Let P be a set of n points in R, and k ≤ n an integer. A sphere σ is k-rich with respect to P if |σ ∩ P | ≥ k, and is η-nondegenerate, for a fixed fraction 0 < η < 1, if no circle γ ⊂ σ contains more than η|σ ∩ P | points of P . We improve the previous bound given in [1] on the number of k-rich η-nondegenerate spheres in 3space with respect to any set of n points inR, from O(n/k+n/k), which hol...

Abstract. An attempt is made to conceptualize the derivation as well as to facilitate the computation of Ohtsuki’s rational invariants λn of integral homology 3-spheres extracted from ReshetikhinTuraev SU(2) quantum invariants. Several interesting consequences will follow from our computation of λ2. One of them says that λ2 is always an integer divisible by 3. It seems interesting to compare th...

In the past ten years, there has been a growing interest in the theory of quantum computation and quantum information.ð1Þ Due to the quantum parallelism, quantum computers have the potential to outperform the classical computers. The quantum computers are based on two-level quantum systems, which are called qubits.ð2Þ In this paper, we will investigate a representation of a single qubit operati...

We develop a Laplace transform method for constructing universal invariants of 3–manifolds. As an application, we recover Habiro's theory of integer homology 3–spheres and extend it to some classes of rational homology 3–spheres with cyclic homology. If |H 1 | = 2, we give explicit formulas for universal invariants dominating the sl 2 and SO(3) Witten–Reshetikhin–Turaev invariants, as well as t...

We develop an exact method for computing the Casimir energy between arbitrary compact objects, either dielectrics or perfect conductors. The energy is obtained as an interaction between multipoles, generated by quantum current fluctuations. The objects' shape and composition enter only through their scattering matrices. The result is exact when all multipoles are included, and converges rapidly...

We present some ideas for a possible Noncommutative Floer Homology. The geometric motivation comes from an attempt to build a theory which applies to practically every 3-manifold (closed, oriented and connected) and not only to homology 3-spheres. There is also a physical motivation: one would like to construct a noncommutative topological quantum field theory. The two motivations are closely r...

We show that by magnetically trapping a superconducting microsphere close to a quantum circuit, it is possible to perform ground-state cooling and prepare quantum superpositions of the center-of-mass motion of the microsphere. Due to the absence of clamping losses and time-dependent electromagnetic fields, the mechanical motion of micrometer-sized metallic spheres in the Meissner state is predi...

We present calculations using a simple model of radiative recombination in 2D slabs, 1D wires, and 0D spheres of hydrogenated amorphous silicon ~a-Si:H! showing a significant size dependence of the photoluminescence. Room-temperature peak emission energies .1.8 eV and efficiencies near unity are possible in a-Si:H spheres with diameters ,20 Å. Broad homogeneous linewidths .0.25 eV are also pred...