Multidimensional van der Corput-type estimates involving Mittag-Leffler functions

The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions E-alpha,E-beta(i lambda phi(x)), x is an element of R-N and E-alpha,E-beta(i(alpha)lambda phi(x)), x is an element of R-N for the various range of alpha and beta. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrodinger equations are considered

Contributions to the theory of interference and diffraction of light and related investigations

The researches described in this thesis are in the main part concerned with Optics and Electromagnetic Theory. They are presented in the form of fifteen papers, published in various scientific journals within the period 1949 -1955, and are grouped together under the following headings: (I) The Optical Image, (II) A General Theory of Interference and Diffraction of Light, (III) Studies in Electromagnetic Theory, and (IV) Related Investigations.In paper 1.1 the light distribution near focus in the three - dimensional aberration -free diffraction image is investigated and formulae are derived for the fraction of the total energy contained within the central core of the diffraction image in any prescribed receiving plane. In paper 1.2 these formulae are used to determine some of the properties of polychromatic star images formed by refracting telescope objectives, taking into account the effect of the secondary spectrum. The next paper is concerned with the effect of the central obstruction of the aperture on the three - dimensional light distribution near focus. An historical and critical survey of Diffraction Theory of Aberrations is presented in paper 1.4, and in paper 1.5 the foundations of the scalar Diffraction Theory of optical image formation are investigated. In the following paper a new aberration function is introduced, which possesses several advantages over aberration functions employed previously, and which may be used in investigations based on Geometrical Optics or Diffraction Theory.In the usual treatments of Interference and Diffraction, the sources are assumed to be of vanishingly small dimensions (point sources), which emit strictly monochromatic light. Numerous problems encountered, especially in applications to Spectroscopy, Microscopy and Astronomy, make it highly desirable to formulate a theory of Interference and Diffraction on a broader basis, taking into account the finite extension of any physical source, as well as the finite frequency range of radiation which any physical source emits. The need for such a generalization became particularly evident when F. Zernike in 1934 established in his pioneering researches on partial coherence a number of important and unsuspected theorems in this field. In Part II of this thesis, a general theory of Interference and Diffraction of Light is presented, which applies to any stationary field. This theory makes it possible to treat directly problems of Interference and Diffraction with polychromatic light from finite sources, and includes the majority of known results on partial coherence as special cases of much more general theorems. An attractive feature of this theory is that it operates with observable quantities only.Part III consists of two papers dealing with Electromagnetic Theory. In paper 3.1 a new representation of any Electromagnetic Field in vacuo is described. The field is represented in terms of a single complex scalar wave function, in terms of which the momentum density and the energy density of the field may be defined by means of formulae strictly analogous to the quantum mechanical formulae for the probability current and the probability density.* In paper 3.2 several new theorems are derived which apply to any Electromagnetic Field in which at least one of the field vectors is linearly polarized.Some related investigations are described in Part IV. In paper 4.1 a systematic derivation is given of the Circle Polynomials of Zernike, which play an important part in some branches of Diffraction Theory. Several new theorems concerning these polynomials are established and a related set of polynomials is investigated. Paper 4.2 is concerned with the Xn and Yn functions of Hopkins which occur in certain extensions of the analysis of Lommel relating to the three -dimensional light distribution near focus. In paper 4.3 the design of the corrector plate of the Schmidt Camera is discussed and a