High Dimensional Model Representations Generated from Low Dimensional Data Samples. I. mp-Cut-HDMR

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input–output system behavior. For a high dimensional system, an output f(x) is commonly a function of many input variables x=|x ₁,x ₂,...,x _n } with n10² or larger. HDMR describes f(x) by a finite hierarchical correlated function expansion in terms of the input variables. Various forms of HDMR can be constructed for different purposes. Cut- and RS-HDMR are two particular HDMR expansions. Since the correlated functions in an HDMR expansion are optimal choices tailored to f(x) over the entire domain of x, the high order terms (usually larger than second order, or beyond pair cooperativity) in the expansion are often negligible. When the approximations given by the first and the second order Cut-HDMR correlated functions are not adequate, this paper presents a monomial based preconditioned HDMR method to represent the higher order terms of a Cut-HDMR expansion by expressions similar to the lower order ones with monomial multipliers. The accuracy of the Cut-HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input–output samples without directly invoking the determination of higher order terms. The mathematical foundations of monomial based preconditioned Cut-HDMR is presented along with an illustration of its applicability to an atmospheric chemical kinetics model.     

Analytical HDMR formulas for functions expressed as quadratic polynomials with a multivariate normal distribution

High Dimensional Model Representation (HDMR) is a general set of quantitative model assessment and analysis tools for systems with many variables. A general formulation for the HDMR component functions with independent and correlated variables was obtained previously. Since the HDMR component functions generally are coupled to one another and involve multi-dimensional integrals, explicit formulas for the component functions are not available for an arbitrary function with an arbitrary probability distribution amongst their variables. This paper presents analytical formulas for the HDMR component functions and the corresponding sensitivity indexes for the common case of a function expressed as a quadratic polynomial with a multivariate normal distribution over its variables. This advance is important for practical applications of HDMR with correlated variables.     

Regularized random-sampling high dimensional model representation (RS-HDMR)

High Dimensional Model Representation (HDMR) is under active development as a set of quantitative model assessment and analysis tools for capturing high-dimensional input–output system behavior. HDMR is based on a hierarchy of component functions of increasing dimensions. The Random-Sampling High Dimensional Model Representation (RS-HDMR) is a practical approach to HDMR utilizing random sampling of the input variables. To reduce the sampling effort, the RS-HDMR component functions are approximated in terms of a suitable set of basis functions, for instance, orthonormal polynomials. Oscillation of the outcome from the resultant orthonormal polynomial expansion can occur producing interpolation error, especially on the input domain boundary, when the sample size is not large. To reduce this error, a regularization method is introduced. After regularization, the resultant RS-HDMR component functions are smoother and have better prediction accuracy, especially for small sample sizes (e.g., often few hundred). The ignition time of a homogeneous H₂/air combustion system within the range of initial temperature, 1000 < T ₀ < 1500 K, pressure, 0.1 < P < 100 atm and equivalence ratio of H₂/O₂, 0.2 < R < 10 is used for testing the regularized RS-HDMR.      

An investigation of the thermal behavior of heterobimetallic species containing copper(II) and tetracyanopalladate(II)

Synthesis, characterization and thermal behavior of four compounds that have the general formula [Cu{Pd(CN)₄}(L)_x]_n, in which en=1,2-diaminoethane and pn=1,3-diaminopropane (L=en, x=1 (I); L=pn, x=1 (II); L=en, x=2 (III); L=pn, x=2 (IV)) were described in this work. The complexes were studied by elemental analysis, infrared spectroscopy (IR), differential thermal analysis (DTA) and thermogravimetry (TG) and the residues of the thermal decomposition were characterized by X-ray powder diffraction and found as a mixture of CuO and PdO. The stoichiometry of the compounds was established via thermogravimetric and elemental analyses and their structures were proposed as coordination polymers based on their infrared spectra. The following thermal stability sequence was found: IV<I=II<III.     

