Almost‐spanning universality in random graphs

A graph G is said to be ‐universal if it contains every graph on at most n vertices with maximum degree at most Δ. It is known that for any and any natural number Δ there exists such that the random graph G(n, p) is asymptotically almost surely ‐universal for . Bypassing this natural boundary, we show that for the same conclusion holds when . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 380–393, 2017     

Semi-classical limit theorems for Hartree-Fock theory

Consider a large number of electrons with Coulomb repulsion moving under the influence of static nuclei. It is assumed the potentials due to the nuclei are Coulombic away from their centers but are smooth at the centers, so no singularities exist. The author shows that the exchange energy for the Hartree-Fock ground state of this system converges in a suitable limit to the formula obtained by Dirac for exchange energy as an integral of the one body density.Research supported by NSF Grant No. MCS8100761     

Cosmological natural selection and the purpose of the universe

The cosmological natural selection (CNS) hypothesis holds that the fundamental constants of nature have been fine‐tuned by an evolutionary process in which universes produce daughter universes via the formation of black holes. Here, we formulate the CNS hypothesis using standard mathematical tools of evolutionary biology. Specifically, we capture the dynamics of CNS using Price's equation, and we capture the adaptive purpose of the universe using an optimization program. We establish mathematical correspondences between the dynamics and optimization formalisms, confirming that CNS acts according to a formal design objective, with successive generations of universes appearing designed to produce black holes. © 2013 Wiley Periodicals, Inc. Complexity 18: 48–56, 2013     

The Coulomb Gas at low temperature and low density

We study the quantum Coulomb Gas ofN particles with HamiltonianH at low temperature and negative values of the chemical potential. If is sufficiently negative the Coulomb gas is approximately a perfect rare gas of charged particles, as expected. The interesting fact is that for higher (but still negative) values of the gas changes to a rare gas of some atom or molecule (which is most likely neutral). The type of molecule is determined by the ground state of the HamiltonianH —N with center of mass motion removed.Dedicated to Roland DobrushinWork partially supported by U.S. National Science Foundation grant DMS 8600748 (J.C.), PHY 85-15288-(A03) (E.L.) and DMS-8601978 and DMS-8806731 (H.-T.Y.)     

Determination of the relative and absolute configuration of the dimethylmyristoyl side chain of pneumocandin B0 by asymmetric synthesis

[reaction: see text] The relative and absolute configuration of the pneumocandin B(0) side chain has been established as (10R,12S)-dimethylmyristoyl by the stereocontrolled synthesis of both antipodes of the side chain acid and their comparison to a sample derived from the natural product.     

TheN 7/5 law for charged bosons

Non-relativistic bosons interacting with Coulomb forces are unstable, as Dyson showed 20 years ago, in the sense that the ground state energy satisfiesE ₀–AN ^7/5. We prove that 7/5 is the correct power by proving thatE ₀–BN ^7/5. For the non-relativistic bosonic, one-component jellium problem, Foldy and Girardeau showed thatE ₀–CN^1/4. This 1/4 law is also validated here by showing thatE ₀–DN^1/4. These bounds prove that the Bogoliubov type paired wave function correctly predicts the order of magnitude of the energy.Work partially supported by U.S. National Science Foundation grant DMS 8600748Work partially supported by U.S. National Science Foundation grant PHY85-15288-A01Work supported by Alfred Sloan Foundation dissertation fellowship     

An upper bound on the critical temperature for a continuous system with short-range interaction

A classical gas with short-range interaction in the grand canonical ensemble is studied. Ifp(, z) denotes the thermodynamic pressure at inverse temperature and activityz, then it follows from the Mayer expansion thatp(, z) is infinitely differentiable provided andz are sufficiently small. Here it is shown that there exists ₀>0 such thatp(, z) is infinitely differentiable if< ₀ andz>0. One can interpret this result as saying that ( ₀)^–1 is an upper bound on the critical temperature for the system.     

Predicting the vibroacoustic response of satellite equipment panels

Modern satellites are constructed of large, lightweight equipment panels that are strongly excited by acoustic pressures during launch. During design, performing vibroacoustic analyses to evaluate and ensure the integrity of the complex electronics mounted on the panels is critical. In this study the attached equipment is explicitly addressed and how its properties affect the panel responses is characterized. FEA and BEA methods are used to derive realistic parameters to input to a SEA hybrid model of a panel with multiple attachments. Specifically, conductance/modal density and radiation efficiency for nonhomogeneous panel structures with and without mass loading are computed. The validity of using the spatially averaged conductance of panels with irregular features for deriving the structure modal density is demonstrated. Maidanik's proposed method of modifying the traditional SEA input power is implemented, illustrating the importance of accounting for system internal couplings when calculating the external input power. The predictions using the SEA hybrid model agree with the measured data trends, and are found to be most sensitive to the assumed dynamic mass ratio (attachments/structure) and the attachment internal loss factor. Additional experimental and analytical investigations are recommended to better characterize dynamic masses, modal densities and loss factors.     

Random Walks Associated with Non-Divergence Form Elliptic Equations

This paper is concerned with the study of the diffusion process associated with a nondivergence form elliptic operator in d dimensions, d2. The authors introduce a new technique for studying the diffusion, based on the observation that the probability of escape from a d–1 dimensional hyperplane can be explicitly calculated. They use the method to estimate the probability of escape from d–1 dimensional manifolds which are C ^1, , and also d–1 dimensional Lipschitz manifolds. To implement their method the authors study various random walks induced by the diffusion process, and compare them to the corresponding walks induced by Brownian motion.