Continuous k-to-1 functions between complete graphs of even order

A function between graphs is k-to-1 if each point in the co-domain has precisely k pre-images in the domain. Given two graphs, G and H, and an integer k ≥ 1, and considering G and H as subsets of R 3, there may or may not be a k-to-1 continuous function (i.e. a k-to-1 map in the usual topological sense) from G onto H. In this paper we review and complete the determination of whether there are finitely discontinuous, or just infinitely discontinuous k-to-1 functions between two intervals, each of which is one of the following: ]0, 1[, [0, 1[and [0, 1]. We also show that for k even and 1 ≤ r < 2s, (r, s) 6= (1, 1) and (r, s) 6= (3, 2), there is a k-to-1 map from K2r onto K2s if and only if k ≥ 2s.peer-reviewe

Early structure formation from cosmic string loops

We examine the effects of cosmic strings on structure formation and on the ionization history of the universe. While Gaussian perturbations from inflation are known to provide the dominant contribution to the large scale structure of the universe, density perturbations due to strings are highly non-Gaussian and can produce nonlinear structures at very early times. This could lead to early star formation and reionization of the universe. We improve on earlier studies of these effects by accounting for high loop velocities and for the filamentary shape of the resulting halos. We find that for string energy scales G\mu > 10^{-7} the effect of strings on the CMB temperature and polarization power spectra can be significant and is likely to be detectable by the Planck satellite. We mention shortcomings of the standard cosmological model of galaxy formation which may be remedied with the addition of cosmic strings, and comment on other possible observational implications of early structure formation by strings.Comment: 22 pages, 10 figures. References adde

A numerical study of steady and unsteady viscoelastic flow past bounded cylinders

We consider two-dimensional, inertia-free, flow of a constant-viscosity viscoelastic fluid obeying the FENE-CR equation past a cylinder placed symmetrically in a channel, with a blockage ratio of 0.5. Through numerical simulations we show that the flow becomes unsteady when the Deborah number (using the usual definition) is greater than De≈1.3, for an extensibility parameter of the model of L2 = 144. The transition from steady to unsteady flow is characterised by a small pulsating recirculation zone of size approximately equal to 0.15 cylinder radius attached to the downstream face of the cylinder. There is also a rise in drag coefficient, which shows a sinusoidal variation with time. The results suggest a possible triggering mechanism leading to the steady three-dimensional Gortler-type vortical structures, which have been observed in experiments of the flow of a viscoelastic fluid around cylinders. The results reveal that the reason for failure of the search for steady numerical solutions at relatively high Deborah numbers is that the two-dimensional flow separates and eventually becomes unsteady. For a lower extensibility parameter, L2 = 100, a similar recirculation is formed given rise to a small standing eddy behind the cylinder which becomes unsteady and pulsates in time for Deborah numbers larger than De≈4.0–4.5

On evolutionary Gamma convergence for gradient systems

In these notes we discuss general approaches for rigorously deriving limits of generalized gradient flows. Our point of view is that a generalized gradient system is defined in terms of two functionals, namely the energy functional Eε and the dissipation potential Rε or the associated dissipation distance. We assume that the functionals depend on a small parameter and the associated gradients systems have solutions uε. We investigate the question under which conditions the limits u of (subsequences of) the solutions uε are solutions of the gradient system generated by the Γ-limits E0 and R0. Here the choice of the right topology will be crucial as well as additional structural conditions. We cover classical gradient systems, where Rε is quadratic, and rate-independent systems as well as the passage from viscous to rate-independent systems. Various examples, such as periodic homogenization, are used to illustrate the abstract concepts and results

