Boundedness of solutions of a system of integro-differential equations

Let b: [?1, 0] →$\text{R}$ be a nondecreasing, strictly convex C²-function with b(? 1) = 0, and let g: $\text{R}$ⁿ → $\text{R}$ⁿ be a locally Lipschitzian mapping, which is the gradient of a function G: $\text{R}$ⁿ → $\text{R}$. Consider the following vector-valued integro-differential equation of the Levin-Nohel type

$\text{x}\text{?}\text{(t)=?∝}{}_{\mathrm{?1}}{}^0\text{b(θ)g(x(t + θ))dθ}$

. (E) This equation is used in applications to model various viscoelastic phenomena. By LaSalle's invariance principle, every bounded solution x(t) goes to a connected set of zeros of g, as time t goes to infinity. It is the purpose of this paper to give several geometric criteria assuring the boundedness of solutions of (E) or some of its components.     

Silicate Xerogels with Dopant-Induced Chirality

A series of silicate xerogels with entrapped chiral amino acids have been obtained via sol-gel technology. The transparent, glassy samples obtained exhibit chirality in the bulk due to the presence of the entrapped asymmetric molecules. Measurements of the optical activity of the doped xerogel samples revealed that the entrapment did not significantly influence the optical activity observed for liquid solutions of the amino acids. Thus, the sol-gel method enables the preparation of amorphous optical materials exhibiting properties of strictly spatially defined molecular systems. Apart from the obvious optical applications, such porous materials with asymmetric centers might find interesting applications in chiral chemical syntheses and separations.     

Effect of temperature and concentration on the structure of N-methylacetamide-water complexes: Near-infrared spectroscopic study

Generalized two-dimensional (2D) FT-NIR correlation spectroscopy and chemometric methods have been used to study temperature-dependent spectral changes in pure N-methylacetamide (NMA) and NMA-water mixtures. We also examined the effect of varying water content on the structure of the mixture. It has been found that the extent of self-association of NMA in CCl4 is very high; the association occurs even at concentration of 0.001 M. In the pure liquid NMA, the population of the monomers is negligible and the structure is dominated by the linear associates. An increase in temperature reduces the number of hydrogen bonds, but in contrast to alcohols their strength remains nearly the same. This reflects a difference in the mechanism of thermal breaking of the associates of NMA and alcohols. The present results reveal that the interaction between NMA and water in the NMA-rich region (X(H2O) < 0.1) does not have a significant effect on the intrinsic structure of NMA. The structure of NMA is dominant, and the molecules of water do not form separate clusters but are dispersed and incorporated into the structure of NMA. We did not observe the presence of the free OH groups in the mixture. This led to the suggestion that each molecule of water forms two hydrogen bonds to two different molecules of NMA. An analysis of the asynchronous spectra reveals that most of the peaks observed in the asynchronous spectra, constructed from the temperature-dependent data, simply result from the frequency shift. This assumption is supported by the simulation studies.     

Single nanocrystal arrays on patterned poly(ethylene glycol) copolymer microstructures using selective wetting and drying

Single nanocrystal arrays were fabricated on sub-microwells of poly(ethylene glycol) (PEG) copolymer using selective wetting on the hydrophilic regions of the exposed substrate surface and subsequent drying. Templates were produced by molding a thin film of a PEG-based random copolymer on hydrophilic substrates such as glass or silicon dioxide. The polymeric microstructures provide a topographical barrier around the well, which makes it possible to create nanocrystal arrays with controlled geometrical features. The size of the nanocrystal was found to decrease with decreasing well size and also decrease with decreasing topological height. A simple empirical equation was derived to predict the size of the crystal as a function of the pattern size and height, which is in good agreement with the experimental data.     

Nanostructured carbon arrays from block copolymers of polyacrylonitrile

Arrays of graphitic carbon nanoclusters were obtained by pyrolysis of nanoscale phase-separated block copolymers of polyacrylonitrile and poly(n-butyl acrylate). Upon heating in an inert atmosphere to temperatures ranging from approximately 400 to 1200 degrees C, polyacrylonitrile domains were converted into carbon nanoclusters, maintaining the overall shape and spacing, whereas the poly(n-butyl acrylate) phase was sacrificed. Preservation of the original nanoscale morphology of a block copolymer was possible only if pyrolysis was preceded by oxidation at temperatures of approximately 230 degrees C, in analogy with thermal stabilization of polyacrylonitrile precursor in the process used in the manufacturing of carbon fibers. Preorganization of the carbon precursor through self-assembly in block copolymers of polyacrylonitrile appears to be an attractive and robust strategy for templated synthesis of well-defined nanostructured carbon materials.     

