Regularity theory for the Isaacs equation through approximation methods

In this paper, we propose an approximation method to study the regularity of solutions to the Isaacs equation. This class of problems plays a paramount role in the regularity theory for fully nonlinear elliptic equations. First, it is a model-problem of a non-convex operator. In addition, the usual mechanisms to access regularity of solutions fall short in addressing these equations. We approximate an Isaacs equation by a Bellman one, and make assumptions on the latter to recover information for the former. Our techniques produce results in Sobolev and Hölder spaces; we also examine a few consequences of our main findings.     

A Protic Ionic Liquid Catalyzes CO2 Conversion at Atmospheric Pressure and Room Temperature: Synthesis of Quinazoline‐2,4(1H,3H)‐diones

The chemical fixation of CO₂ under mild reaction conditions is of significance from a sustainable chemistry viewpoint. Herein a CO₂‐reactive protic ionic liquid (PIL), [HDBU⁺][TFE^?], was designed by neutralization of the superbase 1,8‐diazabicyclo[5.4.0]undec‐7‐ene (DBU) with a weak proton donor trifluoroethanol (TFE). As a bifunctional catalyst for simultaneously activating CO₂ and the substrate, this PIL displayed excellent performance in catalyzing the reactions of CO₂ with 2‐aminobenzonitriles at atmospheric pressure and room temperature, thus producing a series of quinazoline‐2,4(1H,3H)‐diones in excellent yields.     

Iron‐Catalyzed 1,2‐Addition of Perfluoroalkyl Iodides to Alkynes and Alkenes

Iron catalysis has been developed for the intermolecular 1,2‐addition of perfluoroalkyl iodides to alkynes and alkenes. The catalysis has a wide substrate scope and high functional‐group tolerance. A variety of perfluoroalkyl iodides including CF₃I can be employed. The resulting perfluoroalkylated alkyl and alkenyl iodides can be further functionalized by cross‐coupling reactions. This methodology provides a straightforward and streamlined access to perfluoroalkylated organic molecules.     

Synthetic strategies to 2′-hydroxy-4′-methylsulfonylacetophenone,a key compound for the preparation of flavonoid derivatives

Different strategies for the synthesis of 2′-hydroxy-4′-methylsulfonylacetophenone are reported in the present paper. This compound is considered as a key synthon for the synthesis of new flavonoid derivatives designed as potential cyclooxygenase-2 inhibitors. The retrosynthetic approach via 3′-methylsulfonylacetophenone, which included three synthetic pathways, did not allow us to obtain the expected compound. However, a synthesis from 3-mercaptophenol led to the desired acetophenone in three steps: thiophenol methylation, Friedel–Crafts acetylation and oxidation of the sulphide to the corresponding sulfone. The desired compound, 2′-hydroxy-4′-methylsulfonylacetophenone, will be used as a synthon for the preparation of novel flavonoid derivatives, such as 2′-hydroxychalcones, flavanones, flavones, and flavonols.     

Vitrification in a 2D Ising model with mobile bonds

A bond-disordered two-dimensional Ising model is used to simulate Kauzmann's mechanism of vitrification in liquids, by a Glauber Monte Carlo simulation. The rearrangement of configurations is achieved by allowing impurity bonds to hop to nearest neighbors at the same rate as the spins flip. For slow cooling, the theoretical minimum energy configuration is approached, characterized by an amorphous distribution of locally optimally arranged impurity bonds. Rapid cooling to low temperatures regularly finds bond configurations of higher energy, which are both a priori rare and severely restrictive to spin movement, providing a simple realization of kinetic vitrification. A supercooled liquid regime is also found, and characterized by a change in sign of the field derivative of the spin-glass susceptibility at a finite temperature. Received 3 August 2000 and Received in final form 9 March 2001     

Statistical physics analysis of the computational complexity of solving random satisfiability problems using backtrack algorithms