High dimensional model representation constructed by support vector regression. I. Independent variables with known probability distributions

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing high dimensional input-output system behavior. In practice, the HDMR component functions are each approximated by an appropriate basis function expansion. This procedure often requires many input-output samples which can restrict the treatment of high dimensional systems. In order to address this problem we introduce svr-based HDMR to efficiently and effectively construct the HDMR expansion by support vector regression (SVR) for a function \(f(\mathbf{x})\). In this paper the results for independent variables sampled over known probability distributions are reported. The theoretical foundation of the new approach relies on the kernel used in SVR itself being an HDMR expansion (referred to as the HDMR kernel ), i.e., an ANOVA kernel whose component kernels are mutually orthogonal and all non-constant component kernels have zero expectation. Several HDMR kernels are constructed as illustrations. While preserving the characteristic properties of HDMR, the svr-based HDMR method enables efficient construction of high dimensional models with satisfactory prediction accuracy from a modest number of samples, which also permits accurate computation of the sensitivity indices. A genetic algorithm is employed to optimally determine all the parameters of the component HDMR kernels and in SVR. The svr-based HDMR introduces a new route to advance HDMR algorithms. Two examples are used to illustrate the capability of the method.     

The effect of orientation of the lateral methyl substituent on the mesophase behaviour of 4-alkoxyphenylazo aryl benzoates

Two homologous series of 4-alkoxyphenylazo 4?-(2?- (and 3?-) methyl-) 4?-substituted benzoates (IIn_a–f and IIIn_a–f, six series each) were prepared and investigated. Within each series, the length of the terminal alkoxy group varies among 6, 8, 10 and 12 carbons, while the other terminal substituent, X, is a polar group that alternatively changes between the electron-donating CH₃O, CH₃, and the electron-withdrawing Br, NO₂ and CN groups, in addition to the un-substituted analogue, X = H, aiming to investigate the effect of the different orientations of the methyl groups substituted on the central benzene ring, on the mesophase behaviour. The mesomorphic properties were discussed in terms of steric and polarisability effects. The mesophase stability was correlated with the polarisability anisotropy of bonds to the terminal substituent X. Comparative studies were made between the prepared isomers with each other and with the previously investigated laterally neat analogues 4-(4?-alkoxyphenylazo) phenyl 4?-substituted benzoates (In_a–f).     

Initiator efficiency of 2,2′‐azobis(isobutyronitrile) in bulk dodecyl acrylate free‐radical polymerizations over a wide conversion and molecular weight range

An Erratum has been published for this article in J. Polym. Sci. Part A: Polym. Chem. (2004) 42(21) 5559 . The initiator efficiency, f, of 2,2′‐azobis(isobutyronitrile) (AIBN) in dodecyl acrylate (DA) bulk free‐radical polymerizations has been determined over a wide range of monomer conversion in high‐molecular‐weight regimes (M_n ? 10⁶ g mol^?1 [? 4160 units of DA)] with time‐dependent conversion data obtained via online Fourier transform near infrared spectroscopy (FTNIR) at 60 °C. In addition, the required initiator decomposition rate coefficient, k_d, was determined via online UV spectrometry and was found to be 8.4 · 10^?6 s^?1 (±0.5 · 10^?6 s^?1) in dodecane, n‐butyl acetate, and n‐dodecyl acetate at 60 °C. The initiator efficiency at low monomer conversions is relatively low (f = 0.13) and decreases with increasing monomer to polymer conversions. The evolution of f with monomer conversion (in high‐molecular‐weight regimes), x, at 60 °C can be summarized by the following functionality: f^{60 °C} (x) = 0.13–0.22 · x + 0.25 · x² (for x ≤ 0.45). The reported efficiency data are believed to have an error of >50%. The ratio of the initiator efficiency and the average termination rate coefficient, 〈k_t±, (f/〈k_t〉) has been determined at various molecular weights for the generated polydodecyl acrylate (M_n = 1900 g mol^?1 (? 8 units of DA) up to M_n = 36,500 g mol^?1 (? 152 units of DA). The (f/〈k_t〉) data may be indicative of a chain length‐dependent termination rate coefficient decreasing with (average) chain length. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 5170–5179, 2004     

