Open sets satisfying systems of congruences

A famous result of Hausdorff states that a sphere with countably many points removed can be partitioned into three pieces A,B,C such that A is congruent to B (i.e., there is an isometry of the sphere which sends A to B), B is congruent to C, and A is congruent to (B union C); this result was the precursor of the Banach-Tarski paradox. Later, R. Robinson characterized the systems of congruences like this which could be realized by partitions of the (entire) sphere with rotations witnessing the congruences. The pieces involved were nonmeasurable. In the present paper, we consider the problem of which systems of congruences can be satisfied using open subsets of the sphere (or related spaces); of course, these open sets cannot form a partition of the sphere, but they can be required to cover "most of" the sphere in the sense that their union is dense. Various versions of the problem arise, depending on whether one uses all isometries of the sphere or restricts oneself to a free group of rotations (the latter version generalizes to many other suitable spaces), or whether one omits the requirement that the open sets have dense union, and so on. While some cases of these problems are solved by simple geometrical dissections, others involve complicated iterative constructions and/or results from the theory of free groups. Many interesting questions remain open.Comment: 44 page

General Scheme for Perfect Quantum Network Coding with Free Classical Communication

This paper considers the problem of efficiently transmitting quantum states through a network. It has been known for some time that without additional assumptions it is impossible to achieve this task perfectly in general -- indeed, it is impossible even for the simple butterfly network. As additional resource we allow free classical communication between any pair of network nodes. It is shown that perfect quantum network coding is achievable in this model whenever classical network coding is possible over the same network when replacing all quantum capacities by classical capacities. More precisely, it is proved that perfect quantum network coding using free classical communication is possible over a network with $k$ source-target pairs if there exists a classical linear (or even vector linear) coding scheme over a finite ring. Our proof is constructive in that we give explicit quantum coding operations for each network node. This paper also gives an upper bound on the number of classical communication required in terms of $k$, the maximal fan-in of any network node, and the size of the network.Comment: 12 pages, 2 figures, generalizes some of the results in arXiv:0902.1299 to the k-pair problem and codes over rings. Appeared in the Proceedings of the 36th International Colloquium on Automata, Languages and Programming (ICALP'09), LNCS 5555, pp. 622-633, 200

Optimal Clustering under Uncertainty

Classical clustering algorithms typically either lack an underlying probability framework to make them predictive or focus on parameter estimation rather than defining and minimizing a notion of error. Recent work addresses these issues by developing a probabilistic framework based on the theory of random labeled point processes and characterizing a Bayes clusterer that minimizes the number of misclustered points. The Bayes clusterer is analogous to the Bayes classifier. Whereas determining a Bayes classifier requires full knowledge of the feature-label distribution, deriving a Bayes clusterer requires full knowledge of the point process. When uncertain of the point process, one would like to find a robust clusterer that is optimal over the uncertainty, just as one may find optimal robust classifiers with uncertain feature-label distributions. Herein, we derive an optimal robust clusterer by first finding an effective random point process that incorporates all randomness within its own probabilistic structure and from which a Bayes clusterer can be derived that provides an optimal robust clusterer relative to the uncertainty. This is analogous to the use of effective class-conditional distributions in robust classification. After evaluating the performance of robust clusterers in synthetic mixtures of Gaussians models, we apply the framework to granular imaging, where we make use of the asymptotic granulometric moment theory for granular images to relate robust clustering theory to the application.Comment: 19 pages, 5 eps figures, 1 tabl

Translating the Cantor set by a random

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructive dimension

Nucleate pool boiling of refrigerant-oil mixtures from tubes

This experimental investigation was conducted to provide design heat transfer data for applications involving the nucleate pool boiling of Refrigerants 11 and 113 from the external surfaces of tubes. Tests were conducted with 0.625-in. o.d. and 1.125-in. o.d. commercial copper tubing; at 1 and 2 atmospheres pressure; and with refrigerant-oil compositions from 0 to 10 percent oil by weight. Results show that each of these parameters can affect the heat transfer coefficient --Abstract, page ii

Two Amino Acid Residues Contribute to a Cation-π Binding Interaction in the Binding Site of an Insect GABA Receptor

Cys-loop receptor binding sites characteristically possess an "aromatic box," where several aromatic amino acid residues surround the bound ligand. A cation-π interaction between one of these residues and the natural agonist is common, although the residue type and location are not conserved. Even in the closely related vertebrate GABA_A and GABA_C receptors, residues in distinct locations perform this role: in GABA_A receptors, a Tyr residue in loop A forms a cation-π interaction with GABA, while in GABA_C receptors it is a loop B residue. GABA-activated Cys-loop receptors also exist in invertebrates, where they have distinct pharmacologies and are the target of a range of pesticides. Here we examine the location of GABA in an insect binding site by incorporating a series of fluorinated Phe derivatives into the receptor binding pocket using unnatural amino acid mutagenesis, and evaluating the resulting receptors when expressed in Xenopus oocytes. A homology model suggests that two aromatic residues (in loops B and C) are positioned such that they could contribute to a cation-π interaction with the primary ammonium of GABA, and the data reveal a clear correlation between the GABA EC_(50) and the cation-π binding ability both at Phe206 (loop B) and Tyr254 (loop C), demonstrating for the first time the contribution of two aromatic residues to a cation-π interaction in a Cys-loop receptor

Application of Design of Experiments to Extrusion Freeform Fabrication (EFF) of Functional Ceramic Prototypes

Extrusion Freeform Fabrication (EFF) is an adaptation of the Stratasys Fused Deposition Modeling (FDM) process for the Solid Freeform Fabrication (SFF) of functional ceramic prototypes. It is a complex process involving many process variables, including parameters that are operation, machine, materials, and geometry specific. A Taguchi factorial Design of Experiments (DOE) technique was utilized to study the effects of machine specific process parameters as well as their interactions based on the mechanical and physical properties of sintered ceramics specimens. Post-processing software was developed to control and modify these parameters. This software interface wasdesigned to mimic the Quickslice™ interface for setting motion parameters based upon the material and the operation. The results of this investigation provided useful information for the experimental analysis of the machine specific process parameters. Suitable parameters were selected for the EFF process for fabricating representative ceramic prototypes. With the optimized parameters, complicated parts were successfully fabricated using both Kyocera SN282 and Starck M-11 silicon nitride powders.Mechanical Engineerin