The computational complexity of solving random 3-Satisfiability (3-SAT) problems is investigated using statistical physics concepts and techniques related to phase transitions, growth processes and (real-space) renormalization flows. 3-SAT is a representative example of hard computational tasks; it consists in knowing whether a set of αN randomly drawn logical constraints involving N Boolean variables can be satisfied altogether or not. Widely used solving procedures, as the Davis-Putnam-Loveland-Logemann (DPLL) algorithm, perform a systematic search for a solution, through a sequence of trials and errors represented by a search tree. The size of the search tree accounts for the computational complexity, i.e. the amount of computational efforts, required to achieve resolution. In the present study, we identify, using theory and numerical experiments, easy (size of the search tree scaling polynomially with N) and hard (exponential scaling) regimes as a function of the ratio α of constraints per variable. The typical complexity is explicitly calculated in the different regimes, in very good agreement with numerical simulations. Our theoretical approach is based on the analysis of the growth of the branches in the search tree under the operation of DPLL. On each branch, the initial 3-SAT problem is dynamically turned into a more generic 2+p-SAT problem, where p and 1 - p are the fractions of constraints involving three and two variables respectively. The growth of each branch is monitored by the dynamical evolution of α and p and is represented by a trajectory in the static phase diagram of the random 2+p-SAT problem. Depending on whether or not the trajectories cross the boundary between satisfiable and unsatisfiable phases, single branches or full trees are generated by DPLL, resulting in easy or hard resolutions. Our picture for the origin of complexity can be applied to other computational problems solved by branch and bound algorithms. Received 10 March 2001     

Cluster-embedding method to simulate large cluster and surface problems

In order to describe processes which are localized on a surface or inside the bulk of a solid, molecular calculations of an inner cluster may be adequate as long as the effect of the outer environment is taken into account via an embedding-method. Using a relativistic density functional method for the self-consistent cluster calculation we have developed a new cluster-embedding scheme here. As an example we have studied the adsorption of Al on an Al(100) surface and we get significant agreement with different methods. This indicates that this embedding-method is reliable enough to simulate an unlimited solid. Received 29 November 2000     

Material characterization of a nanocrystal based photovoltaic device

Nanocomposites have shown promise as the active layer for photovoltaic energy conversion. One example is the CdSe nanocrystal \polymer composite demonstrated by Hyunh and Greenham [#!Ref1!#,#!Ref2!#]. In this paper we investigate the baseline properties of the materials used in such a device. We present surface chemical information for CdSe nanocrystals and chemical analysis for poly-(3-hexylthiophene) (P3HT) polymer. Received 30 November 2000     

Electronic structures in coupled two quantum dots by 3D-mesh Hartree-Fock-Kohn-Sham calculation

To study the electronic structures of quantum dots in the framework of self-interaction-free including three dimensional effects, we adopt the theory of nonlocal effective potential introduced by Kohn and Sham [#!ks65!#]. For utilizing the advantageous point of the real space (3D) mesh method to solve the original nonlinear and nonlocal Hartree-Fock-Kohn-Sham (HFKS)-equation, we introduce a linearization of the equation in the local form by introducing the local Coulomb potentials which depend on explicitly the two single particle states. In practice, for solving the local form HFKS-equation, we use the Car-Parrinello-like relaxation method and the Coulomb potentials are obtained by solving the Poisson equation under proper boundary conditions. Firstly the observed energy gap between triplet- and singlet-states of N = 4 in DBS [#!tarucha96!#] is discussed to reproduce the addition energies and chemical potentials depending the magnetic field. Next the coupling between two-quantum dots in TBS [#!aht97!#] is studied by adding the square barrier between two dots. The spin-degeneracy [#!aht97!#] measured in gate-voltage depending on magnetic field is well reproduced in the limit of small mismatch. Finally, the electronic states in the ring structure are calculated and discussed how the ring size and magnetic field affect to the structures. Received 30 November 2000     

Adjoint Equation-Based Methods for Control Problems in Incompressible, Viscous Flows

A review of adjoint equation-based methodologies for viscous,incompressible flow control and optimization problems is given and illustrated by a drag minimization example. A number of approaches to ameliorating the high storage and CPU costs associated with straightforward implementations of adjoint equation based methodologies are discussed. Other issues, including the relative merits of the differentiate-then-discretize and discretize-then-differentiate approaches to deriving discrete adjoint equations, the incorporation of side constraints into adjoint equation-based methodologies, and inaccuracies that occur due to differentiations at the boundary, are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.