Gauge parameters for extrapolation of oscillator strengths ofE1 transitions along the boron isoelectronic sequence

The oscillator strengthsf forE1 transitions along an isoelectronic sequence can be written asf=aK ²+bK+c whereK is a gauge parameter representing the gauge condition of the electromagnetic field. The coefficientsa,b, andc are functions of length (f _l) and velocity (f _v) values of the oscillator strengths at the Hartree-Fock level. We have shown by making a perturbation expansion of oscillator strengthsf,f _l andf _v that the gauge parameterK is independent of the nuclear charge. This property has been exploited to extrapolatef values along the isoelectronic sequence of Boron for some representativeE1 transitions within then=2 complex. We obtain good agreement between the extrapolated results with the configuration interaction results.     

Evidence of quasi-intramolecular redox reactions during thermal decomposition of ammonium hydroxodisulfitoferriate(III), (NH4)2[Fe(OH)(SO3)2]·H2O

Synthesis of ammonium hydroxodisulfitoferriate(III), (diammonium catena-{bis(μ ²-sulfito-κO,κO)-μ ²-hydroxo-κ²O}ferrate(III) monohydrate) (NH₄)₂[Fe(OH)(SO₃)₂]·H₂O (compound 1) and its thermal behavior is reported. The compound is stable in air. Its thermal decomposition proceeds without the expected quasi-intramolecular oxidation of sulfite ion with ferric ions. The disproportionation reaction of the ammonium sulfite, formed from the evolved NH₃, SO₂ and H₂O in the main decomposition stage of 1, results in the formation of ammonium sulfate and ammonium sulfide. The ammonium sulfide is unstable at the decomposition temperature of 1 (150 °C) and transforms into NH₃ and H₂S which immediately forms elementary sulfur by reaction with SO₂. The formation and decomposition of other intermediate compounds like (NH₄)₂S_nO_x (n = 2, x = 3 and n = 3, x = 6) results in the same decomposition products (S, SO₂ and NH₃). Two basic iron sulfates, formed in different ratios during synthesizing experiments performed under N₂ or in the presence of air, have been detected as solid intermediates which contain ammonium ions. The final decomposition product was proved to be α-Fe₂O₃ (mineral name hematite).

     

Amino-Modified Silicate Xerogels Complexed with Cu(II) as Catalyst Precursors. Coordination State and Thermal Decomposition

The sol-gel process is a useful method for preparing two series of organically and co-ordinately modified xerogels of the types [CuN _n ]·N _5–n ·5xSiO_4/2 (n < 4) and [Cu(N–N)_n]·(N–N)_2–n ·2x SiO_4/2(n 2), where N = NH₂(CH₂)₃ SiO_3/2, N–N = NH₂(CH₂)₂NH·(CH₂)₃SiO_3/2 and x = [SiO_4/2]/[N] or [SiO_4/2]/[N–N]. The amino groups in the materials are coordinately active and participate partly in the coordination sphere of Cu(II) ions. The composition of the coordination sphere can be varied with the SiO_4/2 content and also as a result of the thermal decomposition of the organic residues at higher temperatures.Because the xerogel materials are considered to be catalyst precursors, this study is focused on their coordination and thermal properties. The prepared xerogels, such as silica, aminated silicates with N and N–N, as well as those entities complexed with Cu(II), were characterised by FT-IR spectroscopy. During gelation and thermal decomposition the materials were analysed by electron paramagnetic resonance (EPR) spectroscopy. The xerogels were additionally studied by UV-Vis absorption spectroscopy. The gaseous products of the thermal decomposition of these materials in an Ar atmosphere were investigated by the use of FT-IR spectroscopy coupled with TG and DTG thermal analysis. These data were complemented by temperature-programmed decomposition (TPDec) in a 2% O₂ + 98% Ar stream coupled with quadrupole mass spectroscopy.     

