On linear and semidefinite programming relaxations for hypergraph matching

The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a well-studied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following:

We consider the standard linear programming relaxation of the problem. We provide an algorithmic proof of a result of Füredi, Kahn and Seymour, showing that the integrality gap is exactly ${k-1+\frac{1}{k}}$ for k-uniform hypergraphs, and is exactly k ? 1 for k-partite hypergraphs. This yields an improved approximation algorithm for the weighted 3-dimensional matching problem. Our algorithm combines the use of the iterative rounding method and the fractional local ratio method, showing a new way to round linear programming solutions for packing problems.

We study the strengthening of the standard LP relaxation by local constraints. We show that, even after linear number of rounds of the Sherali-Adams lift-and-project procedure on the standard LP relaxation, there are k-uniform hypergraphs with integrality gap at least k ? 2. On the other hand, we prove that for every constant k, there is a strengthening of the standard LP relaxation by only a polynomial number of constraints, with integrality gap at most ${\frac{k+1}{2}}$ for k-uniform hypergraphs. The construction uses a result in extremal combinatorics.

We consider the standard semidefinite programming relaxation of the problem. We prove that the Lovász ${\vartheta}$ -function provides an SDP relaxation with integrality gap at most ${\frac{k+1}{2}}$ . The proof gives an indirect way (not by a rounding algorithm) to bound the ratio between any local optimal solution and any optimal SDP solution. This shows a new connection between local search and linear and semidefinite programming relaxations.

     

Synthesis and structure characterization of ladder-like polymethylsilsesquioxane (PMSQ) by isolation of stereoisomer

The stereo cyclic siloxane compounds having tetra hydroxyl and tetra methyl group were synthesized by hydrolysis of tetrahydrido tetramethyl cyclosiloxane consisting of four stereoisomers. Among the stereo cyclic isomers, cis-trans-cis tetrahydroxyl tetramethyl cyclosiloxane (2) was separated by recrystallization and was used as a monomer species for preparation of ladder-like polymethylsilsesquioxane (PMSQ). The isolated isomer was directly polymerized in THF with potassium carbonate (K₂CO₃) by systematic ring condensation. The hydrolyzed stereoisomers were characterized by ¹H and ²⁹Si NMR, and HPLC. The structure of the synthesized PMSQs were characterized by size exclusion chromatography (SEC), MALDI-TOF mass, ²⁹Si NMR, and small angle X-ray scattering (SAXS).     

Crystal growth, structure, and physical properties of LnCu2(Al,Si)5 (Ln = La and Ce)

LnCu(2)(Al,Si)(5) (Ln = La and Ce) were synthesized and characterized. These compounds adopt the SrAu(2)Ga(5) structure type and crystallize in the tetragonal space group P4/mmm with unit cell dimensions of a ≈ 4.2 ? and c ≈ 7.9 ?. Herein, we report the structure as obtained from single crystal X-ray diffraction. Additionally, we report the magnetic susceptibility, magnetization, resistivity, and specific heat capacity data obtained for polycrystalline samples of LnCu(2)(Al,Si)(5) (Ln = La and Ce).     

Spin-adapted density matrix renormalization group algorithms for quantum chemistry

We extend the spin-adapted density matrix renormalization group (DMRG) algorithm of McCulloch and Gulacsi [Europhys. Lett. 57, 852 (2002)] to quantum chemical Hamiltonians. This involves using a quasi-density matrix, to ensure that the renormalized DMRG states are eigenfunctions of S?(2), and the Wigner-Eckart theorem, to reduce overall storage and computational costs. We argue that the spin-adapted DMRG algorithm is most advantageous for low spin states. Consequently, we also implement a singlet-embedding strategy due to Tatsuaki [Phys. Rev. E 61, 3199 (2000)] where we target high spin states as a component of a larger fictitious singlet system. Finally, we present an efficient algorithm to calculate one- and two-body reduced density matrices from the spin-adapted wavefunctions. We evaluate our developments with benchmark calculations on transition metal system active space models. These include the Fe(2)S(2), [Fe(2)S(2)(SCH(3))(4)](2-), and Cr(2) systems. In the case of Fe(2)S(2), the spin-ladder spacing is on the microHartree scale, and here we show that we can target such very closely spaced states. In [Fe(2)S(2)(SCH(3))(4)](2-), we calculate particle and spin correlation functions, to examine the role of sulfur bridging orbitals in the electronic structure. In Cr(2) we demonstrate that spin-adaptation with the Wigner-Eckart theorem and using singlet embedding can yield up to an order of magnitude increase in computational efficiency. Overall, these calculations demonstrate the potential of using spin-adaptation to extend the range of DMRG calculations in complex transition metal problems.     

Selective surface reactions for Janus ORMOSIL particles with multiple functional groups using an ordered monolayer film at liquid-liquid interface

Monodispersed, submicron-sized Janus ORMOSIL particles with multiple functional groups were prepared by the selective surface reaction of a monolayer film formed at a hexane-water interface. A well-ordered monolayer film was obtained by self-assembly of ORMOSIL particles with multiple functional groups at hexane-water interface. The photopolymerization of an ordered monolayer containing ORMOSIL particles yields a rigid film strong enough to maintain its integrity for transfer and further chemical reaction. The chemical reaction of this ordered film with organic and inorganic functional groups produced Janus ORMOSIL particles with multiple functional groups. The morphologies, structures, and chemical compositions of monolayer films and Janus ORMOSIL particles were characterized by FT-IR, solid state NMR, X-ray diffraction (XRD), optical microscopy (OM), electron microscopies (SEM and TEM), and confocal laser scanning microscopy.     

