Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation

Noncommutative elliptic theory. Examples

We study differential operators, whose coefficients define noncommutative algebras. As algebra of coefficients, we consider crossed products, corresponding to action of a discrete group on a smooth manifold. We give index formulas for Euler, signature and Dirac operators twisted by projections over the crossed product. Index of Connes operators on the noncommutative torus is computed.Comment: 23 pages, 1 figur

Assessing awareness in severe Alzheimer’s disease

There is an urgent need to understand the nature of awareness in people with severe Alzheimer’s disease (AD) to ensure effective person-centered care. Objective biomarkers of awareness validated in other clinical groups (e.g., anesthesia, minimally conscious states) offer an opportunity to investigate awareness in people with severe AD. In this article we demonstrate the feasibility of using Transcranial magnetic stimulation (TMS) combined with EEG, event related potentials (ERPs) and fMRI to assess awareness in severe AD. TMS-EEG was performed in six healthy older controls and three people with severe AD. The perturbational complexity index (PCIST) was calculated as a measure of capacity for conscious awareness. People with severe AD demonstrated a PCIST around or below the threshold for consciousness, suggesting reduced capacity for consciousness. ERPs were recorded during a visual perception paradigm. In response to viewing faces, two patients with severe AD provisionally demonstrated similar visual awareness negativity to healthy controls. Using a validated fMRI movie-viewing task, independent component analysis in two healthy controls and one patient with severe AD revealed activation in auditory, visual and fronto-parietal networks. Activation patterns in fronto-parietal networks did not significantly correlate between the patient and controls, suggesting potential differences in conscious awareness and engagement with the movie. Although methodological issues remain, these results demonstrate the feasibility of using objective measures of awareness in severe AD. We raise a number of challenges and research questions that should be addressed using these biomarkers of awareness in future studies to improve understanding and care for people with severe AD

Topological Expansion and Exponential Asymptotics in 1D Quantum Mechanics

Borel summable semiclassical expansions in 1D quantum mechanics are considered. These are the Borel summable expansions of fundamental solutions and of quantities constructed with their help. An expansion, called topological,is constructed for the corresponding Borel functions. Its main property is to order the singularity structure of the Borel plane in a hierarchical way by an increasing complexity of this structure starting from the analytic one. This allows us to study the Borel plane singularity structure in a systematic way. Examples of such structures are considered for linear, harmonic and anharmonic potentials. Together with the best approximation provided by the semiclassical series the exponentially small contribution completing the approximation are considered. A natural method of constructing such an exponential asymptotics relied on the Borel plane singularity structures provided by the topological expansion is developed. The method is used to form the semiclassical series including exponential contributions for the energy levels of the anharmonic oscillator.Comment: 46 pages, 22 EPS figure

Simulation of a GOx-GCH4 Rocket Combustor and the Effect of the GEKO Turbulence Model Coefficients

In this study, a single injector methane-oxygen rocket combustor is numerically studied. The simulations included in this study are based on the hardware and experimental data from the Technical University of Munich. The focus is on the recently developed generalized k–w turbulence model (GEKO) and the effect of its adjustable coefficients on the pressure and on wall heat flux profiles, which are compared with the experimental data. It was found that the coefficients of ‘jet’, ‘near-wall’, and ‘mixing’ have a major impact, whereas the opposite can be deduced about the ‘separation’ parameter Csep, which highly influences the pressure and wall heat flux distributions due to the changes in the eddy-viscosity field. The simulation results are compared with the standard k–# model, displaying a qualitatively and quantitatively similar behavior to the GEKO model at a Csep equal to unity. The default GEKO model shows a stable performance for three oxidizer-to-fuel ratios, enhancing the reliability of its use. The simulations are conducted using two chemical kinetic mechanisms: Zhukov and Kong and the more detailed RAMEC. The influence of the combustion model is of the same order as the influence of the turbulence model. In general, the numerical results present a good or satisfactory agreement with the experiment, and both GEKO at Csep = 1 or the standard k–# model can be recommended for usage in the CFD simulations of rocket combustion chambers, as well as the Zhukov–Kong mechanism in conjunction with the flamelet approach

Asymptotic Improvement of Resummation and Perturbative Predictions in Quantum Field Theory

