Nonoscillatory solutions of linear differential equations with deviating arguments

Summary The equation to be considered is of the form (1) x(ⁿ)(t)+p(t)x(g(t))=0 (t>a), where =±1, p(t) > 0 for ta and g(t) as t. It is well- known that a nonoscillatory solution x(t) of (1) satisfies (2) x(t)x(ⁱ)(t)>0 (0il), (–1)^i–lx(t)x(ⁱ)(t)>0 (lin) for some integer l, 0ln, (–1)^n–l–1=1. In this paper, for a given l such that 0n–l–1=1, necessary conditions and sufficient conditions are found for (1) to have a solution x(t) which satisfies (2), and a necessary and sufficient condition is established in order that for every >0 the equation x(ⁿ)(t)+p(t)x(g(t))=0 (t>a) has a solution x(t) which satisfies (2). Related results are also contained.     

Synthesis of optically active (+)-19-nortestosterone by asymmetric bis-annulation reaction

Michael reaction of 1,7-pctadien-3-one with 2-methylcyclopentane-1,3-dione, followed by intramolecular aldol condensation promoted by L-amino acids produced the optically active (+)-4-(3-butenyl)-7a-methyl-5,6,7,7a-tetrahydroindane-1,5-dione in high chemical and optical yields. The PdCl₂-catalyzed oxidation of the terminal double bond gave the methyl ketone, which had 76% optical purity and was made 100% optically pure by recrystallization. Then aldol condensation afforded the tricyclic ketone, which was alkylated with 3-butenyl iodide to afford (?)-3β-t-butoxy-2,3,3a,4,5,7,8,9,9aβ,9bα-decahydro-6-(3-butenyl)-3aβ-methyl-1H-benz[e]inden-7-one. The synthesis of this compound means the total synthesis of (+)-19-nortestosterone.     

On the photolysis of homo- and copolymers of phenylisopropenylketone

Poly(phenylisopropenylketone (PPIK) and copolymers of PIK and methylmethacrylate (MMA) or styrene (St) were irradiated in benzene solution at 30° with 313 nm light (stationary irradiations) or with 347 nm light (flash photolysis experiments). Homo PPIK undergoes main chain degradation (β-scission) with ø(S) ≈ 0.05. The quantum yield for α-scission is ø(α) = 0.3. For copolymers of MMA and PIK (1 to 15 mol%), ø(S) is 0.04 independent of the copolymer composition. With copolymers of St and PIK, ø(S) was found to increase with decreasing PIK content and to approach 0.15 for very small PIK contents. The flash photolysis experiments showed: (a) the carbonyl triplet decay rate constant k_T(6 × 10⁶ sec^?1) for CP-MMA-PIK samples is independent of copolymer composition but is lower than for homo PPIK (1 × 10⁷ sec^?1). In CP-St-PIK samples k_T decreases with decreasing PIK content [from 8 × 10⁶ sec^?1 (12 mol% PIK) to 3 × 10⁶ sec^?1 (1 mol% PIK)]; (b) the transient spectra of triplet decay products indicate the formation of benzoyl radicals in the cases of PPIK and CP-MMA-PIK, and the formation of various different species in the case of CP-St-PIK.The results are consistent with the following concept. In homo PPIK and CP-MMA-PIK, α-scission (Norrish type I) is the dominant chemical route of triplet deactivation. In CP-St-PIK, however, type II processes become more and more important as the PIK content decreases.     

On the Lp–Lq maximal regularity for the linear thermoelastic plate equation in a bounded domain

This paper is concerned with the thermoelastic plate equations in a domain Ω: subject to the boundary condition: u|=D_νu|=θ|=0 and initial condition: (u, u_t, θ)|_t=0=(u₀, v₀, θ₀). Here, Ω is a bounded domain in ?ⁿ(n≧2). We assume that the boundary ?Ω of Ω is a C⁴ hypersurface. We obtain an L_p–L_q maximal regularity theorem. Copyright © 2008 John Wiley & Sons, Ltd.     

On the rate of convergence of solutions in free boundary problems via penalization

The rate of convergence of approximate solutions via penalization for free boundary problems are concerned. A key observation is to obtain global bounds of penalized terms which give necessary estimates on integrations by the nonlinear adjoint method by L.C. Evans.     

Fredholm operators,evolution semigroups,and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ  -periodic evolutionary process _{{U(t,s)}t≥s}${\{U(t,s)\}}_{t\ge s}$ on the space _Pτ(X)$P_{\tau }(X)$ of all τ-periodic continuous X  -valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−?F(p,?)$Lp=-?F(p,?)$ under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0)$U(\tau ,0)$. The proof is based on a new decomposition of the space _Pτ(X)$P_{\tau }(X)$ by constructing a right inverse of L.     

Characterizations of linear Volterra integral equations with nonnegative kernels

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.     

Construction of Perfect Crystals Conjecturally Corresponding to Kirillov-Reshetikhin Modules over Twisted Quantum Affine Algebras

Assuming the existence of the perfect crystal bases of Kirillov-Reshetikhin modules over simply-laced quantum affine algebras, we construct certain perfect crystals for twisted quantum affine algebras, and also provide compelling evidence that the constructed crystals are isomorphic to the conjectural crystal bases of Kirillov-Reshetikhin modules over twisted quantum affine algebras.