3. Electronic Theses and Dissertations (ETDs) - All submissions
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Item Characterization of operator spaces.(1993) Kalaichelvan, RajendraThis research report serves as an introduction to the concept of Operator Spaces which has gained considerable momentum in its acknowledgement and research interest in the last few decades. It will highlight a very important breakthrough on the characterization of Operator spaces which occurred in the !ast few years brought about by Z.J. Ruan. It investigates the relationship of this space in relation to Banach space theory by looking at an extension theorem for linear functionals,Item Inverse eigenvalue problem(2016-08-16) Boamah, Edward KwasiThis work is concerned with the Inverse Eigenvalue Problem for ordinary differential equations of the Sturm-Liouville type in the general form --dd ( 7' ()xdll(t\,:rI)) + {(q) x - t\p:(r )} u (A, Xl, = 0, .1' c.r (I :::: .7' S; b. The central problem considered ill this research is the approximate reC011- struction of the unknown coefficient function q(:l') in the Sturm-Liouville equation JOIl Irom a given finite spectral data set ~i(q), for i = 1 : n . A solution is sought using a finite element discretization method. The method works br solving the non-Iinear system arising out of the difference between the eigenvalues A,(q) of the Sturm-Liouville differential equation and the given spectral data ~i(q). Numerical results me presented to illustrate the effectiveness of the discretization method ill question.Item Near-rings and their modules(2016-07-18) Berger, Amelie JulieAfter an introduction defining basic structutral aspects of near-rings, this report examines how the ring-theoretic notions of generation and cogeneration can be extended from modules over a ring to modules over a near-ring. Chapter four examines matrix near-rings and connections between the J2 and JS radicals of the near-ring and the corresponding matrix near-ring. By extending the concepts of generation and cogeneration from the ring modules to near-ring modules we are investigating how important distribution and an abelian additive structure are to these two concepts. The concept of generation faces the obstacle that the image of a near-ring module homomorphism is not necessarily a subrnodule of the image space but only a subgroup, while the sum of two subgroups need not even be a subgroup. In chapter two, generation trace and socle are defined for near-ring modules and these ideas are linked with those of the essential and module-essential subgroups. Cogeneration, dealing with kernels which are always submodules proved easier to generalise. This is discussed in chapter three together with the concept of the reject, and these ideas are Iinked to the J1/2 and J2 radicals. The duality of the ring theory case is lost. The results are less simple than in the ring theory case due to the different types of near-ring module substructures which give rise to several Jacobson-type radicals. A near-ring of matrices can be obtained from an arbitrary near-ring by regarding each rxr matrix as a mapping from Nr to Nr where N is the near-ring from which entries are taken. The argument showing that the near-ring is 2-semisimple if and only if the associated near-ring of matrices is 2-semisimple is presented and investigated in the case of s-semisimplicity. Questions arising from this report are presented in the final chapter.Item Radial dynamics of the large N limit of multimatrix models(2016-01-22) Masuku, MthokozisiMatrix models, and their associated integrals, are encoded with a rich structure, especially when studied in the large N limit. In our project we study the dynamics of a Gaussian ensemble of m complex matrices or 2m hermitian matrices for d = 0 and d = 1 systems. We rst investigate the two hermitian matrix model parameterized in \matrix valued polar coordinates", and study the integral and the quantum mechanics of this system. In the Hamiltonian picture, the full Laplacian is derived, and in the process, the radial part of the Jacobian is identi ed. Loop variables which depend only on the eigenvalues of the radial matrix turn out to form a closed subsector of the theory. Using collective eld theory methods and a density description, this Jacobian is independently veri ed. For potentials that depend only on the eigenvalues of the radial matrix, the system is shown to be equivalent to a system of non-interacting (2+1)-dimensional \radial fermions" in a harmonic potential. The matrix integral of the single complex matrix system, (d = 0 system), is studied in the large N semi-classical approximation. The solutions of the stationary condition are investigated on the complex plane, and the eigenvalue density function is obtained for both the single and symmetrically extended intervals of the complex plane. The single complex matrix model is then generalized to a Gaussian ensemble of m complex matrices or 2m hermitian matrices. Similarly, for this generalized ensemble of matrices, we study both the integral of the system and the Hamiltonian of the system. A closed sector of the system is again identi ed consisting of loop variables that only depend on the eigenvalues of a matrix that has a natural interpretation as that of a radial matrix. This closed subsector possess an enhanced U(N)m+1 symmetry. Using the Schwinger-Dyson equations which close on this radial sector we derive the Jacobian of the change of variables to this radial sector. The integral of the system of m complex matrices is evaluated in the large N semi-classical approximation in a density description, where we observe the emergence of a new logarithmic term when m 2. The solutions of the stationary condition of the system are investigated on the complex plane, and the eigenvalue density functions for m 2 are obtained in the large N limit. The \fermionic description" of the Gaussian ensemble of m complex matrices in radially invariant potentials is developed resulting in a sum of non-interacting Hamiltonians in (2m + 1)-dimensions with an induced singular term, that acts on radially anti-symmetric wavefunctions. In the last chapter of our work, the Hamiltonian of the system of m complex matrices is formulated in the collective eld theory formalism. In this density description we will study the large N background and obtain the eigenvalue density function.Item The large-N limit of matrix models and AdS/CFT(2014-06-12) Mulokwe, MbavhaleloRandom matrix models have found numerous applications in both Theoretical Physics and Mathematics. In the gauge-gravity duality, for example, the dynamics of the half- BPS sector can be fully described by the holomorphic sector of a single complex matrix model. In this thesis, we study the large-N limit of multi-matrix models at strong-coupling. In particular, we explore the significance of rescaling the matrix fields. In order to investigate this, we consider the matrix quantum mechanics of a single Hermitian system with a quartic interaction. We “compactify” this system on a circle and compute the first-order perturbation theory correction to the ground-state energy. The exact ground-state energy is obtained using the Das-Jevicki-Sakita Collective Field Theory approach. We then discuss the multi-matrix model that results from the compactification of the Higgs sector of N = 4 SYM on S4 (or T S3). For the radial subsector, the saddle-point equations are solved exactly and hence the radial density of eigenvalues for an arbitrary number of even Hermitian matrices is obtained. The single complex matrix model is parametrized in terms of the matrix valued polar coordinates and the first-order perturbation theory density of eigenstates is obtained. We make use of the Harish-Chandra- Itzykson-Zuber (HCIZ) formula to write down the exact saddle-point equations. We then give a complementary approach - based on the Dyson-Schwinger (loop) equations formalism - to the saddle-point method. We reproduce the results obtained for the radial (single matrix) subsector. The two-matrix integral does not close on the original set of variables and thus we map the system onto an auxiliary Penner-type two matrix model. In the absence of a logarithmic potential we derive a radial hemispherical density of eigenvalues. The system is regulated with a logarithm potential, and the Dobroliubov-Makeenko-Semenoff (DMS) loop equations yield an equation of third degree that is satisfied by the generating function. This equation is solved at strong coupling and, accordingly, we obtain the radial density of eigenvalues.