Dynamical behavior of endomorphisms on certain invariant sets

We study the Hausdorff dimension of the intersection between local stable manifolds and the respective basic sets of a class of hyperbolic polynomial endomorphisms on the complex projective space ?². We consider the perturbation (z ² +?z +b?w ², w ²) of (z ², w ²) and we prove that, for b sufficiently small, it is injective on its basic set Λ_? close to Λ:= {0} × S ¹. Moreover we give very precise upper and lower estimates for the Hausdorff dimension of the intersection between local stable manifolds and Λ _? , in the case of these maps.     

Simultaneous thermal and contraction/expansion analyses of cast iron solidification process

Journal of Thermal Analysis and Calorimetry - Simultaneous thermal and contraction/expansion analysis of hypoeutectic grey (lamellar graphite) and ductile (nodular graphite) cast irons...     

Chaotic dynamics of driven flux drops: a superconducting "dripping faucet"

When a current is applied to a type-I superconducting strip containing a narrow channel across its width, magnetic flux spots nucleate at the edge and are then driven along the channel by the current. These flux "drops" are reminiscent of water drops dripping from a faucet, a model system for studying low-dimensional chaos. We use a novel high-bandwidth Hall probe to detect in real time the motion of individual flux spots moving along the channel. Analyzing the time series consisting of the intervals between successive flux drops, we find distinct regions of chaotic behavior characterized by positive Lyapunov exponents, indicating that there is a close analogy between the dynamics of the superconducting and water drop systems.     

Influence of spin–orbit interaction,Zeeman effect and light polarisation on the electronic and optical properties of pseudo-elliptic quantum rings under magnetic field

ABSTRACT

We theoretically investigated the influences of the magnetic field and light polarisation on the electronic and optical properties of a GaAs/GaAlAs pseudo-elliptic quantum ring, modelled by an outer ellipsis and an inner circle, in the presence of the Rashba and Dresselhaus spin–orbit interactions and Zeeman effect. We show that Aharonov-Bohm oscillations of the energy spectrum are not affected by the presence of the Zeeman effect alone but, in the presence of Rashba and Dresselhaus spin–orbit couplings, the periodicity of certain levels becomes hardly definite. The Zeeman effect generally enhances/diminishes the separation levels produced by Rashba/Dresselhaus interactions (SOI) and when both types of SOI are considered, the effect depends on their relative strength. The magnetic field can trigger spin-flip for each type of spin–orbit interaction and Zeeman effect or their combination through anticrossings in the energy spectra. Our results reveal that the absorption spectra are very sensitive to the magnetic field and light polarisation. For all polarisations considered, the magnetic field increment leads to the redshift or blueshift of some particular peaks (an effect of this ring geometry) and a better separation of the peaks. The x-polarised light determines spectra with many small, but separated peaks while the circular polarised light leads to spectra with large peaks of high amplitude.     

On the shape of interlayer vortices in the Lawrence-Doniach model

We consider the Lawrence-Doniach model for layered superconductors, in which stacks of parallel superconducting planes are coupled via the Josephson effect. To model experiments in which the superconductor is placed in an external magnetic field oriented parallel to the superconducting planes, we study the structure of isolated vortices for a doubly periodic problem. We consider a singular limit which simulates certain experimental regimes in which isolated vortices have been observed, corresponding to letting the interlayer spacing of the superconducting planes tend to zero and the Ginzburg-Landau parameter simultaneously, but at a fixed relative rate.

     

A simple DNase model system comprising a dinuclear Zn(II) complex in methanol accelerates the cleavage of a series of methyl aryl phosphate diesters by 10(11)-10(13)

