共轭虚相干态|{iZ_j^*}〉_q(j=1,2,…q)和多模真空态|{0_j}〉_q线性叠加所组成的第Ⅲ类三态叠加多模叠加态光场|ψ₃⁽³⁾〉_q.利用多模压缩态理论,研究了态|ψ₃⁽³⁾〉_q中广义电场分量(即第二正交相位分量)的广义非线性等幂次N次方H压缩特性.发现:态|ψ₃⁽³⁾〉_q是一种典型的三态叠加多模非经典光场;当腔模总数q与压缩次数N之积q·N为偶数时,在一定条件下,态|ψ₃⁽³⁾〉_q的广义电场分量可分别呈现出周期性变化的广义非线性等幂次奇数模-偶数次、偶数模-奇数次、偶数模-偶数次N次方H压缩效应.     

态|ψ3(3)〉q中广义磁场分量的等幂次高次和压缩

根据量子力学中态的线性叠加原理,构造了由多模复共轭相干态|{Z_j^*〉_q(j=1,2,……q)、多模复共轭虚相干态|{iZ_j^*〉_q和多模真空态|{0_j}〉_q的线性叠加组成的第Ⅲ类三态叠加多模叠加态光场|ψ₃⁽³⁾〉_q.利用多模压缩态理论,研究了态|ψ₃⁽³⁾〉_q广义磁场分量(即第一正交相位分量)的广义非线性等幂次高次和压缩特性结果发现:当腔模总数q与压缩次数N之积qN取偶数时,态|ψ₃⁽³⁾〉_q广义磁场分量在一定条件下可分别呈现出周期性变化的、广义非线性等幂次奇数模—偶数次、偶数模—奇数次或偶数模—偶数次的高次和压缩效应.     

Tomography of Nonclassical States

A review of the symplectic tomography method is presented. Superpositions of different types of photon states are considered within the framework of the tomography approach. Such nonclassical photon states as even and odd coherent states, crystallized Schrödinger cat states, and other superposition states are studied using the construction of symplectic tomograms (tomographic symbols) and the star-product formalism for tomograms.     

第Ⅵ类两态叠加多模叠加态光场的等阶N次方Y压缩特性研究

本文构造了由多模相干态|{Z_j}〉q与多模虚相干态|{iZ_j}〉q这两者的线性叠加所组成的第Ⅵ类两态叠加多模叠加态光场|Ψ₆⁽²⁾〉q.利用多模压缩态理论,研究了态|Ψ₆⁽²⁾〉q的广义非线性等阶N次方Y压缩特性.结果发现:1)在压缩阶数N＝2p且p＝2l(l＝1,2,3,…,…)的条件下,无论各模的初始相位φ_j(j＝1,2,3,…,q)、态间的初始相位差(θ_pq^(R)-θ_pq^(I))以及各单相干态光场的平均光子数之和 R_j²等如何变化,态|ψ₆⁽²⁾〉q总是恒处于偶数阶等阶N-Y最小测不准态.2)在压缩阶数N＝2p且p＝2l+1(l＝0,1,2,3,…,…)的条件下,当各模的初始相位φj、态间的初始相位差(θ_pq^(R)-θ_pq^(I))以及各单模相干态光场的平均光子数之和 R_j²等分别满足一定的量子化条件(或者在特定的区间内连续取值)时,态|ψ₆⁽²⁾〉q的第一及第二这两个正交分量总可分别呈现出周期性变化的偶数阶的等阶N次方Y压缩效应.     

Isothermal viscoelastic properties of PMMA and LDPE over 11 decades of frequency and time: a test of time–temperature superposition

Many instruments used to measure viscoelastic properties are only capable of subjecting a sample to a limited range of loading frequencies. For thermorheologically simple materials, it is assumed that a change in temperature is equivalent to a shift of the viscoelastic behavior on the log frequency or time axis. For many materials, time–temperature superposition appears to work well for modulus or compliance curves over three decades of time or frequency, but some deviations are known if the window is expanded to five or six decades. To apply a more stringent test of the validity of time–temperature superposition, broadband viscoelastic spectroscopy is used to isothermally study polymethylmethacrylate and low-density polyethylene at several temperatures in the glassy region. Shear modulus and damping (tan δ) are measured isothermally over a wide range (up to 11 decades) of time and frequency. Results indicate that, while modulus curves can be approximately superimposed, the damping (tan δ) curves change in height and shape with temperature.     

叠加qs相干态的非经典性质

构造了叠加qs相干态,并研究了它的压缩性质、反聚束效应等非经典性质.数值计算了叠加系数和形变参数q与s对非经典性质的影响.奇偶qs相干态的有关结果作为特例包含在本文的一般结论之中.     

一类q-变形叠加态光场的振幅k次方压缩及高阶反聚束特性

构造了一种由q变形相干态与q变形真空态叠加而成的态,研究了它的振幅k次方压缩性质和高阶反束特性,借助数值计算方法讨论了叠加系数、变形参数对这些特性的影响。     

The classification of travelling wave solutions and superposition of multi-solutions to Camassa-- Holm equation with dispersion

Under the travelling wave transformation, the Camassa--Holm equation with dispersion is reduced to an integrable ordinary differential equation (ODE), whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa--Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa--Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa--Holm equation with dispersion.