Odd Order Derivatives Research Articles

Complementary Lidstone Interpolation and Boundary Value Problems

We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial of degree , which involves interpolating data at the odd-order derivatives. For we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a th order differential equation and the complementary Lidstone boundary conditions.

Optimal waveform design for ultra-wideband communication based on Gaussian derivatives

Ultra-wideband (UWB) radios have attracted great interest for their potential application in short-range high-data-rate wireless communications. High received signal to noise ratio and compliance with the Federal Communications Commissions (FCC) spectral mask call for judicious design of UWB pulse shapers. In this paper, even and odd order derivatives of Gaussian pulse are used respectively as base waveforms to produce two synthesized pulses. Our method can realize high efficiency of spectral utilization in terms of normalized effective signal power (NESP). The waveform design problem can be converted into linear programming problem, which can be efficiently solved. The waveform based on even order derivatives is orthogonal to the one based on odd order derivatives.

Focus on spectroscopy

The usual Euler-Maclaurin expansion [1], [21 contains odd-order derivatives. The one which is proposed here contains only even-order derivatives.

Solutions of two-point boundary value problems for even-order differential equations

The existence of solutions of the two-point boundary value problems consisting of the even-order differential equations x ( 2 n ) ( t ) = f ( t , x ( t ) , x ′ ( t ) , … , x ( 2 n − 2 ) ( t ) ) + r ( t ) , 0 < t < 1 , and the boundary value conditions α i x ( 2 i ) ( 0 ) − β i x ( 2 i + 1 ) ( 0 ) = 0 , γ i x ( 2 i ) ( 1 ) + δ i x ( 2 i + 1 ) ( 1 ) = 0 , i = 0 , 1 , … , n − 1 , is studied. Sufficient conditions for the existence of at least one solution of above BVPs are established. It is interesting that the nonlinearity f in the equation depends on all lower derivatives, especially, odd order derivatives, and the growth conditions imposed on f are allowed to be super-linear (the degrees of phases variables are allowed to be greater than 1 if it is a polynomial). The results are different from known ones since we do not apply the Green's functions of the corresponding problem and the method to obtain a priori bounds of solutions is different from known ones. Examples that cannot be solved by known results are given to illustrate our theorems.

On sensitivity kernels for ‘wave-equation’ transmission tomography

SUMMARY We combine seismological scattering theory with the theory of distributions to study some properties of sensitivity kernels for finite frequency seismic delay times. The theory to be used for calculating the kernels depends on the way the measurements are made. For example, the sensitivity to the traveltime proper, that is the time to instantaneous onset of the phase arrival as defined in geometrical ray theory, has a non-zero value on the ray and zero elsewhere, whereas measurements of the smooth later part of the wave excitation can have more complicated kernels. Analysis based upon the Born approximation reveals that the behaviour of such kernels is determined by the regularization of the nth derivative of the Dirac delta supported on the unperturbed source-receiver ray, where n is the spatial dimension. Such regularization induces approximately the desired band-limitation and the associated Fresnel zone averaging. If the regularization is symmetric about the singular support of the delta its odd order derivatives vanish on this support, but the even orders render non-zero values. This explains why 3-D finite frequency kernels for body wave delay times seem—in specific, rather restrictive circumstances—to have zero sensitivity on the ray (as in the so called ‘banana-doughnut kernels’). Indeed, a regularization of the kernel can have a zero on the unperturbed ray, in the absence of caustics (that is, the medium must be simple or quasi-homogeneous) and only if the exact source signature (e.g., source time function) is known and used; both conditions can be met in synthetic cases but not—in general—in applications to real data and heterogeneous media. In general, one will not know where the oscillatory kernels have zero values, which has implications for the use of such kernels for the linearization of transmission tomography. We briefly clarify these observations by invoking a multiresolution analysis of sensitivity kernels, connected with a time-frequency analysis of phase arrival times.

