Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes

A uniform random vector over a simplex is generated. An explicit expression for the first moment of its largest spacing is derived. The result is used in a proposed diagnostic tool which examines the validity of random number generators. It is then shown that the first moment of the largest uniform spacing is related to the dependence measure of random vectors following any extreme value distribution. The main result is proved by a geometric proof as well as by a probabilistic one.     

The effect of mechanochemical treatments of sepiolite with CsCl on the calcination products

Calcination of sepiolite and of two sepiolite/CsCl mixtures, unground and air-ground was investigated by thermo-XRD-analysis. At 200 °C sepiolite, neat, mixed or air-ground with CsCl lost interparticle and zeolitic water. The framework of sepiolite persisted during the dehydration but became defected, mainly in the air-ground mixture, less in the unground mixture and little in the neat clay. At 500 °C, with the loss of bound water, the neat clay was folded and transformed into sepiolite anhydride. In sepiolite/CsCl mixtures the dehydrated variety persisted but the degree of crystal-imperfection increased in the air-ground mixture more than in the unground mixture. At 700 °C the neat clay remained crystallized, but the CsCl mixtures became amorphous. Some crystalline dehydrated sepiolite or sepiolite anhydride persisted in the unground and air-ground CsCl mixtures, respectively. At 850 °C, the neat clay crystallized into protoenstatite with some enstatite and clinoenstatite. The amorphous fraction of sepiolite in the unground sepiolite/CsCl mixtures crystallized into pollucite and forsterite and the crystalline fraction was transformed into enstatite, protoenstatite, and clinoenstatite. In the air-ground mixture, the amorphous phase was transformed into pollucite with some forsterite and the crystalline fraction into enstatite.     

Finding the Homology of Submanifolds with High Confidence from Random Samples

Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high-dimensional spaces. We consider the case where data are drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data are “noisy” and lie near rather than on the submanifold in question. The main results of this paper were first presented at a conference in honor of John Franks and Clark Robinson at Northwestern University in April 2003. These results were formally written as Technical Report No. TR-2004-08, Department of Computer Science, University of Chicago.     

Mechanics of composites with two families of finitely extensible fibers undergoing large deformations

Motivated by experimental measurements of the uniaxial response of collagen fibers, a new strain energy-density function for a solid which is finitely extensible along a preferred direction is proposed. Next, by application of a homogenization technique, the macroscopic strain energy-density function of composites with one and two families of fibers of this material embedded in a neo-Hookean matrix are determined. The resulting explicit expressions for the macroscopic strain energy-density functions are given in terms of the behaviors of the individual phases and their volume fractions. The responses of a few specific composites are investigated, revealing the lock-up surfaces of these composites. Finally, we examine the ability of the macroscopic model to capture experimentally measured stress-strain curves for the medial layer of a human abdominal aorta.     

β-Amino alcohol selectors for enantioselective separation of amino acids by ligand-exchange capillary zone electrophoresis in a low molecular weight organogel

A new family of copper ligand-exchange selectors, L- or D-β-amino alcohols, is employed for the chiral separation of D,L-dansyl-amino acids, unmodified amino acid racemates, phenylalanine and tryptophan, and β-blocker L,D-propranolol by SDS-micellar electrokinetic chromatography and by electrophoretic chromatography in a low molecular weight organogel (LMOG)-filled capillary. The LMOG comprised a self-assembled fibrillar gel of trans-(1S,2S)-1,2-bis-(dodecylamido) cyclohexane in methanol. The di-L-valinol-copper complex exhibited the best performance on LMOG-CE compared with all other β-amino alcohol-copper selectors. The dependence of chiral resolution on the pH*, the ratio between the copper and the L-valinol ligand and the concentration of added selector complex in the run buffer were investigated revealing a marked difference between the activity of the copper-valinol and the previously studied copper-valine selector. The optimal separation conditions were achieved using a 2:1 valinol/copper ratio, in accordance with the 2:1 structure of the complex, which was proven by single crystal and powder X-ray diffractions and by elemental analysis. Unlike the copper-valine selectors that could be used only under acidic conditions (pH* 3.5), the copper-valinol selectors could be used under near-neutral conditions and even at pH* 9.1. A comparison between SDS-micellar electrokinetic chromatography and LMOG-CE under otherwise identical conditions revealed a significant superior separation on the LMOG-filled capillaries.     

N-fold integer programming and nonlinear multi-transshipment

The multi-transshipment problem is NP-hard already for two commodities over bipartite networks. Nonetheless, using our recent theory of n-fold integer programming and extensions developed herein, we are able to establish the polynomial time solvability of the problem in two broad situations. First, for any fixed number of commodities and number of suppliers, we solve the problem over bipartite networks with variable number of consumers in polynomial time. This is very natural in operations research applications where few facilities serve many customers. Second, for every fixed network, we solve the problem with variable number of commodities in polynomial time.     

Stabilizing stick-slip friction

Even the most regular stick-slip frictional sliding is always stochastic, with irregularity in both the intervals between slip events and the sizes of the associated stress drops. Applying small-amplitude oscillations to the shear force, we show, experimentally and theoretically, that the stick-slip periods synchronize. We further show that this phase locking is related to the inhibition of slow rupture modes which forces a transition to fast rupture, providing a possible mechanism for observed remote triggering of earthquakes. Such manipulation of collective modes may be generally relevant to extended nonlinear systems driven near to criticality.     

Kilohertz laser frequency sensing with Brillouin mutually modulated cross-gain modulation

Using mutually modulated cross-gain modulation, Stokes optical frequency changes are converted into modulation phase changes with high sensitivity. In the slow-light transition regime, we demonstrate kilohertz sensitivity to the Stokes optical carrier frequency. The sensitivity is inversely proportional to the modulation frequency of the pump and Stokes beams.