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.)     

Influence of branch length asymmetry on the electrophoretic mobility of rigid rod-like DNA

The electrophoretic mobility of three-arm asymmetric star DNA molecules, produced by incorporating a short DNA branch at the midpoint of rigid-rod linear DNA fragments, is investigated in polyacrylamide gels. We determine how long the added branch must be to separate asymmetric star DNA from linear DNA with the same total molecular weight. This work focuses on two different geometric progressions of small DNA molecules. First, branches of increasing length were introduced at the center of a linear DNA fragment of constant length. At a given gel concentration, we find that relatively small branch lengths are enough to cause a detectable reduction in electrophoretic mobility. The second geometric progression starts with a small branch on a linear DNA fragment. As the length of this branch is increased, the DNA backbone length is decreased such that the total molar mass of the molecule remains constant. The branch length was then increased until the asymmetric branched molecule becomes a symmetric three-arm star polymer, allowing the effect of molecular topology on mobility to be studied independent of size effects. DNA molecules with very short branches have a mobility smaller than linear DNA of identical molar mass. The reason for this change in mobility when branching is introduced is not known, however, we explore two possible explanations in this article. (i) The branched DNA could have a greater interaction with the gel than linear DNA, causing it to move slower; (ii) the linear DNA could have modes of motion or access to pores that are unavailable to the branched DNA.     

Topological effect on MO energies,IV. The total π-electron energy ofS- andT-isomers

A novel general property of theS- andT-isomers (a concept which has been introduced and elaborated elsewhere^{1, 2}) of alternant hydrocarbons is demonstrated, namely that due to the HMO total -electron energy theS-isomer should always be more stable than theT-isomer. Some other classes of conjugated isomers are also constructed, for which similar inequalities are derived.

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Topologischer Effekt bei MO-Energien, 4. Mitt.: Die -Elektronen-Gesamtenergie vonS- undT-Isomeren

Zusammenfassung Es wird allgemein gezeigt, daß bei einemS-T-Isomerenpaar (beschrieben in^{1, 2}) eines alternierenden Kohlenwasserstoffes infolge der HMO -Elektronen-Gesamtenergie dasS-Isomere stets stabiler sein sollte als dasT-Isomere. Weitere Klassen konjugierter Isomere werden konstruiert und ähnliche Ungleichungen angegeben.

     

The topology of energy hypersurfaces III. The fundamental group of reaction mechanisms on potential energy hypersurfaces

The family of all possible reaction mechanisms on a potential surface has an algebraic structure with potential applications in quantum chemical molecular design and synthesis planning.Transformation properties and equivalence relations of reaction paths on potential energy hypersurfaces lead to a topological definition of reaction mechanisms. The family of all fundamental reaction mechanisms on the hypersurface has a group structure,the fundamental group of an appropriately defined topological space. Isomorphism and homomorphism relations between fundamental groups of reaction mechanisms are used to characterize the chemically important topological properties of various subsets of a hypersurface, or those of different excited state hypersurfaces.     

Topological Space of the Chemical Elements and its Properties

We carried out a mathematical study of 72 chemical elements taking advantage of the chemotopological method. We selected 128 properties to define the elements (physico-chemical, geochemical and chemical properties). Then, we looked for correlated properties and we reduced the number of them to 90. In this way we defined each element as a 90-tuple. Afterwards, we applied principal component analysis and cluster analysis (4 similarity functions and 5 grouping methodologies). Then, we calculated a consensus tree for the 20 dendrograms generated by the CA. Afterwards, we extracted the similarity relationships from the consensus tree and built up a basis for a topology on the set of chemical elements. Finally, we calculated some topological properties (closures, derived sets, boundaries, interiors and exteriors) of several subsets of chemical elements. We found that alkali metals, alkaline earth metals and noble gases appear not related to the rest of the elements. Also, we found that the boundary of non-metals are the semimetals with a stair-shape on the periodic table     

Simulations of the active transport of a neutral solute based on a kinase-channel-phosphatase topology

Simulations of coupled interactions involving two opposite enzymatic reactions, solute diffusions, and electrostatic interactions between membrane charges and charged solutes were conducted under a fixed kinase-channel-phosphatase (KCP) topology oriented from the outside to the inside of a porous membrane structure. Depending on the kinase and phosphatase locations, we recently demonstrated that an active transport of a phosphorylated substrate may occur via the opposite topology, that is, a PCK topology. The present analysis demonstrates that, under a KCP membrane topology, which also behaves as a specific ATP-dependent transporter, the active transport of a neutral substrate may occur. This analogous active transport appears to be dependent on the phosphatase location and on the membrane surface potentials. A broad analysis of the role played by the main parameters taken into account in the model was conducted in order to define precisely the physico-chemical conditions and the membrane topology needed for the highest active transports within the shortest time.     

