Binding modes of cucurbit[6]uril and cucurbit[7]uril with a tetracationic bis(viologen) guest

Binding behaviors of two cucurbit[n]urils (CB[n]) hosts with the [CH3bpy(CH2)6bpyCH3]4+ (bpy = 4,4'-bipyridinium) guest were investigated by 1H NMR and MALDI-TOF-MS experiments. While the CB[6] and CB[7] form [2]pseudorotaxanes with the host located over the hexamethylene chain of the guest, only the CB[7] forms a [3]pseudorotaxane with both host molecules residing over the bipyridinium groups. The initial CB[7] host vacates the inclusion of the hexamethylene chain as a result of the electrostatic and steric repulsions that would arise in simultaneous binding of adjacent aliphatic and aromatic portions of the guest.     

Carbolithiation of o-Amino-(E)-stilbenes: diastereoselective electrophile substitution with applications to quinoline synthesis

A regioselective carbolithiation of o-amino-(E)-stilbenes has been achieved with a series of alkyllithiums when THF is employed as the reaction solvent. The use of other solvents, such as diethyl ether or hydrocarbons, leads to a pronounced loss in regioselectivity. Moreover, high levels of diastereoselectivity have been obtained following reaction of the lithiated intermediate in THF with different electrophiles such as MeOD, CO2, and Bu3SnCl. It was shown that diastereoselectivity was influenced by the ortho-amino substituent and the alkyllithium utilized for carbolithiation with N-Boc substituent and t-BuLi proving optimal. In the case of carbolithiated intermediate 3a, obtained from the reaction of N-Boc substituted stilbene with t-BuLi, 1H and 13C NMR analysis revealed predominantly one diastereoisomer which was stable at room temperature. Application of the carbolithiation/electrophile reaction methodology to the synthesis of six-membered quinoline ring systems is demonstrated with substituted 3,4-dihydroquinolin-2-ones, 1,2,3,4-tetrahydroquinolines, 1,4-dihydroquinolines, and quinolines all prepared by a common cascade route.     

Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functions are pointed out. The equilibrium of the generalized proportional Caputo fractional model and its generalized exponential stability are defined, and sufficient conditions for the generalized exponential stability and asymptotic stability of the equilibrium are obtained. As a special case, the stability of the equilibrium of the Caputo fractional model is discussed. Several examples are provided to illustrate our theoretical results and the influence of the type of fractional derivative on the stability behavior of the equilibrium.     

Well-posedness for generalized quasi-variational inclusion problems and for optimization problems with constraints

In this paper, well-posedness of generalized quasi-variational inclusion problems and of optimization problems with generalized quasi-variational inclusion problems as constraints is introduced and studied. Some metric characterizations of well-posedness for generalized quasi-variational inclusion problems and for optimization problems with generalized quasi-variational inclusion problems as constraints are given. The equivalence between the well-posedness of generalized quasi-variational inclusion problems and the existence of solutions of generalized quasi-variational inclusion problems is given under suitable conditions.     

Twin Solutions to Singular Dirichlet Problems

The existence of two nonnegative solutions to Dirichlet second order boundary value problems is established in this paper. Our nonlinearity may be singular at y = 0, t = 0, and/or t = 1.     

Deterministic and random fixed points for maps on extension type spaces

We present a variety of deterministic and random fixed point results for very general spaces and classes of maps.     

Intersection Theorems with Applications in Optimization

In this paper, we establish two intersection theorems which are useful in considering some optimization problems (complementarity problems, variational inequalities, minimax inequalities, saddle point problems).     

On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses

In this article, we investigate the orbital Hausdorff continuous dependence of the solutions to integer order and fractional nonlinear non-instantaneous differential equations. The concept of orbital Hausdorff continuous dependence is used to characterize the relations of solutions corresponding to the impulsive points and junction points in the sense of the Hausdorff distance. Then, we establish sufficient conditions to guarantee this specific continuous dependence on their respective trajectories. Finally, two examples are given to illustrate our theoretical results.