Integral Derivative Research Articles

On the interaction of a moving hollow vortex with an aerofoil, with application to sound generation

A hollow vortex in the form of a straight tube, parallel to the z-axis, and of radius a, moves in a uniform stream of fluid with velocity U in the x-direction, with U small compared with the sound speed c. This steady flow is disturbed by the presence of a thin symmetric fixed aerofoil. With a change of x-coordinate, the problem is equivalent to that of a moving aerofoil cutting through an initially fixed vortex in still fluid. The aim of this work is to determine the resulting perturbed flow, and to estimate the distant sound field. A detailed calculation is given for the perturbed velocity potential in the incompressible flow case, for the linearized equations in the limit of small aerofoil thickness. A formally exact solution involves a four-fold integral and an infinite sum over all mode numbers. For the important special case where the vortex tube has small radius a compared with the aerofoil width, the deformed vortex is characterized by a hypothetical vortex filament located at the ‘mean centre’ x¯(z, t), y¯(z, t) of the tube. Explicit results are given for x¯(z, t), y¯(z, t) for the case where the aerofoil has the elementary rectangular profile; results can then be obtained for more general and realistic cylindrical aerofoils by a single integral weighted with the derivative of the aerofoil thickness function. Finally the distant sound field is estimated, representing the aerofoil by a distribution of moving monopole sources and representing the effect of the deformed vortex in terms of compressible dipoles along the mean centre of the vortex.

An extension of Ito's formula for elliptic diffusion processes

We prove an extension of Itô's formula for F( X t , t), where F( x, t) has a locally square integrable derivative in x that satisfies a mild continuity condition in t, and X is a one-dimensional diffusion process such that the law of X t has a density satisfying some properties. Following the ideas of Föllmer, et al. (1995), where they prove an analogous extension when X is the Brownian motion, the proof is based on the existence of a backward integral with respect to X. For this, conditions to ensure the reversibility of the diffusion property are needed. In a second part of this paper we show, using techniques of Malliavin calculus, that, under certain regularity on the coefficients, the extended Itô's formula can be applied to strongly elliptic and elliptic diffusions.

Limit of the quadratic risk in density estimation using linear methods

We prove here that when estimating a density, using a kernel or a linear wavelet estimate, one can choose the smoothing parameter such that the limit when n tends to infinity of ?????? may be arbitrarily small for every density having a square integrable derivative. This choice consists in starting from the usual rate n −1/3 and then operate an oversmoothing proportional to the limit of the risk we want to obtain. Looking at the limit of the risk is another way of looking at the performances of estimators: We introduce here the maximal functional space where the results still stand. We show that this space contains the Sobolev spaces for instance. We also give a comparison with the standard minimax theory.

Semiempirical parametric method in the theory of vibronic spectra of polyatomic molecules

A semiempirical parametric method for calculating the vibrational structure of the electronic spectra of polyatomic molecules is developed; the method is based on the adiabatic molecular model and uses a single parametric system for all excited states. Within the model approach, simplified analytical expressions for potential surface variation during molecular excitation are derived; the expressions include the principal terms according to the order of magnitude. The first and second derivatives of Coulomb and resonance one-electron integrals with respect to natural coordinates in a basis set of hybrid atomic orbitals are used as parameters. It is shown that the parameters possess distinct locality, are transferable in molecular series, and may be easily ranked according to absolute values; describing a molecular model requires few most significant parameters. Excitation-induced variations of potential surfaces and absorption spectra of some molecules (butadiene, hexatriene, octatetraene) are calculated using only two parameters, which are the same for all molecules. The results of calculations are in good agreement with the experimental data.

Periodic output feedback stabilization of neutral systems

A new feedback control technique called periodic output feedback is investigated in the context of infinite-dimensional linear systems modeled by neutral functional differential equations. In this method, discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control. This work focuses on stabilization of neutral systems with delayed control modeled in the state space W/sub 2//sup (1)/ ([-r,0];R/sup n/)/spl times/L/sup 2/([-r,0],R/sup p/), where W/sub 2//sup (1)/([-r,0];R/sup n/) denotes the Sobolev space of R/sup n/-valued, absolutely continuous functions with square integrable derivatives on [-r,0]. We show that a class of these systems can be stabilized by periodic output feedback, even though their input operators are unbounded. We overcome this difficulty by representing the system state using an abstract integral variation of constants formula. An algorithm is presented at the end of this paper to construct a periodic output feedback gain function. An example is provided to illustrate the construction of the gain function.

Differentiation of the Martinelli‐Bochner Integrals and the Notion of Hyperderivability

AbstractThe notion of hyperderivability is introduced and discussed which allows us to obtain, in particular, formulas for the derivatives of the Martinelli‐Bochner integrals.

