Chemical applications of topology and group theory. 17. An information theoretical approach to polyhedral symmetry [1]

Information theoretic parameters are described which measure the asymmetry of polyhedra based on partitions of their vertices, faces, and edges into orbits under action of their symmetry point groups. Such asymmetry parameters are all zero only for the five regular polyhedra and are all unity for polyhedra having no symmetry at all, i.e. belonging to the C ₁ symmetry point group. In all other cases such asymmetry parameters have values between zero and unity. Values for such asymmetry parameters are given for all topologically distinct polyhedra having five, six, and seven vertices; all topologically distinct eight-vertex polyhedra having at least six symmetry elements; and selected polyhedra having from nine to twelve vertices. Effects of polyhedral distortions on these asymmetry parameters are examined for the tetrahedron, trigonal bipyramid, square pyramid, and octahedron. Such information theoretic asymmetry parameters can be used to order site partitions which are incomparable by the chirality algebra methods of Ruch and co-workers.     

A novel heterobidentate chiral phosphine and its coordination chemistry in transition metal clusters

The optically active ligand R,R-PHAZAN (1,3-bis[(1R)-1-Phenylethyl]-2-(2-thienyl)-1,3,2-diazaphospholane) has been prepared and the products resulting from the reactions with Rh₆(CO)₁₅NCMe, H₃RhOs₃(CO)₁₂, and H₄Ru₄(CO)₁₂ have been investigated by X-ray crystallography and a variety of multinuclear NMR methods. X-ray studies show that PHAZAN can behave as a bidentate ligand in Rh₆(CO)₁₄(μ₂,κ²-R,R-PHAZAN) (with coordination through P and S) or a monodentate ligand (through P coordination) in H₄Ru₄(CO)₁₁(κ¹-R,R-PHAZAN) and NMR studies show that these structures are retained in solution. In Rh₆(CO)₁₄(μ₂,κ²-R,R-PHAZAN), edge-bridging coordination of PHAZAN results in the formation of an additional two novel chiral centres and these are observed in solution. Reaction of PHAZAN with H₃RhOs₃(CO)₁₂ results in cleavage of the thienyl group and formation of the phosphido cluster, H₂RhOs₃(CO)₁₁(μ₂-PNN), (PNN = 1,3-bis-(1-phenylethyl)-[1,3,2]diazaphospholidine-2-yl). A variety of NMR measurements show that the hydride site-occupancies in the solid state are retained in solution and there is evidence for interaction of an ortho-phenyl hydrogen and a hydride through “dihydrogen” bonding.     

Effizienz kommerziell erwerblicher Silikagele (dp=5 μm und 10 μm) in der Flüssigkeits-Chromatographie

Zusammenfassung 18 kommerziell erwerbliche Kieselgele mit Teilchengrößen (d _p) von 10 m und 5 m wurden systematisch untersucht. Die chromatographisch bestimmte mittlere Teilchengröße (d) ist etwa um 35% größer als die mit Hilfe des Coulter Counter bestimmte Größe (d _c). Die von den Herstellern angegebenend _p-Werte liegen meistens zwischend undd _c. Died-Werte streuten-für die mitd _p=10 m angegebenen Silikagele-zwischen 7,6 m und 12,2 m. Die Konstanten der vereinfachten (10 m Teilchen) bzw der vollständigen (5 m Teilchen) van Deemter Gleichung bei verschiedenen Kapazitäts verhältnissen (k=0–3) und die Asymmetriefaktoren (As) für die Inertsubstanzen wurden gemessen und tabelliert. Beiu=1 mm/s waren die reduzierten Bodenhöhen (h/d) kleiner als 3 für Substanzen mit 0k<3 fürd _p=10 m. Für sämtliche untersuchten Kieselgele waren die Kapazitätsverhältnisse der einzelnen Proben an einer gegebenen stationären Phase mit einer Genauigkeit von ±0.05 reproduzierbar. DieA, B undC Terme der van Deemter Gleichung (d _p=5 m) waren innerhalb von ±10% reproduzierbar, auch dann, wenn die Säule eine Woche trocken gelagert wurde. Die Abnahme vonU _min mit zunehmendenk-Werten hängt sowohl von der Qualität (z.B. geometrischer Form, Enge der Siebfraktion, etc.) der stationären Phase als auch von der Güte der Packungsmethode ab. Für kleinere Teilchen werden die reduzierten Bodenhöhen (h/d)_min größer. Da die Permeabilität der Säulen gemäß der Definition vond immer zud ² proportional ist, nimmt der Druckabfall an der Säule mit abnehmender Teilchengröße wesentlich zu. Die Lebensdauer von Säulen nimmt erfahrungsgemäß mit zunehmendem Druck ab. Die Bandenverbreiterung außerhalb der Säule ist oft bei kommerziell erwerblichen Geräten so groß, daß 10 cm lange Säule (i.D.=4 mm), die mit 5 m. Teichen gepackt wurden, identischen, jedoch mit 10 m Kieselgelen gepackten Säulen, in der Effizienz kaum überlegen sind. Obwohl die Definition vond sphärische Teilchen begünstigt, wenn reduzierte Bodenhöhen errechnet werden, ist die Effizienz von Säulen, die mit sphärischen und unregelmäßigen Teilchen gepackt sind, sehr ähnlich. die Bevorzugung der sphärischen Packungsmaterialien gegenüber den preiswerteren unregelmäßigen Silikagelen scheint experimentell nicht belegbar zu sein. DieA-Terme nehmen mit zunehmendenk-Werten immer zu, obwohl dies von der Theorie hier nicht zu erwarten ist. DieC Terme scheinen überwiegend von der Massentransportgeschwindigkeit in der Eluentenphase bestimmt zu sein. Diek-Abhängigkeit derC-Terme ist mit Hilfe des ausschließlichk-abhängigen Faktors derC _m-Terme der Golay-Gleichung gut zu beschreiben, fallsk>0,75 ist. Allerdings ist die Zunahme derC-Terme vonk=0 aufk=0,75 etwa um einen Faktor 3 kleiner als theoretisch errechenbar ist. Die Stabilität der Säulen wird demonstriert. Obwohl die 20 cm langen Säulen drei Monate lang trocken gelagert wurden, nahm die Effizienz nur vonn=17.000 aufn=15.000 ab. Mit 27 bar Druckabfall wurden 130 theoretische Böden pro Sekunde erreicht. Mittlerer Porendurchmesser, Porenvolumen und spezifische Oberfläche der kommerziell erwerblichen Kieselgele beeinflußten diek-Werte der Probesubstanzen. Obwohl die Variationsbreite dieser Faktoren bei den 18 untersuchten Kieselgelen nicht allzu eng war, beeinflußten diese Parameter die Effizienz der Säulenkaum.

