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Biometrics Of Bilateral Symmetry In Plants., T. Davis Dr.
Biometrics Of Bilateral Symmetry In Plants., T. Davis Dr.
Doctoral Theses
During the past ten deendes, observation rolating to ayune try, asymmatry, dis-oyame try, radlal symmetry, rotatiÉ”nal Bymne try, bilateral aymme try and ambidextry in the fields of bo tany, Boology, geology, chemiatry, eryatellography, physics, astronomy, engineering, mathematics, sunio, poetry and art vere recorded by various scientists. The prevent treatise is a sua- mary of an elaborato study on bilaterai syame try or microrinage symse try occurring in vari us plant organs. Unlike moet of the earlior reports, the problem here has been takled unified on a quantitative basis, thereby ereating situations where sophisti-cated techniques of statietios could be effectively employed. …
Contributions To Measure Theory., Merepalli Bhaskara Rao Dr.
Contributions To Measure Theory., Merepalli Bhaskara Rao Dr.
Doctoral Theses
This thesis 1s devoted to a study of measures vith emplas sis on nonatonie nea sures. Me briefrly desoribe here the work carried out in various chapters. In Chapter 1, we study various aspects of nonatomic nea sures based on some characterisations obtained early in the Chapter. In Chapter 2, the problem - when a mixture of nonatoftic mea sures is nonatomic - is exanined. An example and rive sufficient conditions are given. In Chapter 3, we examine when a mixture of 1nvariant non-ergorlie mesures is non-ergodic. An example and three sufficient eonditions are given. In Chapter 4, we study …
Some Contribution To The Theory Applications And Computations Of Generalized Inverses Of Matrices., Pochiraju Bhimasankaram Dr.
Some Contribution To The Theory Applications And Computations Of Generalized Inverses Of Matrices., Pochiraju Bhimasankaram Dr.
Doctoral Theses
The origin of the concept of a generalized inverse dates back to as early as 1920 when Moore defined the generalized inverse of matrix which is equivalent toDefinition 1 (Moore) : Let A be a m >< n matrix over the field of complex numbers. Then a is the generalized Inverse of A if AG is the orthogonal projection operator projecting arbitrary vectors onto the column space of A and GA is the orthogonal projection operator projecting arbitrary sectors onto, the column space of G.Mod re (1935) discussed this concept and its properties in some detail. Tsong (1949a, 1949b, 1956) discussed about generalized 1nverses of operators in more general spaces and Bjerhammer (1951) discussed the generalized inverse of a matrix in connection with an application to geodetic calculations. Unaware of the work of Hoore and others, Penrose (1955) defined a generalized inverse of a matrix as follows :Definition 2 (Penrosel) : Let A be a m *n -matrix over the field of complex numbers. Then G is a generalized inverse of a if (i) AFA= A; (ii) GAG=G; (iii) (AG)*=AG and (iv) (GA)*-GA.Penrose (1955,1956) showed that for every matrix there exists a unique generalized inverse, discussed several of its important properties, gave applications to solution of matrix equations and suggested a practical method of computation of the generalised inverse.As was pointed out by Rado (1956) Moo res definition of generalized inverse is equivalent to that of Penrose, Such generalized inverse is called the Moore-Penrose inverse and A is used to denote the Moore-Penrose inverse of A.Rao (1955), unaware of the earlier or contemporary Work, constructed a pseudo-inverse of a matrix which he used in some least squares computations, In a paper in 1962, he defined a generalized inverse (g-inverse) as follows, proved some interesting properties and gave applications of g-inverses to Mathomatical Statistics.Definition 3 (Rao) : Lot A be am x n matrix, Then a n >< m matrix. Then a n >< m matrix G is a g-inverse of A if x = Gy is a solution of the linear system Ax = y whenever it is consistent.A g-inverse if u matrix (in the sense of Rao) is in general not unique, As 1s easily observed (from definitions 2 and 4) the class of all g-inverses of a matrix A contains A*. Rạo (1965, 1967) developed a calculus of g-inverses, classified the g-inverses according to their use and according to the proporties they possess similar to those of the inverse of a nonsingular matrix and suggested further applications to Mathematical Statistics. Mitra (1968a, 1968b) gave an equivalem definition of a g-inverse, developed further calculus of z-inverses, used g-invorses to solve some matrix equations of interest and explored the possibilities of some new classes of g-inverses with applications. In a series of papers, and a monograph Mitra and Rao (1968, 1970) pursued the research on generalized inverses of matrix's and their applications to various scientific disciplines.