On the isoperimetric inequality for the buckling of a clamped plate

In two dimensions, we prove the existence of a domain minimizing the buckling load of a clamped plate among all domains of given measure. In order to prove that the optimal set is the disk, we discuss possible ways of applying an idea of Willms and Weinberger if the regularity of the optimal set is not a priori assumed.     

Optimal bounds for ratios of eigenvalues of one-dimensional Schrödinger operators with Dirichlet boundary conditions and positive potentials

Consider the Schrödinger equation –u+V(x)u=u on the intervalI, whereV(x)0 forxI and where Dirichlet boundary conditions are imposed at the endpoints ofI. We prove the optimal bound

Sharp Upper Bound to the First Nonzero Neumann Eigenvalue for Bounded Domains in Spaces of Constant Curvature

The main result of this paper is that for a domain containedin a hemisphere of the n-dimensional sphere Sⁿ the first nonzeroNeumann eigenvalue µ₁() is less than or equal to the firstnonzero Neumann eigenvalue µ₁(D) where D is a geodesicball in Sⁿ of the same measure as . Equality occurs if and onlyif is isometric to D. This result generalizes old results ofSzegö and Weinberger which gave the corresponding upperbound for µ₁() in the Euclidean case, and a result ofChavel for domains in Sⁿ which restricted to lie in a geodesicball of radius when n = 2and to even smaller geodesic balls for larger n. The techniquesused are analogous to those for our recent proof of the Payne-Pólya-Weinbergerconjecture: rearrangement inequalities and properties of specialfunctions are the key elements. The general approach is a directextension of Weinberger's for domains in Rⁿ.     

The S.M.A.R.T. (small mass,affordable, rapid,transfer-less) digestion method for heavy metal determinations

The S.M.A.R.T. (small mass, affordable, rapid, transfer-less) digestion method was developed to determine heavy metal concentrations in small sample masses. The S.M.A.R.T. digestion method is a hot water bath digestion where sample digestion and dilution are performed in the original sample tube. This method is faster than the typical methods used and reduces potential sources of error. Masses as small as 0.01 g have been digested and analysed using this method. The preparation and digestion time is reduced from 10 h to less than 4 h. Acid volumes are reduced from millilitres to microlitres and the only disposable supplies needed are sample tubes and pipette tips. Method accuracy was determined by digesting seven replicates of two standard reference materials using the S.M.A.R.T. method and analysing samples by inductively coupled plasma mass spectrometry. The S.M.A.R.T. digestion method was found to provide excellent recoveries for Al (76 ± 2.7%), Mn (99 ± 11%), Co (92 ± 17%), Ni (93 ± 28%), Cu (109 ± 33%), Zn (97 ± 7.1%), As (108 ± 20%), Sr (90 ± 12%), Mo (84 ± 23%), Ag (91 ± 1.8%), Cd (95 ± 6.2%), Sn (139 ± 52%) and Pb (95 ± 22%). This study has successfully developed an efficient and reproducible digestion method for heavy metal determination in limited biomass samples.     

Spectral theory for perturbed Krein Laplacians in nonsmooth domains

We study spectral properties for HK,Ω, the Krein-von Neumann extension of the perturbed Laplacian −Δ+V defined on , where V is measurable, bounded and nonnegative, in a bounded open set Ω⊂Rn belonging to a class of nonsmooth domains which contains all convex domains, along with all domains of class C1,r, r>1/2. In particular, in the aforementioned context we establish the Weyl asymptotic formula

     

Contrasting nonaqueous against aqueous solvation on the basis of scaled-particle theory

Normal hexane is adopted as a typical organic solvent for comparison with liquid water in modern theories of hydrophobic hydration, and detailed results are worked-out here for the C-atom density in contact with a hard-sphere solute, rhoCG(R), for the full range of solute radii. The intramolecular structure of an n-hexane molecule introduces qualitative changes in G(R) compared to scaled-particle models for liquid water. Also worked-out is a revised scaled-particle model implemented with molecular simulation results for liquid n-hexane. The classic scaled-particle model, acknowledging the intramolecular structure of an n-hexane molecule, is in qualitative agreement with the revised scaled-particle model results, and is consistent in sizing the methyl/methylene sites which compose n-hexane in the simulation model. The classic and revised scaled-particle models disagree for length scales greater than the radius of a methyl group, however. The liquid-vapor surface tension of n-hexane predicted by the classic scaled-particle model is too large, though the temperature variation is reasonable; this contrasts with the classic scaled-particle theory for water which predicts a reasonable magnitude of the water liquid-vapor surface tension, but an incorrect sign for the temperature derivative at moderate temperatures. Judging on the basis of the arbitrary condition that drying is indicated when G(R)<1, hard spheres dry at smaller sizes in n-hexane than in liquid water.     

A weight-function approach for determining watershed initial conditions for combustion waves

In many generic combustion models, one finds that a combustionwave will develop with a specific wave speed. However, thereare possible initial temperature profiles which do not evolveinto such waves, but rather die out to the ambient temperature.There can exist, in some models, a clear distinction betweenthose initial conditions that do evolve into combustion wavesand those that do not; this is sometimes referred to as thewatershed initial condition. When fuel consumption is consideredto be negligible, analytical methods can be used to obtain theexact watershed. In this paper, we consider the problem of determiningpseudo-watersheds and ascertaining the relationship betweenthese pseudo-watersheds and the exact watersheds. In the processa novel weight-function approach for infinite spatial domainsis developed.