Edge-Orders

Canonical orderings and their relatives such as st-numberings have been used as a key tool in algorithmic graph theory for the last decades. Recently, a unifying concept behind all these orders has been shown: they can be described by a graph decomposition into parts that have a prescribed vertex-connectivity. Despite extensive interest in canonical orderings, no analogue of this unifying concept is known for edge-connectivity. In this paper, we establish such a concept named edge-orders and show how to compute (1,1)-edge-orders of 2-edge-connected graphs as well as (2,1)-edge-orders of 3-edge-connected graphs in linear time, respectively. While the former can be seen as the edge-variants of st-numberings, the latter are the edge-variants of Mondshein sequences and non-separating ear decompositions. The methods that we use for obtaining such edge-orders differ considerably in almost all details from the ones used for their vertex-counterparts, as different graph-theoretic constructions are used in the inductive proof and standard reductions from edge- to vertex-connectivity are bound to fail. As a first application, we consider the famous Edge-Independent Spanning Tree Conjecture, which asserts that every k-edge-connected graph contains k rooted spanning trees that are pairwise edge-independent. We illustrate the impact of the above edge-orders by deducing algorithms that construct 2- and 3-edge independent spanning trees of 2- and 3-edge-connected graphs, the latter of which improves the best known running time from O(n^2) to linear time

BIOMOLECULE LOCALIZATION AND SURFACE ENGINEERING WITHIN SIZE TUNABLE NANOPOROUS SILICA PARTICLES

Mesoporous silica materials are versatile platforms for biological catalysis, isolation of small molecules for detection and separation applications. The design of mesoporous silica supports for tailored protein and biomolecule interactions has been limited by the techniques to demonstrate biomolecule location and functionality as a function of pore size. This work examines the interaction of proteins and lipid bilayers with engineered porous silica surfaces using spherical silica particles with tunable pore diameters (3 – 12 nm) in the range relevant to biomolecule uptake in the pores, and large particle sizes (5 - 15 µm) amenable to microscopy imaging The differentiation of protein location between the external surface and within the pore, important to applications requiring protein protection or catalytic activity in pores, is demonstrated. A protease / fluorescent protein system is used to investigate protein location and protection as a function of pore size, indicating a narrow pore size range capable of protein protection, slightly larger than the protein of interest and approaching the protease dimensions. Selective functionalization, in this case exterior-only surface functionalization of mesoporous particles with amines, is extended to larger pore silica materials. A reaction time dependent functionalization approach is demonstrated as the first visually confirmed, selective amine functionalization method in protein accessible supports. Mesoporous silica nanoparticles are effective supports for lipid bilayer membranes and membrane associated proteins for separations and therapeutic delivery, although the role of support porosity on membrane fluidity is unknown. Transport properties of bilayers in lipid filled nanoparticles as a function of pore diameter and location in the particle are measured for the first time. Bilayer diffusivity increases with increasing pore size and is independent of bilayer location within the core, mid or cap of the particle, suggesting uniform long range bilayer mobility in lipid filled pores. Application of lipid bilayers on mesoporous silica was examined for membrane associated proteins A unique method to adhere functional proteins in lipid bilayers on mesoporous silica particles is established using vesicles derived from cell plasma membranes and their associated proteins. This method of membrane protein investigation retains proteins within native lipid membranes, stabilizing proteins for investigation on supports

Non-perturbative approaches to magnetism in strongly correlated electron systems

The microscopic basis for the stability of itinerant ferromagnetism in correlated electron systems is examined. To this end several routes to ferromagnetism are explored, using both rigorous methods valid in arbitrary spatial dimensions, as well as Quantum Monte Carlo investigations in the limit of infinite dimensions (dynamical mean-field theory). In particular we discuss the qualitative and quantitative importance of (i) the direct Heisenberg exchange coupling, (ii) band degeneracy plus Hund's rule coupling, and (iii) a high spectral density near the band edges caused by an appropriate lattice structure and/or kinetic energy of the electrons. We furnish evidence of the stability of itinerant ferromagnetism in the pure Hubbard model for appropriate lattices at electronic densities not too close to half-filling and large enough $U$. Already a weak direct exchange interaction, as well as band degeneracy, is found to reduce the critical value of $U$ above which ferromagnetism becomes stable considerably. Using similar numerical techniques the Hubbard model with an easy axis is studied to explain metamagnetism in strongly anisotropic antiferromagnets from a unifying microscopic point of view.Comment: 11 pages, Latex, and 6 postscript figures; Z. Phys. B, in pres

