Asymptotics of eigenvalue clusters for Schrödinger operators on the Sierpinski gasket

In this note we investigate the asymptotic behavior of spectra of Schrödinger operators with continuous potential on the Sierpinski gasket . In particular, using the existence of localized eigenfunctions for the Laplacian on we show that the eigenvalues of the Schrödinger operator break into clusters around certain eigenvalues of the Laplacian. Moreover, we prove that the characteristic measure of these clusters converges to a measure. Results similar to ours were first observed by A. Weinstein and V. Guillemin for Schrödinger operators on compact Riemannian manifolds.

     

Endpoints of set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and strict contractions in uniform spaces

In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones.     

Comparison of Boolean algebras

In this work, some results related to superatomic Boolean interval algebras are presented, and proved in a topological way. Let x be an uncountable cardinal. To each I x, we can associate a superatomic interval Boolean algebra B _I of cardinality x in such a way that the following properties are equivalent: (i) I I x, (ii) B _I is a quotient algebra of B _J, and (iii) there is an homomorphism f from B _J into B _I such that for every atom b of B _I, there is an atom a of B _J satisfying f(a)=b. As a corollary, there are 2 ^x isomorphism types of superatomic interval Boolean algebras of cardinality x. This case is quite different from the countable one.     

Generalization of Strang's Preconditioner with Applications to Toeplitz Least Squares Problems

In this paper, we propose a method to generalize Strang's circulant preconditioner for arbitrary n-by-n matrices A_n. The th column of our circulant preconditioner Sn is equal to the th column of the given matrix A_n. Thus if A_n is a square Toeplitz matrix, then Sn is just the Strang circulant preconditioner. When Sn is not Hermitian, our circulant preconditioner can be defined as . This construction is similar to the forward-backward projection method used in constructing preconditioners for tomographic inversion problems in medical imaging. We show that if the matrix A_n has decaying coefficients away from the main diagonal, then is a good preconditioner for An. Comparisons of our preconditioner with other circulant-based preconditioners are carried out for some 1-D Toeplitz least squares problems: min ∥ b - Ax∥₂. Preliminary numerical results show that our preconditioner performs quite well, in comparison to other circulant preconditioners. Promising test results are also reported for a 2-D deconvolution problem arising in ground-based atmospheric imaging.     

Simulation of the inherent turbulence and wake interaction inside an infinitely long row of wind turbines

The turbulence in the interior of a wind farm is simulated using large eddy simulation and the actuator line technique implemented in the Navier–Stokes equations. The simulations are carried out for an infinitely long row of turbines simulated by applying cyclic boundary conditions at the inlet and outlet. The simulations investigate the turbulence inherent to the wind turbines as no ambient turbulence or shear is added to this idealised case. The simulated data give insight into the performance of the wind turbines operating in the wake of others as well as details on key turbulent quantities. One of the key features of wakes behind wind turbines is the dynamic wake meandering, which is shown to be related to the wind turbine spacing and the vortex shedding from the turbine as a bluff body. The flow is analysed and reconstructed by applying proper orthogonal decomposition.     

Sintering of a bcc structure of spherical particles of equal and different sizes

Solid-state sintering of a bcc structure of spherical particles has been studied numerically by use of simple shape parameters to describe the state of the unit cell. Both free and pressure-assisted sintering of particles of equal and different sizes for various ratios of boundary and surface diffusion have been considered. At low relative densities, the evolution of particle radius, interparticle junction radius and shrinkage predicted by the bcc model are rather similar to sintering of a simple row of particles. At higher densities, the porosity closes up; that is, the junctions start to interact, or the next-nearest neighbours in the particle structure attain contact. Quantative measures of the density at the transition from open pore space between the particles to closed porosity as well as the density at which the next-nearest neighbours start to touch are derived. Furthermore, different dihedral angles and the evolution of relative density and sintering stresses are studied.     

A solution to Schrödinger's problem of non-linear integral equations

A solution of Schrödinger's system of non-linear integral equations determines the rate function of a large deviation principle for kernels with prescribed marginal distributions. This kind of large deviation principle has some meaning in quantum mechanics.Diffusion equations associated with Schrödinger equations have typically transition functions with singular creation and killing. Hence they provide measurable non-negative generally unbounded kernels which may vanish on sets with positive measure and which can possess infinite mass.For Schrödinger systems with such kernels, a solution is proved to exist uniquely in terms of a product measure. It is obtained from a variational principle for the local adjoint of a product measure endomorphism. The generally unbounded factors of the solution are characterized by integrability properties.     

Implications of population change on social insurance systems providing old-age benefits

General agreement, although not completely universal, exists in North America that cost analysis of programs providing old-age retirement benefits should be conducted over the long range, and not merely over the next few years. The actuarial cost estimates for what is now the U.S. Old-Age, Survivors, and Disability Insurance program have always been made over a long period of future years. Initially, for the 1935 Act, this period was 44 years (up through 1980), but most subsequent valuations of the program have used a 75-year period. The justification for this length of time is that it covers essentially the entire lifetime of every person in covered employment on the valuation date.It must be recognized that outside of North America — but with the exception of the United Kingdom — long-range social insurance programs providing retirement benefits are not generally considered over long future periods. The argument made in support of this short-sighted approach is that it is impossible to predict with any precision such long-range future operations. Countering this is the point that an estimate prepared by well-qualified persons is far better for policy planning and analysis than nothing at all.     

Asymptotic distribution of the increase of the largest canonical correlation when one of the vectors is augmented

Let r₁ > r₂ > … be the sample canonical correlations in a sample of size n from a multivariate normal population partitioned into two subvectors with population canonical correlations ₁ > ₂ > …. Let one of the subvectors be augmented by adding one or more variables to it. For the increase in the largest canonical correlation, Δr in the sample and Δ in the population, it is shown that √n(Δr − Δ) → ^DN(0, σ²) and a formula for σ² is derived.