Two point problems and analyticity of solutions of abstract parabolic equations

Fort ∈ [a, b], letA(t) be the unbounded operator inH ^0,p (G) associated with an elliptic-boundary value problem that satisfies Agmon’s conditions on the rays λ=±iτ, τ ≥0. Existence and uniqueness results are obtained for weak and strict solutions of two-point problems of the type (du/dt)−A(t) u(t) =f(t),E ₁(α)u (α)=u _α,E ₂ (β)u (β)=u _β. Here [α, β) χ- [a, b],E ₁ (α) andE ₂ (β) are spectral projections associated withA(α) andA(β) respectively, andA(α)E ₁ (α) and =A (β)E ₂ (β) are infinitesimal generators of analytic semigroups. WhenA(t) andf(t) are analytic in a convex, complex neighborhoodO of [a, b] we show that for someθ _i ,i=1,2, any solution ofdu/dt =A(t)u (t)=f(t) in [a, b] is analytic and satisfies the above equation in the setO∩{t; t ≠ a, t ≠ b, | arg (t −a) | <θ ₁, | arg (b −t) |θ ₂}. Research partially supported by N. N. F. grant at Brandeis University.     

Topology and measure in logics for region-based theories of space

Space, as we typically represent it in mathematics and physics, is composed of dimensionless, indivisible points. On an alternative, region-based approach to space, extended regions together with the relations of ‘parthood’ and ‘contact’ are taken as primitive; points are represented as mathematical abstractions from regions.Region-based theories of space have been traditionally modeled in regular closed (or regular open) algebras, in work that goes back to [5] and [21]. Recently, logics for region-based theories of space were developed in [3] and [19]. It was shown that these logics have both a nice topological and relational semantics, and that the minimal logic for contact algebras, ${\mathbb{L}}_{min}^{cont}$ (defined below), is complete for both.The present paper explores the question of completeness of ${\mathbb{L}}_{min}^{cont}$ and its extensions for individual topological spaces of interest: the real line, Cantor space, the rationals, and the infinite binary tree. A second aim is to study a different, algebraic model of logics for region-based theories of space, based on the Lebesgue measure algebra (or algebra of Borel subsets of the real line modulo sets of Lebesgue measure zero). As a model for point-free space, the algebra was first discussed in [2]. The main results of the paper are that ${\mathbb{L}}_{min}^{cont}$ is weakly complete for any zero-dimensional, dense-in-itself metric space (including, e.g., Cantor space and the rationals); the extension ${\mathbb{L}}_{min}^{cont}+(Con)$ is weakly complete for the real line and the Lebesgue measure contact algebra. We also prove that the logic ${\mathbb{L}}_{min}^{cont}+(Univ)$ is weakly complete for the infinite binary tree.     

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.     

Cellular properties and population asymptotics in the population balance equation

Proliferating cell populations at steady-state growth often exhibit broad protein distributions with exponential tails. The sources of this variation and its universality are of much theoretical interest. Here we address the problem by asymptotic analysis of the population balance equation. We show that the steady-state distribution tail is determined by a combination of protein production and cell division and is insensitive to other model details. Under general conditions this tail is exponential with a dependence on parameters consistent with experiment. We discuss the conditions for this effect to be dominant over other sources of variation and the relation to experiments.     

Theory of three-dimensional alignment by intense laser pulses

We introduce a theoretical framework for study of three-dimensional alignment by moderately intense laser pulses and discuss it at an elementary level. Several features of formal interest are noted and clarified. Our approach is nonperturbative, treating the laser field within classical and the material system within quantum mechanics. The theory is implemented numerically using a basis set of rotational eigenstates, transforming the time-dependent Schrodinger equation to a set of coupled differential equations where all matrix elements are analytically soluble. The approach was applied over the past few years to explore different adiabatic and nonadiabatic three-dimensional alignment approaches in conjunction with experiments, but its formal details and numerical implementation were not reported in previous studies. Although we provide simple numerical examples to illustrate the content of the equations, our main goal is to complement previous reports through an introductory discussion of the underlying theory.     

Quantum mechanics/molecular mechanics investigation of the chemical reaction in Dpo4 reveals water-dependent pathways and requirements for active site reorganization

The nucleotidyl-transfer reaction coupled with the conformational transitions in DNA polymerases is critical for maintaining the fidelity and efficiency of DNA synthesis. We examine here the possible reaction pathways of a Y-family DNA polymerase, Sulfolobus solfataricus DNA polymerase IV (Dpo4), for the correct insertion of dCTP opposite 8-oxoguanine using the quantum mechanics/molecular mechanics (QM/MM) approach, both from a chemistry-competent state and a crystal closed state. The latter examination is important for understanding pre-chemistry barriers to interpret the entire enzyme mechanism, since the crystal closed state is not an ideal state for initiating the chemical reaction. The most favorable reaction path involves initial deprotonation of O3'H via two bridging water molecules to O1A, overcoming an overall potential energy barrier of approximately 20.0 kcal/mol. The proton on O1A-P(alpha) then migrates to the gamma-phosphate oxygen of the incoming nucleotide as O3' attacks P(alpha), and the P(alpha)-O3A bond breaks. The other possible pathway in which the O3'H proton is transferred directly to O1A on P(alpha) has an overall energy barrier of 25.0 kcal/mol. In both reaction paths, the rate-limiting step is the initial deprotonation, and the trigonal-bipyramidal configuration for P(alpha) occurs during the concerted bond formation (O3'-P(alpha)) and breaking (P(alpha)-O3A), indicating the associative nature of the chemical reaction. In contrast, the Dpo4/DNA complex with an imperfect active-site geometry corresponding to the crystal state must overcome a much higher activation energy barrier (29.0 kcal/mol) to achieve a tightly organized site due to hindered O3'H deprotonation stemming from larger distances and distorted conformation of the proton acceptors. This significant difference demonstrates that the pre-chemistry reorganization in Dpo4 costs approximately 4.0 to 9.0 kcal/mol depending on the primer terminus environment. Compared to the higher fidelity DNA polymerase beta from the X-family, Dpo4 has a higher chemical reaction barrier (20.0 vs 15.0 kcal/mol) due to the more solvent-exposed active site.     

