A partition-based relaxation for Steiner trees

The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V, E), a set of terminals \({R\subseteq V}\) , and non-negative costs c _e for all edges \({e \in E}\) . Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The vertices \({V \backslash R}\) are called Steiner vertices. The best approximation algorithm known for the Steiner tree problem is a greedy algorithm due to Robins and Zelikovsky (SIAM J Discrete Math 19(1):122–134, 2005); it achieves a performance guarantee of \({1+\frac{\ln 3}{2}\approx 1.55}\) . The best known linear programming (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math Program 60:145–166, 1993) and achieves an approximation ratio of 2?2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primal-dual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the well-known bidirected cut relaxation. An instance is b-quasi-bipartite if each connected component of \({G \backslash R}\) has at most b vertices. We show that Robins’ and Zelikovsky’s algorithm has an approximation ratio better than \({1+\frac{\ln 3}{2}}\) for such instances, and we prove that the integrality gap of our LP is between \({\frac{8}{7}}\) and \({\frac{2b+1}{b+1}}\) .     

On the heat flow equation of surfaces of constant mean curvature in higher dimensions

In this paper, we consider the heat flow for the Hsystem with constant mean curvature in higher dimensions. We give sufficient conditions on the initial data such that the heat flow develops finite time singularity. We also provide a new set of initial data to guarantee the existence of global regular solution to the heat flow, that converges to zero in W 1,n with the decay rate t 2/(2-n) as time goes to infinity.     

Extension of the TOPSIS method for decision making problems under interval-valued intuitionistic fuzzy environment

TOPSIS is one of the well-known methods for multiple attribute decision making (MADM). In this paper, we extend the TOPSIS method to solve multiple attribute group decision making (MAGDM) problems in interval-valued intuitionistic fuzzy environment in which all the preference information provided by the decision-makers is presented as interval-valued intuitionistic fuzzy decision matrices where each of the elements is characterized by interval-valued intuitionistic fuzzy number (IVIFNs), and the information about attribute weights is partially known. First, we use the interval-valued intuitionistic fuzzy hybrid geometric (IIFHG) operator to aggregate all individual interval-valued intuitionistic fuzzy decision matrices provided by the decision-makers into the collective interval-valued intuitionistic fuzzy decision matrix, and then we use the score function to calculate the score of each attribute value and construct the score matrix of the collective interval-valued intuitionistic fuzzy decision matrix. From the score matrix and the given attribute weight information, we establish an optimization model to determine the weights of attributes, and construct the weighted collective interval-valued intuitionistic fuzzy decision matrix, and then determine the interval-valued intuitionistic positive-ideal solution and interval-valued intuitionistic negative-ideal solution. Based on different distance definitions, we calculate the relative closeness of each alternative to the interval-valued intuitionistic positive-ideal solution and rank the alternatives according to the relative closeness to the interval-valued intuitionistic positive-ideal solution and select the most desirable one(s). Finally, an example is used to illustrate the applicability of the proposed approach.     

Highly efficient hydrogen-bonding catalysis of the Diels-Alder reaction of 3-vinylindoles and methyleneindolinones provides carbazolespirooxindole skeletons

Carbazolespirooxindole derivatives were synthesized in a high-yielding, atypically rapid, stereocontrolled Diels-Alder reaction catalyzed by a C(2)-symmetric bisthiourea organocatalyst. Simple precursors and mild conditions were used to construct carbazolespirooxindole derivatives with high enantiopurity and structural diversity under H-bonding catalysis. The practical approach recycles the organocatalyst and solvent. This simple and efficient operational procedure will allow diversity-oriented syntheses of this intriguing class of compounds.     

Double isocyanide cyclization: a synthetic strategy for two-carbon-tethered pyrrole/oxazole pairs

A new strategy for the construction of the compounds with two different heterocycles, linked by a C(2)-tether via a domino process involving [5 + 1] annulation, ring-opening, and subsequent double isocyanide cyclization, from the reaction of ethyl isocyanoacetate with divinyl ketones (DVKs) has been developed. The chemoselective fragmentation of the cyclohexanone intermediate is the key for the formation of not only the C(2)-tether but also the two different heterocycles.     

An advance in proline ligation

Native chemical ligation (NCL) is widely applicable for building proteins in the laboratory. Since the discovery of this method, many strategies have been developed to enhance its capability and efficiency. Because of the poor reactivity of proline thioesters, ligation at a C-terminal proline site is not readily accomplished. Here, we demonstrate that ligation at an N-terminal protein is feasible using the combined logic of NCL and metal-free dethiylation (MFD).     

A novel 2,6-dicarbonylpyridine-based fluorescent chemosensor for Co2+ with high selectivity and sensitivity

A novel fluorescence chemosensor based on 2,6-dicarbonylpyridine was designed and synthesized and its photophysical properties were characterized. Upon coordination of Co(2+) by the central 2,6-dicarbonylpyridinyl functional group, the chemosensor 2,6-bis(4-diphenylamino-styrylcarbonyl)pyridine (PhPy) showed nearly complete fluorescence quenching, while no fluorescence response was seen towards other competing cations. The experimental results show the chemosensor is highly selective and sensitive towards Co(2+) in the presence of competing ions, even in the ppb range. Job plot analysis was carried out and the results suggested that the binding of PhPy and Co(2+) was probably a 2?:?1 stoichiometry.     

Terbium hybrid particles with spherical shape as luminescent probe for detection of Cu2+ and Fe3+ in water

A novel green emissive terbium inorganic-polymeric hybrid particle was designed and this material could detect cations in water. Polyvinyl alcohol as an amphiphilic surfactant rendered the powders dispersible in water with regular round shape (10-20 μm). Interestingly, we noticed that not only Cu(2+) (detection limit 10(-4)M) but also Fe(3+) (detection limit 10(-4) M) can give rise to emission quenching to this target material in comparison with K(+), Na(+), Fe(2+), Mn(2+), Pd(2+), Cd(2+) and Co(2+) (10(-3) mol L(-1)). We regarded that the coordination interactions between ligand and metal ions resulted in these quenching processes. Additionally, it was found that the sensing material can be repeatedly used at least 5 cycles. More importantly, this novel material demonstrated higher thermal-stability in aqueous media than pure silica hybrid material.     

A novel eudesmene sesquiterpenoid from <Emphasis Type="Italic">Schisandra sphenanthera</Emphasis> stems

Chemical investigation on the stems of Schisandra sphenanthera has afforded a novel eudesmene-type sesquiterpenoid, schisansphene A (1), and a known compound, alismol (2). Their structures and configurations were elucidated by spectroscopic methods, including 2D NMR techniques.     

Synthesis of 2‐Arylbenzoxazoles from Flash Vacuum Pyrolysis of 2‐Methoxy‐N‐(arenylidene)anilines

Flash vacuum pyrolysis of 2‐methoxy‐N‐(arenylidene)anilines 2a‐g at 700 °C and 1 × 10^‐2 Torr gave the corresponding 2‐arylbenzoxazoles 1a‐g .