Integral Derivative Research Articles

A unified scheme for the calculation of differentiated and undifferentiated molecular integrals over solid-harmonic Gaussians

Utilizing the fact that solid-harmonic combinations of Cartesian and Hermite Gaussian atomic orbitals are identical, a new scheme for the evaluation of molecular integrals over solid-harmonic atomic orbitals is presented, where the integration is carried out over Hermite rather than Cartesian atomic orbitals. Since Hermite Gaussians are defined as derivatives of spherical Gaussians, the corresponding molecular integrals become the derivatives of integrals over spherical Gaussians, whose transformation to the solid-harmonic basis is performed in the same manner as for integrals over Cartesian Gaussians, using the same expansion coefficients. The presented solid-harmonic Hermite scheme simplifies the evaluation of derivative molecular integrals, since differentiation by nuclear coordinates merely increments the Hermite quantum numbers, thereby providing a unified scheme for undifferentiated and differentiated four-center molecular integrals. For two- and three-center two-electron integrals, the solid-harmonic Hermite scheme is particularly efficient, significantly reducing the cost relative to the Cartesian scheme.

Differentiability of Lieb functional in electronic density functional theory

AbstractA solid understanding of the Lieb functional FL is important because of its centrality in the foundations of electronic density functional theory. A basic question is whether directional derivatives of FL at an ensemble‐V‐representable density are given by (minus) the potential. A widely accepted purported proof that FL is Gâteaux differentiable at EV‐representable densities would say, “yes.” But that proof is fallacious, as shown here. FL is not Gâteaux differentiable in the normal sense, nor is it continuous. By means of a constructive approach, however, we are able to show that the derivative of FL at an EV‐representable density ρ0 in the direction of ρ1 is given by the potential if ρ0 and ρ1 are everywhere strictly greater than zero, and they and the ground state wave function have square integrable derivatives through second order. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007

Fourier expansions of functions with bounded variation of several variables

In the first part of the paper we establish the pointwise convergence as t → +∞ for convolution operators ∫ Rd t d K (ty) φ(x - y)dy under the assumptions that φ(y) has integrable derivatives up to an order a and that |K(y)| ≤ c(1 + |y|) -β with α+β > d. We also estimate the Hausdorff dimension of the set where divergence may occur. In particular, when the kernel is the Fourier transform of a bounded set in the plane, we recover a two-dimensional analog of the Dirichlet theorem on the convergence of Fourier series of functions with bounded variation. In the second part of the paper we prove an equiconvergence result between Fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds, which allows us to transfer some of the results proved for Fourier integrals to eigenfunction expansions. Finally, we present some examples of different behaviors between Fourier integrals, Fourier series and spherical harmonic expansions.

Modified Dyadic Integral and Fractional Derivative on ℝ+

For functions from the Lebesgue space L(ℝ+), we introduce the modified strong dyadic integral J α and the fractional derivative D (α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(ℝ+). We find a countable set of eigenfunctions of the operators D (α) and J α, α > 0. We also prove the relations D (α)(J α(f)) = f and J α(D (α)(f)) = f under the condition that $$\smallint _{\mathbb{R}_ + } f(x)dx = 0$$ . We show the unboundedness of the linear operator $$J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$$ , where L J α is its natural domain of definition. A similar assertion is proved for the operator $$D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$$ . Moreover, for a function f ∈ L(ℝ+) and a given point x ∈ ℝ+, we introduce the modified dyadic derivative d (α)(f)(x) and the modified dyadic integral j α(f)(x). We prove the relations d (α)(J α(f))(x) = f(x) and j α(D (α)(f)) = f(x) at each dyadic Lebesgue point of the function f.

Derivation of the supersymmetric Harish-Chandra integral for UOSp(k1/2k2)

The previous supersymmetric generalization of the unitary Harish-Chandra integral prompted the conjecture that the Harish-Chandra formula should have an extension to superspaces. We prove this conjecture for the unitary orthosymplectic supermanifold UOSp(k1/2k2). To this end, we construct and solve an eigenvalue equation.

Dual-broadband rotational CARS modelling of nitrogen at pressures up to 9 MPa. I. Inter-branch interference effect

An improved model for pure rotational coherent anti-Stokes Raman scattering (CARS) has been developed for the dual-broadband approach. This model takes into account the fact that coherent anti-Stokes as well as Stokes radiation are generated simultaneously, and these two branches overlap and interact at high pressures giving rise to an inter-branch interference effect. In combustion studies, the anti-Stokes branch is used to determine temperature and concentrations of key species, of which nitrogen is the most widely used, and we have found that the interference effect on the anti-Stokes side is of such magnitude that it clearly needs to be taken into account at pressures of a few MPa. The necessary modifications to the total third-order susceptibility are presented and a derivation of the convolution integral for a finite-bandwidth pump source is given.

