Algebraic topological methods for the supercritical Q-curvature problem

We study the problem of existence of conformal metrics with prescribed Q-curvature on closed four-dimensional Riemannian manifolds. This problem has a variational structure, and in the case of interest here, it is noncompact in the sense that accumulations points of some noncompact flow lines of a pseudogradient of the associated Euler–Lagrange functional, the so-called true critical points at infinity of the associated variational problem, occur. Using the characterization of the critical points at infinity of the associated variational problem which is established in [42], combined with some arguments from Morse theory, some algebraic topological methods, and some tools from dynamical system originating from Conley's isolated invariant sets and isolated blocks theory, we derive a new kind of existence results under an algebraic topological hypothesis involving the topology of the underling manifold, stable and unstable manifolds of some of the critical points at infinity of the associated Euler–Lagrange functional.     

Infinitely many homoclinic orbits for a class of second‐order damped differential equations

We investigate the existence and multiplicity of homoclinic orbits for the second‐order damped differential equations For Equation 1 where L(t) and W(t,u) are neither autonomous nor periodic in t. Under certain assumptions on g, L, and W, we get infinitely many homoclinic orbits for superquadratic, subquadratic and concave–convex nonlinearities cases by using fountain theorem and dual fountain theorem in critical point theory. These results generalize and improve some existing results in the literature. Copyright © 2015 JohnWiley & Sons, Ltd.     

Q‐symmetry and conditional Q‐symmetries for Drinfel'd–Sokolov–Wilson system

We study in this paper the Q‐symmetry and conditional Q‐symmetries of Drinfel'd–Sokolov–Wilson equations. The solutions which we obtain in this paper take the form of convergent power series with easily computable components. Copyright © 2014 John Wiley & Sons, Ltd.     

Universal C*-algebras defined by completely bounded unital homomorphisms

Suppose A is a unital C*-algebra and r 1.In this paper,we define a unital C*-algebra C_(cb)*(A,r) and a completely bounded unital homomorphism α_r:A → C_(cb)*(A,r)with the property that C_(cb)*(A,r)=C*(α_r(A))and,for every unital C*-algebra B and every unital completely bounded homomorphism φ:A→ B,there is a(unique)unital *-homomorphism π:C_(cb)*(A,r)→B such thatφ=πoα_r.We prove that,if A is generated by a normal set {t_λ:λ∈Λ},then C_(cb)*(A,r)is generated by the set {α_r(t_λ):λ∈Λ}.By proving an equation of the norms of elements in a dense subset of C_(cb)*(A,r)we obtain that,if Β is a unital C*-algebra that can be embedded into A,then C_(cb)*(B,r)can be naturally embedded into C(cb)*(A,r).We give characterizations of C_(cb)*(A,r)for some special situations and we conclude that C_(cb)*(A,r)will be "nice" when dim(A)≤ 2 and "quite complicated" when dim(A)≥ 3.We give a characterization of the relation between K-groups of A and K-groups of C_(cb)*(A,r).We also define and study some analogous of C_(cb)*(A,r).     

Optimal Groomings with Grooming Ratios Six and Seven

Grooming uniform all‐to‐all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C‐groomings has been considered for , and completely solved for . For , it has been shown that the lower bound for the drop cost of an optimal C‐grooming can be attained for almost all orders with 5 exceptions and 308 possible exceptions. For , there are infinitely many unsettled orders; especially the case is far from complete. In this paper, we show that the lower bound for the drop cost of a 6‐grooming can be attained for almost all orders by reducing the 308 possible exceptions to 3, and that the lower bound for the drop cost of a 7‐grooming can be attained for almost all orders with seven exceptions and 16 possible exceptions. Moreover, for the unsettled orders, we give upper bounds for the minimum drop costs.     

Rational Perturbation of the Fluorescence Quantum Yield in Emission‐Tunable and Predictable Fluorophores (Seoul‐Fluors) by a Facile Synthetic Method Involving C?H Activation

Fluorescence imaging enables the uniquely sensitive observation of functional‐ and molecular‐recognition events in living cells. However, only a limited range of biological processes have been subjected to imaging because of the lack of a design strategy and difficulties in the synthesis of biosensors. Herein, we report a facile synthesis of emission‐tunable and predictable Seoul‐Fluors, 9‐aryl‐1,2‐dihydrolopyrrolo[3,4‐b]indolizin‐3‐ones, with various R¹ and R² substituents by coinage‐metal‐catalyzed intramolecular 1,3‐dipolar cycloaddition and subsequent palladium‐mediated C H activation. We also showed that the quantum yields of Seoul‐Fluors are controlled by the electronic nature of the substituents, which influences the extent of photoinduced electron transfer. On the basis of this understanding, we demonstrated our design strategy by the development of a Seoul‐Fluor‐based chemosensor 20 for reactive oxygen species that was not accessible by a previous synthetic route.     

Polymerization of Ethylene by Silica‐Supported Dinuclear CrIII Sites through an Initiation Step Involving C?H Bond Activation

The insertion of an olefin into a preformed metal–carbon bond is a common mechanism for transition‐metal‐catalyzed olefin polymerization. However, in one important industrial catalyst, the Phillips catalyst, a metal–carbon bond is not present in the precatalyst. The Phillips catalyst, CrO₃ dispersed on silica, polymerizes ethylene without an activator. Despite 60 years of intensive research, the active sites and the way the first Cr C bond is formed remain unknown. We synthesized well‐defined dinuclear Cr^II and Cr^III sites on silica. Whereas the Cr^II material was a poor polymerization catalyst, the Cr^III material was active. Poisoning studies showed that about 65 % of the Cr^III sites were active, a far higher proportion than typically observed for the Phillips catalyst. Examination of the spent catalyst and isotope labeling experiments showed the formation of a Si–(μ‐OH)–Cr^III species, consistent with an initiation mechanism involving the heterolytic activation of ethylene at Cr^III O bonds.