Prescreening of two‐electron integral derivatives in SCF gradient and Hessian calculations

AbstractThe definition and implementation of a rigorous two‐electron integral bound based on Schwarz' inequality both for gradient and hessian calculations is presented. Tests demonstrate the advantages of this easily implemented and effective bound.

General formula evaluation of the electron-repulsion integrals and the first and second integral derivatives over Gaussian-type orbitals

Recurrence relations between auxiliary functions to calculate the electron-repulsion integrals (ERIs) over Cartesian Gaussian-type orbitals (GTOs) can be derived. With the use of the Rys polynomials, this serves as a general formula method for evaluating ERIs and the ERI analytical derivatives. The computer code of the present method is quite general (i.e., it has no limitation for the shell type of GTOs), very compact (i.e., it has no coding difficulty), and extremely rapid (e.g., 2.1 μs per one ERI, 0.51 μs per one ERI first derivative, and 0.40 μs per one ERI second derivative for the [II‖II] shell block). It is found that the present method is especially efficient for evaluating ERIs including g-type or higher GTOs, the first derivatives including f-type or higher GTOs, and the second derivatives including d-type or higher GTOs.

On the derivation of first integrals for similarity solutions

In two recent papers the authors have obtained a number of first integrals for similarity solutions of nonlinear diffusion and of general high-order nonlinear evolution equation. Such integrals exist only for special parameter values and are obtained via integration of the ordinary differential equation, which results when the functional form of the solution is substituted into the governing partial differential equation. In this paper we show that these special parameter values also occur in a natural way when we utilize the first order partial differential equation instead of the explicit functional form and we ask under what conditions can a first integral with respect to either of the independent variables x or t be deduced. This simple procedure generates all previous results and presents the idea of similarity solutions in an entirely new light. That is, the significant features of similarity solutions for partial differential equations are not necessarily the explicit functional form and subsequent reduction to an ordinary differential equation but rather that the solutions sort are common to two partial differential equations. The process is illustrated with reference to an extensive number of examples including nonlinear diffusion, general diffusion equations containing a number of parameters and high-order nonlinear evolution equations. In addition a new exact solution for nonlinear diffusion is obtained which is illustrated graphically.

Studies on the three-dimensional weight function method for elliptical crack problems. (1st Report. Analytical solution for the first order variation of the displacement field due to geometrical changes in an elliptical crack).

In the present paper, based on the general solution for an elliptical crack in an infinite solid, subjected to arbitrary tractions (the VNA solution), the analytical solution of the first order variation of displacement field due to geometrical changes in the shape of an elliptical crack is derived. The analytical solution is related to the partial derivatives of the displacement field with respect to the lengths of the major and minor axes of the ellipse, respectively. Since the VNA solution involves the generic elliptic integrals. The partial derivatives of the generic elliptic integrals with respect to the lengths of the major and minor axes of the ellipse are required. A systematic procedure for the evaluation of these derivatives is also developed in this paper. The presently developed analytical solution will be effectively utilized to establish the three-dimensional weight function method for elliptical crack problems, in the forthcoming paper.

Application of cholesky‐like matrix decomposition methods to the evaluation of atomic orbital integrals and integral derivatives

AbstractWhen viewed as a square two‐indexed matrix, the array of atomic orbital‐based, two‐electron integrals (ij|kl) is a positive semidefinite array. Beebe and Linderberg showed, in 1977, that actual or near linear dependencies often exist within the types of atomic orbital basis sets employed in conventional quantum chemical calculations. In fact, large (i.e., higher quality) bases were shown to be substantially more redundant than smaller or more spatially separated bases. In situations where there exists significant basis near redundancy, the rank (r) of the (ij|kl) ≡ Vl,J matrix of integrals will be significantly smaller than the matrix dimension M. When this occurs, it proves computationally tractable to decompose the M‐dimensional matrix V into components L (V = LLT) which contain all of the information needed to form the full V matrix. The Cholesky algorithm allow such a decomposition to be carried out and forms the basis of the work described here. The method is found to be highly successful in reducing the number of integrals and integral derivatives that must actually be calculated. In particular, results on the C2 molecule indicate that the algorithm can be superior to traditional methods of integral derivative generation if the orbital basis is large enough to contain appreciable near redundancy. In contrast, results on benzene with a more spatially delocalized basis show that conventional methods are preferred whenever substantial basis (near) redundancy is not present.