Constructive Inversion of Vadose Zone GPR Observations

To predict the of Earth system dynamics, observations of the vadose zone structure and water content are of vital interest. A suited measurement technique is ground penetrating radar (GPR). In this dissertation, the constructive inversion of surface GPR data is introduced. It relies on a parameterized model of the subsurface structure and distribution of dielectric permittivity. With it, GPR measurements are simulated by numerically solving Maxwell’s equations. After detecting signals in the measured and simulated data, the residuals of the signals’ traveltime and amplitude is iteratively minimized to estimate the subsurface parameters. Then, water content is computed from dielectric permittivity. The method was applied to measurements obtained on a testbed, providing ground-truth data. A comparison with the estimation results showed an agreement for the structure within ±5 cm and for the water content, a difference less than 2 % vol. A further evaluation of field data demonstrated the method’s applicability, when representing structure and permittivity by spline functions. Additionally a time-series was evaluated with assuming a constant structure, which enabled to interpret water dynamics. Besides providing accurate information on water content distribution and subsurface structure, the method allows the future attempt to estimate hydraulic properties

Hamilton-Jacobi theory for connecting equilibrium magnetic fields across a toroidal surface supporting a plasma pressure discontinuity

This thesis explores the conditions for connecting magnetic fields separated by an infinitely thin current sheet in a plasma. A condition of force balance strongly constrains the allowable continuations of the magnetic field across the surface. As this study is principally motivated by the need to calculate plasma equilibria in non-axisymmetric (3-D) toroidal fusion devices, the current sheet is assumed to be a 2-torus, and there is a focus on current sheets supporting a jump in pressure. Such surfaces serve as interfaces separating volumes of constant-pressure Taylor relaxation in MRXMHD. The topology of the problem allows the adaptation of dynamical systems criteria for the existence of invariant tori in Hamiltonian phase space to infer a necessary criterion for valid continuation. To this end a Hamiltonian-Jacobi equation is derived and termed the pressure jump Hamiltonian system. Given the metric of the toroidal surface, two approaches to applying the Hamilton-Jacobi method are adopted: (a) a continuation approach, in which the magnetic vector field is taken to be known on one side of the interface; and (b) an inverse problem approach, in which the magnetic vector fields on both sides of the interface are taken to be initially unknown, only scalar data on the fluctuation of the square of the field being given. An appeal to the Birkhoff theorem resolves the question as to whether Hamiltonian trajectories map homeomorphically to actual magnetic field lines. A corollary of this reconciliation implies that the winding number of the Hamiltonian orbit is indeed conserved under the mapping as the rotational transform of the corresponding field line. Thus for a given pressure jump and desired rotational transform of the magnetic field proposed for continuation or connection, the criterion used for feasibility is that there exists an invariant torus in the phase space of the pressure-jump Hamiltonian at a given energy (pressure jump) with the same winding number as the desired rotational transform. The criterion is applied computationally by solving for orbits and determining if they result in an ergodic covering of an invariant surface using Greene's residue. This thesis also describes a computer code written by the author that solves the pressure jump Hamiltonian and applies tests based on Greene's residue criterion to create robustness plots that visualise regimes of parameters for which connection is allowed. Conditions for connection are explored for three cases: 1) a simplified version of the pressure jump Hamiltonian to better understand the effects of the perturbative variables, 2) prescribed, smooth interfaces typical of the type that would be used as interfaces in the equilibrium code SPEC, and 3) a flux surface extracted from a partially chaotic volume from a SPEC equilibrium. Continuation across rotational transform discontinuities is also investigated. In some cases energy healing is observed in which increasing energy can disallow a connection, then allow it again at a higher energy. This healing implies that solutions to the problem are allowed (or forbidden) in energy "bands". The implications of all results to MRXMHD and SPEC are then articulated

Potential-based Formulations of the Navier-Stokes Equations and their Application

