Sound transmission into a thick hollow cylinder with the fixed-end boundary condition

In this paper sound transmission through the air filled finite thick cylinders exposed to the different incident acoustic wave is studied. The effect of end boundary conditions on the noise reduction of finite cylinders is evaluated. The uniform incident wave and the wave radiated from monopole and dipole sources are used in this study. Three positions are considered for the dipole source. Every position for the dipole source causes symmetric or antisymmetric pressure distributions on the external surface of the cylinder in tangential or axial direction. For the purpose of sound transmission analysis the linear three-dimensional theory of elasticity utilizing the technique of variables separation for the infinite circular cylinders is used to analyze the vibration of finite circular cylinder. In these analyzes the stress continuity condition on the inner and outer surfaces of the cylinder is satisfied using orthogonalization technique and velocity continuity condition is exactly satisfied on the interfacial surfaces. The sound transmission evaluation is carried out for cylinders with various half-length to outer-radius ratios. The results show that in the case of the fixed-end cylinder, the effect of boundary conditions on the noise reduction can be neglected for the half-length to outer-radius ratio of more than 10. Comparing between the obtained results from different acoustic sources shows that the obtained noise reductions from the uniform acoustic wave are less than those obtained from the monopole and dipole sources.     

MILU-CG方法和绕旋转圆柱流动的数值研究

本文采用以修正的不完全LU分解作预处理器的共轭梯度法(MILU-CG),结合高阶隐式差分格式,改进了作者(1992)提出的基于区域分解、有限差分法与涡法杂交的数值方法(HDV).系统地研究了雷诺数Re=1000,200,旋转速度比α∈(0.5,3.25)范围内,绕旋转圆柱从突然起动到充分发展,长时间内尾流旋涡结构和阻力、升力系数的变化规律.计算所得流线与实验流场显示相比,完全吻合.首次揭示了临界状态时的旋涡结构特性,并指出最佳升阻比就在该状态附近得到.     

On Factorisations and Generators in Transformation Semigroups

It is shown that the classical decomposition of permutations into disjoint cycles can be extended to more general mappings by means of path-cycles, and an algorithm is given to obtain the decomposition. The device is used to obtain information about generating sets for the semigroup of all singular selfmaps of $X_{n} = \{1, 2, \dots, n\}$. Let $T_{n,r} = S_{n}\cup K_{n,r}$, where $S_{n}$ is the symmetric group and $K_{n,r}$ is the set of maps $\alpha\,:\, X_{n} \to X_{n}$ such that $|im(\alpha)| \le r$. The smallest number of elements of $K_{n,r}$ which, together with $S_{n}$, generate $T_{n,r}$ is $p_{r}(n)$, the number of partitions of $n$ with $r$ terms.     

Least Common Multiple of Path, Star with Cartesian Product of Some Graphs

A graph $G$ without isolated vertices is a least common multiple of two graphs $H_1$ and $H_2$ if $G$ is a smallest graph, in terms of number of edges, such that there exists a decomposition of $G$ into edge disjoint copies of $H_1$ and $H_2$. The collection of all least common multiples of $ H_1 $ and $ H_2 $ is denoted by $ \LCM (H_1, H_2) $ and the size of a least common multiple of $ H_1 $ and $ H_2 $ is denoted by $ \lcm (H_1, H_2) $. In this paper $\lcm ( P_4, P_m\ \square\ P_n) $, $\lcm (P_4, C_m \ \square\ C_n)$ and $\lcm (K_{1,3}, K_{1,m}\ \square\ K_{1,n}) $ are determined.     

ON A PROBLEM OF SUMS OF MIXED POWERS （II）

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Development of a high order discretization scheme for solving fully nonlinear magnetohydrodynamic equations

We have developed fully fourth order accurate compact finite difference discretization scheme for the Navier-Stokes equations coupled with Maxwell''s equations. The implementation is done in cylindrical polar geometry. Due to the full-MHD modeling of physical flow, the modeled equations are fully nonlinear coupled hydrodynamic equations which are again coupled with Maxwells equations. In our computations, we have accounted for the induced magnetic field in the flow of an electrically conducting fluid in an external magnetic field. The code is tested against available experimental and theoretical data where applicable. It is observed that a smaller grid of $64 \times 64$ is sufficient for weakly nonlinear problems and higher grids up to $512 \times 512$ are needed as the degree of nonlinearities grow in the modeled equation. In the absence of magnetic field, a discontinuity of total drag coefficient and separation length is noted for $Re=73$ which is in agreement with literature. When the magnetic Reynolds number $Rm<1$ separation length decreases linearly with strength of magnetic field on a log-log scale whereas if $Rm>1$, it decreases nonlinearly, at a much faster rate. Thermal boundary layer thickness decreases as the strength of magnetic field increases and it forces the thermal convection to take place in a laminar structure as observed from thermal contour lines. Finally, using divided differences, we establish that the accuracy of the proposed numerical scheme is in fact fourth order.     

