Positive and Nontrivial Solutions for the Urysohn Integral Equation

We establish new criteria for the existence of either positive or nonzero solutions of the Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigenfunction for the Urysohn integral operator. Among others, we employ techniques based on fixed point index theory for compact maps, which are new for this type of equation.     

Quotient probabilistic normed spaces and completeness results

We introduce the concept of quotient in PN spaces and give some examples. We prove some theorems with regard to the completeness of a quotient.     

Existence and boundary behavior for singular nonlinear differential equations with arbitrary boundary conditions

Existence theory is developed for the equation ?(u)=F(u), where ? is a formally self-adjoint singular second-order differential expression and F is nonlinear. The problem is treated in a Hilbert space and we do not require the operators induced by ? to have completely continuous resolvents. Nonlinear boundary conditions are allowed. Also, F is assumed to be weakly continuous and monotone at one point. Boundary behavior of functions associated with the domains of definitions of the operators associated with ? in the singular case is investigated. A special class of self-adjoint operators associated with ? is obtained.     

Leggett-Williams norm-type theorems for coincidences

We study the existence of positive solutions to the operator equation Lx = Nx, where L is a linear Fredholm mapping of index zero and N is a nonlinear operator. Using the properties of cones in Banach spaces and Leray-Schauder degree for completely continuous operators, k-set contractions and condensing mappings, we obtain some refinements of the results established in [3] and [14]. Received: 9 July 2005; revised: 12 January 2006     

Efficient synthesis of aryl vinyl ethers exploiting 2,4,6-trivinylcyclotriboroxane as a vinylboronic acid equivalent

The synthesis of functionalized aryl vinyl ether derivatives can be readily achieved utilizing a room-temperature copper(II) acetate mediated coupling of substituted phenols with 2,4,6-trivinylcyclotriboroxane-pyridine complex in the presence of a suitable base. The scope of the procedure was demonstrated by the generation of an array of substituted aryl vinyl ethers. The reaction was seen to be tolerant of a diverse range of functional groups yielding products in high isolated yields. We have shown that one role of an amine base in the reaction sequence is the in situ generation of an amine coordinated boroxine ring. An X-ray crystal structure and low temperature (11)B NMR study of 2,4,6-trivinylcyclotriboroxane-pyridine complex demonstrated the nature of the tetracoordinate boron species, which may have a key role to play within the reaction sequence.     

Positive solutions for a class of boundary value problems on infinite intervals

This paper studies the existence of nonegative solutions to the two point boundary value , 0<t< with appropriate boundary conditions.     

Synthesis and properties of monodisperse oligofluorene-functionalized truxenes: highly fluorescent star-shaped architectures

This paper describes the strategy toward novel monodisperse, well-defined, star-shaped oligofluorenes with a central truxene core and from monofluorene to quaterfluorene arms. Introduction of solubilizing n-hexyl groups at both fluorene and truxene moieties results in highly soluble, intrinsically two-dimensional nanosized macromolecules T1-T4. The radius for the largest oligomer of ca. 3.9 nm represents one of the largest known star-shaped conjugated systems. Cyclic voltammetry experiments reveal reversible or quasi-reversible oxidation and reduction processes (Eox = +0.74 to 0.80 V, Ered = -2.66 to 2.80 eV vs Fc/Fc+), demonstrating excellent electrochemical stability toward both p- and n-doping, while the band gaps of the oligomers are quite high (EgCV = 3.20-3.40 eV). Close band gaps of 3.05-3.29 eV have been estimated from the electron absorption spectra. These star-shaped macromolecules demonstrate good thermal stability (up to 400-420 degrees C) and improved glass transition temperatures with an increase in length of the oligofluorene arms (from Tg = 63 degrees C for T1 to 116 degrees C for T4) and show very efficient blue photoluminescence (lambdaPL = 398-422 nm) in both solution (PhiPL = 70-86%) and solid state (PhiPL = 43-60%). Spectroelectrochemical experiments reveal that compounds T1-T4 are stable electrochromic systems which change their color reversibly from colorless in the neutral state (approximately 340-400 nm) to colored (from red to purple color; approximately 500-600 nm) in the oxidized state.     

Assembly and Dissociation of Α-Cyclodextrin [2]Pseudorotaxanes with α,ω-Bis(N-(N,N′-dimethylethylenediamine)alkane Threads

A series of novel bischelate bridging ligands, CH₃NH(CH₂)₂N(CH₃)(CH₂) _n N(CH₃)(CH₂)₂NHCH₃ (n = 9, 10, 11, and 12) were synthesized as hydrochloride salts and characterized by elemental analyses, electrospray mass spectrometry, and ¹H and ¹³C NMR spectroscopy. These ligands form [2]pseudorotaxanes with α-cyclodextrin (α-CD) and the stability constants have been determined from ¹H NMR titrations in D₂O. The kinetics and mechanism of the assembly and dissociation of a [2]pseudorotaxane in which α-CD has been threaded by the CH₃NH₂(CH₂)₂N(CH₃)(CH₂)₁₂N(CH₃)(CH₂)₂NH₂CH ₃ ²⁺ ligand were determined in aqueous solution using ¹H NMR spectroscopy. A weak inclusion of the dimethylethylenediamine end group precedes the passage of the α-CD onto the hydrophobic dodecamethylene chain.     

New organolithium addition methodology to diversely functionalized indoles

A novel synthetic approach to diversely functionalized indoles is described. Boc-protected ortho-aminostyrenes undergo an alkyllithium addition reaction, thereby generating a lithiated intermediate, which upon reaction with specific electrophiles sets up a cascade reaction process between the reacted electrophile and ortho-amino substituent, facilitating an in situ ring closure, followed by dehydration, to generate an indole ring system. This methodology is demonstrated by the synthesis of a range of 3,5-, 1,3,5-, and 2,3,5-substituted indoles.