An integral-equation approach to problems of vibrating beams

In this paper integral equations are applied for the calculation of the normal modes of vibrating beams. Both exact and approximate methods of solving the integral equation are considered. The Green's function, or kernel, of the integral equation is constructed for both uniform and nonuniform beams. Solutions for the normal modes of a uniform cantilever are given. A nonuniform, naturally-twisted turbine blade is studied in detail and the first and second normal modes are calculated by the integral-equation method.     

A photo-electric integraph

A machine for the purpose of facilitating mathematical solution of problems requiring the evaluation of an integral in which the integrand involves a variable parameter is described in this article. It involves the use of an optical system in which the transmission of light is limited in a definite manner by apertures having the shape of the area under curves representing mathematical functions. The accuracy of the machine itself is from 2 to 5 per cent. Examples of the use of the device in the solution of the superposition theorem integral and a form of integral equation are given. Its application to other problems including the Fourier transform, correlation analysis, and periodogram analysis is discussed, and its limitations are considered.     

Solution of integral equations using a set of block pulse functions

A set of the block pulse functions is applied to solve the Fredholm's and the Volterra's integral equations of the second kind. An algebraic equation in matrix form which is equivalent to the solution of the integral equation is developed. The approximate results are easily obtained by a few computations. An accurate solution canbe evaluated in a digital computer by solving the algebraic equation. Two examples are given.     

小垂度单根悬索方程的改进

推导了小垂度单根悬索在纵向均布荷载和横向线性分布荷载作用下的悬索公式，获得了一个解析的更为精确的单索方程．并通过实例计算与传统方程进行了对比，指出了这一改进方程在理论上和实践上的双重意义。     

On the absorption of the hard component of the cosmic radiation

A phenomenological theory of absorption of mesotrons in the atmosphere and in dense materials at and below sea-level is given. The calculations are based on the following assumptions: (1) absorption due entirely to energy loss caused by electric interactions of charged particles and to spontaneous decay, (2) the localization of mesotron production in an equivalent layer 10 per cent. below the top of the homogeneous atmosphere, and (3) neglect of scattering. From the absorption of vertically incident mesotrons in thick layers of rock the energy distribution in the formation layer is deduced. The energy distribution at sea-level is then determined, without approximations in the analysis, and compared with observations. It is shown that fair agreement is obtained for any reasonable value of the proper lifetime but that a value of about 2 × 10^?6 sec. allows the most favorable comparison of the calculated results and the data. The integral energy distribution and the angular distribution at sea-level are obtained. For the latter it is assumed, as is plausible, that the initial angular distribution is isotropic and that the primary energy distribution is independent of the original direction of motion. The agreement between calculated and observed angular distributions is very satisfactory. The absorption in lead and in light materials for vertically incident mesotrons at sea-level is considered. The need for a more precise definition of equivalent thicknesses is pointed out and the stopping powers of lead and light substances are compared. The absorption in lead is calculated and comparison with the data indicates a slightly larger absorption than is found theoretically. However, the absorption at large lead thicknesses (over 100 cm.) and the qualitative behavior of the absorption coefficient with increasing depth below sea-level are in accord with experiment. The consideration of absorption above sea-level indicates that there should be appreciable mesotron production at elevations comparable to that of Mt. Evans (4,300 meters). This is borne out by recent observations.     

采样定理的截断误差分析

讨论带限函数的Shannon采样定理截断误差的点态、一致和积分3种估计．对于点态 情形,用Dirichilet核的计算方法算出截断误差的阶为O(1/N);对于一致情形和积分情形,当 函数满足一定的衰减条件时,可分别得到收敛阶的一致及积分估计．     

Transmittance tuning by particle chain polarization in electrowetting-driven droplets

