Let f and F be members of H. The function f is said to be subordinate to F, or F is said to be superordinate to f , if there exists a functionw analytic in U, withw 0 0 and |w z | < 1, and such that f z F w z . In such a case, we write f ≺ F or f z ≺ F z . If the function F is univalent in U, then we have f ≺ F if and only if f 0 F 0 and f U ⊂ F U cf. 1 . Definition 1.1 see 1 . Let φ : C2 → C a...

Starting from the scheme given by Hudson and Parthasarathy 7, 11] we extend the conservation integral to the case where the underlying operator does not commute with the time observable. It turns out that there exist two extensions, a left and a right conservation integral. Moreover, It^ o's formula demands for a third integral with two integrators. Only the left integral shows similar continui...

In this paper, integral plus double derivative (IDD) in coordination with Thyristor Controlled Phase Shifter (TCPS) is applied for the first time in automatic generation control (AGC) system. Comparative studies on the performance of different classical controllers viz. integral (I), proportional plus integral (PI), proportional plus integral plus derivative (PID) and integral plus double deriv...

In the study of functions of two variables f(x, y) defined over a region in the plane, the double integral has two interpretations: the signed volume between the graph of z = f(x, y) and the xy-plane or the total mass of a lamina with density determined by the function f(x, y). In the case of a function of three variables f(x, y, z), the graph w = f(x, y, z) is a subset of four-dimensional spac...

We shall study a weak solution in the Sobolev space of the transmission problem for the Laplace equation using the integral equation method. First we use the indirect integral equation method. We look for a solution in the form of the sum of the double layer potential corresponding to the skip of traces on the interface and a single layer potential with an unknown density. We get an integral eq...

We propose an efficient computation method for the infinite integral ∫∞ 0 xdx/(1+ x 6 sin x), whose integrand contains a series of spikes, approximately π apart, growing taller and narrower as x increases. Computing the value of this integral has been a problem since 1984. We herein demonstrate a method using the Hilbert transform for changing this type of singular function into a smooth functi...

Hilbert’s proof, apart from the determination of the best possible constant π csc(π/p), was published by Weyl [7]. The calculation of the constant, and the integral analogue of Hilbert’s double series theorem (for p = 2) are due to Schur [6]. The generalizations to other p′s of both the discrete and integral versions of this result were discovered later on by Hardy and Riesz and published by Ha...

Green’s Theorem is another higher dimensional analogue of the fundamental theorem of calculus: it relates the line integral of a vector field around a plane curve to a double integral of “the derivative” of the vector field in the interior of the curve. It admits two different but completely equivalent formulations, a “flux” version for normal line integrals and a “circulation” version for tang...

We propose an extension to the double iteration method, the Multipower Defect Correction method, which is used to refine approximate eigenelements of integral operators. It consists of an inner/outer iteration where, inside a defect correction iteration, p steps of a power iteration are used. The approximate inverse used in the defect correction is built with an approximation to the reduced res...