Coexistence of solutions in dynamical mean-field theory of the Mott transition

In this paper, I discuss the finite-temperature metal-insulator transition of the paramagnetic Hubbard model within dynamical mean-field theory. I show that coexisting solutions, the hallmark of such a transition, can be obtained in a consistent way both from Quantum Monte Carlo (QMC) simulations and from the Exact Diagonalization method. I pay special attention to discretization errors within QMC. These errors explain why it is difficult to obtain the solutions by QMC close to the boundaries of the coexistence region.Comment: 3 pages, 2 figures, RevTe

Quantum Monte Carlo calculation of the finite temperature Mott-Hubbard transition

We present clear numerical evidence for the coexistence of metallic and insulating dynamical mean field theory(DMFT) solutions in a half-filled single-band Hubbard model with bare semicircular density of states at finite temperatures. Quantum Monte Carlo(QMC) method is used to solve the DMFT equations. We discuss important technical aspects of the DMFT-QMC which need to be taken into account in order to obtain the reliable results near the coexistence region. Among them are the critical slowing down of the iterative solutions near phase boundaries, the convergence criteria for the DMFT iterations, the interpolation of the discretized Green's function and the reduction of QMC statistical and systematic errors. Comparison of our results with those of other numerical methods is presented in a phase diagram.Comment: 4 pages, 5 figure

Landau Theory of the Finite Temperature Mott Transition

In the context of the dynamical mean-field theory of the Hubbard model, we identify microscopically an order parameter for the finite temperature Mott endpoint. We derive a Landau functional of the order parameter. We then use the order parameter theory to elucidate the singular behavior of various physical quantities which are experimentally accessible.Comment: 4 pages, 2 figure

Surface Instabilities and Magnetic Soft Matter

We report on the formation of surface instabilities in a layer of thermoreversible ferrogel when exposed to a vertical magnetic field. Both static and time dependent magnetic fields are employed. Under variations of temperature, the viscoelastic properties of our soft magnetic matter can be tuned. Stress relaxation experiments unveil a stretched exponential scaling of the shear modulus, with an exponent of beta=1/3. The resulting magnetic threshold for the formation of Rosensweig-cusps is measured for different temperatures, and compared with theoretical predictions by Bohlius et. al. in J. Phys.: Condens. Matter., 2006, 18, 2671-2684.Comment: accepted to Soft Matte

Linked Cluster Expansion Around Mean-Field Theories of Interacting Electrons

A general expansion scheme based on the concept of linked cluster expansion from the theory of classical spin systems is constructed for models of interacting electrons. It is shown that with a suitable variational formulation of mean-field theories at weak (Hartree-Fock) and strong (Hubbard-III) coupling the expansion represents a universal and comprehensive tool for systematic improvements of static mean-field theories. As an example of the general formalism we investigate in detail an analytically tractable series of ring diagrams that correctly capture dynamical fluctuations at weak coupling. We introduce renormalizations of the diagrammatic expansion at various levels and show how the resultant theories are related to other approximations of similar origin. We demonstrate that only fully self-consistent approximations produce global and thermodynamically consistent extensions of static mean field theories. A fully self-consistent theory for the ring diagrams is reached by summing the so-called noncrossing diagrams.Comment: 17 pages, REVTEX, 13 uuencoded postscript figures in 2 separate file

Challenges in Simulation of Aerodynamics, Hydrodynamics, and Mooring-Line Dynamics of Floating Offshore Wind Turbines

This paper presents the current major modeling challenges for floating offshore wind turbine design tools and describes aerodynamic and hydrodynamic effects due to rotor and platform motions and usage of non-slender support structures