317 research outputs found
A sausage body is a unique solution for a reverse isoperimetric problem
We consider the class of -concave bodies in ; that
is, convex bodies with the property that each of their boundary points supports
a tangent ball of radius that lies locally (around the boundary
point) inside the body. In this class we solve a reverse isoperimetric problem:
we show that the convex hull of two balls of radius (a sausage
body) is a unique volume minimizer among all -concave bodies of given
surface area. This is in a surprising contrast to the standard isoperimetric
problem for which, as it is well-known, the unique maximizer is a ball. We
solve the reverse isoperimetric problem by proving a reverse quermassintegral
inequality, the second main result of this paper.Comment: 1 figur
Small ball probability for the condition number of random matrices
Let be an random matrix with i.i.d. entries of zero mean,
unit variance and a bounded subgaussian moment. We show that the condition
number satisfies the small ball probability estimate
where may only depend on the subgaussian moment.
Although the estimate can be obtained as a combination of known results and
techniques, it was not noticed in the literature before. As a key step of the
proof, we apply estimates for the singular values of , obtained (under some additional assumptions) by Nguyen.Comment: Some changes according to the Referee's comment
The Thermodynamics of Fluid-Phase Benzene via Molecular Simulation
Accurate values for thermodynamic properties throughout the fluid phase are a requirement for the design of separation processes. To date, very few pure substances have been completely characterized because of time and monetary constraints. Low cost computing power now permits complete determination of the thermodynamic properties of pure substances via molecular simulation. Molecular simulation is computational statistical mechanics. Benzene is an important industrial chemical and pharmaceutical precursor. It is the prototypical, symmetric, hexagonal molecule and is an ideal candidate for molecular simulation. The molecular models of three researchers in the field are submitted for Monte Carlo simulation in the virtual laboratories at All claim that their models best represent real benzene. The MC code used for experimentation measures 12 thermodynamic properties with associated errors, and derivatives of the residual Helmholtz energy with respect to density and temperature to order 4. The thermodynamic properties are used to generate a multiparameter fundamental equation of state that represents the model throughout the fluid phase. Thermodynamic properties from the three models are compared to the values from the Goodwin equation of state for benzene. A single model is chosen as the best representative of real benzen
The Thermodynamics of Fluid-Phase Benzene via Molecular Simulation
Accurate values for thermodynamic properties throughout the fluid phase are a requirement for the design of separation processes. To date, very few pure substances have been completely characterized because of time and monetary constraints. Low cost computing power now permits complete determination of the thermodynamic properties of pure substances via molecular simulation. Molecular simulation is computational statistical mechanics. Benzene is an important industrial chemical and pharmaceutical precursor. It is the prototypical, symmetric, hexagonal molecule and is an ideal candidate for molecular simulation. The molecular models of three researchers in the field are submitted for Monte Carlo simulation in the virtual laboratories at All claim that their models best represent real benzene. The MC code used for experimentation measures 12 thermodynamic properties with associated errors, and derivatives of the residual Helmholtz energy with respect to density and temperature to order 4. The thermodynamic properties are used to generate a multiparameter fundamental equation of state that represents the model throughout the fluid phase. Thermodynamic properties from the three models are compared to the values from the Goodwin equation of state for benzene. A single model is chosen as the best representative of real benzen
The production, properties and applications of the zinc imidazolate, ZIF-8.
Venna, Carreon, and Jasinski produced and characterized the first samples of the zinc imidazolate framework ZIF-8 at the University of Louisville in 2010. In this dissertation the production, properties, and applications of this unique metal-organic framework are explored. Previously, only minute laboratory amounts (1/4 gram), of ZIF-8 were produced via time-consuming and expensive processes. Production quantities have been synthesized via both a continuous and a batch process using a spray drying operation to effect separation of the solid product (ZIF-8) from the mother liquor. Approximately 85% of the mother liquor (methanol), can be recovered from the spray dryer resulting in magnitude-of-order savings in time and money. Before any engineering applications could be suggested it was necessary to quantify important physical properties of ZIF-8 not currently available. The density, thermal conductivity, specific heat, and BET surface area were measured via strict ASTM procedures and reported. It was hoped that the massive surface area of ZIF-8 (~ 1300 m2/g), would effect enhanced heat transfer in engineering applications. The Heat Transfer Laboratories at the University of Louisville, served as the testing site for the use of the microparticle ZIF-8 as an agent for enhanced heat transfer when mixed in small vol% in synthetic oil. Unfortunately ZIF-8 delivered no such enhancement
-Steiner quermassintegrals
Inspired by an Steiner formula for the affine surface area proved
by Tatarko and Werner, we define, in analogy to the classical Steiner formula,
-Steiner quermassintegrals. Special cases include the classical mixed
volumes, the dual mixed volumes, the affine surface areas and the mixed
affine surface areas. We investigate the properties of the -Steiner
quermassintegrals. In particular, we show that they are rotation and reflection
invariant valuations on the set of convex bodies with a certain degree of
homogeneity. Such valuations seem new and not have been observed before
Random polytopes obtained by matrices with heavy tailed entries
Let be an random matrix with independent entries and
such that in each row entries are i.i.d. Assume also that the entries are
symmetric, have unit variances, and satisfy a small ball probabilistic estimate
uniformly. We investigate properties of the corresponding random polytope
in (the absolute convex hull of rows of
). In particular, we show that where depends only on parameters in small ball inequality. This
extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion
is equivalent to so-called -quotient property and plays an important
role in compressive sensing (see \cite{KKR} and references therein).Comment: Last version, to appear in Communications in Contemporary Mathematic
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