THERMAL STRESSES IN JOINED SEMI-INFINITE SOLIDS

The elastic stress field caused by an ellipsoidal inclusion with uniform dilatational eigenstrains in one of two perfectly bonded semi-infinite solids is investigated. The thermal stress in domain z induced by the temperature variation j T could be simulated effectively by pure dilatational eigenstrain. The solutions are obtained using the method of dilatation centers. The potential functions for the problem solved are the harmonic potential functions of attracting matter filling the element of volume and expressed in terms of derivatives of elliptic integrals. Numerical examples are given to show the normal and tangential stresses along the boundary of the ellipsoidal inclusion and the maximum principle stresses along the major and minor axes in the inclusion that are important to the fracture problems. The effects of the inclusion depth from the interface, the ratio of elastic moduli of joined semi-infinite solids, the shapes of the inclusion and the rotation angle of the ellipsoidal inclusion are also studied. They are of great significance in physical applications pertaining to thermal stress problems.

Symmetry Analysis of Differential Equations Using MATHLIE

A general procedure for solving ordinary differential equations of arbitrary order is discussed. The method used is based on symmetries of a differential equation. The known symmetries allow the derivation of first integrals of the equation. The knowledge of at least r symmetries of an ordinary differential equation of order n with r ≥ n is the basis for deriving the solution. Our aim is to show that Lie's theory is a useful tool for solving ordinary differential equations of higher orders. Bibliography: 12 titles.

Stochastic integral representations, stochastic derivatives and minimal variance hedging

The stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale. In relation to this, a certain stochastic derivative is defined. It is shown that this derivative determines the integrand in the stochastic integral which serves as the best L 2 - approximation to the random variable considered. For a general Lévy process as integrator some specification of the suggested stochastic derivative is given. In the case of the Wiener process the considered specification reduces to the well-known Clark-Haussmann-Ocone formula. This result provides a general solution to the problem of minimal variance hedging in incomplete markets.

On fuzzy correlations

A general framework for dealing with numerical measurements in an approximate, uncertain, or fuzzy environment is presented. A fuzzy measurement is defined. It possesses several unique properties, which arise from its physical nature and distinguish it from concepts such as the fuzzy number. These properties, which include the fuzzy correlation term and the fuzzy Equality relation, follow directly from physical considerations. The introduction of the fuzzy correlation term provides a mathematical tool for representing any correlation relations, which may exist between different fuzzy measurements. The main function of the fuzzy correlation term is to eliminate, or filter out, measurement values that are unlikely, given other fuzzy measurements. Thus, using the fuzzy correlation term, the range of possible measurement values is limited by physical realities. The information represented by the fuzzy correlation term is shown to be of great value in providing a wider picture of reality than it is possible to obtain by simply considering individual fuzzy measurements. Arithmetic operations on fuzzy measurements and functions of fuzzy measurements are also discussed, leading to the derivation of the fuzzy Riemann integral and its applications.

Output-Feedback Control of Reactive Batch Distillation Columns

In this work, an output-feedback control for the regulation of distillate purity via manipulations of the reflux ratio in reactive batch distillation is designed. The approach is based on an approximate model of the composition dynamics and makes use of a reduced-order observer to estimate the modeling error. An input/output linearizing feedback is proposed where the estimated modeling error is included to achieve robust tracking of a composition reference. It is shown that the resulting controller has the structure of a proportional−integral derivative (PID) controller with antireset windup. The controller performance is tested using a simulation example including strong uncertainties in the reaction model. An interesting finding is that the required reflux ratio policy to reach asymptotically a constant reference resembles the reflux ratio policy obtained from posing an optimization technique (Mujtaba, I. M.; Macchietto, S. Ind. Eng. Chem. Res. 1997, 36, 2287−2295).

On the integral means and schwarzian derivative

Let fbe locally univalent in the unit disc D. We study the integral means of powers |f′| p and obtain upper bounds for them in terms of a solution of an algebraic equation of degree 4. The coefficients of this algebraic equation depend on the parameter p and on the hyperbolic norms of the pre-Schwarzian f″/f′ and Schwarzian derivative Sf Wepresent some results derived from numerical analysis of the polynomial equation.

On the regularity of spectral densities of continuous-time completely linearly regular processes11This research was done while the author was at the University of Chicago. The manuscript was prepared using computer facilities supported in part by The University of Chicago Block Fund.

