Maximal Classes of Topological Spaces and Domains Determined by Function Spaces

In this paper the question of what classes A of T ₀-spaces should be paired with classes of domains in order that all function spaces [AB] for AA and B are -compact domains is considered. It is shown that core compact spaces are paired with bounded complete domains and a class of topological spaces called RW-spaces (with finitely many components) is paired with the class of -compact pointed L-domains (L-domains).     

Growth of Betti numbers

We discuss growth rates of Betti numbers in a family of coverings of a compact cell complex X, when the corresponding L² Betti number of X is zero. We show that the Betti numbers are bounded by a function, sub-linear in the order of the covering. If the appropriate Novikov-Shubin invariant of X is positive, the rate bounds are improved. For well behaved families (such as congruence covers of symmetric spaces), if the L² spectrum of X? has a gap at zero then the growth rate is bounded by the order of the covering raised to a power less than one.     

A New Family of Cyclic Difference Sets with Singer Parameters in Characteristic Three

We construct a new family of cyclic difference sets with parameters ((3 ^d – 1)/2, (3 ^{d – 1} – 1)/2, (3 ^{d – 2} – 1)/2) for each odd d. The difference sets are constructed with certain maps that form Jacobi sums. These new difference sets are similar to Maschietti's hyperoval difference sets, of the Segre type, in characteristic two. We conclude by calculating the 3-ranks of the new difference sets.     

Ramsey partitions of integers and pair divisions

Ifk ₁ andk ₂ are positive integers, the partitionP = (₁,₂,..., _n ) ofk ₁+k ₂ is said to be a Ramsey partition for the pairk ₁,k ₂ if for any sublistL ofP, either there is a sublist ofL which sums tok ₁ or a sublist ofP –L which sums tok ₂. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork ₁,k ₂ having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork ₁ andk ₂.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok ₁k ₂. This can be done trivially usingk ₁+k ₂-1 cuts. However, every Ramsey partition ofk ₁+k ₂ also yields a fair division algorithm. This method yields fewer cuts except whenk ₁=1 andk ₂=1, 2 or 4.     

Lamb wave tomography of pipe-like structures

Lamb waves are guided ultrasonic plate waves that can follow the curvature of pipe-like structures. By transmitting and receiving many helically propagating Lamb waves via longitudinal transducers in contact with the surface of a pipe, crosshole tomographic geometries can be mimicked and tomographic reconstructions performed in order to locate and size flaws. We describe here a meridional-array scheme which mimics a single line of transducers along the exterior surface of the pipe in the axial direction, and show proof of concept results on a pipe sample with an internal wall-thinning. We also demonstrate improved reconstructions for the other helical ultrasound tomography geometry where the transmitters and receivers lie along parallel circumferential rings. We find frequency compounding smoothes out some of the noise and artifacts that appear in the reconstructions.     

Mode beating in spot-size converter lasers

An anomalous modulation in the wavelength spectrum has been observed in lasers with spot-size converters. This intensity modulation is shown to be caused by beating between the fundamental lasing mode and radiation modes in the taper. This results in a periodic modulation in the net gain spectrum, which causes wavelength jumps between adjacent net gain maxima, and a drive current dependent spectral width that is expected to affect system performance. The amplitude of this spectral modulation is reduced significantly by either using an angled rear-facet which reflects the beating radiation modes away from the laser axis, or by using a nonlinear, adiabatic taper.     

Interface interactions in modulated phases,and upsilon points

Certain features in Frenkel-Kontorova and other models of phases with a one-dimensional modulation can be analyzed by assuming parallel interfaces separating sets of lattice planes belonging to two different phases, and treating the free energy to create interfaces, as well as the interaction of two, three, or more interfaces, as phenomenological parameters. A strategy employed by Fisher and Szpilka for interacting defects can be extended to the case of interfaces, allowing a systematic study of the phase diagram by ignoring all interface interactions, and then successively taking into account pair, triple, and higher-order terms. The possible phase diagrams which can occur near the point where =0 include: various sorts of endpoints analogous to critical endpoints, an accumulation point of first-order transitions and triple points, and a self-similar structure which we call an upsilon point, which turns out to be an accumulation point of an infinite number of segments of first-order transition lines, each of which terminates in two upsilon points.     

The superfluid transition temperature of liquid 3He

The superfluid transition temperature, T_c, is calculated from a scattering amplitude (constructed from the polarization potentials of Aldrich and Pines) which has been shown to yield agreement with experiment for transport properties. Our calculated values agree with experiment at all pressures up to 34 atm.     

Crystallization of Semiconductor Nanorods into Perfectly Faceted Hexagonal Superstructures

Super crystallization of ligand‐capped nanocrystals into defined periodic solids from solution is the definitive demonstration of their self‐organizing properties. To date, this has been mainly limited to spherical nanocrystals where organization emulates atom or molecule packing in regular crystals with the most thermodynamically stable arrangement being eventually preferred. Here, the crystallization of wurtzite CdS nanorods into micrometer‐sized CdS superstructures with regular hexagonal symmetry is demonstrated by fine‐tuning the nanorod dispersibility over time. It is shown that the supercrystals have a long nucleation stage to form monolayer hexagons followed by a relatively faster growth stage both occurring rod by rod (in‐plane) and layer by layer (out of plane). The perfectly symmetrical hexagon shape of the final structure is mapped from the wurtzite crystal structure of each individual nanorod where they pack in side by side and end to end arrangements. These well‐defined superstructures are highly attractive for applications that collectively exploit electronic or optical properties that are synthetically tunable through the size and shape of each nanorod building block.