-Some new oscillation criteria for the even order damped functional differential equation (a(t)z(n-1)(t)) ' ÷ p(t)lz(n-1)(Ol#x(n-a)(O ÷ q(t)f(xIaa(t)] . . . . . =Lq.~(t)]) = 0 are established, where ~ _> O. These criteria are an extension of some of the known results. 1. I N T R O D U C T I O N Recently, Grace and Lalli [1] discussed the oscillation of the nth-order functional differential equa...

Abmucr-The least mean square pattern synthesis method is extended to include constraints uch as pattern nulls or patternderivative nulls at a given set of angles. The problem is formulated as a constrained approximation problem which is solved exactly, and a clear geometrical interpretation of the solution in a multidimensional vector space is given. The relation of the present method to ...

We present an general algorithm for the evaluation of the nth derivatives (with respect to the nuclear Cartesian coordinates) of two-electron-repulsion integrals (ERIs) over Gaussian basis functions. The algorithm is a generalization of our recent synthesis of the McMurchie/Davidson and Head-Gordon/Pople methodologies for ERI generation. Any ERI nth derivative may be viewed as an inner product ...

The eigenvalues of Sturm-Liouville (SL) problems depend not only continuously but smoothly on boundary points. The derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. This for arbitrary (separated or coupled) self-adjoint regular boundary conditions. In addition, as the length of the interval shrinks to zero all higher eigenvalues march o...

Here a proof will be given by a variational method which already has been used to prove Sendov’s conjecture (s. [2]). Let n > 1 be fixed and define Fn as the class of nth degree monic complex polynomials p with p(0) = 0, p(0) 6= 0 and p(ζ) 6= 0 for all derivative zeros ζ of p. Obviously it suffices to consider polynomials p ∈ Fn in order to give a proof of Smale’s conjecture. For such p we defi...

Let f : R→ R have an nth derivative of finite variation Vf(n) and a locally absolutely continuous (n− 1)st derivative. Denote by E±(f, δ)p the error of onesided approximation of f (from above and below, respectively) by entire functions of exponential type δ > 0 in Lp(R)–norm. For 1 ≤ p ≤ ∞ we show the estimate E±(f, δ)p ≤ C n π1/pVf(n)δ −n− 1 p , with constants Cn > 0.

We develop the nth order Fourier-Bessel series expansion of 1-D functions in the interval (0,α). Hence we establish the sampling theorem for a function with α-bandlimited nth order Hankel transform. The latter statement implies that the function is also Fourier transform αbandlimited. The samples’ locations are given by the roots of nth order Bessel functions. In addition, the sampling distance...

The aim of this paper is to find the class of continuous pointwise transformations (as general as possible) in the framework of which Kummer’s transformation z(t) = g(t)y(h(t)) represents the most general pointwise transformation converting every linear homogeneous differential equation of the nth order into an equation of the same type. Further, some forms of these equations having certain sub...

This work presents the nth-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (nth-CASAM-N), which enables most efficient computation of exactly determined expressions arbitrarily high-order sensitivities generic nonlinear system responses with respect to model parameters, uncertain boundaries, and internal interfaces in model’s phase space. The mathematical fram...