Determination of the Jump of a Function of Generalized Bounded Variation by the Derivatives of a Trigonometric Interpolation Polynomial

We obtain formulas expressing the value of the jump of a bounded periodic function of harmonic bounded variation in a neighborhood of the point under consideration via the derivatives of odd order of the Lagrange trigonometric interpolation polynomial with equidistant nodes and via the derivatives of even order of the conjugate polynomial.

An inverse problem for a layered medium with apoint source

Suppose q(z) is a smoothfunction on [0, ∞)whose odd order derivatives are zero at z = 0. Take r = (x, y, z) andlet U(r, t)be the solution of the IBVP We show thatq(z) may be recovered froma knowledge of U(a, b, 0, t)for t varying over aninterval and fixed a, b,by reducing the problem to a one-dimensional inverse problem. This reduction is not trivial sinceU(r, t)depends on rand t andnot just on zand t.

Euler Polynomials and the Related Quadrature Rule

The use of Euler polynomials and Euler numbers allows us to construct a quadrature rule similar to the well-known Euler–MacLaurin quadrature formula, using Euler (instead of Bernoulli) numbers, and even (instead of odd) order derivatives of a given function evaluated at the extrema of the considered interval. An expression of the remainder term and numerical examples are also given.

Sharp inequalities connected with estimates for approximations of periodic functions by means of moduli of continuity of their derivatives of odd order with different steps

Some inequalities similar to the Jackson theorem and the Landau-Hadamard-Kolmogorov theorem for the best approximations and moduli of continuity of the derivatives of odd order with different steps are established. The inequalities are sharp if the norm is uniform. Bibliography: 4 titles.

HIGHER-DERIVATIVE TWO-DIMENSIONAL MASSIVE FERMION THEORIES

We consider the canonical quantization of a generalized two-dimensional massive fermion theory containing higher odd-order derivatives. The requirements of Lorentz invariance, hermiticity of the Hamiltonian and absence of tachyon excitations suffice to fix the mass term, which contains a derivative coupling. We show that the basic quantum excitations of a higher-derivative theory of order 2N+1 consist of a physical usual massive fermion, quantized with positive metric, plus 2N unphysical massless fermions, quantized with opposite metrics. The positive-metric Hilbert subspace, which is isomorphic to the space of states of a massive free fermion theory, is selected by a subsidiary-like condition. Employing the standard bosonization scheme, the equivalent boson theory is derived. The results obtained are used as a guideline to discuss the solution of a theory including a current–current interaction.

Canonical transformations in a higher-derivative field theory

It has been suggested that the chiral symmetry can be implemented only in classical Lagrangians containing higher covariant derivatives of odd order. Contrary to this belief, it is shown that one can construct an exactly soluble two-dimensional higher-derivative fermionic quantum field theory containing only derivatives of even order whose classical Lagrangian exhibits chiralgauge invariance. The original field solution is expressed in terms of usual Dirac spinors through a canonical transformation, whose generating function allows the determination of the new Hamiltonian. It is emphasized that the original and transformed Hamiltonians are different because the mapping from the old to the new canonical variables depends explicitly on time. The violation of cluster decomposition is discussed and the general Wightman functions satisfying the positive-definiteness condition are obtained.

Necessary optimality conditions for a singular surface in the form of synthesis

For a linear control problem using the traditional open-loop approach, a new representation for the singular control and generalized, invariant conditions for optimality are found. The phase portrait of a nonlinear control problem is considered in the neighborhood of singular trajectories. The singular paths form a hypersurface, approached by regular paths from both sides. The Bellman function for this problem is a classical (smooth) solution to a first-order PDE with nonsmooth Hamiltonian over two smooth (regular) branches, related to the halfneighborhoods of the surface. These solutions are at least twice differentiable and have first discontinuous derivatives of odd order. The invariant form for these necessary conditions is found in terms of Jacobi (Poisson) brackets, consisting of several equalities and inequalities. The latter relations guarantee the validity of the Kelley condition as well as the geometrical constraints for the singular control variables. Thus, the Kelley condition appears to be just a certain property of a smooth solution to a first-order PDE with nonsmooth Hamiltonian. All the relations, including the Hamiltonian equations of singular motion, do not use singular controls; they are based on regular Hamiltonians depending only upon the state vector and the gradient of the Bellman function (adjoint vector).