solution is obtained for the design of a plate which leads to an optimum performance over the field taken as a whole, with light covering a given spectral range.The greater part of the work described in this thesis was carried out whilst I was a Research Assistant to Professor Max Born at Edinburgh University. I wish to acknowledge my sincere gratitude to Professor Born for allowing me to spend much time on work of my own interest and for many stimulating discussions.In accordance with the regulations I state that the work described in those papers here submitted, which were published under my own authorship, was done entirely by myself, and that none of the work reported in this thesis was carried out under supervision. The papers of Part II are presented as the main contribution.PART I THE OPTICAL IMAGE • 1.1 Light Distribution near Focus in an Error-Free Diffraction Image (Reprinted from Proc. Roy. Soc., A, 1951, 204, 533). • 1.2 On Telescopic Star Images (With E. H. Linfoot, Reprinted from Mon. Not. Roy. Astr. Soc., 1952, 112, 452). • 1.3 Diffraction Images in Systems with an Annular Aperture (With E. H. Linfoot, Reprinted from Proc. Phys. Soc., B, 1953, 66, 145). • 1.4 The Diffraction Theory of Aberrations (Reprinted from Rep. Progr. Phys., 1951, 14, 95) • 1.5 On the Foundation of the Scalar Diffraction Theory of Optical Imaging (With O. Theimer and G. D. Wassermann, Reprinted from Proc. Roy. Soc., A, 1952, 212, 426). • 1.6 On a New Aberration Functioh of Optical Instruments (Reprinted from J. Opt. Soc. Amer., 1952, 42, 547). • • PART II A GENERAL THEORY OF INTERFERENCE AND DIFFRACTION OF LIGHT • 2.1 A Macroscopic Theory of Interference and Diffraction of Light from Finite Sources. I. Fields with a Narrow Spectral Range (Reprinted from Proc. Roy. Soc., A, 1954, 225, 96). • 2.2 A Macroscopic Theory of Interference and Diffraction of Light from Finite Sources. II. Fields with a Spectral Range of Arbitrary Width (Reprinted from Proc. Roy. Soc., A, 1955, 230, 246). • 2.3 Optics in Terms of Observable Quantities (Reprinted from Il Nuovo Cimento, 1954, 12, 884). • • PART III STUDIES IN ELECTROMAGNETIC THEORY • 3.1 A Scalar Representation of Electromagnetic Fields (With H. S. Green, Reprinted from Proc. Phys. Soc., A, 1953, 66, 1129). • 3.2 On Linearly Polarized Electromagnetic Waves of Arbitrary Form (With A. Nisbet, Reprinted from Proc. Cambr. Phil. Soc., 1954, 614). • • PART IV RELATED INVESTIGATIONS • 4.1 On the Circle Polynomials of Zernike and Related Orthogonal Sets (With A. B. Bhatia, Reprinted from Proc. Cambr. Phil. Soc., 1954, 121 40). • 4.2 The X and Y Functions of Hopkins, occurring in the Theory of Diffraction (Reprinted from J. Opt. Soc. Amer., 1953, Al, 218). • 4.3 On the Corrector Plates of Schmidt Cameras (With E. H. Linfoot, Reprinted from J. Opt. Soc. Amer., 1949, 3.42_, 752). • 4.4 Microwave Optics (Reprinted from Nature, 1953, 172, 615)

Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity

We discuss strong local and global well-posedness for the three-dimensional NLS equation with nonlinearity concentrated on S2 . Precisely, local well-posedness is proved for any C 2 power-nonlinearity, while global well-posedness is obtained either for small data or in the defocusing case under some growth assumptions. With respect to point-concentrated NLS models, widely studied in the literature, here the dimension of the support of the nonlinearity does not allow a direct extension of the known techniques and calls for new ideas

Asymptotic Estimates for Some Dispersive Equations on the Alpha-modulation Space

The alpha-modulation space is a function space developed by Grobner in 1992. The alpha-modulation space is a generalization of the modulation space and Besov space. In this thesis we obtain asymptotic estimates for the Cauchy Problem for dispersive equation, a generalized half Klein-Gordon, and the Klein-Gordon equations. The wave equations will also be considered in this thesis too. These estimates were found by using standard tools from harmonic analysis. Then we use these estimates with a multiplication algebra property of the alpha-modulation space to prove that there are unique solutions locally in time for a nonlinear version of these partial differential equations in the function space of continuous function in time and alpha-modulation in the spatial component. These results are obtained by using the fixed point theorem