The improvement of resummation algorithms for divergent perturbative expansions in quantum field theory by asymptotic information about perturbative coefficients is investigated. Various asymptotically optimized resummation prescriptions are considered. The improvement of perturbative predictions beyond the reexpansion of rational approximants is discussed.Comment: 21 pages, LaTeX, 3 tables; title shortened; typographical errors corrected; minor changes of style; 2 references adde

Poincare isomorphism in K-theory on manifolds with edges

The aim of this paper is to construct the Poincare isomorphism in K-theory on manifolds with edges. We show that the Poincare isomorphism can naturally be constructed in the framework of noncommutative geometry. More precisely, to a manifold with edges we assign a noncommutative algebra and construct an isomorphism between the K-group of this algebra and the K-homology group of the manifold with edges viewed as a compact topological space.Comment: 15 pages, no figure

The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects

We study the dynamics of four dimensional gauge theories with adjoint fermions for all gauge groups, both in perturbation theory and non-perturbatively, by using circle compactification with periodic boundary conditions for the fermions. There are new gauge phenomena. We show that, to all orders in perturbation theory, many gauge groups are Higgsed by the gauge holonomy around the circle to a product of both abelian and nonabelian gauge group factors. Non-perturbatively there are monopole-instantons with fermion zero modes and two types of monopole-anti-monopole molecules, called bions. One type are "magnetic bions" which carry net magnetic charge and induce a mass gap for gauge fluctuations. Another type are "neutral bions" which are magnetically neutral, and their understanding requires a generalization of multi-instanton techniques in quantum mechanics - which we refer to as the Bogomolny-Zinn-Justin (BZJ) prescription - to compactified field theory. The BZJ prescription applied to bion-anti-bion topological molecules predicts a singularity on the positive real axis of the Borel plane (i.e., a divergence from summing large orders in peturbation theory) which is of order N times closer to the origin than the leading 4-d BPST instanton-anti-instanton singularity, where N is the rank of the gauge group. The position of the bion--anti-bion singularity is thus qualitatively similar to that of the 4-d IR renormalon singularity, and we conjecture that they are continuously related as the compactification radius is changed. By making use of transseries and Ecalle's resurgence theory we argue that a non-perturbative continuum definition of a class of field theories which admit semi-classical expansions may be possible.Comment: 112 pages, 7 figures; v2: typos corrected, discussion of supersymmetric models added at the end of section 8.1, reference adde

Key Role of Polyphosphoinositides in Dynamics of Fusogenic Nuclear Membrane Vesicles

The role of phosphoinositides has been thoroughly described in many signalling and membrane trafficking events but their function as modulators of membrane structure and dynamics in membrane fusion has not been investigated. We have reconstructed models that mimic the composition of nuclear envelope precursor membranes with naturally elevated amounts of phosphoinositides. These fusogenic membranes (membrane vesicle 1(MV1) and nuclear envelope remnants (NER) are critical for the assembly of the nuclear envelope. Phospholipids, cholesterol, and polyphosphoinositides, with polyunsaturated fatty acid chains that were identified in the natural nuclear membranes by lipid mass spectrometry, have been used to reconstruct complex model membranes mimicking nuclear envelope precursor membranes. Structural and dynamic events occurring in the membrane core and at the membrane surface were monitored by solid-state deuterium and phosphorus NMR. “MV1-like” (PC∶PI∶PIP∶PIP2, 30∶20∶18∶12, mol%) membranes that exhibited high levels of PtdIns, PtdInsP and PtdInsP2 had an unusually fluid membrane core (up to 20% increase, compared to membranes with low amounts of phosphoinositides to mimic the endoplasmic reticulum). “NER-like” (PC∶CH∶PI∶PIP∶PIP2, 28∶42∶16∶7∶7, mol%) membranes containing high amounts of both cholesterol and phosphoinositides exhibited liquid-ordered phase properties, but with markedly lower rigidity (10–15% decrease). Phosphoinositides are the first lipids reported to counterbalance the ordering effect of cholesterol. At the membrane surface, phosphoinositides control the orientation dynamics of other lipids in the model membranes, while remaining unchanged themselves. This is an important finding as it provides unprecedented mechanistic insight into the role of phosphoinositides in membrane dynamics. Biological implications of our findings and a model describing the roles of fusogenic membrane vesicles are proposed