DISTRIBUTION OF THE（0，∞）ACCUMULATIVE LINES OF MEROMORPHIC FUNCTIONS

Suppose that f(z)is a meromorphic function of order λ（0&lt;λ&lt;+∞）and of lower order μ in the plane.Let ρ be a positive number such that μ≤ρ≤λ.(1)If f^(l)(z)(0≤l&lt;+∞）has p(1≤p&lt;+∞）finite nonzero deficient valnes αi(i=1,…，p)with deficiencies δ（αi,f^(l)),then f(z)has a (0,∞）accumulative line of order ≥ρin any angular domain whose vertex is at the origin and whose magnitude is larger than max(π/ρ，2π-4/ρ ∑i=1^p arcsin √δ（αi,f^(l))/2).(2)If f(z) has only p(0&lt;p&lt;+∞）（0,∞）,accumulative lines of order≥ρ：arg z=θk(0≤θ1&lt;θ2&lt;…&lt;θp&lt;2π，θp+1=θ1+2π）,then λ≤π/ω，where ω=min I≤k≤p(θk+1-θk),provided that f^(l)(z)(0≤l&lt;+∞）has a finite nonzero deficient value.     

On the Number of Integral Ideals in Two Different Quadratic Number Fields

Let K be an algebraic number field of finite degree over the rational field Q,and a K(n) the number of integral ideals in K with norm n. When K is a Galois extension over Q, many authors contribute to the integral power sums of a K(n),Σn≤x a K(n)~l, l = 1, 2, 3, ···.This paper is interested in the distribution of integral ideals concerning different number fields. The author is able to establish asymptotic formulae for the convolution sum Σn≤x aK_1(n~j)~laK_2(n~j)~l, j = 1, 2, l = 2, 3, ···,where K_1 and K_2 are two different quadratic fields.     

The generalized triangular decomposition

Given a complex matrix , we consider the decomposition , where is upper triangular and and have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where is an upper triangular matrix with the eigenvalues of on the diagonal. We show that any diagonal for can be achieved that satisfies Weyl's multiplicative majorization conditions:

where is the rank of , is the -th largest singular value of , and is the -th largest (in magnitude) diagonal element of . Given a vector which satisfies Weyl's conditions, we call the decomposition , where is upper triangular with prescribed diagonal , the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

     

Partial Differential Equations of Higher Order and Symmetric Positive Systems

In this paper we study the relation between symmetric positive systems and equations of higher order. The main result is: Theorem 1. An equation of second order $L\phi =f$ can be transformed into a symmetric positive system by introducing new unknown functions $u_i=\sum\limits_{j=0}^n {\alpha_ij \varphi _j(i=0,1,\cdots,n),\varphi_0=\varphi_2,\varphi_j=\partial \varphi /\partial x_j}$ iff there exists L_1 of order 1 such that $Re(L_1 \varphi \cdot \bar {L\varphi})=\sum\limits_{i=1}^n{\frac{\partial}{\partial x_i}}+B(\varphi,\varphi)$, where P_i(\varphi,\varphi)(i=1,2,\cdots,n),B(\varphi,\varphi) are differential quadarlic forms and B(\varphi,\varphi) is positive definite. This Theorem can be extended into equations of higher order. Some examples of deducing equations of higher order into symmetric positive systems are given. Finally, we give a counter example which shows that a boundary problem of a symmetric positive system deduced from an equation of higher order is admissible, but its corresponding bounbary problem of the original equation is not well-posed.     

A computational approach for solving

We present a computational approach for finding all integral solutions of the equation for even values of . By reducing this problem to that of finding integral solutions of a certain class of quartic equations closely related to the Pell equations, we are able to apply the powerful computational machinery related to quadratic number fields. Using our approach, we determine all integral solutions for assuming the Generalized Riemann Hypothesis, and for unconditionally.

     

Minimizers for the embedding of Besov spaces

Using the profile decomposition, we will show the relatively compactness of the minimizing sequence to the critical embeddings between Besov spaces, which implies the existence of minimizer of the critical embeddings of Besov spaces $\dot{B}^{s_1}_{p_1,q_1}\hookrightarrow \dot{B}^{s_2}_{p_2,q_2}$ in $d$ dimensions with $s_1-d/p_1=s_2-d/p_2$, $s_1>s_2$ and $1 \leq q_1相似文献   

Domain decomposition algorithms for mixed methods for second-order elliptic problems

In this paper domain decomposition algorithms for mixed finite element methods for linear second-order elliptic problems in and are developed. A convergence theory for two-level and multilevel Schwarz methods applied to the algorithms under consideration is given. It is shown that the condition number of these iterative methods is bounded uniformly from above in the same manner as in the theory of domain decomposition methods for conforming and nonconforming finite element methods for the same differential problems. Numerical experiments are presented to illustrate the present techniques.

     

ON THE CONSTRUCTION OF INDECOMPOSABLE POSITIVE DEFINITE QUADRATIC FORMS OVER Z

The aim of this paper is to construct indecomposable positive definite Z-lattices withgiven rank n(2≤n≤9) and discriminant a,a being composite and square-free,but for afinite number of exceptions.And all exceptional cases for 3≤n≤9 are determined.Theseare unsettled cases of a paper of O’Meara in 1975.     

