A Coupled System of Integrodifferential Equations Arising in Liquidity Risk Model

We study the mathematical aspects of the portfolio/consumption choice problem in a market model with liquidity risk introduced in (Pham and Tankov, Math. Finance, 2006, to appear). In this model, the investor can trade and observe stock prices only at exogenous Poisson arrival times. He may also consume continuously from his cash holdings, and his goal is to maximize his expected utility from consumption. This is a mixed discrete/continuous time stochastic control problem, nonstandard in the literature. We show how the dynamic programming principle leads to a coupled system of Integro-Differential Equations (IDE), and we prove an analytic characterization of this control problem by adapting the concept of viscosity solutions. This coupled system of IDE may be numerically solved by a decoupling algorithm, and this is the topic of a companion paper (Pham and Tankov, Math. Finance, 2006, to appear).     

A new regularized method for two dimensional nonhomogeneous backward heat problem

We consider the problem of finding, from the final data u(x,y,T)=g(x,y), the initial data u(x,y,0) of the temperature function u(x,y,t),(x,y)I=(0,π)×(0,π),t[0,T] satisfying the following system

The problem is severely ill-posed. In this paper a simple and convenient new regularization method for solving this problem is considered. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.     

Solving polynomial optimization problems via the truncated tangency variety and sums of squares

Let f,g_i,i=1,…,l,h_j,j=1,…,m, be polynomials on Rⁿ and S?{x∈Rⁿ∣g_i(x)=0,i=1,…,l,h_j(x)≥0,j=1,…,m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.     

Er3+/Yb3+-activated silica-hafnia planar waveguides for photonics fabricated by rf-sputtering

95.8SiO₂–4.2HfO₂ planar waveguide activated by 0.2 mol% Er and 0.2 mol% Yb was fabricated by multi-target rf-sputtering technique. The optical parameters were measured by an m-line apparatus operating at 543.5, 632.8, 1319 and 1542 nm. The waveguide compositions were investigated by energy dispersive spectroscopy. The waveguide exhibits a single propagation mode at 1.3 and 1.5 μm with an attenuation coefficient of 0.2 dB/cm at 1.5 μm. The emission of ⁴I_13/2 → ⁴I_15/2 transition of Er³⁺ ion, with a 42 nm bandwidth was observed upon TE₀ mode excitation at 980 and 514.5 nm. Photoluminescence excitation spectroscopy was used to obtain information about the effective excitation efficiency of Er³⁺ ions by co-doping with Yb³⁺ ions. Channel waveguide in rib configuration were fabricated by wet etching process in the active film.     

Periodic Motions of Stokes and Navier–Stokes Flows Around a Rotating Obstacle

We prove the existence and uniqueness of periodic motions to Stokes and Navier–Stokes flows around a rotating obstacle \({D \subset \mathbb{R}^3}\) with the complement \({\Omega = \mathbb{R}^3 \backslash D}\) being an exterior domain. In our strategy, we show the C _b -regularity in time for the mild solutions to linearized equations in the Lorentz space \({L^{3,\infty}(\Omega)}\) (known as weak-L ³ spaces) and prove a Massera-typed Theorem on the existence and uniqueness of periodic mild solutions to the linearized equations in weak-L ³ spaces. We then use the obtained results for such equations and the fixed point argument to prove such results for Navier–Stokes equations around a rotating obstacle. We also show the stability of such periodic solutions.     

Efficient and continuous monoacylation with superior selectivity of symmetrical diamines in microreactors

Efficient and continuous monoacylation of symmetrical diamines performed in microreactors yielded superior selectivity to that predicted by statistical considerations. It is highly valuable that the kinetically controlled product in high yields was achieved without any special catalyst at ambient temperature.     

Structural role of counterions adsorbed on self-assembled peptide nanotubes

Among noncovalent forces, electrostatic ones are the strongest and possess a rather long-range action. For these reasons, charges and counterions play a prominent role in self-assembly processes in water and therefore in many biological systems. However, the complexity of the biological media often hinders a detailed understanding of all the electrostatic-related events. In this context, we have studied the role of charges and counterions in the self-assembly of lanreotide, a cationic octapeptide. This peptide spontaneously forms monodisperse nanotubes (NTs) above a critical concentration when solubilized in pure water. Free from any screening buffer, we assessed the interactions between the different peptide oligomers and counterions in solutions, above and below the critical assembly concentration. Our results provide explanations for the selection of a dimeric building block instead of a monomeric one. Indeed, the apparent charge of the dimers is lower than that of the monomers because of strong chemisorption. This phenomenon has two consequences: (i) the dimer-dimer interaction is less repulsive than the monomer-monomer one and (ii) the lowered charge of the dimeric building block weakens the electrostatic repulsion from the positively charged NT walls. Moreover, additional counterion condensation (physisorption) occurs on the NT wall. We furthermore show that the counterions interacting with the NTs play a structural role as they tune the NTs diameter. We demonstrate by a simple model that counterions adsorption sites located on the inner face of the NT walls are responsible for this size control.     

Folding pentapeptides into left and right handed alpha helices

Left or right handed alpha helicity can be induced in a pentapeptide (ANGYG) by appending left or right handed helical cycles as chiral templates. This sequence corresponds to a rare left handed helix found in the protein alanine racemase. Circular dichroism spectra reveal that pentapeptide ANGYG has no detectable structure in aq phosphate buffer, that it is an ambidextrous peptide in that it can be directed to fold into either a left handed or right handed alpha helix in water, with greater propensity for the uncommon left handed than the normal right handed conformation. A helix-inducing cyclic peptide at both ends of this peptide was more effective at inducing alpha helicity than a single cyclic peptide at one end. The alpha helical cyclic peptides provide novel tools for folding short peptides into thermodynamically unstable helices in water, and for studying factors that control chirality and helix induction.