A Boundary Integral Equation Formulation in Derivative Unknowns for Two-Dimensional Potential Problems

A boundary integral equation called Derivative BIE is developed for two-dimensional potential problems in terms of tangential and normal derivatives of the potential on the boundary, by integrating by parts the Cauchy formula. The potential values on the boundary can be calculated by integration after the solution is obtained. The primary unknowns in this formulation can be of direct interest in a shape design sensitivity analysis where the tangential derivatives of the potential are also required. The method is applied to several test problems, and the results show better accuracy than those by the conventional boundary element method, not only for the derivatives of the potential but also for the potential itself.

On path integrals in spherical coordinates

An alternative derivation of the path integrals in spherical coordinates is presented for the purpose of discussing the nature of the quantum mechanical terms appearing in the action. It is shown that the Jacobian resulting from the transformation of the functional measure from cartesian coordinates changes the action by surface terms which then lead to the correct ordering contributions.

Shape Optimization and Identification of Solid Geometries Considering Discontinuities

Shape sensitivity analysis of heat-conducting bodies is performed in general terms incorporating interface conditions and boundary singularities. Adjoint variables and the material derivative concept are utilized to obtain the material derivatives of volume and surface integrals of temperature and heat flux. Two illustrative examples are then analyzed by iterative numerical techniques incorporating the boundary element method of discretization. In the first problem, the interface position in a nonhomogeneous material is optimized for a minimum of total surface heat flow. The second problem involves the determination of the solidification interface shape in the so-called steady-state one-phase Stefan problem. Numerical results, checked by exact solutions, where available, indicate that the proposed solution procedure is suitable for free boundary problems in heat transfer.

Integral derivations

Integral derivations

Path-integral derivation of the complex ABCD Huygens integral

The Huygens integral, or Green’s function, for paraxial beam propagation in a generalized lenslike medium is derived as a Feynman path integral involving a summation over real rays. This formulation provides a physically intuitive understanding of the structure of the Green’s function. The path integral is evaluated by introducing in a consistent manner the notion of complex rays, and the familiar result for the Huygens integral in terms of the complex ABCD matrix of the medium is obtained.

Hartree–Fock energy derivatives with respect to basis set exponents. Integral derivatives using Rys polynomials

The Rys polynomial method has been used to evaluate the first derivatives of the two electron integrals and the nuclear electron attraction integrals with respect to the Gaussian exponents. Gradients for the sulfur 6-31 G(d) basis set in HS−, H2S, H3S+, H2S2, H3CSH, H2CS, CS, H2SO, SO2 and SO3 have been evaluated to illustrate the effect of charge, multiple bonding and hypervalency on the basis set exponents.

Theoretical analysis of non-thermal pump effects in broadband CARS spectroscopy

The effect of finite-bandwidth multimode pump lasers in CARS is normally accounted for by an intensity convolution integral based on gaussian (thermal) pump sources. This paper presents a derivation of the convolution integral for non-gaussian pump sources derived from frequency doubling. The modified convolution integral makes it possible to identify regions of insensitivity to pump stochastic properties.

Molecular Hessians for large-scale MCSCF wave functions

The calculation of molecular Hessians for large-scale multiconfiguration self-consistent-field (MCSCF) functions is described. The formalism is based on exponential parametrization of the wave function and symmetric orthonormalization of the molecular orbitals. Extensive use is made of one-index transformations of the molecular integrals, both to construct the gradient vectors that appear in the linear MCSCF response equations, and to perform the multiplication of the trial vectors on the electronic Hessian in the iterative, direct solution of the response equations. No element of the electronic Hessian is ever calculated explicitly, allowing for use of large configuration expansions. Efficient methods are developed for obtaining the solution vectors of the linear response equations. The accuracy of the molecular Hessian is analyzed in terms of the accuracy of these solution vectors. To allow for large basis sets Fock matrices are used to minimize transformations and integrals are recalculated to minimize storage requirements. Integral derivatives are calculated following the McMurchie–Davidson scheme. A simplified algorithm for calculation of derivatives of integrals involving one-center overlap distributions is described. Sample calculations involving several thousand configurations are reported.

Direct construction of path integrals in the lattice-space multiband dynamics of electrons in a solid.

It is suggested that complex problems in ultrasubmicrometer electronics research may greatly benefit from use of the path-integral technique. The use of the Weyl-Wigner formalism of the quantum dynamics of electrons in solids provides a rigorous and straightforward derivation of the path integral in solid-state physics, both from the single-particle and from the many-body field-theoretical description of electron dynamics, without the need to postulate a priori the isomorphism between quantum operators and c-numbers of the base field. A rigorous construction of the path integral in many-body solid-state band theory necessitates a two-stage Weyl correspondence between quantum operators and c-numbers of the base field, namely, the Weyl correspondence of the base field of ''lattice-space'' particle-dynamical variables and that of the continuum many-body field-dynamical variables.