Microchips as Controlled Drug-Delivery Devices

Controlled-release systems are common in a number of product areas, including foods, cosmetics, pesticides, and paper. Microencapsulated systems, for example, are used for the release of flavors and vitamins in foods, fragrances in perfumes, and inks in carbonless copy paper. Controlled-release systems for drug delivery first appeared in the 1960s and 1970s. In the past three decades, the number and variety of controlled release systems for drug-delivery applications has increased dramatically. Many of these use polymers having particular physical or chemical characteristics such as biodegradability, biocompatibility, or responsiveness to pH or temperature changes. However, recent advances in the field of microfabrication have created the possibility of a new class of controlled-release systems for drug delivery, namely, that of small, programmable devices. Their small size, potential for integration with microelectronics, and ability to store and release chemicals on demand could make controlled-release microchips useful in a number of areas, including medical diagnostics, analytical chemistry, chemical detection, industrial process monitoring and control, combinatorial chemistry, microbiology, and fragrance delivery. More importantly, drug-delivery microchips resulting from this convergence of controlled release and microfabrication technologies may provide new treatment options to clinicians in their fight against disease.     

Trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations

Consider the following nonlinear Dirichlet boundary value problems:

$\text{D}{}_{\mathrm{t}}\text{u(t,x)=Lu(t,x)+f(u(t,x)),}\text{t?0,x∈Ω}\text{u(t,x)=0,}\text{t?0,x∈?Ω}$

(1.1)

$\text{D}{}_{\mathrm{t}}\text{D}{}_{\mathrm{t}}\text{u(t,x)=Lu(t,x)+}\text{α}\text{L(D}{}_{\mathrm{t}}\text{u(t,x))+f(u(t,x)),}\text{α}\text{>0,t?0,x∈Ω}\text{u(t,x)=0,}\text{t?0,x∈?Ω}$

(1.2)

$\text{Lu(x)+f(u(x)),}\text{x∈Ω}\text{u(x)=0,}\text{x∈?Ω}$

. (1.3) In all of these equations, f: R → R is a locally Lipschitzian asymptotically linear function with positive asymptotic slope, f(0) = 0, and L is a self-adjoint, negativedefinite and strongly elliptic second-order differential operator on a smooth domain Ω in Rⁿ. The solutions of (1.1) and (1.2) generate semiflows which are not pointdissipative and whose equilibria are determined by solutions of (1.3). In this paper, using an extension (due to the present author) of Conley's Morse index theory to noncompact spaces, we prove not only the existence of positive solutions of (1.3) (a result shown earlier by Peitgen and Schmitt using different methods), but also show the existence of (nonconstant) heteroclinic orbits of (1.1) and (1.2) joining two sets of equilibria.     

The formation of propylene fumarate oligomers for use in bioerodible bone cement composites

Poly(propylene fumarate) (PPF) oligomers were synthesized by step polymerization using bis(2-hydroxypropyl fumarate) or propylene bis(hydrogen maleate) as starting materials. Oligomers possessing identical degrees of polymerization (DP), but varying in their end group character (either hydroxyl or carboxyl) were first prepared and characterized, then used as part of a bone cement preparation consisting of oligomer, tricalcium phosphate, calcium carbonate, and methyl methacrylate. Compressive strength of the resulting composite appeared to be dependent on both the degree of polymerization of the PPF, and the nature of the oligomers' end groups.     

Entropy-enthalpy compensations in solutions of dual character molecules with polymeric chromatographic liquid phases

Careful gas chromatographic studies provide thermodynamic data for insights into solution processes in nonvolatile solvents. Using 24 solutes and five stationary phases, several entropy-enthalpy compensation effects in the thermodynamics of solution were identified. Despite solute structure differences, when excess enthalpy and entropy of solutions were examined, entropy-enthalpy compensation effects were found in solvents dominated by single types of interaction: squalane and, to some extent, methoxy poly(ethylene oxide) (PEO). The main reason for the absence of linearity in other solvents is pure solute state interactions in the reference state and the multicharacter nature of solvents. In this study, consideration of solute state interactions was removed through examination of the thermodynamics of transfer between solvent pairs. It was found that solute transfers from squalane to poly[methyl(trifluoropropyl)siloxane] (QF-1) and to poly(methylphenyl) (DC-550) also gave linear relationships. The former system contains a second correlation for ester type solutes. The transfer data for squalane to poly(methylsiloxane) (DC-200) had smaller ranges and were more scattered. The effects of derivatizing groups on the transfer enthalpy and entropy were treated as a summation of hydrocarbon cores with the derivative groups. The group properties of transfer then also show entropy-enthalpy compensation effects. Many solution effects could be explained on the basis of solvent composition and local interactions with solutes.