A tiny droplet containing nano∕microparticles commonly handled in digital microfluidic lab-on-a-chip is regarded as a micro-optical component with tunable transmittance at programmable positions for the application of micro-opto-fluidic-systems. Cross-scale electric manipulations of droplets on a millimeter scale as well as suspended particles on a micrometer scale are demonstrated by electrowetting-on-dielectric (EWOD) and particle chain polarization, respectively. By applying electric fields at proper frequency ranges, EWOD and polarization can be selectively achieved in designed and fabricated parallel plate devices. At low frequencies, the applied signal generates EWOD to pump suspension droplets. The evenly dispersed particles reflect and∕or absorb the incident light to exhibit a reflective or dark droplet. When sufficiently high frequencies are used on to the nonsegmented parallel electrodes, a uniform electric field is established across the liquid to polarize the dispersed neutral particles. The induced dipole moments attract the particles each other to form particle chains and increase the transmittance of the suspension, demonstrating a transmissive or bright droplet. In addition, the reflectance of the droplet is measured at various frequencies with different amplitudes.     

Initial Value Treatment for Torsion of an Elastic Layer

An initial value treatment is presented for a mixed boundary value problem of axisymmetric torsion of an elastic layer. The problem is expressed in terms of dual integral equations which are first reduced to a Fredholm integral equation of second kind with a continuous kernel. An initial value problem is derived for this Fredholm integral equation using invariant imbedding. This constitutes a new analytical approach to a large class of torsion problems.     

A Walsh function method for a non-linear Volterra integral equation

A new method for the numerical solution of linear and non-linear Volterra integral equations, using the discontinuous wavelet packets known as the Walsh functions, is proposed and investigated. Sufficient conditions for the method to converge are derived and a priori error estimates are obtained. Given sufficient regularity conditions on the integral equation, the method is shown to be locally of order two.     

刚性地基梁非线性分析的积分方程法

提出了刚性地基梁非线性分析的积分方程法。以地基反力为未知量,导出了刚性地基梁弯曲的积分方程。通过引入梁的边界条件,建立起一组刚性地基梁关于地基反力和挠度的线性互补方程。利用Lemke方法求解得到梁上任一点处的挠度、转角、弯矩和剪力。最后给出了一些数值结果,表明本文数值解与解析解良好吻合。     

On the existence and the magnitude of electronic charges

In order to quantize Dirac's classical point electron¹ we supplement Einstein's classical equation (E/c)² ? p² = b2 with a reciprocal classical equation (CΔt)² ? (Δr)² = a² where b = mc and a is Dirac's signal radius. Δt is the time saved by a light signal in various states of motion of the electron, and a/c is the rest time saved. Our former efforts² of obtaining an integral equation for the probability amplitude have been rectified by Born.³ There is no solution of the integral equation, however, unless advanced and retarded phases are introduced simultaneously, along with Dirac's advanced and retarded potentials. We have obtained a transcendental equation for the eigen-value μ = αγ where α is the Sommerfeld fine-structure constant, and γ is the numerical factor in Dirac's signal radius a = γe²/mc². The smallest eigen-value is μ = 0.0299.That is, ab = hγ = h/210.     

Prescribed performance bipartite consensus for nonlinear agents with antagonistic interactions: A PI transformation approach

The leaderless, prescribed performance consensus problem for groups of agents with antagonistic interactions is addressed for the first time in this paper. We consider agents modeled by pure feedback nonlinear systems with unknown dynamics and an agent communication network described by a signed digraph with a directed spanning tree. A new proportional and integral (PI) variable transformation is proposed that enables the solution of the problem of leaderless bipartite consensus with prescribed performance by recasting it into a regulation problem with prescribed performance, which in turn we solve by a low complexity distributed control law. The algorithm guarantees uniform boundedness of all closed-loop signals and prescribed performance for the bipartite consensus error. Simulations verify the validity of our theoretical analysis.     

