Nonlocal fractional elliptic equations and applications

The L_p-coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in L_p space and also is a generator of an analytic semigroup. Moreover, by using the L_p-separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established.     

Bounding the consistency strength of a five element linear basis

In [13] it was demonstrated that the Proper Forcing Axiom implies that there is a five element basis for the class of uncountable linear orders. The assumptions needed in the proof have consistency strength of at least infinitely many Woodin cardinals. In this paper we reduce the upper bound on the consistency strength of such a basis to something less than a Mahlo cardinal, a hypothesis which can hold in the constructible universe L. A crucial notion in the proof is the saturation of an Aronszajn tree, a statement which may be of broader interest. We show that if all Aronszajn trees are saturated and PFA(ω ₁) holds, then there is a five element basis for the uncountable linear orders. We show that PFA(ω ₂) implies that all Aronszajn trees are saturated and that it is consistent to have PFA(ω ₁) plus every Aronszajn tree is saturated relative to the consistency of a reflecting Mahlo cardinal. Finally we show that a hypothesis weaker than the existence of a Mahlo cardinal is sufficient to force the existence of a five element basis for the uncountable linear orders. The first author acknowledges a fellowship granted by the French ministry of research. The research of the second author was partially supported by the Centre de Rercerca Matemàtica of the Universitat Autònoma de Barcelona, and by NSF Grant DMS-0401603. The second author would also like to thank the third and fourth authors for bringing him to Boise and Paris respectively for further discussions. The third author was supported by NSF grants DMS-0401893 and DMS-0200671. The second and fourth authors would like to thank CIRM in Luminy for hosting them during a petit group de travaille, and to thank the others participants, Ralf Schindler and Ernest Schimmerling, for discussions on this topic.     

Anisotropic elliptic equations with VMO coefficients

This paper presents the study of maximal regularity properties for anisotropic differential-operator equations with VMO (vanishing mean oscillation) coefficients. We prove that the corresponding differential operator is separable and is a generator of analytic semigroup in vector-valued L^p spaces. Moreover, discreetness of spectrum and completeness of root elements of this operator is obtained.     

Separable anisotropic elliptic operators and applications

This paper focuses on boundary value problems for anisotropic differential-operator equations of high order with variable coefficients in the half plane. Several conditions are obtained which guarantee the maximal regularity of anisotropic elliptic and parabolic problems in Banach-valued L _p -spaces. Especially, it is shown that this differential operator is R-positive and is a generator of an analytic semigroup. These results are also applied to infinite systems of anisotropic type partial differential equations in the half plane.     

Dual-polarization interferometry for quantification of small molecules using aptamers

An interferometry-based method was developed for detection of a small molecule, argininamide. The quantification of argininamide was demonstrated using aptamers immobilized on silicone oxynitride sensor surface via avidin–biotin binding. The aptamers formed a thin film over avidin layer corresponding to a thickness of 1.2 nm, consistent with a molecular positioning of multipoint attachment to the surface. The binding of argininamide did not cause any significant changes in the thickness of the aptamer film, suggesting that the specific binding did not affect the overall conformation of the aptamer molecules after adaptive rearrangement of aptamer molecules. However, the binding results in clearly detectable changes in mass calculated from multiple parameters determined by mass deposition and structural changes. The limit of detection of the developed sensor was determined to be 5 μM. The sensor can monitor real-time changes in argininamide concentrations with high reliability and sensitivity. The model system demonstrated that a combined measurement considering structural and mass changes through interferometry-based techniques can overcome one of the major problems associated with real-time monitoring of small mass analytes.     

Synthesis,spectroscopic characterization and EPR studies on electron transfer reactions of bis[N-(2,5-di-tert-butylphenyl)salicylaldiminato]copper complexes with PPh3

New bidentate N-(2,5-di-tert-butylphenyl)salicylaldimines bearing X = H, HO, CH3O, Br, NO2, 3,5-di-Br, 3-NO2-5-Br and 5,6-benzo substituents on the salicylaldehyde moiety, LxH, and their mononuclear bis[N-(2,5-di-tert-butylphenyl)salicylaldiminato]copper(II) complexes, Cu(Lx)2, have been prepared and investigated by IR, UV-Visible, 1H NMR, ESR spectroscopy, magnetic measurements, as well as reactions of Cu(LX)2 with PPh3 were studied. It has been found that some complexes with X = HO and CH3O unlike their electron-withdrawing and unsubstituted analogues are readily reduced by PPh3 via intramolecular electron transfer from ligand to copper(II) to give Cu-stabilized radical intermediates. The spectra of the primary radicals interpreted in terms of couplings of unpaired electron with (63,65)Cu, 31P, 14N nuclei and aromatic protons.     

A Fast Robust Algorithm for Computing Discrete Voronoi Diagrams

We describe an algorithm for the construction of discretized Voronoi diagrams on a CPU within the context of a large scale numerical fluid simulation. The Discrete Voronoi Chain (DVC) algorithm is fast, flexible and robust. The algorithm stores the Voronoi diagram on a grid or lattice that may be structured or unstructured. The Voronoi diagram is computed by alternatively updating two lists of grid cells per particle to propagate a growth boundary of cells from the particle locations. Distance tests only occur when growth boundaries from different particles collide with each other, hence the number of distance tests is effectively minimized. We give some empirical results for two and three dimensions. The method generalizes to any dimension in a straight forward manner. The distance tests can be based any metric.