On the Regularity of Optimal Transportation Potentials on Round Spheres

In this paper the regularity of optimal transportation potentials defined on round spheres is investigated. Specifically, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satisfied on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper's calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (A3) condition for any case where the cost-fuction of the associated optimal transportation problem can be expressed as a function of the geodesic distance between points on a round sphere. With this, several examples of such cost-functions are then analysed to see whether or not they satisfy this (A3) condition.Comment: 24 pages, 4 figure

Fundamental limits of super-resolution microscopy by dielectric microspheres and microfibers

In recent years, optical super-resolution by microspheres and microfibers emerged as a new paradigm in nanoscale label-free and fluorescence imaging. However, the mechanisms of such imaging are still not completely understood and the resolution values are debated. In this work, the fundamental limits of super-resolution imaging by high-index barium-titanate microspheres and silica microfibers are studied using nanoplasmonic arrays made from Au and Al. A rigorous resolution analysis is developed based on the object's convolution with the point-spread function that has width well below the conventional (∼λ/2) diffraction limit, where λ is the illumination wavelength. A resolution of ∼λ/6-λ/7 is demonstrated for imaging nanoplasmonic arrays by microspheres. Similar resolution was demonstrated for microfibers in the direction perpendicular to the fiber axis with hundreds of times larger field-of-view in comparison to microspheres. Using numerical solution of Maxwell's equations, it is shown that extraordinary close point objects can be resolved in the far field, if they oscillate out of phase. Possible super-resolution using resonant excitation of whispering gallery modes is also studied. Keywords: Optical super-resolution; near-field microscopy; confocal microscop

Two problems related to prescribed curvature measures

Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's $C^2$ estimates for prescribed curvature equation in \cite{I1, I}.Comment: 12 pages, Corrected typo

Photonic band structure of highly deformable, self-assembling systems

We calculate the photonic band structure at normal incidence of highly deformable, self-assembling systems - cholesteric elastomers subjected to external stress. Cholesterics display brilliant reflection and lasing owing to gaps in their photonic band structure. The band structure of cholesteric elastomers varies sensitively with strain, showing new gaps opening up and shifting in frequency. A novel prediction of a total band gap is made, and is expected to occur in the vicinity of the previously observed de Vries bandgap, which is only for one polarisation

Self-Assembly of Supramolecular Triblock Copolymer Complexes

Four different poly(tert-butoxystyrene)-b-polystyrene-b-poly(4-vinylpyridine) (PtBOS-b-PS-b-P4VP) linear triblock copolymers, with the P4VP weight fraction varying from 0.08 to 0.39, were synthesized via sequential anionic polymerization. The values of the unknown interaction parameters between styrene and tert-butoxystyrene and between tert-butoxystyrene and 4-vinylpyridine were determined from random copolymer blend miscibility studies and found to satisfy 0.031<χS,tBOS<0.034 and 0.39<χ4VP,tBOS<0.43, the latter being slightly larger than the known 0.30<χS,4VP≤0.35 value range. All triblock copolymers synthesized adopted a P4VP/PS core/shell cylindrical self-assembled morphology. From these four triblock copolymers supramolecular complexes were prepared by hydrogen bonding a stoichiometric amount of pentadecylphenol (PDP) to the P4VP blocks. Three of these complexes formed a triple lamellar ordered state with additional short length scale ordering inside the P4VP(PDP) layers. The self-assembled state of the supramolecular complex based on the triblock copolymer with the largest fraction of P4VP consisted of alternating layers of PtBOS and P4VP(PDP) layers with PS cylinders inside the latter layers. The difference in morphology between the triblock copolymers and the supramolecular complexes is due to two effects: (i) a change in effective composition and, (ii) a reduction in interfacial tension between the PS and P4VP containing domains. The small angle X-ray scattering patterns of the supramolecules systems are very temperature sensitive. A striking feature is the disappearance of the first order scattering peak of the triple lamellar state in certain temperature intervals, while the higher order peaks (including the third order) remain. This is argued to be due to the thermal sensitivity of the hydrogen bonding and thus directly related to the very nature of these systems.

A glimpse into the differential topology and geometry of optimal transport

This note exposes the differential topology and geometry underlying some of the basic phenomena of optimal transportation. It surveys basic questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It also establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.Comment: 27 page

Absolute Configuration of β-Hydroxy-β-phenylpropionic acid*

(+)-Methyl β-methoxy-β-phenylpropionate (VIII) was prepared from (+)-mandelic acid and from (+)-β-hydroxy-β-phenylpropionic acid (I). In this way the configuration of I was correlated with that of mandelic acid

Optical Characterization of Silver Nanorod Thin Films Grown Using Oblique Angle Deposition

Nanorods are metamaterial structures that have been shown to have wide application, ranging from biomedical uses to photovoltaic materials. These materials have unique optical characteristics. In this paper, two silver (Ag) nanorod thin-film samples are created using Glancing Angle Deposition (GLAD) at both near-room temperature (~300 K) and cryogenic temperature (~100 K). Generalized ellipsometry is used to measure the optical constants of the samples. The strong difference between the optical constants of the constituent materials and those of these thin films shows that the characteristics of the samples are due to how their metamaterial structures are defined. The principle optical axes of the films align well with the morphological characteristics of the nanostructures. The axis with the greatest index of refraction remains aligned to the principle axes but shifts orientation with respect to morphological characteristics between samples. Experimental results show differences in both magnitude and characteristics of the nanorod indexes. Reflectance and transmittance measurements are performed to extract absorptance data. The room temperature deposited sample shows a higher overall absorptance, while the cryogenic sample shows a clear orientation-dependent absorptance. Polarization data is analyzed to show that the 100 K thin film exhibits polarization-dependent absorptance, while the 300 K sample’s absorptance has a strong orientation dependence

\u3csup\u3e26\u3c/sup\u3eAl-Containing Acidic and Basic Sodium Aluminum Phosphate Preparation and Use in Studies of Oral Aluminum Bioavailability from Foods Utilizing \u3csup\u3e26\u3c/sup\u3eAl as an Aluminum Tracer

We synthesized 26Al-containing acidic and basic (alkaline) sodium aluminum phosphates (SALPs) which are FDA-approved leavening and emulsifying agents, respectively, and used them to determine the oral bioavailability of aluminum incorporated in selected foods. We selected applicable methods from published syntheses (patents) and scaled them down (∼3000- and 850-fold) to prepare ∼300–400 mg of each SALP. The 26Al was incorporated at the beginning of the syntheses to maximize 26Al and 27Al equilibration and incorporate the 26Al in the naturally-occurring Al-containing chemical species of the products. Near infrared spectroscopy (NIR) and X-ray powder diffraction (XRD) were used to characterize the two SALP samples and some intermediate samples. Multi-elemental analysis (MEA) was used to determine Na, Al and P content. Commercial products were included for comparison. Satisfactory XRD analyses, near infrared spectra and MEA results confirmed that we synthesized acidic and basic SALP, as well as some of the syntheses intermediates. The 26Al-containing acidic and basic SALPs were incorporated into a biscuit material and a processed cheese, respectively. These were used in oral bioavailability studies conducted in rats in which the 26Al present in blood after its oral absorption was quantified by accelerator mass spectrometry. The results showed oral Al bioavailability from acidic SALP in biscuit was ∼0.02% and from basic SALP in cheese ∼0.05%, lower than our previous determination of Al bioavailability from drinking water, ∼0.3%. Both food and water can appreciably contribute to the Al absorbed from typical human Al intake