Swelling behavior and mechanical properties of endlinked poly (dimethylsiloxane) networks and randomly crosslinked polyisoprene networks

Measurements of the equilibrium degree of swelling and of the equilibrium modulus were performed on poly(dimethylsiloxane) networks (PDMS) and on polyisoprene vulcanizates. The results support the concept that topological interactions between network chains, e.g. entanglements or the like, have a large influence on the rubber elastic behavior, at least within a certain range of network densities.PDMS networks having network chains of different lengths and varying functionlities of the crosslinks were prepared in bulk by endlinking fractionated ,-divinyl PDMS via multifunctional hydrogen-siloxanes (f=3 to 22). Natural rubber (NR) and synthetic liquid polyisoprene (IR) were cured in bulk with various amounts of dicumyl peroxide to give randomly crosslinked samples.The experimentally determined moduli and degrees of swelling were compared with theoretical predictions based on the phantom network theory and affine network theory, taking into account only chemical crosslinks. The observed discrepancies can be traced back to a contribution of topological interactions (trapped entanglements) to the total effective network density. The modulus and swelling data are consistent, thus ruling out non-equilibrium effects.     

Study on the Intramolecular Hydrogen Bonds of Dibenzofurans, Xanthones and Anthraquinones with One or Two Positions Substituted by Hydroxyls

<正>The thermodynamic properties of dibenzofurans (DFs), xanthones (XTs) and an- thraquinones (AQs) with one and two positions substituted with hydroxyls in the ideal gas state at 298.15 K and 1.013×10~5 pa were calculated at the B3LYP/6-311G~* level using Gaussian 03 program. The isodesmic reactions were designed to calculate the standard free energy of formation (Δ_fG~e). Three types of hydrogen bonds exist in the three kinds of chemicals and their bond energies were ascertained as 7-15, 15-23 and 49-58 kJ·mol~(-1) respectively by comparing the Δ_fG~e values. Electronic density topology analysis was applied to validate the strength of bond.     

Uncovering the Hidden Landscape of Tris(4-pyridyl) Ligands: Topological Complexity Derived from the Bridgehead

Supramolecular main group chemistry is a developing field which parallels the conventional domain of metallo-organic chemistry. Little explored building blocks in this area are main group metal-based ligands which have the appropriate donor symmetry to build desired molecular or extended arrangements. Tris(pyridyl) main group ligands (E(py)₃, E=main group metal) are potentially highly versatile building blocks since shifting the N-donor arms from the 2- to the 3-positions and 4-positions provides a very simple way of changing the ligand character from mononuclear/chelating to multidentate/metal-bridging. Here, the coordination behaviour of the first main group metal tris(4-pyridyl) ligands, E(4-py)₃ (E=Sb, Bi, Ph−Sn) is explored, as well as their ability to build metal-organic frameworks (MOFs). The complicated topology of these MOFs shows a marked influence on the counter anion and on the ability of the E(4-py)₃ ligands to switch coordination mode, depending on the steric and donor character of the bridgehead. This structure-directing influence of the bridgehead provides a potential building strategy for future molecular and MOF design in this area.     

Two coordination polymers based on 2,2′-biimidazoles and d10 metal ions: syntheses,structures, and luminescence

Two d¹⁰ metal coordination polymers, [Zn(µ-Me₂biim)Cl₂] _n (1) and [Cd₃(MeHbiim)₂(1,4-BDC)₃] _n (2) (Me₂biim?=?N,N′-dimethyl-2,2′-biimidazole, MeHbiim?=?N-methyl-2,2′-biimidazole, 1,4-BDC?=?1,4-benzenedicarboxylate), were synthesized under hydrothermal conditions and characterized by elemental analysis, infrared spectroscopy, thermogravimetric analysis, and single-crystal X-ray crystallography. Complex 1 features an infinite neutral zigzag 1-D chain. Interchain hydrogen-bonding interactions further extend the 1-D arrangement to generate a 2-D supramolecular architecture. Complex 2 features a 3-D coordination polymer with α-Po net topology, based on linear trinuclear {Cd₃O₁₄N₄} clusters. Both complexes have high thermal stability and exhibit strong luminescence at room temperature.