Über eine vereinfachte Methode des self-consistent field für Atome

Zesummenfassung In erster Näherung werden die zur gleichen Hauptquantenzahln gehörenden Einelektronen-Quantenzustände durch eine einzige radiale Eigenfunktionf _n vom Slaterschen Typ beschrieben und mit dem Variationsverfahren aus dem Minimumprinzip der Energie bestimmt, wobei die Orthogonalisierung der Eigenfunktionen auf die der tieferen Zustände durch ein statistisches Pseudopotential ersetzt wird. In der zweiten Näherung werden dann die Eigenfunktionen der ersten Näherungf ₁,f ₂, ...,f _n , ... nach der Nebenquantenzahll aufgespalten, und zwar in der Weise, daß man diese Eigenfunktionen mit dem Schmidtschen Verfahren orthogonalisiert und die so gewonnenen Eigenfunktionen den Quantenzuständenn, l zuordnet, wobei sich eine ausgezeichnete Übereinstimmung mit den Hartreeschen und Fockschen Eigenfunktionen ergibt.

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As a first approximation all one-electron states of principal quantum numbern are described by a single radial Slater-type functionf _n , the variation parameters of which are calculated by minimisation of energy. The orthogonalization of the functions to those of the lower energy states is replaced by a statistical pseudopotential. As a second approximation, using a Schmidt procedure thef _n are split to functionsf _nl corresponding to the quantum statesn, l. The agreement with Hartree and Fock functions is very good.

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Resume En première approximation, nous décrivons les états monoélectroniques appartenant au nombre quantique principaln, par une seule fonction fn de Slater déterminée par variation d'énergie où l'orthogonalisation des fonctions sur celles des états plus basses est substituée par un pseudo-potentiel statistique. En seconde approximation, ces fonctionsf ₁,f ₂, ...,f _n sont dédoublées suivant le nombre quantiquel, par une orthogonalization de Schmidt. Les fonctions obtenues sont attribuées aux états quantiquen, l et s' accordent tres bien aux fonctions de Hartree et de Fock.

     

Zur Ermittlung und rechnerischen Überprüfung thermodynamischer Daten aus experimentell gefundenen Werten, 8. Mitt.: Zur Programmierung desRedlich-Kisterschen Rechenansatzes für binäre Systeme

Zusammenfassung Bei der Auswertung experimenteller thermodynamischer Daten ist derRedlich-Kistersche Ansatz dem älteren äquivalenten Ansatz vonMargules wesentlich überlegen. Für die Beziehungen zwischen den Koeffizienten der beiden Ansätze werden allgemeine Formeln angegeben.Die thermodynamischen Größeny ₁,f ₁,f ₂,P (ber.) sowie die FunktionG ^E /x ₁ x ₂, dannG ^E ,H ^E ,TS ^E werden mit Hilfe einer Siemens-2002-Rechenanlage errechnet. Das Programm läßt sich innerhalb der Speicherkapazität der Maschine bis auf 12 ParameterA ausweiten.Für eine Isotherme des Systems n-Butan/Methylamin werden die Fehlerkurven (RP in Abhängigkeit vonx) dargestellt. Aus den Abbildungen folgt, daß die Dampfdruckisothermen dieses Systems mit fünf Parametern (n=6) am besten dargestellt werden können.Zur Berechnung der thermodynamischen Zusatzfunktionen wird ein Differenzenverfahren entwickelt und mit einem Beispiel belegt.

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TheRedlich-Kister statement is more convenient for evaluating of experimental thermodynamic data then the former statement ofMargules. The correlations between the two statements are given by means of generalized formulas.For calculation of the essential thermodynamic quantitiesy ₁,y ₂,f ₁,f ₂,P(calc.),G ^E /x ₁ x ₂, thenG ^E ,H ^E andTS ^E aSiemens-2002-computer is used. Within the capacity of the computer the programm can be extended up to 12 parameters.In the system n-butane/methylamine differences between experimental and calculated values ofP(=RP) are plotted as function of the mole fractionx for one temperature (15°C). Five parameters (n=6) describe the isothermal lines of this system in the optimal manner. This is demonstrated by means of Fig. 1 and 2.The thermodynamic excess functions are calculated with a difference-table-method. This method is demonstrated with one example.