The di-Zn(II) complex of 1,3-bis[ N1, N1'-(1,5,9-triazacyclododecyl)]propane with an associated methoxide ( 3:Zn(II) 2: (-)OCH 3) was prepared and its catalysis of the methanolysis of a series of fourteen methyl aryl phosphate diesters ( 6) was studied at s (s)pH 9.8 in methanol at 25.0 +/- 0.1 degrees C. Plots of k obs vs [ 3:Zn(II) 2: (-)OCH 3] free for all members of 6 show saturation behavior from which K(M) and kcat (max) were determined. The second order rate constants for the catalyzed reactions (kcat (max)/K(M)) for each substrate are larger than the corresponding methoxide catalyzed reaction (k 2 (-OMe)) by 1.4 x 10(8) to 3 x 10 (9)-fold. The values of k cat (max) for all members of 6 are between 4 x 10(11) and 3 x 10(13) times larger than the solution reaction at s (s)pH 9.8, with the largest accelerations being given for substrates where the departing aryloxy unit contains ortho-NO 2 or C(O)OCH 3 groups. Based on the linear Br?nsted plots of k cat (max) vs s (s)pKa of the phenol, beta lg values of -0.57 and -0.34 are determined respectively for the catalyzed methanolysis of "regular" substrates that do not contain the ortho-NO 2 or C(O)OCH 3 groups, and those substrates that do. The data are consistent with a two step mechanism for the catalyzed reaction with rate limiting formation of a catalyst-coordinated phosphorane intermediate, followed by fast loss of the aryloxy leaving group. A detailed energetics calculation indicates that the catalyst binds the transition state comprising [CH 3O (-): 6], giving a hypothetical [ 3:Zn(II) 2:CH 3O (-): 6] complex, by -21.4 to -24.5 kcal/mol, with the strongest binding being for those substrates having the ortho-NO 2 or C(O)OCH 3 groups.     

Large ‐ or ‐Minors in 3‐Connected Graphs

There are numerous results bounding the circumference of certain 3‐connected graphs. There is no good bound on the size of the largest bond (cocircuit) of a 3‐connected graph, however. Oporowski, Oxley, and Thomas (J Combin Theory Ser B 57 (1993), 2, 239–257) proved the following result in 1993. For every positive integer k, there is an integer such that every 3‐connected graph with at least n vertices contains a ‐ or ‐minor. This result implies that the size of the largest bond in a 3‐connected graph grows with the order of the graph. Oporowski et al. obtained a huge function iteratively. In this article, we first improve the above authors' result and provide a significantly smaller and simpler function . We then use the result to obtain a lower bound for the largest bond of a 3‐connected graph by showing that any 3‐connected graph on n vertices has a bond of size at least . In addition, we show the following: Let G be a 3‐connected planar or cubic graph on n vertices. Then for any , G has a ‐minor with , and thus a bond of size at least .     

Rapid three-step cleavage of RNA and DNA model systems promoted by a dinuclear Cu(II) complex in methanol. energetic origins of the catalytic efficacy

A dinuclear Cu(II) complex of 1,3-bis-N(1)-(1,5,9-triazacyclododecyl)propane with an associated methoxide (2-Cu(II)(2):(-OCH(3))) was prepared, and its kinetics of reaction with an RNA model (2-hydroxypropyl-p-nitrophenyl phosphate (1, HPNPP)) and two DNA models (methyl p-nitrophenyl phosphate (3) and iso-butyl p-chlorophenyl phosphate (4)) were studied in methanol solution at (s)(s)pH 7.2 +/- 0.2. X-ray diffraction structures of 2-Cu(II)(2):(-OH)(H(2)O)(CF(3)SO(3)-)(3):0.5CH(3)CH(2)OCH(2)CH(3) and 2-Cu(II)(2):(-OH)((C(6)H(5)CH(2)O)(2)PO(2)-)(CF(3)SO(3)-)2 show the mode of coordination of the bridging -OH and H(2)O between the two Cu(II) ions in the first complex and bridging -OH and phosphate groups in the second. The kinetic studies with 1 and 3 reveal some common preliminary steps prior to the chemical one of the catalyzed formation of p-nitrophenol. With 3, and also with the far less reactive substrate (4), two relatively fast events are cleanly observed via stopped-flow kinetics. The first of these is interpreted as a binding step which is linearly dependent on [catalyst] while the second is a unimolecular step independent of [catalyst] proposed to be a rearrangement that forms a doubly Cu(II)-coordinated phosphate. The catalysis of the cleavage of 1 and 3 is very strong, the first-order rate constants for formation of p-nitrophenol from the complex being approximately 0.7 s(-1) and 2.4 x 10(-3) s(-1), respectively. With substrate 3, 2-Cu(II)(2):(-OCH(3)) exhibits Michaelis-Mentin kinetics with a k(cat)/K(M) value of 30 M(-1) s(-1) which is 3.8 x 10(7)-fold greater than the methoxide promoted reaction of 3 (7.9 x 10(-7) M(-1) s(-1)). A free energy calculation indicates that the binding of 2-Cu(II)(2):(-OCH(3)) to the transition states for 1 and 3 cleavage stabilizes them by -21 and -24 kcal/mol, respectively, relative to that of the methoxide promoted reactions. The results are compared with a literature example where the cleavage of 1 in water is promoted by a dinuclear Zn(II) catalyst, and the energetic origins of the exalted catalysis of the 2-Cu(II)(2) and 2-Zn(II)(2) methanol systems are discussed.     

