A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

Synthesis and optical resolution of a series of inherently chiral calix[4]crowns with cone and partial cone conformations

A series of inherently chiral calix[4]arenes with cone and partial cone conformations and with crown ether moieties of variable size have been readily synthesized. By taking advantage of the carboxy appendage on the lower rim, these were condensed with the chiral auxiliary (S)-BINOL to form diastereomers which, in most cases, could be separated by preparative TLC, or more desirably, by column chromatography on silica gel (diastereomeric excess >99 % based on HPLC analysis). Seven enantiopure antipodes of inherently chiral calix[4]crowns were obtained after hydrolysis. It has been found that both the size of the crown moiety and alkylation of the last phenolic hydroxy group (accompanied with or without a change in the conformation) affect the separation of the diastereomers.     

Selective Transport of Lead(II) and Strontium(II) Through a Crown Ether-Based Polymer Inclusion Membrane Containing Dialkylnaphthalenesulfonic Acid

New polymer inclusion membranes (PIMs) containing 18-membered crownethers and dialkylnaphthalenesulfonic acid are proposed for Sr2+ and Pb2+removal from nitric acid solutions. The influence of source phasecomposition and stripping agents was characterized and permeabilitycoefficients were calculated. The PIMs are easy to prepare and may be usefulin separation and concentration procedures for these cations from complexmixtures such as nuclear waste. Long-term stability was obtained for atleast several weeks of constant use during which no significant change ofpermeability was observed.     

The structure of 1-formyl-3-phenyl-Δ-pyrazoline in the gas phase (DFT calculations), in solution (NMR) and in the solid state (X-ray crystallography)

The molecular structure of 1-formyl-3-phenyl-Δ²-pyrazoline was determined by X-ray crystallography (triclinic, P-1). The geometry thus obtained was compared with that obtained by DFT calculations. The GIAO method was used to calculate absolute shieldings, which agree conveniently with those measured by ¹³C and ¹⁵N NMR. The title compound appears to be an essentially planar molecule.     

Molecular structures of benzoylbenzo-12-crown-4 and diphenylacetylbenzo-12-crown-4 and possible reasons for different conformational states of their macrocycles

The molecular structures of (benzoyl)benzo-12-crown-4 (a=22.387,b=4.503,c=16.167 Å,d _calc=1.34 g cm^–3,Z=4, space groupP2₁ nc, DAR-UM, Cu-K,R=0.056) and (diphenylacetyl)benzo-12-crown-4 (a=8.866,b=23.337,c=10.737 Å;d _calc=1.25 g cm^–3,Z=4, space groupP2₁2₁2₁, DAR-UM, Cu-K, R=0.056) have been investigated. The differences in the conformations of the macrocycles and the degree of conjugation between the benzene ring orbitals and the lone pairs on the adjacent oxygen atoms in the macrocycle are discussed.Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 11, pp. 1906–1911, November, 1993.     

SYNTHESIS AND CHARACTERIZATION OF PHTHALAZINONE POLY(ARYL ETHER SULFONE KETONE)WITH CARBONYL GROUP*

A kind of novel heat-resistant, high performance engineering thermoplastic phthalazinone poly(aryl ether sulfoneketone) (PPESK) containing a carboxyl group in its side chain was prepared by the nucleophilic displacement reaction of 4-(4-hydroxylphenyl)-1(2H)-phthalazinone with di(4-chlorophenyl) sulfone, 4,4'-difluoro-benzophenone and phenolphthalin insulfolane in the presence of K_2CO_3 to produce high molecular weight polymers which can be dissolved in some polarsolvents such as chloroform and nitrobenzene at room temperature and can be easily can into flexible, yellowish andtransparent films. PPESK is an amorphous polymer having a decomposition temperature above 400℃, which indicates that ithas high thermal stability. At the same time, the thermal properties of PPESKs with dicyandiamide (DICY) as curing agentindicated that the heat-resistance properties of the PPESKs are improved after curing. The apparent activation energy (ΔE) ofthe cross-linking reaction and the reaction order (n) of PPESK/DICY were found to be 52.2 kJ/mol and ca. 1.0, respectively.Therefore, the cross-linking reaction is approximately a first order reaction.     

Synthesis and extraction properties of new chromogenic azo-calix[4]dibenzothiacrown ethers

Two novel chromogenic cone calix[4]dibenzothiacrown ethers 3 and 4 in which nitrophenylazo groups attached at the phenyl ring of dibenzothiacrown unit were described. The extraction properties of 3 and 4 toward different transition metal ions have been studied using conductometric technique and found to exhibit Cu²⁺ and Hg²⁺ selectivity with very high stability constants range from log K _assoc = 5.19 to log K _assoc = 8.72.     

Optimization of alkali-alkalkine earth cation separation on weak cation-exchangers

Summary The elution behavior of alkali and earth alkaline cations has been studied on a newly synthesized weak cationexchanger (WCX). Applying this stationary phase isocratic separation of these cations is possible. The dependence of retention on competing ion concentration is described via a simple model. The influence of crown ethers as well as organic eluent additives on selectivity and peak efficiency is demonstrated. With optimized separation such cations can be determined with detection limits of about 0.5 ppm using a conductivity detector without ion suppression.     

Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

Stochastic Navier-Stokes equations

We construct a solution to stochastic Navier-Stokes equations in dimension n4 with the feedback in both the external forces and a general infinite-dimensional noise. The solution is unique and adapted to the Brownian filtration in the 2-dimensional case with periodic boundary conditions or, when there is no feedback in the noise, for the Dirichlet boundary condition. The paper uses the methods of nonstandard analysis.The research of this author was supported by an SERC Grant.