Boundedness of Solutions for Duffing Equation with Low Regularity in Time

It is shown that all solutions are bounded for Duffing equation x+ x~(2n+1)+2∑i=nPj(t)x~j= 0, provided that for each n + 1 ≤ j ≤ 2 n, P_j ∈ C~y(T~1) with γ 1-1/n and for each j with 0 ≤ j ≤ n, Pj ∈ L(T~1) where T~1= R/Z.     

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考虑半参数回归模型$y_i=x_i\beta+g(t_i)+V_i$ $(1\le i\len)$, 其中$(x_i,t_i)$是已知的设计点, 斜率参数$\beta$是未知的,$g(\cdot)$是未知函数, 误差$V_i=\tsm^\infty_{j=-\infty}c_je_{i-j}$,$\tsm^\infty_{j=-\infty}|c_j|<\infty$并且$e_i$是负相关的随机变量.在适当的条件下, 我们研究了$\beta$与$g(\cdot)$小波估计量的强收敛速度.结果显示$g(\cdot)$的小波估计量达到最优收敛速度. 同时,对$\beta$小波估计量也作了模拟研究.     

The Geometry of the Phase Diffusion Equation

$ \tau(|{{\vec k}}|) \mbox{\bf $\Theta$}_T = -\nabla\cdot (B(|{{\vec k}}|)\cdot {{\vec k}}), \,\, {{\vec k}} = \nabla \mbox{\bf $\Theta$},$ and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order ``self-dual' equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. Received on October 30, 1998; final revision received July 6, 1999     

增益图与其底图的正惯性指数之间的关系

设$\mathbb{T}$是模为1的复数乘法子群.图$G=(V,E)$,这里$V,E$分别表示图的点和边.增益图是将底图中的每条边赋于$\mathbb{T}$中的某个数值$\varphi(v_iv_j)$,且满足$\varphi(v_iv_j) =\overline{\varphi(v_jv_i)}$.将赋值以后的增益图表示为$(G,\varphi)$.设$i_+(G,\varphi)$和$i_+(G)$分别表示增益图与底图的正惯性指数,本文证明了如下结论: $$ - c( G ) \le {i_ + } ( {G,\varphi } ) - {i_ + }( G ) \le c( G ), $$ 这里$c(G)$表示圈空间维数,并且刻画了等号成立时候的所有极图.     

Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations

In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter $0 < \varepsilon≤ 1.$ In fact, the solution to this equation propagates waves with wavelength at $O(\varepsilon^2)$ when $0<\varepsilon≪1,$ which brings significantly numerical burdens in practical computation. We rigorously establish two independent error bounds for the two MTIs at $O(\tau^2/\varepsilon^2)$ and $O(\varepsilon^2)$ for $\varepsilon ∈ (0,1]$ with $\tau > 0$ as step size, which imply that the two MTIs converge uniformly with linear convergence rate at $O(\tau)$ for $ε ∈ (0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $ε=O(1)$ or $0<ε≤\tau.$ Thus the meshing strategy requirement (or $ε$-scalability) of the two MTIs is $\tau =O(1)$ for $0<ε≪1,$ which is significantly improved from $\tau =O(ε^3)$ and $\tau =O(ε^2)$ requested by finite difference methods and exponential wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards better understanding on the convergence and resolution properties of the two MTIs. In addition, numerical results support the two error bounds very well.     

DECOMPOSITION OF BMO FUNCTIONS AND FACTORIZATION OF $\[{A_p}\]$ WEIGHTS IN MARTINGALE SETTING

Let $\[(\Omega ,F,\mu )\]$ be a probabilty space with an increasing family $\[{\{ {F_t}\} _{t > 0}}\]$ of sub-fields satisfying the usual conditions. The following results are obtained: for $\[f \in BMO\]$, we have $\[f = g - h\]$ with $\[g,h \in BLO\]$; if in addition, f satisfies then for $\[s > 0\]$ arbitrary, g,h can be chosen such that $\[g,h \in BLO\]$, and $$\[E({\varepsilon ^{(a - \varepsilon )(g - {g_t})}}|{F_t}) \le {C_{a,\beta ,\varepsilon }},E({\varepsilon ^{(\beta - \varepsilon )(h - {h_t})}}|{F_t}) \le {C_{a,\beta ,\varepsilon }}\]$$ and for weights z, we have $\[z \in {A_p} \cap S \Leftrightarrow z = {z_1}z_2^{1 - p}\]$ with $\[{z_i} \in {A_i} \cap S,i = 1,2\]$, where $\[S = \{ \begin{array}{*{20}{c}} {weight}&{z:C{z_{{T^ - }}} \le {z_T} \le C{z_{{T^ - }}}} \end{array}\} \]$, $\[\forall \]$ stopping times T, outside a null set }.