Optimal design of nanoplasmonic materials using genetic algorithms as a multiparameter optimization tool

An optimal control approach based on multiple parameter genetic algorithms is applied to the design of plasmonic nanoconstructs with predetermined optical properties and functionalities. We first develop nanoscale metallic lenses that focus an incident plane wave onto a prespecified, spatially confined spot. Our results illustrate the mechanism of energy flow through wires and cavities. Next we design a periodic array of silver particles to modify the polarization of an incident, linearly polarized plane wave in a desired fashion while localizing the light in space. The results provide insight into the structural features that determine the birefringence properties of metal nanoparticles and their arrays. Of the variety of potential applications that may be envisioned, we note the design of nanoscale light sources with controllable coherence and polarization properties that could serve for coherent control of molecular, electronic, or electromechanical dynamics in the nanoscale.     

Octahydroxypyridine[4]arene self-assembles spontaneously to form hexameric capsules and dimeric aggregates

In recent years it has been observed that resorcin[4]arenes and pyrogallol[4]arenes form hydrogen-bonded hexameric capsules in nonpolar solvents. In the present study we have used NMR spectroscopy, with an emphasis on diffusion NMR, to investigate the self-assembly and the aggregation mode of solutions of octahydroxypyridine[4]arene (1 b) in chloroform. In spectroscopic studies, the hexameric capsule of C-undecylresorcin[4]arene (2 b) was used as a reference compound and in some cases also as an internal reference. The current diffusion NMR spectroscopy study shows, in contrast to a previous report, that compound 1 b self-assembles spontaneously into hexameric and dimeric aggregates in solutions in chloroform. The (1)H NMR and diffusion NMR spectroscopic studies on a solution of 1 b in CHCl(3) show the presence of new upfield-shifted peaks, which diffuse with the same diffusion coefficient as the hexameric peaks in the spectrum. Therefore, these new upfield-shifted peaks were attributed to encapsulated CHCl(3) molecules. Interestingly, diffusion NMR measurements showed that the addition of trifluoroacetic acid (6.7 equiv), which had no effect on the hexameric capsules of 2 b, led to the disassembly of the hexamer and the dimer of 1 b into its monomers. Therefore, we conclude that compound 1 b self-assembles spontaneously into hexameric capsules in nonpolar organic solvents, as do resorcin[4]arenes 2 b and 2 c and pyrogallol[4]arenes 3 a and 3 b.     

Self-assembled ionophores from isoguanosine: diffusion NMR spectroscopy clarifies cation's and anion's influence on supramolecular structure

Cation-templated self-assembly of the lipophilic isoguanosine (isoG 1) with different monovalent cations (M(+)=Li(+), Na(+), K(+), NH(4) (+), and Cs(+)) was studied in solvents of different polarity by using diffusion NMR spectroscopy. Previous studies that did not use diffusion NMR techniques concluded that isoG 1 forms both pentamers (isoG 1)(5)M(+) and decamers (isoG 1)(10)M(+) in the presence of alkali-metal cations. The present diffusion NMR studies demonstrate, however, that isoG 1 does not form (isoG 1)(5)M(+) pentamers. In fact, the diffusion NMR data indicates that both doubly charged decamers of formula (isoG 1)(10)2 M(+) and singly charged decamers, (isoG 1)(10)M(+), are formed with lithium, sodium, potassium, and ammonium tetraphenylborate salts (LiB(Ph)(4), KB(Ph)(4), NaB(Ph)(4) and NH(4)B(Ph)(4)), depending on the isoG 1:salt stoichiometry of the solution. In the presence of CsB(Ph)(4), isoG 1 affords only the singly charged decamers (isoG 1)(10)Cs(+). By monitoring the diffusion coefficient of the B(Ph)(4) (-) ion in the different mixtures of solvents, we also concluded that the anion is more strongly associated to the doubly charged decamers (isoG 1)(10)2 M(+) than to the singly charged decamers (isoG 1)(10)M(+). The (isoG 1)(10)2 M(+) species can, however, exist in solution without the mediation of the anion. This last conclusion was supported by the finding that the doubly charged decamers (isoG 1)(10)2 M(+) also prevail in 1:1 CD(3)CN:CDCl(3), a solvent mixture in which the B(Ph)(4) (-) ion does not interact significantly with the self-assembled complex. These diffusion measurements, which have provided new and improved structural information about these decameric isoG 1 assemblies, demonstrate the utility of combining diffusion NMR techniques with conventional NMR methods in seeking to characterize labile, multicomponent, supramolecular systems in solution, especially those with high symmetry.