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Efficiency of commerically available silicas in HPLC

Summary Eighteen commercially available silicas having nominal particle sizes (d _p) of 10 m and 5 m, have been systematically studied. The chromatographically determined average particle size (d) is about 35% greater than the average particle size as estimated using a Coulter Counter (d _c). The particle sizes quoted by the manufacturers mostly lie betweend andd _c. Thed values for the silicas with a quotedd _p of 10 m lay between 7.6 m and 12.2 m. The constants of the simplified (for 10 m particles) or the complete (for 5 m particles) van Deemter equation were measured and tabulated for varying capacity ratios (k=0–3) as were the asymmetry factorsAs for an inert substance. When u=1 mm/s the reduced plate height (h/d) was less than 3 for substances with 0k<3 whend _p=10 m. The capacity ratios of individual samples on a given stationary phase were reproducible with an accuracy of ±0.05 for all the silicas studied. TheA, B andC terms of the van Deemter equation (d _p=5 m) were reproducible to within ±10%, even when the column was stored dry for a week. The decrease inu _min with increasingk value depended both on the quality of the stationary phase (e.g., its geometrical form, the narrowness of the sieve fraction) and on the packing method. The reduced plate heights (h/d)_min were bigger for smaller particles. In consequence of the definition ofd, the permeability of a column is always proportional tod ² and the pressure drop in columns with smaller size particles increased substantially. It is a matter of experience that the column life decreases with increasing pressure drop. The extracolumn band broadening of commercial instruments is often so large that 10 cm long columns (i.d.=4 mm) packed with 5 m particles are scarcely more efficient than identical columns packed with 10 m particles. Although the definition ofd favours spherical particles if reduced plate heights are to be calculated, the efficiencies of columns packed with spherical particles are very similar to those packed with irregular particles. The preference for spherical packing material over the cheaper irregular silica does not seems to be supported by experiment. TheA term always increases with increasingk values, although this is theoretically unexpected. TheC terms seems to be overwhelmingly determined by the speed of mass transport in the mobile phase. The dependence of theC term onk is, whenk is >0.75, well described using the exclusivelyk dependent factions of theC _m term of the Golay equation. However, the increase in theC term whenk is increased from 0 to 0.75 is about a factor three less than the theoretically calculated value. The stability of the columns has been demonstrated. Although columns (20 cm long in length) were stored dry for three months the efficiency only decreased slightly. With a 26 bar pressure drop 130 theoretical plates per second were obtained. The average pore diameters, pore volumes and specific surface areas of the commerical silicas influenced thek values of the sample substances. These factors however scarcely affected the column efficiencies even though their range of variation was not very narrow in the 18 silicas studied.

