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Reaction and Diffusion on the Sierpinski Gasket
No full textIn this thesis we study non-linear dynamical systems on complex domains. Although the systems we consider are mathematical abstractions, our motivation is to gain insights into neurobiological systems. The mathematical techniques we employ concern analysis on a particular class of fractal sets. This theory allows one to construct a Laplacian and to study the spectrum and eigenfunctions given a variety of boundary conditions. This thesis uses these results to define and study the cable equation and the FitzHugh-Nagumo system on the Sierpinski Gasket
Uniform decision problems for automatic semigrouos
No full textWe consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which are necessary to make the problem well defined, the uniform word problem for semigroups described by automatic structures is decidable. Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix together with an automatic structure for its maximal subgroup. On the other hand, we show that it is undecidable in general whether a given element of a given automatic monoid has a right inverse
Sylow theory for p=0 in solvable groups of finite Morley rank
No full textThe algebraicity conjecture for simple groups of finite Morley rank, also known as the Cherlin–Zil'ber conjecture, states that simple groups of finite Morley rank are simple algebraic groups over algebraically closed fields. In the last fifteen years, the main line of attack on this problem has been Borovik's program of transferring methods from finite group theory, which has led to considerable progress; however, the conjecture itself remains completely open. In Borovik's program, groups of finite Morley rank are divided into four types, odd, even, mixed, and degenerate, according to the structure of their Sylow 2-subgroup. For even and mixed type the algebraicity conjecture has been proven
Isomorphism classes for Banach vector bundle structures of second tangents
No full textOn a smooth Banach manifold M, the equivalence classes of curves that agree up to acceleration form the second order tangent bundle T^2M of M. This is a vector bundle in the presence of a linear connection on M and the corresponding local structure is heavily dependent on the choice of connection. In this paper we study the extent of this
dependence and we prove that it is closely related to the notions of conjugate connections and second order differentials. In particular, the vector bundle structure on T^2M remains invariant under conjugate connections with respect to diffeomorphisms of M
Exact solutions of the 3-wave resonant interaction equation
Get PDFThe Darboux–Dressing Transformations are applied to the Lax pair associated to the system of nonlinear equations describing the resonant interaction of three waves in 1+1 dimensions. We display explicit solutions featuring localized waves whose profile vanishes at the spacial boundary |x|=∞, and which are not pure soliton solutions. These solutions depend on an arbitrary function and allow us to deal with collisions of waves with various profiles
Convergence Estimates of Krylov Subspace Methods for the Approximation of Matrix Functions Using Tools from Potential Theory
Get PDFThis diploma thesis from 2006 reviews various definitions of matrix functions and polynomial Krylov methods for their approximation. Relations to polynomial interpolation and best approximation problems are made. The convergence behavior of Ritz values associated with Hermitian matrices is investigated. A new algorithm for the solution of the constrained energy problem with a measure supported in the complex plane is developed. This algorithm is then used to study Ritz values associated with a normal non-Hermitian matrix
Acceleration bundles on Banach and Fréchet manifolds
Get PDFThe second order tangent bundle T^2M of a
smooth manifold M consists of the equivalence classes of curves on M that agree up to their acceleration. Dodson and Radivoiovici showed that in the case of a finite
n-dimensional manifold M, T^2M becomes a vector bundle over
M if and only if M is endowed with a linear connection.
We have extended this result to M modeled on an arbitrary
Banach space and more generally to those Fréchet manifolds
which can be obtained as projective limits of Banach manifolds. Various structural properties have been deduced
Cross-Subtype T-Cell Immune Responses Induced by a Human Immunodeficiency Virus Type 1 Group M Consensus Env Immunogen
Get PDFThe genetic diversity among globally circulating human immunodeficiency virus type 1 (HIV-1) strains is a
serious challenge for HIV-1 vaccine design. We have generated a synthetic group M consensus env gene (CON6)
for induction of cross-subtype immune responses and report here a comparative study of T-cell responses to
this and natural strain env immunogens in a murine model. Three different strains of mice were immunized
with CON6 as well as subtype A, B, or C env immunogens, using a DNA prime-recombinant vaccinia virus boost
strategy. T-cell epitopes were mapped by gamma interferon enzyme-linked immunospot analysis using five
overlapping Env peptide sets from heterologous subtype A, B, and C viruses. The CON6-derived vaccine was
immunogenic and induced a greater number of T-cell epitope responses than any single wild-type subtype A,
B, and C env immunogen and similar T-cell responses to a polyvalent vaccine. The responses were comparable
to within-clade responses but significantly more than between-clade responses. The magnitude of the T-cell
responses induced by CON6 (measured by individual epitope peptides) was also greater than the magnitude
of responses induced by individual wild-type env immunogens. Though the limited major histocompatibility
complex repertoire in inbred mice does not necessarily predict responses in nonhuman primates and humans,
these results suggest that synthetic centralized env immunogens represent a promising approach for HIV-1
vaccine design that merits further characterization
Superdecomposable pure-injective modules and integral group rings
No full textWe prove that if G is a non-trivial finite group, then the integral group ring possesses a superdecomposable pure-injective module
Quantum mushroom billiards
Get PDFWe report the first calculations of eigenmodes (quantum states) of a mushroom billiard of the type proposed by L. Bunimovich in this journal. The phase space of this mixed system has a single regular region and a single ergodic region, and no KAM hierarchy. For a symmetric mushroom with a square foot, we find: i) low-eigenvalue modes with very high relative eigenvalue accuracy of order , and ii) high-eigenvalue modes at mode number around . We outline the simple but highly-efficient mesh-free boundary collocation methods which make such calculations tractable. We test Percival's conjecture that almost all modes localize either to regular or ergodic regions, report the relative frequencies of such modes, and examine Husimi distributions on the Poincaré surface of section