Based on a Clebsch-like velocity representation and a combination of classical variational principles for the special cases of ideal and Stokes flow a novel discontinuous Lagrangian is constructed; it bypasses the known problems associated with non-physical solutions and recovers the classical Navier-Stokes equations together with the balance of inner energy in the limit when an emerging characteristic frequency parameter tends to infinity. Additionally, a generalized Clebsch transformation for viscous flow is established for the first time. Next, an exact first integral of the unsteady, three-dimensional, incompressible Navier-Stokes equations is derived; following which gauge freedoms are explored leading to favourable reductions in the complexity of the equation set and number of unknowns, enabling a self-adjoint variational principle for steady viscous flow to be constructed. Concurrently, appropriate commonly occurring physical and auxiliary boundary conditions are prescribed, including establishment of a first integral for the dynamic boundary condition at a free surface. Starting from this new formulation, three classical flow problems are considered, the results obtained being in total agreement with solutions in the open literature. A new least-squares finite element method based on the first integral of the steady two-dimensional, incompressible, Navier-Stokes equations is developed, with optimal convergence rates established theoretically. The method is analysed comprehensively, thoroughly validated and shown to be competitive when compared to a corresponding, standard, primitive-variable, finite element formulation. Implementation details are provided, and the well-known problem of mass conservation addressed and resolved via selective weighting. The attractive positive definiteness of the resulting linear systems enables employment of a customized scalable algebraic multigrid method for efficient error reduction. The solution of several engineering related problems from the fields of lubrication and film flow demonstrate the flexibility and efficiency of the proposed method, including the case of unsteady flow, while revealing new physical insights of interest in their own right

Calculus I

Preface: The goal of this text is to help students learn to use calculus intelligently for solving a wide variety of mathematical and physical problems. This book is an outgrowth of our teaching of calculus at Berkeley, and the present edition incorporates many improvements based on our use of the first edition...

Space(time) oddity: dualities, holography and branes

In this Thesis, we study (several) aspects of three and five dimensional non-supersymmetric gauge theories. Using non-perturbative techniques, such as known strong-weak coupling dualities and holography, we present new results concerning their dynamics and phase diagrams. The thesis is divided into six Chapters. In the first Chapter, we start reviewing some general aspects of non-supersymmetric three dimensional theories, focusing on the dynamics of gauge theory both in the absence and in presence of Chern-Simons terms. We then focus on known dualities among three dimensional theories, such as particle-vortex and bosonization duality. Thanks to these tools, we discuss what is known about the phase diagram of QCD3, namely the three dimensional analog of four dimensional quantum chromodynamics, for various ranges of its parameters. In Chapter two, we introduce the basics of holography, starting by reviewing the AdS/CFT correspondence. We then generalize the discussion to the case of non-conformal field theories, with particular emphasis on the description of confining theories. Finally, we review the holographic construction of four dimensional and three dimensional gauge theories, and, focusing on the latter case, we construct the gravity dual of QCD3. In Chapter three, we show new results regarding the phase diagram of QCD3 in presence of flavor-breaking mass deformation. The corresponding theory, namely QCD3 with two sets of flavors, is studied in detail, thanks to the conjectured infrared dualities characterizing gauge theories with matter in three dimensions, namely boson-fermion dualities. In particular, the low-energy phase diagram is charted, and its consistency gives additional support to the conjectured phase diagram of QCD3. Moreover, new non-perturbative phases are observed, together with peculiar phase transitions among them, which are novel to QCD3 with two flavors. In Chapter four, we study the phase diagram of large N QCD3 through its holographic dual. This novel study shows perfect agreement with the field theory analysis, giving a simple explanation of the observed peculiarity of its phase diagram, together with an holographic evidence of the validity of boson-fermion dualities. In Chapter five, we review the main aspects of five dimensional theories. Firstly, we focus on general properties of supersymmetric gauge theories, their BPS spectrum, and their moduli spaces of vacua. Then, we study their non-perturbative dynamics using string constructions, both in type I’ and in type IIB string theory. The latter type of construction, known as the pq-web or brane web construction, gives us the possibility of studying in detail many non-perturbative phenomena characterizing these theories and their superconformal ultraviolet fixed points, such as global symmetry enhancement and continuation past infinite coupling