THE CAUCHY PROBLEM FOR ODD-ORDER QUASILINEAR EQUATIONS

A nonlocal Cauchy problem for multidimensional quasilinear evolution equations containing a linear differential operator L(t, x, Dx) with leading derivatives of odd order is considered. The conditions on the nonlinear terms are chosen so that they are subordinate to the operator L. The Korteweg-de Vries equation is a special case of such equations. No smoothness conditions are imposed on the initial function u0(x) (u0(x) L2(Rn)). Theorems on the existence, uniqueness, and continuous dependence on the initial data of generalized solutions are established. Bibliography: 20 titles.

A set of exact and explicit solitary wave solutions for a class of generalized KdV equations with arbitrary odd order derivatives (A sieving method in a closed function class)

Recent higher order KdV equations are interesting for physics, for example when the surface tension approaches close to a certain value. In this paper using the method of sieving in a closed function class, a set of exact and explicit solitary wave solutions with a unified form are given to higher order or generalized higher order KdV equations with 7th, 9th or arbitrary odd order derivatives, when their parameters satisfy some suitable restricted conditions.

Second and third derivatives of variational energy expressions: Application to multiconfigurational self-consistent field wave functions

General analytical expressions are given for the second and third derivatives of constrained variational energy expressions. It is pointed out that variational energy expressions and odd-order derivatives have a distinct advantage over nonvariational (e.g., perturbative) energy expressions and even-order derivatives. In particular, the first-order wave function suffices to determine the derivatives of the variational energy up to third order. The coupled-perturbed multiconfigurational SCF (MC-SCF) equations, obtained from the general results, are equivalent, with minor corrections, to the ones very recently presented by Osamura, Yamaguchi, and Schaefer. Explicit expressions are given for the second and third derivatives of the MC-SCF energy. Computational implementation is briefly discussed.

Lebesgue summability of double trigonometric series

We formulate a definition of symmetric derivatives of odd order for functions of two variables. Our definition is based on expanding in a Taylor’s series a weighted average of the function about circles. The definition is applied to derive results on Lebesgue summability for spherically convergent double trigonometric series.

Lebesgue Summability of Double Trigonometric Series

We formulate a definition of symmetric derivatives of odd order for functions of two variables. Our definition is based on expanding in a Taylor’s series a weighted average of the function about circles. The definition is applied to derive results on Lebesgue summability for spherically convergent double trigonometric series.

A Variant of the Euler-Maclaurin Formula

The usual Euler-Maclaurin expansion [1], [21 contains odd-order derivatives. The one which is proposed here contains only even-order derivatives.

Finite Difference Accuracy in Structural Analysis

An accuracy study is made of central finite difference methods for solving boundary value problems in structural analysis which are governed by equations with variable coefficients leading to odd order derivatives. Two methods are studied through application to beam-columns with nonuniform inplane loads and nonuniform stiffness. Definitive expressions for the error in each method are obtained by using Taylor series to derive the differential equations which exactly represent the finite difference approximations. The resulting differential equations are accurately solved by a perturbation technique which yields the error directly. A half station method, which corresponds to making finite difference approximations before expanding derivatives of function production in the beam-column differential equations, is found clearly superior to a whole station method which corresponds to expanding such products first.

Time-delay networks for an analog computer

Time-delay networks suitable for an analog computer are designed by considering the location of poles and zeros in their transfer functions. The curve of phase shift against frequency should be a straight line. The negative slope of this curve is the time delay. Even-order derivatives of the curve automatically vanish at zero frequency. Roots of the transfer function are chosen to make vanish similarly as many as possible of the derivatives of odd order, higher than the first. Data are given for networks with one, two, three, and four pairs of roots.