Mathematical physics

Boundary layer control on a circular cylindrical body through oscillating Lorentz
forcing is studied by means of numerical simulation of the vorticity-stream
function formulation of the Navier-Stokes equations. The model problem
considers axisymmetric seawater flow along an infinite cylinder controlled by an
idealized radially directed Lorentz force oscillating spatially and temporally.
Under optimum forcing parameters, it is shown that sustainable Lorentz induced
vortex rings can travel along the cylinder at a speed equivalent to the phase
speed of forcing . Wall stress is shown to locally change sign in the region
adjacent to the vortex, considerably decreasing net viscous drag . Adverse flow
behaviors are revealed as a result of studying the effects of the Reynolds
numbers, strength of the Lorentz force, and phase speed of forcing for boundary
layer control. Adverse flow behaviors inc I ude complex vortex configurations
found for suboptimal forcing resulting in a considerable increase in wall stress.

Spatially continuous networks with heterogeneous connections are ubiquitous in
biological systems, in part icular neural systems. To understand the mutual effects
of locally homogeneous and globally heterogeneous connectivity, the st ability of the
steady state activity of a neural field as a fun ction of its connectivity is investigated.
The variation of the connectivity is operationalized through manipulation of a heterogeneous
two-point connection embedded into the otherwise homogeneous connectivity
matrix and by variation of connectivity strength and transmission speed. A detailed
discussion of the example of the real Ginzburg-Land au equation with an embedded
two-point heterogeneous connection in addition to the homogeneous coupling due to
the diffusion term is performed. The system is reduced to a set of delay differential
equations and the stability di agrams as a function of the time delay and the local and
global coupling strengths are computed. The major finding is that the stability of
the heterogeneously connected elements with a well-defined velocity defines a lower
bound for the stabil ity of the entire system . Diffusion and velocity dispersion always
result in increased stability. Various other local architectures represented by exponentially
decaying connectivity fun ctions are also discussed. The analysis shows that
developmental changes such as the myelination of the cortical large-scale fib er system generally result in the stabilization of steady state activity via oscillatory instabilities
independent of the local connectivity. Non-oscillatory (Turing) instabilities are shown
to be independent of any influences of time delay.

The new formalism for quantization of gauge systems based on the concept of the
dynamical Hamiltonian recently introduced as a basis for the canonical theory of
quantum gravity was considered in the context of general gauge theories. This and
other Hamiltonian methods, that include Dirac's theory of extended Hamiltonian
and the Hamiltonian reduction formalism were critically examined. It was established
that the classical theories of constrained gauge systems formulated within the
framework of either of the approaches are equivalent. The central to the proof of
equivalence was the fact that the gauge symmetries resuIt in the constraints of the
first class in Dirac's terminology that Iead to redundancy of equations of motion
for some of the canonica variables. Nevertheless, analysis of the quantum theories
showed that in general, the quantum theory of the dynamical Hamiltonian is inequivalent
to those of the extended Hamiltonian and the Hamiltonian reduction. The
new method of quantization was applied to a number of gauge systems, including
the theory of relativistic particle, the Bianchi type IX cosmological model and spinor electrodynamics along side with the traditional methods of quantization. In all of the
cases considered the quantum theory of the dynamical Hamiltonian was found to be
well-defined and to possess the appropriate classical limit. In particular, the quantization
procedure for the Bianchi type IX cosmological spacetime did not run into
any of the known problems with quantizing the theory of General Relativity. On the
other hand, in the case of the quantum electrodynamics the dynamical Hamiltonian
approach led to the quantum theory with the modified self-interaction in the matter
sector. The possible consequence of this for the quantization of the full theory of
General Relativity including the matter fields are discussed.

In this dissertation the dynamics of general relativity is studied via the spin-foam approach to quantum gravity. Spin-foams are a proposal to compute a transition amplitude from a triangulated space-time manifold for the evolution of quantum 3d geometry via path integral. Any path integral formulation of a quantum theory has two important parts, the measure factor and a phase part. The correct measure factor is obtained by careful canonical analysis at the continuum level. The basic variables in the Plebanski-Holst formulation of gravity from which spin-foam is derived are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one usually sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two-form appears, and in which the conne ction degrees of freedom have been integrated out. Calculating the measure factor for Plebanksi Holst formulation without the connection degrees of freedom is one of the aims of this dissertation. This analysis is at the continuum level and in order to be implemented in spin-foams one needs to properly discretize and quantize this measure factor. The correct phase is determined by semi-classical behavior. In asymptotic analysis of the Engle-Pereira-Rovelli-Livine spin-foam model, due to the inclusion of more than the usual gravitational sector, more than the usual Regge term appears in the asymptotics of the vertex amplitude. As a consequence, solutions to the classical equations of motion of GR fail to dominate in the semi-classical limit. One solution to this problem has been proposed in which one quantum mechanically imposes restriction to a single gravitational sector, yielding what has been called the “proper” spin-foam model. However, this revised model of quantum gravity, like any proposal for a theory of quantum gravity, must pass certain tests. In the regime of small curvature, one expects a given model of quantum gravity to reproduce the predictions of the linearized theory. As a consistency check we calculate the graviton two-point function predicted by the Lorentzian proper vertex and examine its semiclassical limit.