Optimal control in unobservable integral Volterra systems

This paper presents solution of the optimal linear-quadratic controller problem for unobservable integral Volterra systems with continuous/discontinuous states under deterministic uncertainties, over continuous/discontinuous observations. Due to the separation principle for integral systems, the initial continuous problem is split into the optimal minmax filtering problem for integral Volterra systems with deterministic uncertainties over continuous/discontinuous observations and the optimal linear-quadratic control (regulator) problem for observable deterministic integral Volterra systems with continuous/discontinuous states. As a result, the system of the optimal controller equations are obtained, including the linear equation for the optimally controlled minmax estimate and two Riccati equations for its ellipsoid matrix (optimal gain matrix of the filter) and the optimal regulator gain matrix. Then, in the discontinuous problems, the equation for the optimal controller and the equations for the optimal filter and regulator gain matrices are obtained using the filtering procedure for deriving the filtering equations over discontinuous observations proceeding from the known filtering equations over continuous ones and the dual results in the optimal control problem for integral systems. The technical example illustrating application of the obtained results is finally given.     

轴对称双倒易边界元法的f函数及其奇性处理

基于右端项为径基函数的三维泊松方程,推导了轴对称情况下双倒易边界元方程求和形式的 f函数通式.通过对f函数取积分平均和根据计算区域特点选取不同类型的 f函数组合,消除了f函数在对称轴上呈现奇性的不足.通过计算轴对称实心圆柱体和球体的非稳态导热问题,很好地验证了对f函数的选择和处理的可行性.     

Direct transformation of variational problems into Cauchy systems—II. scalar-semi-quadratic case

This series of papers addresses three interrelated problems: the solution of a variational problem, the solution of integral equations, and the solution of an initial valued system of integrodifferential equations. It will be shown that a large class of variational problems requires the solution of a nonlinear integral equation. It has also been shown that the solution of a nonlinear integral equation is identical to the solution of a Cauchy system. In this paper, we by-pass the nonlinear integral equations and show that the minimization problems directly imply a solution of the Cauchy system. This second paper in the series looks at semi-quadratic functional and scalar functions.     

关于信息立法的有关问题研究

本文从发生信息纠纷案件的广泛性、信息纠纷存在的各部门的分散性和要求信息统一立法的迫切性三方面论述了信息立法的重要性并提出建议。     

Radiation fields from polygonal sources

The calculation of the radiation field D(x, y, z) produced by a uniform polygonal source is of considerable practical importance. Fock showed that there are advantages in replacing the traditional surface integral by a contour integral, using Stokes' theorem. This innovation, however, has not yet penetrated very deeply. It seemed worthwhile, therefore, to systematize the procedure and to show how easily results can be obtained by the Fock method. An incidental conclusion of the paper is that the radiation field from a polygonal source often differs by less than one per cent from the field produced by a circular disk of the same area. Thus the simple disk formula can often be substituted for the more complicated polygonal formula.     

多元函数介值定理的推广

介值定理是数学分析的一个重要定理,对研究函数方程根的存在性、不动点和积分中值定理等问题起到重要作用。在多元函数中推广介值定理,并且将只有第一类间断点的函数的介值定理推广运用到积分中值定理中,推广了文[4]的结论。     

Chebyshev polynomial solution of nonlinear integral equations

In the current work, the Chebyshev collocation method is adopted to find an approximate solution for nonlinear integral equations. Properties of the Chebyshev polynomials and operational matrix are used in the integral equation of a system consisting of nonlinear algebraic equations with the unknown Chebyshev coefficients. Numerical examples are presented to illustrate the method and results are discussed.     

Bilinear system identification by block pulse functions

A new method is employed to identify the unknown parameters of a bilinear system. This method expands the system input and output by block pulse functions and reduces the original identification problem to an algebraic form. Furthermore, the dyad formed by block pulse functions and its integral are in diagonal forms, whereas the integration of the “triple-product” matrix can be reduced to the upper triangular form. Consequently, only very few calculations are required to find the solution for the algebraic equation. Two examples are given to show that the use of this method is considerably more economical in computation time than the use of Walsh function expansion.