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Mit 3 Abbildungen

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1.–6. Mitt.:F. Gölles, Mh. Chem.92, 981 (1961);93, 191, 201 (1962);94, 1108 (1963);95, 1656 (1964);96, 1366 (1965); 7. Mitt.:F. Gölles undA. Höpfner, Mh. Chem.97, 368 (1966).     

On the concept of “Partitions” in the perturbation theory of n-electron systems

This paper deals with the perturbation theory of an n-electron Hamiltonian of the general form H = ∑ⁿ ?(i) + λ∑ⁿ g(i, j) = H (f, g; n). In comparison to the Brueckner–Goldstone diagrammatic perturbation theory, we adopt the more general standpoint of admitting, for the construction of an n-particle state, component states of 1, 2, 3, and more particles [O. Sinanoglu, Phys. Rev. 122 , 493 (1961) and C. D. H. Chisholm and A. Dalgarno, Proc. R. Soc. (London) Sec. A 292 , 264 (1966)]. We show that this leads to the concept of a “partition” of a perturbational eigenstate (or energy) of H. A “partition” is a natural decomposition which: (i) is finite; (ii) relates the eigenvalue problem of the system H = H (f, g; n) to those of certain subsystems H (f, g; n₁)(n₁ < n); (iii) uses “nonseparable” components. We domonstrate (under the preliminary assumption of “strict” nondegeneracy) the second-order energy to possess a “partition.” The components therein are second-order energies of two- and three-particle states. The proof uses an extension of Racah's concept of the fractional-parentage expansion.     

Isobaric thermal behavior of glasses during uniform cooling and heating: Dependence of the characteristic temperatures on the relative contributions of temperature and structure to the rate of recovery. II. A one-parameter model approach

Isobaric variations of the characteristic temperatures T_g and T_max, obtained on uniform cooling and heating of glasses, are investigated in terms of their dependence on the relevant experimental variables, using a single retardation time model. The corresponding partial derivatives of T_g and T_max are derived as functions of the partition parameter x (ranging between zero and unity), which determines the relative contributions of temperature and structure to the retardation time. It is shown that the variation of T_g with the cooling rate is independent of x. In contrast, T_max critically depends on x, and its value as well as those of its three partial derivatives are linear functions of x^?1. The variations of T_max are analyzed in terms of a set of reduced variables, leading to simple reduction rules between any two of the experimental variables when the third is kept invariant. The reduction rules are further substantiated by investigating the behavior of glasses in two-step thermal cycles, which result in a unique set of inter-relationships between any pair of the partial derivatives of T_max, whatever the value of x. The results are discussed in terms of their relevance to the behavior of real glasses.     

Thermodynamic study of glasses in Ag-As-Se system using EMF method with Ag<Subscript>4</Subscript>RbI<Subscript>5</Subscript> solid electrolyte

The EMF method with Ag₄RbI₅ solid electrolyte was used to study silver solubility in Ag-As-Se glasses on the basis of the cross-sections of (I) Ag-As_0.25Se_0.75, (II) Ag-As_0.33Se_0.67, (III) Ag-As_0.4Se_0.6, and (IV) Ag-As_0.5Se_0.5. It is found that silver solubility reaches 30 and 40 at % in sections (I), (IV) and (II), (III), accordingly. The data of EMF measurements were used as a basis for calculation of partial polar functions of Ag in glasses. The Gibbs-Duhem equations were integrated to calculate thermodynamic functions of silver dissolution in vitreous As _x Se_{1 − x} (x = 0.25; 0.33; 0.4; 0.5), from which the corresponding data for the latter were used to obtain the standard integral thermodynamic functions of the mixing of glasses. The obtained results were compared with the thermodynamic data for crystalline silver selenoarsenites.     

