Sequentially continuous non-linear fundamental systems of solutions of affine equations in locally convex spaces

According to the Mickael's selection theorem any surjective continuous linear operator from one Fr\'echet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if $E$ is a Fr\'echet space and $T:E\to E$ is a continuous linear operator such that the Cauchy problem $\dot x=Tx$, $x(0)=x_0$ is solvable in $[0,1]$ for any $x_0\in E$, then for any $f\in C([0,1],E)$, there exists a continuos map $S:[0,1]\times E\to E$, $(t,x)\mapsto S_tx$ such that for any $x_0\in E$, the function $x(t)=S_tx_0$ is a solution of the Cauchy problem $\dot x(t)=Tx(t)+f(t)$, $x(0)=x_0$ (they call $S$ a fundamental system of solutions of the equation $\dot x=Tx+f$). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Fr\'echet spaces and strong duals of Fr\'echet--Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.

Improving neural network training solutions using regularisation

This paper describes the application of regularisation to the training of feedforward neural networks, as a means of improving the quality of solutions obtained. The basic principles of regularisation theory are outlined for both linear and nonlinear training and then extended to cover a new hybrid training algorithm for feedforward neural networks recently proposed by the authors. The concept of functional regularisation is also introduced and discussed in relation to MLP and RBF networks. The tendency for the hybrid training algorithm and many linear optimisation strategies to generate large magnitude weight solutions when applied to ill-conditioned neural paradigms is illustrated graphically and reasoned analytically. While such weight solutions do not generally result in poor fits, it is argued that they could be subject to numerical instability and are therefore undesirable. Using an illustrative example it is shown that, as well as being beneficial from a generalisation perspective, regularisation also provides a means for controlling the magnitude of solutions. (C) 2001 Elsevier Science B.V. All rights reserved.

Nutrient regulation of pancreatic beta-cell function in diabetes: problems and potential solutions

increasing prevalence of obesity combined with longevity will produce an epidemic of Type 2 (non-insulin-dependent) diabetes in the next 20 years. This. disease is associated with defects in insulin secretion, specifically abnormalities of insulin secretory kinetics and pancreatic beta-cell glucose responsiveness. Mechanisms underlying beta-cell dysfunction include glucose toxicity, lipotoxicity and beta-cell hyperactivity. Defects at various sites in beta-cell signal transduction pathways contribute, but no single lesion can account for the common form of Type 2 diabetes. Recent studies highlight diverse beta-cell actions of GLP-1 (glucagon-like peptide-1) and GIP (glucose-dependent insulinotropic polypeptide). These intestinal hormones target the beta-cell to stimulate glucose-dependent insulin secretion through activation of protein kinase A and associated pathways. Both increase gene expression and proinsulin biosynthesis, protect against apoptosis and stimulate replication/neogenesis of beta-cells. Incretin hormones therefore represent an exciting future multi-action solution to correct beta-cell defect in Type 2 diabetes.

Density-viscosity product of small-volume ionic liquid samples using quartz crystal impedance analysis

Quartz crystal impedance analysis has been developed as a technique to assess whether room-temperature ionic liquids are Newtonian fluids and as a small-volume method for determining the values of their viscosity-density product, rho eta. Changes in the impedance spectrum of a 5-MHz fundamental frequency quartz crystal induced by a water-miscible room-temperature ionic liquid, 1-butyl-3-methylimidazolium. trifluoromethylsulfonate ([C(4)mim][OTf]), were measured. From coupled frequency shift and bandwidth changes as the concentration was varied from 0 to 100% ionic liquid, it was determined that this liquid provided a Newtonian response. A second water-immiscible ionic liquid, 1-butyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide [C(4)mim][NTf2], with concentration varied using methanol, was tested and also found to provide a Newtonian response. In both cases, the values of the square root of the viscosity-density product deduced from the small-volume quartz crystal technique were consistent with those measured using a viscometer and density meter. The third harmonic of the crystal was found to provide the closest agreement between the two measurement methods; the pure ionic liquids had the largest difference of similar to 10%. In addition, 18 pure ionic liquids were tested, and for 11 of these, good-quality frequency shift and bandwidth data were obtained; these 12 all had a Newtonian response. The frequency shift of the third harmonic was found to vary linearly with square root of viscosity-density product of the pure ionic liquids up to a value of root(rho eta) approximate to 18 kg m(-2) s(-1/2), but with a slope 10% smaller than that predicted by the Kanazawa and Gordon equation. It is envisaged that the quartz crystal technique could be used in a high-throughput microfluidic system for characterizing ionic liquids.

The electrochemical oxidation of hydrogen at activated platinum electrodes in room temperature ionic liquids as solvents

The oxidation of hydrogen was studied at an activated platinum micro-electrode by cyclic voltammetry in the following ionic liquids: [C(2)mim][NTf2], [C(4)mim][NTf2], [N-6.2.2.2][NTf2], [P-14.6.6.6][NTf2], [C(4)mim][OTf], [C(4)mim][BF4] [C(4)mim][PF6], [C(4)mim][NO3], [C(6)mim]Cl and [C(6)mim][FAP] (where [C(n)mim](+) = 1-alkyl-3-methylimidazolium, [N-6,N-2,N-2,N-2](+) = n-hexyltriethylammonium, [P-14,P-6,P-6,P-6](+) = tris(n-hexyltetradecyl) phosphonium, [NTf2](-) = bis(trifluoromethylsulfonyl)amide, [OTf] = trifluoromethlysulfonate and [FAP](-) = tris(perfluoroethyl)trifluorophosphate). Activation of the Pt electrode was necessary to obtain reliable and reproducible voltammetry. After activation of the electrode, the H-2 oxidation waves were nearly electrochemically and chemically reversible in [C(n)mim][NTf2] ionic liquids, chemically irreversible in [C(6)mim]Cl and [C(4)mim][NO3], and showed intermediate characteristics in OTf-, [BF4](-), [PF6](-), [FAP](-) and other [NTf2](-)-based ionic liquids. These differences reflect the contrasting interactions of protons with the respective RTIL anions. The oxidation peaks are reported relative to the half-wave potential of the cobaltocenium/cobaltocene redox couple in all ionic liquids studied, giving an indication of the relative proton interactions of each ionic liquid. A preliminary temperature study (ca. 298-333 K) has also been carried out in some of the ionic liquids. Diffusion coefficients and solubilities of hydrogen at 298 K were obtained from potential-step chronoamperometry, and there was no relationship found between the diffusion coefficients and solvent viscosity. RTILs possessing [NTf2](-) and [FAP](-) anions showed the highest micro-electrode peak currents for the oxidation in H-2 saturated solutions, with[C(4)mim][NTf2] toeing the most sensitive. The large number of available RTIL anion/cation pairs allows scope for the possible electrochemical detection of hydrogen gas for use in gas sensor technology. (c) 2008 Elsevier B.V. All rights reserved.