Abstract available on page 4 of the document / Thesis (PhD (Mathematics))--University of Pretoria, 2007. / Mathematics and Applied Mathematics / unrestricted

<p>We are study the spaces of convolutors and multipliers in the spaces of<br />tempered ultradistributions. There given theorems which gives us the characteri-zation of all the elements which belongs to spaces of convolutors and multipliers.<br />Structural theorem for ultradistribution semigroups and exponential ultradistri-bution semigroups is given. Fourier hyperfunction semigroups and hyperfunction<br />semigroups with non-densely dened generators are analyzed and also structural<br />theorems and spectral characterizations give necessary and sucient conditions<br />for the existence of such semigroups generated by a closed not necessarily densely<br />dened operator A. The abstract Cauchy problem is considered in the Banach<br />valued weighted Beurling ultradistribution setting and given some applications on<br />particular equations.</p> / <p>U disertaciji se proučavaju prostor konvolutora i multiplikatora na prostorima temperiranih ultradistribucija. Dokazane su&nbsp;teoreme koji karakteri&scaron;u elemente prostora konvolutora i multiplikatora. Date su strukturne teoreme za ultradistribucione &nbsp;polugrupe&nbsp;i eksponenecijalne polugrupe. Furijeve huperfunkciske polugrupe i&nbsp;hiperfunkciske polugrupe sa generatorima koji su negusto definisani&nbsp;<br />su analizirani, takođe su date strukturne teoreme i spektralne karakterizacije kao i dovoljni uslovi za postojenje na takvih polugrupa&nbsp;za operator A koji ne mora biti gust. Apstraktni Ko&scaron;ijev problem je&nbsp;proučavan za težinske Banahove prostore kao i za odgovarujuće prostora ultradistribucija. Takođe su date i primene za određene klase<br />jednačina.</p>

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Earth and Planetary Sciences, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND LINDGREN. / Includes bibliographies. / by Keiichiro Ikeda. / Ph.D.

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / Ph.D.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18042号 / 理博第3920号 / 新制||理||1566(附属図書館) / 30900 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 加藤 毅, 教授 上田 哲生 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

We study a Cauchy problem arising in uid mechanics, involving the socalled stationary generalized Stokes system, where one should recover the ow from boundary measurements. The problem is ill-posed in the sense that the solution does not depend continuously on data. Two iterative procedures for solving this problem are proposed and investigated. These methods are regularizing and in each iteration one solves a series of well-posed problems obtained by changing the boundary conditions. The advantage with this approach, is that these methods place few restrictions on the domain and on the coefficients of the problem. Also the structure of the equation is preserved. Well-posedness of the problems used in these procedures is demonstrated, i.e., that the problems have a unique solution that depends continuously on data. Since we have numerical applications in mind, we demonstrate well-posedness for the case when boundary data is square integrable. We give convergence proofs for both of these methods.

We look at the Cauchy problem with an accretive Operator on a Banach space. We give an upper bound for the norm of the difference of two solutions of Euler schemes with this accretive operator. This concrete estimate also works for the problem with a non-zero right-hand side in the Cauchy problem and is a generalization of a famous result by Kobayashi. We also show, how this result gives direct proofs for existence, uniqueness, stability and regularity of Euler solutions of the Cauchy problem and also the rate of convergence of solutions of Euler schemes. The results concerning regularity and rate of convergence are generalized for problem data in interpolation sets.:1. Accretive operators 1.1. Thebracket. 1.2. Accretive operators 1.3. The Cauchy problem and Euler solutions 2. A priori estimates for solutions of implicit Euler schemes 2.1. An implicit upper bound 2.2. Properties of the density 2.3. An explicit upper bound 3. Applications 3.1. Wellposedness of the Cauchy problem 3.2. Interpolation theory A. Functions of bounded variation

In this paper we consider approximation schemes and questions of well-posedness for a general class of functional differential equations of neutral-type (NFDE) where the difference operator does not have an atom at zero. Equations of this type occur in the modeling of certain aeroelastic control problems and include many singular integro-differential equations. We obtain general necessary and sufficient conditions for the well-posedness of functional differential equations of neutral-type on the Banach-spaces R<sup>n</sup?xL_p. As an example of the well-posedness of the non-atomic NFDE-system that arises in the study of aeroelasticity is established on R<sup>n</sup?xL_p, 1≤p<2. Employing the equivalence between generalized solutions of NFDEs and mild solutions of the “corresponding” abstract Cauchy-problems, we make use of general approximation results for well-posed Cauchy-problems to establish and analyze the convergence of the “averaging projection” scheme on the Banach spaces R<sup>n</sup?xL_p, 1<p<∞, for a class of problems with atomic difference operators. / Ph. D.

The present work in the area of soliton theory studies two problems which arise when seeking analytic solutions to certain nonlinear partial differential equations. In the first one, Lax pairs associated with prolonged eigenfunctions and prolonged squared eigenfunctions (prolonged squares) are derived for a Schrödinger equation with a potential depending polynomially on the spectral parameter (of degree N) and its respective hierarchy of nonlinear evolution equations (here named generalized polynomial Korteweg-de Vries equations or GPKdV). It is shown that the prolonged squares satisfy the linearized GPKdV equations. On that basis, the Cauchy problem for the linearized GPKdV has been solved by finding a complete set of such prolonged squares and applying an expansion formula derived in another work by the author. The results are a generalization of the ones by Sachs (SIAM J. Math. Anal. 14, 1983, 674-683). Moreover, a condition on the so-called recursion operator A is derived which generates the whole hierarchy of Lax pairs associated with the prolonged squares. As for the second part of the work, it developed a scheme for deriving gauge transformations between different linear spectral problems. Then the scheme is applied to obtain all known Darboux transformations for a quadratic pencil (the spectral problem considered in the first part at N = 2), Schrödinger equation (N = 1), Ablowitz-Kaup-Newell-Segur (AKNS) system and also derive the Jaulent-Miodek transformation. Moreover, the scheme yields a large family of new transformations of the above types. It also gives some insight on the structure of the transformations and emphasizes the symmetry with respect to inversion that they possess. / Ph. D.

In order to better describe dense gases, a smooth attractive tail arising from a Coulomb-type potential is added to the hard core repulsion of the Enskog equation, along with a velocity diffusion. By choosing the diffusing term of Fokker-Planck type with or without dynamical friction forces. The Cauchy problem for the Diffusive-Vlasov-Poisson-Enskog equation (DVE) and the Cauchy problem for the Fokker-Planck-Vlasov-Poisson-Enskog equation (FPVE) are addressed. / Ph. D.