水在C12-EOx-C12•2Br/正己醇/正庚烷W/O微乳中的状态

以FT-IR方法研究了水/C₁₂-EO_x-C₁₂•2Br/正己醇/正庚烷形成的W/O微乳中水的状态. 结果表明, 其中的水存在4种状态, 分别为阳离子头基结合水、反离子结合水、类似本体的水以及束缚在微乳栅栏层中的水. 由解卷积技术分解FT-IR谱图, 进而获得每个表面活性剂分子对应于这4种状态水分子的数目n_N＋, n_Br-, n_b和n_f. 随着水含量(W₀)增加, n_b急剧增大, n_N＋少许上升, 而n_f和n_Br-维持不变, 这说明微乳水核逐渐长大, 且在所考察W₀范围内, 表面活性剂头基解离度保持不变.     

N-representability of diagonal elements of second-order reduced density matrices

The problem of pure-state N-representability of the two-particle spin-dependent density function ρ(x₁, x₂) is considered for an N-electron system, and a procedure for finding an N-representable ρ(x₁, x₂) is advanced. The problem is formulated in the framework of a family of N × N matrices formed from integrals of auxiliary two-particle functions θ_n(x₁, x₂) converging at n → ∞ to ρ(x₁, x₂)/[N(N−1)]. The simple requirement of positive definiteness of these matrices is shown to play a decisive role in finding an N-representable ρ(x₁, x₂). The results obtained may open new possibilities for using ρ(x₁, x₂) in the density-functional theory. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 127–142, 1997     

Coordination diversity in tin compounds with bis(benzoxazole)phenol as a polydentate ligand: Synthesis and crystal structure studies

Abstract

The reaction of 2,6-bis(benzoxazolyl)-4-tert-butylphenol (HL) with [ⁿBu_xSnCl_4?x] (x?=?0, 1) in 1:1 stoichiometry yielded the tin coordination complexes [(HL)SnⁿBu_xCl_4?x] [x?=?0 (1); x?=?1 (2)]. Deprotonation of HL was performed using reagents having groups with high basicity such as ⁿBuLi or [Sn{N(SiMe₃)₂}₂]. These basic reagents prompted the coordination of the ligand in its anionic form, yielding the complexes [(thf)₂Li(L)SnCl₄] (3) and [(L)Sn{N(SiMe₃)₂}] (4), respectively. The molecular structure of HL displayed an intramolecular hydrogen bond OH—N and a planar arrangement of the bis(benzoxazolyl)phenolic system. In the molecular structures of both complexes containing HL an intramolecular hydrogen bond of NH—O type was also present. The coordination of the ligand in either neutral or anionic form is described by a κO,κN chelate mode toward Sn. All complexes displayed bis(benzoxazolyl)phenolic moieties close to planar; the least planar system was observed in 4 that was also studied by DFT methods.     

L-Proline catalyzed three component synthesis of pyrano[2,3-c]pyrazole-5-carbonitrile derivatives and in vitro antimalarial evaluation

An efficient one-pot synthesis of 6-amino-4-(2-chloroquinolin-3-yl)-3-methyl-2, 4-dihydro-pyrano[2,3-c]pyrazole-5-carbonitrile derivatives (4a–f)(5a–f) by three component reactions of 2-chloroquinolin-3-carbaldehyde derivatives, malanonitrile, and 3-methyl pyrazolin-5-one derivatives catalyzed by L-proline in ethanol medium under mild conditions is established. The synthesized compounds were evaluated for antimalarial activity and the LC₅₀/LC₉₀ values were described. Compounds 4d, 5d, and 5f exhibits good antimalarial activity when compared to other pyrano[2,3-c]pyrazole scaffolds.     

Fast evaluation of molecular auxiliary functions <Emphasis Type="Italic">A</Emphasis><Subscript>α</Subscript> and <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> by analytical relations

Analytical relations through the initial values are derived for the molecular auxiliary functions A _α (x) and B _n (x), where α =n+ɛ, 0⩽ ɛ < 1 and n=0,1,2,.... These relations are useful in the fast calculation of multicenter molecular integrals over integer and noninteger n Slater type orbitals. It is shown that the formulas obtained are numerically stable for all values of n,ɛ and x.PACS No: 31.15.+q, 31.20.EjAMS subject classification: 81-V55, 81-V45