Processes of flats induced by higher dimensional processes III

Stationary processes of k-flats in ^d can be thought of as point processes on the Grassmannian _k ^d of k-dimensional subspaces of ^d . If such a process is sampled by a (d–k+ j)-dimensional space F, it induces a process of j-flats in F. In this work we will investigate the possibility of determining the original k-process from knowledge of the intensity measures of the induced j-processes. We will see that this is impossible precisely when 1<k<d–1 and j=0,...,2[r/2]–1, where r is the rank of the manifold _k ^d . We will show how the problem is equivalent to the study of the kernel of various integral transforms, these will then be investigated using harmonic analysis on Grassmannian manifolds.The research of the first and third authors was supported in part by NSF grants DMS-9207019 and DMS-9304284. The research of the second author was supported in part by NFR contract number R-RA 4873-306 and the Swedish Academy of Sciences.     

The Boolean Prime Ideal Theorem Plus Countable Choice Do Not Imply Dependent Choice

Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2 and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second. Mathematics Subject Classification: 03E25, 03E35, 04A25.     

Boundary and Lens Rigidity of Lorentzian Surfaces

Let be a Lorentzian metric on the plane that agrees with the standard metric outside a compact set and so that there are no conjugate points along any time-like geodesic of . Then and are isometric. Further, if and are two dimensional compact time oriented Lorentzian manifolds with space--like boundaries and so that all time-like geodesics of maximize the distances between their points and and are ``boundary isometric', then there is a conformal diffeomorphism between and and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

     

The selective reductive acylation of a-keto imines

Reaction of α-keto imines with the $\text{in}$$\text{situ}$ generated acetylcobalt tetracarbonyl occurs only at the carbon—nitrogen double bond to give β-keto amides. An unexpected by-product was formed in several instances.     

Star-shaped azo-based dipolar chromophores: design, synthesis, matrix compatibility, and electro-optic activity

Three new azo-benzene-based push-pull chromophores with dendritic architecture were synthesized as active materials for electro-optic applications. These chromophores were synthesized in six or seven synthetic steps with an overall yield of around 80% per step and high purity. UV-vis spectroscopy showed significant influence of the transient dipole moment on the observed r(33) values. The chromophores were stable to photochemical oxidation in ambient light and air. The electrical poling conditions were optimized for each chromophore as the T(g) of the composite material varied significantly. The highest EO coefficient achieved was 22-25 pm/V at 1550 nm wavelength. STEM analysis of the blends enabled the correlation of the activity of these large chromophores with the blend morphology. An amorphous polycarbonate host effectively disperses the chromophores in 2-20 nm aggregates in the active materials. However, macrophase separation into 200-500 nm aggregates was observed in a methacrylate host matrix.     

Syntheses,structure, and photoluminescence properties of the 1-dimensional chain compounds [(TPA)(2)Au][Au(CN)(2)] and (TPA)AuCl (TPA = 1,3,5-triaza-7-phosphaadamantane)

The structures and temperature-dependent photoluminescence properties of the one-dimensional compounds [(TPA)(2)Au][Au(CN)(2)], 1, and (TPA)AuCl, 2, are reported. An extended linear chain with weak Au.Au interactions along the c-axis is evident in the structure of 1, and a helical chain with a pitch of 3.271 A is seen for 2. The intrachain Au...Au separation is 3.457(1) and 3.396(2) A in 1 and 2, respectively. As a result of this weak Au...Au interaction, the physical properties of these compounds are anisotropic. Scanning electron microscopy (SEM) studies indicate that single crystals of both compounds are noninsulating. Single crystals of 1 do not luminesce visibly, but grinding the crystals finely initiates a strong green emission under UV irradiation at room temperature. Further interesting optical properties include the dependence of the emission profile of the powder on the exciting wavelength and luminescence thermochromism. When excited at wavelengths < 360 nm, the powder exhibits a blue emission at 425 nm while excitation with longer wavelengths leads to a green emission near 500 nm. While the green emission dominates at ambient temperature, cooling to cryogenic temperatures leads to the dominance of the blue emission. Fibers of 2 are luminescent at 78 K with an emission band centered at 580 nm. Compound 1 crystallizes in the orthorhombic space group Cccm (No. 66), with Z = 2, a = 6.011(1) A, b = 23.877(6) A, c = 6.914(1) A, V = 992.3(3) A(3), and R = 0.0337. Compound 2 crystallizes in the trigonal space group R3 (No. 148), with Z = 18, a = 22.587(2) A, b = 22.587(2) A, c = 9.814(2) A, V = 4336 A(3), and R = 0.0283.     

One-dimensional rigorous hole theory of fluids: Internally constrained ensembles

A hole in a fluid is specified in a well-defined manner. The concentration of holes is a thermodynamic property of the fluid and we derive this concentration in three different ensembles for a one-dimensional fluid of hard rods. The thermodynamics of these rigorously defined holes is developed, and the properties of holes are explored. The ensemble in which the concentration of holes is maintained fixed exhibits dramatic properties. Finally, pair correlation functions for hard rods in the various ensembles are computed. Contrary to a frequently made assumption, the equilibrium number of holes is found to never be proportional to the probability of finding a single hole in the fluid. Constraining the concentration of holes as well as the density leads to dramatic structural effects prominently displayed by the pair correlation function. The ensemble in which the concentration of holes is fixed is an example of an internally constrained metastable system.     

Nitrato- and fluoroboracites M3B7O13NO3 and M3B7O13F

New boracites containing nitrato- or fluoroanions that appear to be true low-pressure phases have been synthesized at superatmospheric pressures. The M₃B₇O₁₃NO₃ compounds (M = Co, Ni, Cu, Zn, Cd) transform rapidly and reversibly in the temperature region 300–500°C between probable orthorhombic and face centered cubic symmetry, while the M₃B₇O₁₃F compounds (M = Mg, Mn, Fe, Co, Zn) appear to maintain rhombohedral symmetry up to their decomposition temperatures of 800–900°C. True high-pressure boracite-like phases containing F and Cr, Mn, Fe, or Co that decompose upon heating to M₃B₇O₁₃F have also been isolated.