Measurement and analysis of forces in bubble and droplet systems using AFM

The use of atomic force microscopy to measure and understand the interactions between deformable colloids - particularly bubbles and drops - has grown to prominence over the last decade. Insight into surface and structural forces, hydrodynamic drainage and coalescence events has been obtained, aiding in the understanding of emulsions, foams and other soft matter systems. This article provides information on experimental techniques and considerations unique to performing such measurements. The theoretical modelling frameworks which have proven crucial to quantitative analysis are presented briefly, along with a summary of the most significant results from drop and bubble AFM measurements. The advantages and limitations of such measurements are noted in the context of other experimental force measurement techniques.     

Mechanism for the reaction of 2-naphthol with N-methyl-N-phenyl-hydrazine suggested by the density functional theory investigations

For the first time the computed mechanisms for the novel reaction of 2-naphthol with N-methyl-N-phenylhydrazine, leading to 1-amino-2-naphthol (Tang et al., J Am Chem Soc 2008, 130, 5840), have been investigated using the density functional theory. Four distinct possible pathways were evaluated: two amination mechanisms with the attack of NH(2) group respectively at the α-position C1 and β-position C3 atoms of 2-naphthol (pathways 1 and 2) as well as two rearrangement processes with displacement of the phenolic hydroxyl group followed by the benzidine-like rearrangement at the α-position C1 and β-position C3 atoms of 2-naphthol, respectively (pathways 3 and 4). Solvent effect has been tested based on the optimized geometries of the stationary points in solution at the B3LYP/PCM/6-31+G(d,p) level of theory with an averaged dielectric constant of binary solvent. Single-point energies of the optimized structures have been calculated using three hybrid density functionals, B3LYP, MPW3LYP, and B3PW91 with the 6-311++G(3df,2p) basis set. Our computed results clearly manifest that pathway 1 (α-amination) has the highest possibility to occur, with the Gibbs free energies being lower by 6 to 20 kcal/mol compared with the other three pathways, which leads to 1-amino-2-naphthol and N-methylaniline as products. It is in excellent agreement with the experimental observation.     

Proteomic analysis reveals platelet factor 4 and beta-thromboglobulin as prognostic markers in severe acute respiratory syndrome

Previously, we reported that proteomic fingerprints were present in sera of patients with severe acute respiratory syndrome (SARS), and could separate patients into subgroups with different prognoses. In the present study, we examined the prognostic values of the SARS-associated proteomic features by biostatistical analysis, and deciphered the identities of those with prognostic values. Data of 20 SARS-associated serum proteomic features and ten serological variables from 38 SARS adult patients before treatment were subjected to multivariate logistic regression. Proteomic features of m/z 6634, m/z 7769, m/z 8635, and m/z 8865 were identified as independent prognostic markers. After purification by cation-exchange chromatography and gel electrophoresis, proteomic features of m/z 7769 and m/z 8865 were found to be platelet factor 4 (PF4) and beta-thromboglobulin (beta-TG) by tandem mass spectrometry, respectively. The associations of decreased serum PF4 and increased serum beta-TG levels with poor prognosis were confirmed by Western blot. Previous studies suggest that PF4 and beta-TG are involved in the pathogenesis of acute respiratory distress syndrome (ARDS) in a negative and positive way, respectively. Our results suggest that PF4 and beta-TG may also play similar roles in the development of ARDS in SARS patients.     

Direct synthesis of aqueous quantum dots through 4,4'-bipyridine-based twin ligand strategy

We report a new class of derivatized 4,4'-bipyridinium ligands for use in synthesizing highly fluorescent, extremely stable, water-soluble CdSe and CdTe quantum dots (QDs) for bioconjugation. We employed an evaporation-condensation technique, also known as solvated metal atom dispersion (SMAD), followed by a digestive ripening procedure. This method has been used to synthesize both metal nanoparticles and semiconductors in the gram scale with several stabilizing ligands in various solvents. The SMAD technique comprised evaporation condensation and stabilization of CdSe or CdTe in tetrahydrofuran. The as-prepared product was then digestively ripened in both water and dimethyl formamide, leading to narrowing of the particle size distributions. The ligands were synthesized by nucleophilic substitution (S(N)2) reactions using 4,4'-bipyridine as a nucleophile. Confocal microscopy images revealed the orange color of the nanocrystalline QDs with diameters of ~5 nm. The size has been confirmed by using transmission electron microscopy. As a part of our strategy, 85% of the 4,4'-bipyridinium salt was synthesized as propionic acid derivative and used to both stabilize the QDs in water and label basic amino acids and different biomarkers utilizing the carboxylic acid functional group. Fifteen percent of the 4,4'-bipyridinium salt was synthesized as N-propyl maleimide and used as a second ligand to label any protein containing the amino acid cysteine by means of a 1,4-Michael addition.     

On parameter estimation of threshold autoregressive models

This paper studies the threshold estimation of a TAR model when the underlying threshold parameter is a random variable. It is shown that the Bayesian estimator is consistent and its limit distribution is expressed in terms of a limit likelihood ratio. Furthermore, convergence of moments of the estimators is also established. The limit distribution can be computed via explicit simulations from which testing and inference for the threshold parameter can be conducted. The obtained results are illustrated with numerical simulations.