     

An experimental study on the asymmetry and the dip form of the Hβ line profiles in microwave produced plasmas at atmospheric pressure

An experimental study on the asymmetry of the Balmer H_β profile in plasmas produced by microwaves at atmospheric pressure is presented. The study is based on the definition of several functions that quantify the asymmetry aspects of the profile. Apart from the asymmetry aspects of the flanks also form-functions are defined that characterize the central part of the profile, the so-called dip or central valley, the combination of the two peaks and the dip in between them. The study shows the experimental dependence of these characteristics on the electron density and control parameters such as the gas flow and the hydrogen admixture ratio. The possible use of these newly introduced profile characteristics to plasma diagnosis is discussed.     

Effect of field and medium asymmetries on enhancement of signal-to-noise ratio

The effect of field and medium asymmetry on improvement of signal-to-noise ratio (SNR) in respect of homodyne detection technique is analysed theoretically. The analysis is applied to the sample of bulk GaAs irradiated by off-resonant nanosecond pulsed laser. The dependence of SNR on electric field amplitude and local oscillator (LO) phase is examined. It is found that SNR is maximum for the combination of asymmetric field and asymmetric medium.      

Improving density forecast by modeling asymmetric features: An application to S&P500 returns

A density forecast is an estimate of the probability distribution of the possible future values of a random variable. From the current literature, an economic time series may have three types of asymmetry: asymmetry in unconditional distribution, asymmetry in conditional distribution, volatility asymmetry. In this paper, we propose three density forecasting methods under two-piece normal assumption to capture these asymmetric features. A GARCH model with two-piece normal distribution is developed to capture asymmetries in the conditional distributions. In this approach, we first estimate parameters of a GARCH model by assuming normal innovations, and then fit a two-piece normal distribution to the empirical residuals. Block bootstrap procedure, and moving average method with two-piece normal distribution are presented for volatility asymmetry and asymmetry in the conditional distributions. Application of the developed methods to the weekly S&P500 returns illustrates that forecast quality can be significantly improved by modeling these asymmetric features.     

第Ⅱ种强度不等的非对称两态叠加多模叠加态光场的任意奇数阶等阶N次方Y压缩

本文利用多模压缩态理论,研究了第Ⅱ种强度不等的非对称两态叠加多模叠加态光场|Ψ_Ⅱ^(ab)>_q的任意奇数阶等阶N次方Y压缩特性.结果发现:在压缩阶数N=2p+1的条件下,无论p=2m还是p=2m+1(m=0,1,2,3,…,…),只要构成态|Ψ_Ⅱ^(ab)>_q的两个不同的量子态|{-Z_j^(a)*}>_q与|{-iZ_j^(b)*}>_q的各对应模的强度(即平均光子数)和初始相位都不相等,亦即Rj(a)≠Rj(b)和φj(a)≠φj(b)(j=1,2,3,…,q),并且 R_j^(a)(2p+1)= R_j^(b)(2p+1),则当各对应模的初始相位φ_j^(a)与φ_j^(b)、各对应模的初始相位差(φ_j^(a)-φ_j^(b)),态间的初始相位差(θ_nq^(aR)-θ_nq^(bI))以及光子干涉项的幅度 =R_j^(a)R_j^(b)等分别满足一定的量子化条件时,态|Ψ_Ⅱ^(ab)>_q的第一及第二这两个正交分量总可分别呈现出周期性变化的、任意奇数阶等阶N次方Y压缩效应.这一结果,与现有报道的结果截然不同.     

A numerical study of bifurcation in laminar flow in curved ducts

The bifurcation phenomenon whereby multiple-vortex secondary flow occurs in place of the normal two-vortex flow in laminar flow in curved ducts has previously been studied numerically by several researchers. However, the various results have been conflicting on many points. The present paper describes a set of numerical experiments conducted to study the effect of numerical accuracy on the solution. The results show that the transition from two- to four-vortex structure depends strongly on the differencing scheme and to a lesser extent on the grid size. The study also shows that as the Reynolds number of the flow increases, a two-vortex structure is re-established via a path which involves strongly asymmetric secondary flow patterns. These results are in agreement, at least qualitatively, with recent experimental theoretical and numerical results.     

Visual angle model for car-following theory

The vast majority of car-following models are lack of the consideration of human drivers’ characteristics. Based on the fact that each driver of a following vehicle perceives closing-in or shying-away a leading vehicle in front of him/her, primarily due to changes in the apparent size of the leading vehicle, we improved the full velocity difference (FVD) model and presented a visual angle car-following model. This model is in view of the stimulus-response framework and uses the visual angle and the change rate of the visual angle as stimulus. Results from linear analysis showed that the neutral stability line is asymmetry and the width of the leading vehicle has a great impact on the stability of traffic flow. Numerical simulations obtained the same results as theoretical analysis clearly such as density wave, shrinking hysteresis, asymmetry and wide scattering. Thus, the introducing of the visual angle can explain some complex nature of traffic flow and contribute to the design of more realistic car-following models.