Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes

We consider the following eigenvalue optimization problem: Given a bounded domain $\Omega\subset\R^n$ and numbers $\alpha\geq 0$, $A\in [0,|\Omega|]$, find a subset $D\subset\Omega$ of area $A$ for which the first Dirichlet eigenvalue of the operator $-\Delta + \alpha \chi_D$ is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some symmetric domains (thin annuli and dumbbells with narrow handle) optimal solutions must possess fewer symmetries than $\Omega$; on the other hand, for convex $\Omega$ reflection symmetries are preserved. Also, we present numerical results and formulate some conjectures suggested by them.Comment: 24 pages; 3 figures (as separate files); (shortened previous version); to appear in Comm. Math. Phy

Inertial manifolds for Burgers' original model system of turbulence

AbstractThe existence of inertial manifolds for Burgers' original mathematical model system of turbulence is investigated. The system consists of two equations and enjoys the characteristic quantity: the Reynolds number. Our object in this article is to express the existence in terms of this Reynolds number. The difficulty of first order derivatives is circumvented by the method originally due to M. Kwak

Studying Diquark Structure of Heavy Baryons in Relativistic Heavy Ion Collisions

We propose the enhancement of $\Lambda_c$ yield in heavy ion collisions at RHIC and LHC as a novel signal for the existence of diquarks in the strongly coupled quark-gluon plasma produced in these collisions as well as in the $\Lambda_c$. Assuming that stable bound diquarks can exist in the quark-gluon plasma, we argue that the yield of $\Lambda_c$ would be increased by two-body collisions between $ud$ diquarks and $c$ quarks, in addition to normal three-body collisions among $u$, $d$ and $c$ quarks. A quantitative study of this effect based on the coalescence model shows that including the contribution of diquarks to $\Lambda_c$ production indeed leads to a substantial enhancement of the $\Lambda_c/D$ ratio in heavy ion collisions.Comment: Prepared for Chiral Symmetry in Hadron and Nuclear Physics (Chiral07), Nov. 13-16, 2007, Osaka, Japa

Explosions inside Ejecta and Most Luminous Supernovae

The extremely luminous supernova SN2006gy is explained in the same way as other SNIIn events: light is produced by a radiative shock propagating in a dense circumstellar envelope formed by a previous weak explosion. The problems in the theory and observations of multiple-explosion SNe IIn are briefly reviewed.Comment: 9 pages, 6 figures, LateX aipproc.cls. A bit more details and color added to Fig.3. The 10th International Symposium on Origin of Matter and Evolution of Galaxies (OMEG07), Sapporo, Japan, December 200

Complex Intergrowths of Non-Stoichiometric Defect-Structured Hibonite and Al-Rich Spinel in an Allende Ca-Al-Rich Inclusion

Hibonite is a primary ultra-refractory mineral occurring in many Ca-Al-rich inclusions (CAIs) and is predicted to condense as the second major phase from a cooling gas of solar composition. Our previous microstructural studies of hibonite in carbonaceous chondrites revealed its unique microstructures consisting of numerous defects that contain Mg-enriched, wider spinel blocks in stoichiometric hibonite. Han, J. et al. [2 items] demonstrated experimentally that defect-structured hibonites can grow easily in the presence of minor Mg and are kinetically more stable than equilibrium assemblages predicted by thermodynamic calculations. However, a thermo-dynamic vs. structural stability of defect-structured hibonite relative to other early-condensed Al-rich phases such as corundum, grossite, and spinel remains poorly constrained. Here, we present the results of atomic resolution TEM (Transmission Electron Microscopy) imaging of hibonite in a compact Type A CAI in the Allende CV3 chondrite in order to better understand the crystal structure and chemistry of defect-structured hibonite and its associated Al-rich phases, especially non-stoichiometric, Al-rich spinel, in the context of the formation of first refractory solids in the early solar nebula

A Mathematical Study of the One-Dimensional Keller and Rubinov Model for Liesegang Bands

Our purpose is to start understanding from a mathematical viewpoint experiments in which regularized structures with spatially distinct bands or rings of precipitated material are exhibited, with clearly visible scaling properties. Such patterns are known as Liesegang bands or rings. In this paper, we study a one-dimensional version of the Keller and Rubinow model and present conditions ensuring the existence of Liesegang bands

Pathophysiology of acute experimental pancreatitis: Lessons from genetically engineered animal models and new molecular approaches

The incidence of acute pancreatitis is growing and worldwide population-based studies report a doubling or tripling since the 1970s. 25% of acute pancreatitis are severe and associated with histological changes of necrotizing pancreatitis. There is still no specific medical treatment for acute pancreatitis. The average mortality resides around 10%. In order to develop new specific medical treatment strategies for acute pancreatitis, a better understanding of the pathophysiology during the onset of acute pancreatitis is necessary. Since it is difficult to study the early acinar events in human pancreatitis, several animal models of acute pancreatitis have been developed. By this, it is hoped that clues into human pathophysiology become possible. In the last decade, while employing molecular biology techniques, a major progress has been made. The genome of the mouse was recently sequenced. Various strategies are possible to prove a causal effect of a single gene or protein, using either gain-of-function (i.e., overexpression of the protein of interest) or loss-of-function studies (i.e., genetic deletion of the gene of interest). The availability of transgenic mouse models and gene deletion studies has clearly increased our knowledge about the pathophysiology of acute pancreatitis and enables us to study and confirm in vitro findings in animal models. In addition, transgenic models with specific genetic deletion or overexpression of genes help in understanding the role of one specific protein in a cascade of inflammatory processes such as pancreatitis where different proteins interact and co-react. This review summarizes the recent progress in this field. Copyright (c) 2005 S. Karger AG, Basel