The Construction of Rational Tetra-Inner Functions

Ph. D. ThesisThe tetrablock is the set E = fx 2 C3 : 1 x1z x2w + x3zw 6= 0 whenever jzj 1; jwj 1g: The closure of E is denoted by E. A tetra-inner function is an analytic map x from the unit disc D to E whose boundary values at almost all points of the unit circle T belong to the distinguished boundary bE of E. There is a natural notion of degree of a rational tetra-inner function x; it is simply the topological degree of the continuous map xjT from T to bE. In this thesis we give a prescription for the construction of a general rational tetra-inner function of degree n. The prescription makes use of a known solution of an interpolation problem for nite Blaschke products of given degree in terms of a Pick matrix formed from the interpolation data. Alsalhi and Lykova proved that if x = (x1; x2; x3) is a rational tetra-inner function of degree n, then x1x2x3 either is equal to 0 or has exactly n zeros in the closed unit disc D, counted with an appropriate notion of multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function x = (x1; x2; x3) consists of the points in D for which x1x2 x3 = 0 and the values of x at these points. We also give a matricial formulation of a criterion for the solvability of a Diag-synthesis problem. The symbol Diag denotes an instance of the structured singular value of 2 2 matrix corresponding to the subspace of diagonal matrices in M2 2(C). Given distinct points 1; :::; n 2 D and target matrices W1; :::;Wn 2 M2 2(C) one seeks an analytic 2 2 matrix-valued function F on D such that F( j) = Wj for j = 1; :::; n; and Diag(F( )) < 1; for all 2 D:Government of Saudi Arabia, Jouf Universit

Optimal control problems governed by a class of nonlinear systems

This article suggested a solution to a flow control problem governed by a class of nonlinear systems called bilinear systems. The problem was initially well-posed, and after it was established that an optimal control solution existed, its characteristics were stated. After that, we demonstrated how to use various bounded feedback controls to make a plate equation's flow close to the required profile. As an application, we resolved the plate equation-governed partial flow control issue. The findings bring up a variety of system applications, which can be employed in engineering advancement

Intraventricular Hemorrhage in Preterm Infants, Review Article

Intraventricular hemorrhage (IVH) or germinal matrix (GM) in other words, is a condition that can occur in premature births and can lead to long-term medical and developmental effects. While GM/IVH can happen in full-term infants, the hemorrhage in this group of infants is different from periventricular hemorrhage (PVH)/IVH in premature infants. Family members and caregivers of preterm infants and those at risk of preterm birth are confronted with two significant uncertainties concerning these newborns: Is the survival of this child likely? Will the child experience long-term sequelae, particularly developmental sequelae, if they survive? The significance of these questions lies in their potential to impact future medical decisions, including the level of intensity in the care provided. Infants born prematurely can suffer from various acquired lesions in the central nervous system (CNS), leading to long-term disability. These lesions include GM/IVH, periventricular white matter injury, hemorrhage, and diffuse injury to the developing brain. GM/IVH continues to be a major contributor to both illness and death in premature newborns.&nbsp; GM/IVH is primarily diagnosed by brain imaging techniques, typically cranial ultrasonography, as depicted below. Screening and serial examinations are essential for diagnosing GM/IVH, as it can occur without any noticeable clinical indications

Higher order ( n , m ) (n,m)(n,m) -Drazin normal operators

Abstract The purpose of this paper is to introduce and study the structure of p-tuple of ( n , m ) $(n,m)$ - D $\mathcal{D}$ -normal operators. This is a generalization of the class of p-tuple of n-normal operators. We consider a generalization of these single variable n- D $\mathcal{D}$ -normal and ( n , m ) $(n,m)$ - D $\mathcal{D}$ -normal operators and explore some of their basic properties