The dinuclear Zn(II) complex catalyzed cyclization of a series of 2-hydroxypropyl aryl phosphate RNA models: progressive change in mechanism from rate-limiting P-O bond cleavage to substrate binding

A methoxide-bridged dinuclear Zn(II) complex of 1,3-[N,N'-bis(1,5,9-triazacyclododecane)]propane (1-Zn(II)2:(-OCH3)) was prepared, and its catalysis of the cyclization of a series of 2-hydroxypropyl aryl phosphates (4a-g) was investigated in methanol at pH 9.8, T = 25degreesC by stopped-flow spectrophotometry. An X-ray diffraction structure of the hydroxide analogue of 1-Zn(II)2:(-OCH3), namely 1-Zn(II)2:(-OH), reveals that each of the Zn(II) ions is coordinated by the three N's of the triazacyclododecane units and a bridging hydroxide. The cyclizations of substrates 4a-g reveal a progressive change in the observed kinetics from Michaelis-Menten saturation kinetics for the poorer substrates (4-OCH3 (4g); 4-H (4f); 3-OCH3 (4e); 4-Cl (4d); 3-NO2, (4c)) to second-order kinetics (linear in 1-Zn(II)2:(-OCH3)) for the better substrates (4-NO2,3-CH3 (4b); 4-NO2, (4a)). The data are analyzed in terms of a multistep process whereby a first formed complex rearranges to a reactive complex with a doubly activated phosphate coordinated to both metal ions. The kinetic behavior of the series is analyzed in terms of change in rate-limiting step for the catalyzed reaction whereby the rate-limiting step for the poorer substrates (4g-c) is the chemical step of cyclization of the substrate, while for the better substrates (4b,a) the rate-limiting step is binding. The catalysis of the cyclization of these substrates is extremely efficient. The kcat/KM values for the catalyzed reactions range from 2.75 x 10(5) to 2.3 x 10(4) M-1 s-1, providing an acceleration of 1 x 10(8) to 4 x 10(9) relative to the methoxide reaction (k2OCH3, which ranges from 2.6 x 10(-3) to 5.9 x 10(-6) M-1 s-1 for 4a-g). At a pH of 9.8 where the catalyst is maximally active, the acceleration for the substrates ranges from (1 - 4) x 10(12) relative to the background reaction at the same pH. Detailed energetics calculations show that the transition state for the catalyzed reaction comprising 1-Zn(II)2, methoxide, and 4 is stabilized by about -21 to -23 kcal/mol relative to the transition state for the methoxide reaction. The pronounced catalytic activity is attributed to a synergism between a positively charged catalyst that has high affinity for the substrate and for the transition state for cyclization, and a medium effect involving a reduced polarity/dielectric constant that complements a reaction where an oppositely charged reactant and catalyst experience charge dispersal in the transition state.     

A reductionist biomimetic model system that demonstrates highly effective Zn(II)-catalyzed cleavage of an RNA model

The cyclization of the RNA model 2-hydroxypropyl p-nitrophenyl phosphate (HPNPP, 1) promoted by Zn2+ alone and the 1,5,9-triazacyclododecane complex of Zn2+ (Zn2+:[12]aneN3) is studied in ethanol in the presence of 0.5 equiv of -OEt/Zn2+ to investigate the effect of a low polarity/dielectric medium on a metal-catalyzed reaction of biological relevance. Ethanol exerts a medium effect that promotes strong binding of HPNPP to Zn2+, followed by a dimerization to form a catalytically active complex (HPNPP:Zn2+)2 in which the phosphate undergoes cyclization with a rate constant of kcat = 2.9 s(-1) at s(s)pH 7.1. In the presence of the triaza ligand:Zn2+ complex, the change from water to methanol and then to ethanol brings about a mechanism where two molecules of the complex, suggested as EtOH:Zn2+:[12]aneN3 and its basic form, EtO-:Zn2+:[12]aneN3, bind to HPNPP and catalyze its decomposition with a rate constant of kcat of 0.13 s(-1) at s(s)pH 7.1. Overall, the acceleration exhibited in these two situations is 4 x 10(14)-fold and 1.7 x 10(12)-fold relative to the background ethoxide-promoted reactions at the respective s(s)pH values. The implications of these findings are discussed within the context of the idea that enzymatic catalysis is enhanced by a reduced effective dielectric constant within the active site.