Pairing-friendly curves and elliptic curves with a trapdoor for the discrete
logarithm problem are versatile tools in the design of cryptographic protocols. We
show that curves having both properties enable a deterministic identity-based signing
with “short” signatures in the random oracle model. At PKC 2003, Choon and Cheon
proposed an identity-based signature scheme along with a provable security reduction.
We propose a modification of their scheme with several performance benefits. In
addition to faster signing, for batch signing the signature size can be reduced, and if
multiple signatures for the same identity need to be verified, the verification can be
accelerated. Neither the signing nor the verification algorithm rely on the availability
of a (pseudo)random generator, and we give a provable security reduction in the
random oracle model to the (`-)Strong Diffie-Hellman problem. Implementing the group arithmetic is a cost-critical task when designing quantum circuits for Shor’s algorithm to solve the discrete logarithm problem. We introduce a tool for the automatic generation of addition circuits for ordinary binary elliptic curves, a prominent platform group for digital signatures. Our Python software generates circuit descriptions that, without increasing the number of qubits or T-depth, involve less than 39% of the number of T-gates in the best previous construction. The software also optimizes the (CNOT) depth for F2-linear operations by means of suitable graph colorings.

This thesis was prepared out of the necessity for an identification
of the concepts of mathematics used in PSSC physics. First,
a survey of past and current educational journals was used to
establish general mathematical prerequisites for PSSC physics. Then
the necessary mathematical concepts as conceived by the authors of
the PSSC textbook are discussed, with no attempt to state the level
of proficiency required in behavioral terms for the identified
topics. A survey of the literature was made in order to suggest
provisions for the teaching of these concepts. These suggestions
take the form of mathematics-physics curriculum changes such as
integrated mathematics-physics courses, fusion of science applications
into the mathematics curriculum, and the institution of a
mathematics course specifically designed to prepare students for the
study of physics.

Despite the high-dimensional nature of the nervous system, humans produce low-dimensional cognitive and behavioral dynamics. How high-dimensional networks with complex connectivity give rise to functionally meaningful dynamics is not well understood. How does a neural network encode function? How can functional dynamics be systematically obtained from networks? There exist few frameworks in the current literature that answer these questions satisfactorily. In this dissertation I propose a general theoretical framework entitled 'Structured Flows on Manifolds' and its underlying mathematical basis. The framework is based on the principles of non-linear dynamical systems and Synergetics and can be used to understand how high-dimensional systems that exhibit multiple time-scale behavior can produce low-dimensional dynamics. Low-dimensional functional dynamics arises as a result of the timescale separation of the systems component's dynamics. The low-dimensional space in which the functi onal dynamics occurs is regarded as a manifold onto which the entire systems dynamics collapses. For the duration of the function the system will stay on the manifold and evolve along the manifold. From a network perspective the manifold is viewed as the product of the interactions of the network nodes. The subsequent flows on the manifold are a result of the asymmetries of network's interactions. A distributed functional architecture based on this perspective is presented. Within this distributed functional architecture, issues related to networks such as flexibility, redundancy and robustness of the network's dynamics are addressed. Flexibility in networks is demonstrated by showing how the same network can produce different types of dynamics as a function of the asymmetrical coupling between nodes. Redundancy can be achieved by systematically creating different networks that exhibit the same dynamics. The framework is also used to systematically probe the effects of lesion

In this work, a two-dimensional model representing the vortices that animals produce, when they are flying/swimming, was constructed. A D{shaped cylinder and an oscillating airfoil were used to mimic these body{shed and wing{generated vortices, respectively. The parameters chosen are based on the Reynolds numbers similar to that which is observed in nature (104). In order to imitate the motion of ying/swimming, the entire system was suspended into a water channel from frictionless air{bearings. The position of the apparatus in the channel was regulated with a linear, closed loop PI controller. Thrust/drag forces were measured with strain gauges and particle image velocimetry (PIV) was used to examine the wake structure that develops. The Strouhal number of the oscillating airfoil was compared to the values observed in nature as the system transitions between the accelerated and steady states... As suggested by previous work, this self-regulation is a result of a limit cycle process that stems from nonlinear periodic oscillations. The limit cycles were used to examine the synchronous conditions due to the coupling of the foil and wake vortices. Noise is a factor that can mask details of the synchronization. In order to control its effect, we study the locking conditions using an analytic technique that only considers the phases.. The results suggest that Strouhal number selection in steady forward natural swimming and flying is the result of a limit cycle process and not actively controlled by an organism. An implication of this is that only relatively simple sensory and control hardware may be necessary to control the steady forward motion of man-made biomimetically propelled vehicles.

In a projective plane (PG(2, K) defined over an algebraically closed field K of characteristic p = 0, we give a complete classification of 3-nets realizing a finite group. The known infinite family, due to Yuzvinsky, arised from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky, comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family and list all possible sporadic examples. If p is larger than the order of the group, the same classification holds true apart from three possible exceptions: Alt4, Sym4 and Alt5.

An algebraic surface defined by an equation of the form z2 = (x+a1y) ... (x + any) (x - 1) is studied, from both an algebraic and geometric point of view. It is shown that the surface is rational and contains a singular point which is nonrational. The class group of Weil divisors is computed and the Brauer group of Azumaya algebras is studied. Viewing the surface as a cyclic cover of the affine plane, all of the terms in the cohomology sequence of Chase, Harrison and Roseberg are computed.