Professor V. Lakshmikantham,Editor Stochastic Analysis and Applications (1983–2012)

We use the stochastic calculus of variations for the fractional Brownian motion to derive formulas for the replicating portfolios for a class of contingent claims in a Bachelier and a Black–Scholes markets modulated by fractional Brownian motion. An example of such a model is the Black–Scholes process whose volatility solves a stochastic differential equation driven by a fractional Brownian motion that may depend on the underlying Brownian motion.     

An alternative approach to selective sea water oxidation for hydrogen production

Sea water electrolysis is one of the promising ways to produce hydrogen since it is available in plentiful supply on the earth. However, in sea water electrolysis toxic chlorine evolution is the preferred reaction over oxygen evolution at the anode. In this work, research has been focused on the development of electrode materials with a high selectivity for oxygen evolution over chlorine evolution. Selective oxidation in sea water electrolysis has been demonstrated by using a cation-selective polymer. We have used a perm-selective membrane (Nafion^®), which electrostatically repels chloride ions (Cl⁻) to the electrode surface and thereby enhances oxygen evolution at the anode. The efficiency and behaviour of the electrode have been characterized by means of anode current efficiency and polarization studies. The surface morphology of the electrode has been characterized by using a scanning electron microscope (SEM). The results suggest that nearly 100% oxygen evolution efficiency could be achieved when using an IrO₂/Ti electrode surface-modified by a perm-selective polymer.     

Efficient evaluation of sampling quality of molecular dynamics simulations by clustering of dihedral torsion angles and Sammon mapping

A direct conformational clustering and mapping approach for peptide conformations based on backbone dihedral angles has been developed and applied to compare conformational sampling of Met-enkephalin using two molecular dynamics (MD) methods. Efficient clustering in dihedrals has been achieved by evaluating all combinations resulting from independent clustering of each dihedral angle distribution, thus resolving all conformational substates. In contrast, Cartesian clustering was unable to accurately distinguish between all substates. Projection of clusters on dihedral principal component (PCA) subspaces did not result in efficient separation of highly populated clusters. However, representation in a nonlinear metric by Sammon mapping was able to separate well the 48 highest populated clusters in just two dimensions. In addition, this approach also allowed us to visualize the transition frequencies between clusters efficiently. Significantly, higher transition frequencies between more distinct conformational substates were found for a recently developed biasing-potential replica exchange MD simulation method allowing faster sampling of possible substates compared to conventional MD simulations. Although the number of theoretically possible clusters grows exponentially with peptide length, in practice, the number of clusters is only limited by the sampling size (typically much smaller), and therefore the method is well suited also for large systems. The approach could be useful to rapidly and accurately evaluate conformational sampling during MD simulations, to compare different sampling strategies and eventually to detect kinetic bottlenecks in folding pathways.     

Transforming a graph into a 1-balanced graph

Let G be a non-trivial, loopless graph and for each non-trivial subgraph H of G, let . The graph G is 1-balanced if γ(G), the maximum among g(H), taken over all non-trivial subgraphs H of G, is attained when H=G. This quantity γ(G) is called the fractional arboricity of the graph G. The value γ(G) appears in a paper by Picard and Queyranne and has been studied extensively by Catlin, Grossman, Hobbs and Lai. The quantity γ(G)−g(G) measures how much a given graph G differs from being 1-balanced. In this paper, we describe a systematic method of modifying a given graph to obtain a 1-balanced graph on the same number of vertices and edges. We obtain this by a sequence of iterations; each iteration re-defining one end-vertex of an edge in the given graph. After each iteration, either the value γ of the new graph formed is less than that of the graph from the previous iteration or the size of the maximal γ-achieving subgraph of the new graph is smaller than that of the graph in the previous iteration. We show that our algorithm is polynomial in time complexity. Further ways to decrease the number of iterations are also discussed.     

Torus quotients of homogeneous spaces — minimal dimensional Schubert varieties admitting semi-stable points

In this paper, for any simple, simply connected algebraic group G of type B,C or D and for any maximal parabolic subgroup P of G, we describe all minimal dimensional Schubert varieties in G/P admitting semistable points for the action of a maximal torus T with respect to an ample line bundle on G/P. We also describe, for any semi-simple simply connected algebraic group G and for any Borel subgroup B of G, all Coxeter elements τ for which the Schubert variety X(τ) admits a semistable point for the action of the torus T with respect to a non-trivial line bundle on G/B.     