This paper deals with the study of the relationship between the complete linear regularity of continuous-time weakly stationary processes and the smoothness of their spectral densities. It is shown that when the coefficient of complete linear regularity behaves like O( τ −( r + μ ) ) as τ → +∞ , for some r ∈ N , μ ∈ (0,1] , then the spectral density has at least r uniformly continuous, bounded, and integrable derivatives, with the r th derivative satisfying a Lipschitz continuity condition of order μ . Conversely, under certain smoothness assumptions on the spectral density, upper bounds on the rate of decay of the coefficient of complete linear regularity are obtained.

The calculation of molecular geometrical properties in the Hellmann—Feynman approximation

The ab initio calculation of molecular geometrical properties in the Hellmann—Feynman approximation is discussed in which the atomic orbitals are fixed at the positions of the nuclei at the reference geometry, thereby avoiding the calculation of derivatives of the molecular integrals with respect to the positions of the atomic orbitals. For the molecular gradient, the molecular Hessian, and the molecular dipole gradient, the convergence of the calculated properties is studied for a large number of basis sets at the Hartree—Fock level and at the CCSD(T)-R12 level. In the Hellmann—Feynman approximation, it is found to be necessary to impose explicitly rotational and translational invariance. Although small basis sets perform poorly in the Hellmann—Feynman approximation (compared with the standard approach where the atomic orbitals are moving with the displaced nuclei), satisfactory convergence is obtained for geometries and harmonic frequencies (to within 1% of the standard approximation) with the larger of the correlation-consistent core—valence cc-pCVXZ basis sets. For the infrared intensities, the agreement with the standard approach is still poor (only within 15% for the largest correlation-consistent basis). The best results are obtained with an R12 basis previously developed for the calculation of energies in the explicitly correlated R12 approximation. In this basis, the geometrical parameters and harmonic frequencies are within 0.5% of the standard approach and the infrared intensities within 5%, suggesting that the Hellmann—Feynman approximation may be useful for applications at the highly accurate MP2-R12, CCSD-R12, and CCSD(T)-R12 levels of theory.

Transformations of index set for Skorokhod integral with respect to Gaussian processes

We consider a Gaussian process {Xt, t ∈ T} with an arbitrary index set T and study consequences of transformations of the index set on the Skorokhod integral and Skorokhod derivative with respect to X. The results applied to Skorokhod SDEs of diffusion type provide uniqueness of the solution for the time‐reversed equation and, to Ogawa line integral, give an analogue of the fundamental theorem of calculus.

Interconnected operation of small-hydroelectric and diesel power systems

The paper deals with some aspects of the interconnected operation of small-hydroelectric and diesel power systems provided with classical controllers. A comprehensive procedure for optimizing the integral controller gain setting using an integral squared error (ISE) technique is suggested. Analysis reveals that the integral-derivative (ID) controller in the small-hydroelectric area and integral controller in the diesel area improves the system performance significantly as compared to that of integral control in both areas. Analysis also reveals that the continuous frequency and tie-power oscillations can completely be eliminated using an ID controller in the small-hydroelectric area. © 1998 John Wiley & Sons, Ltd.

Further comments on the reciprocity integral for magnetic recording

A straightforward and physically motivated derivation of the reciprocity integral in magnetic recording theory is given for the quasi-static limit, where electromagnetic radiation and retardation effects are neglected, but frequency dispersion in the read head from all sources, including eddy currents, is specifically accounted for. This work attempts to correct some errors or misconception on this subject matter that remain in the literature.

Continuity of some anticipating integral processes

In this paper we provide some sufficient conditions for the Skorohod integral process to have a continuous version. A first set of conditions require the existence of two square integrable derivatives and that the process has moments of order β > 2. Secondly, we prove that the existence of the second derivative can be replaced by an integrability condition of the first derivative on the diagonal.

The Controllability of Systems Governed by Parabolic Differential Equations

The main result of this paper is a new, weaker condition for the exact controllability of linear parabolic partial differential equations. The result is derived through an examination of a related optimal control problem. It is shown that if the terminal state belongs to the Sobolev space of functions having two square integrable derivatives, then the parabolic equation is exactly controllable.

Morin singularities and global geometry in a class of ordinary differential operators

(∗) u′(t) + f(t, u(t)) = g(t), where the unknown u is a real function on S1 and the nonlinearity f : S1×R → R can assume a number of forms. Our approach is to study the global geometry of the operator F : B1 → B0, u → u′ + f(t, u) where the domain is either C1(S1) (the Banach space of periodic functions with continuous derivatives) or the Hilbert space H1(S1) of periodic functions with square integrable derivative. Ideally, we search for global changes of variables in both domain and image taking the operator F to a simple normal form. This goal has been achieved in previous occasions, starting with the seminal work of A. A. Ambrosetti and G. Prodi ([AP]) and its geometric interpretation by M. S. Berger and P. T. Church ([BC]), who showed that the operator associated to a certain nonlinear Dirichlet problem gives rise to a global fold between infinite