Nonlinear boundary value problems and operators TT1

We consider nonlinear boundary value problems of the type $\text{L? + N? = 0}$ for the existence of solutions. It is assumed that L is a 2nth-order linear differential operator in the real Hilbert space S = L²[a, b] which admits a decomposition of the form $\text{L =}{}^{\mathrm{TT1}}$ where T is an nth-order linear differential operator and N is a nonlinear operator defined on a subspace of S. The decomposition of L induces a natural decomposition of the generalized inverse of L. Using the method of “alternative problems,” we split the boundary value problem into an equivalent system of two equations. The theory of monotone operators and the theory of nonlinear Hammerstein equations are then utilized to consider the solvability of the equivalent system.     

Organometallic chemistry of chiral diphosphazane ligands: Synthesis and structural characterisation

The diphosphazane ligands of the type, (C₂₀H₁₂O₂)PN(R)P(E)Y₂ (R = CHMe₂ or (S)-*CHMePh; E = lone pair or S; Y₂ = O₂C₂₀H₁₂ or Y = OC₆H₅ or OC₆H₄Me-4 or OC₆H₄OMe-4 or OC₆H₄Bu^t-4 or C₆H₅) bearing axially chiral 1,1'-binaphthyl-2,2′-dioxy moiety have been synthesised. The structure and absolute configuration of a diastereomeric palladium complex, [PdCl₂{ηsu2}-((O₂C₂₀H₁₂)PN((S)-*CHMePh)PPh₂] has been determined by X-ray crystallography. The reactions of [CpRu(PPh₃)₂Cl] with various symmetrical and unsymmetrical diphosphazanes of the type, X₂PN(R)PYY′ (R = CHMe₂ or (S)-*CHMePh; X = C₆H₅ or X₂ = O₂C₂₀H₁₂; Y=Y′= C₆H₅ or Y = C₆H₅, Y′ = OC₆H₄Me-4 or OC₆H₃Me₂-3,5 or N₂C₃HMe₂-3,5) yield several diastereomeric neutral or cationic half-sandwich ruthenium complexes which contain a stereogenic metal center. In one case, the absolute configuration of a trichiral ruthenium complex, viz. [Cp*Ruη₂-Ph₂PN((S)-*CHMePh)*PPh (N₂C₃HMe₂-3,5)Cl] is established by X-ray diffraction. The reactions of Ru₃(CO)₁₂ with the diphosphazanes (C₂₀H₁₂O₂)PN(R)PY₂ (R = CHMe₂orMe; Y₂=O₂C₂₀H₁₂or Y= OC₆H₅ or OC₆H₄Me-4 or OC₆H₄OMe-4 or OC₆H₄Bu^t-4 or C₆H₅) yield the triruthenium clusters [Ru₃(CO)₁₀{η-(O₂C₂₀H₁₂)PN(R)PY₂}], in which the diphosphazane ligand bridges two metal centres. Palladium allyl chemistry of some of these chiral ligands has been investigated. The structures of isomeric η³-allyl palladium complexes, [Pd(η³-l,3-R′₂-C₃H₃){η²-(rac)-(0₂C₂₀H₁₂)PN(CHMe₂)PY₂}](PF₆) (R′ = Me or Ph; Y = C₆H₅ or OC₆H₅) have been elucidated by high field two-dimensional NMR spectroscopic and X-ray crystallographic studies.     

Monotone iterative methods for nonlinear equations involving a noninvertible linear part

Summary We study monotone iterative methods for the numerical solution of a class of nonlinear elliptic boundary value problems by applying a theorem of Ortega and Rheinboldt. This generalizes the work of earlier authors to the case of nonlinear perturbations of linear problems with a nontrivial kernel.     

Structure,refinement, and function of human carbonic anhydrase-B

The three dimensional structure of human carbonic anhydrase-B has been model fitted to electron density maps in an interactive graphics display and improved by real space refinement and restrained least-squares refinement. The crystallographic R factor for the 5 to 3 Å data dropped from 41.5% to 36.5% after four cycles of least-squares refinement. The important residues involved in the function of the enzyme showed improved positional parameters after the refinement. Thus GLU106 and THR199 oxygen atoms are within hydrogen bond distance of each other after the refinement. Whereas they were very close to each other before the refinement. The procedures involved in the refinement and the implication of the structure to the mechanism of action of the enzyme are brought out.