Asymptotics for stationary very nearly unit root processes

Abstract. This article considers a mean zero stationary first‐order autoregressive (AR) model. It is shown that the least squares estimator and t statistic have Cauchy and standard normal asymptotic distributions, respectively, when the AR parameter ρ_n is very near to one in the sense that 1 ? ρ_n = o(n^?1).     

Identification of Persistent Cycles in Non‐Gaussian Long‐Memory Time Series

Abstract. Asymptotic distribution is derived for the least squares estimates (LSE) in the unstable AR(p) process driven by a non‐Gaussian long‐memory disturbance. The characteristic polynomial of the autoregressive process is assumed to have pairs of complex roots on the unit circle. In order to describe the limiting distribution of the LSE, two limit theorems involving long‐memory processes are established in this article. The first theorem gives the limiting distribution of the weighted sum, is a non‐Gaussian long‐memory moving‐average process and (c_n,k,1 ≤ k ≤ n) is a given sequence of weights; the second theorem is a functional central limit theorem for the sine and cosine Fourier transforms      

Performance Assessment of Some Isomers of Saturated Polycyclic Hydrocarbons for Use as Jet Fuels

The performance of jet fuel depends on the density (ρ), condensed phase heat of formation (▵_fH°(c)), and specific impulse (I_SP). Exo‐tricyclo[5.2.1.0(2,6)]decane (C₁₀H₁₆) or JP‐10 is now used as a suitable synthetic liquid jet fuel because it has the approximated values of ρ=1.1 g cm⁻³ and ▵_fH°(c)=− 123 kJ mol⁻¹ and a broad range between the melting and boiling points, i.e. T_bp–T_mp=196.2 K. This work introduces a suitable pathway for calculation of the values of ρ, ▵_fH°(c), and I_SP of 13 well‐known isomers of JP‐10 and a series of saturated polycyclic hydrocarbons with general formula of C_nH_n′ (5≤n≤12) in order to specify high performance jet fuels. Although 13 compounds have larger values of I_SP*ρ than JP‐10, only two compounds, tetraspiro[2.0.0.0.2.1.1.1]undecane and tetracyclo[3.2.0.0(2,7).0(4,6)]heptane, are suitable as jet fuels.     

On processes with hyperbolically decaying autocorrelations

We discuss some relations between autocorrelations (ACFs) and partial autocorrelations (PACFs) of weakly stationary processes. First, we construct an extension of a process ARIMA(0,d,0) for d ∈ (?∞, 0), which enjoys non‐summable partial autocorrelations and autocorrelations decaying as rapidly as ρ_n ? n^?1+2d. Such a situation is impossible if the absolute sum of autocorrelations is sufficiently small. We show that then the PACF is less than the ACF up to a multiplicative constant. Our second result complements a similar result of Baxter (1962).     

Direct oxidation of sodium borohydride on Pt, Ag and alloyed Pt–Ag electrodes in basic media: Part II. Carbon-supported nanoparticles

We studied the borohydride oxidation reaction (BOR) by voltammetry in 0.1 M NaOH/10⁻³ M BH₄⁻ on carbon-supported Pt, Ag and alloyed PtAg nanoparticles (here-after denoted as Pt/C, Ag/C and Pt–Ag/C). In order to compare the different electrocatalysts, we measured the BOR kinetic parameters and the number of electrons exchanged per BH₄⁻ anion (faradaic efficiency). The BOR kinetics is much faster for Pt/C than for Ag/C (i_Pt=0.15, i_Ag=3.1×10⁻⁴ A cm⁻² at E=−0.65 V vs. NHE at 25 °C), but both materials present similar Tafel slope values. The n value involved in the BOR depends on the thickness of the active layer of electrocatalysts. For a “thick layer” (approximately 3 m), n is nearly 8 on Pt/C and 4 on Ag/C, whereas n decreases for thinner Pt/C active layers (n2 for thickness <1 m). These results are in favour of the sequential BH₄⁻ hydrolysis (yielding H₂) followed by hydrogen oxidation reaction (HOR), or direct sequential BOR on Pt/C, whereas Ag/C promotes direct but incomplete BOR (Ag has no activity regarding hydrogen evolution reaction, HER). The n value close to 8 for the thick Pt/C layer displays the sufficient residence time of the molecules formed (H₂ by heterogeneous hydrolysis or BOR intermediates) within the active layer, which favours the complete HOR and/or BOR. Two PtAg/C nanoparticles alloys have been tested (noted APVES-4C and APVES-E1). They show different behavior; the borohydride oxidation reaction kinetics is faster on APVES-E1 than on APVES-4C (b=0.15, and b=0.31 V dec⁻¹,  A cm⁻², respectively, at 25 °C), but the n values are higher on APVES-4C than APVES-E1 (nearly 8 vs. 3, respectively, at 25 °C). These discrepancies probably originate from the heterogeneity of such bimetallic materials, as observed from physicochemical characterizations.     

Significance of logarithmic throwing index: a theoretical approach

An expression for the metal distribution ratio in electroplating systems as a function of the primary current density ratioL in the formM=L [W(1–r)/(1+K)] is derived.W,r andK are three dimensionless parameters related to the current efficiency ratio, the concentration polarization and activation polarization during the metal discharge. The function [W(1–r)/1+K] is compared with 1/A, the logarithmic throwing index empirically determined by Chin. The metal distribution ratio calculated by the use of the above formula is compared with the experimentally observed values. The close agreement between the two within an accuracy of 10% proves the validity of the equation derived. The logarithmic throwing power of electroplating systems is thus confirmed on theoretical grounds.Nomenclature A Logarithmic Throwing Index —inverse of the slope of the plot of logM versus logL - b Tafel slope. Slope of the equation =a + b logi - d_n Current efficiency in percent for metal deposition at near cathode - d _f Current efficiency in percent for metal deposition at far cathode - E The overall cell potential - E _n The potential drop in the electrolyte between the anode and near cathode - E _f The potential drop in the electrolyte between the anode and far cathode - e _a Dynamic anode potential - e _n Dynamic potential at the near cathode at a current densityi _n - e _f Dynamic potential at the far cathode at a current densityi _n - f a fraction = - i The average current density (A dm^–2) - i _n The primary current density at the near cathode when there is no polarization(A dm^–2) - i _f The primary current density at the far cathode when there is no polarization(A dm^–2) - i _n The secondary current density at the near cathode (A dm^–2) - i _f The secondary current density at the far cathode (A dm^–2) - i _{H n} The partial cathode current density at the near cathode for parallel cathodic reactions other than metal discharge (A dm^–2) - i _{H f} The partial cathode current density at the far cathode for parallel cathodic reactions other than metal discharge (A dm^–2) - i _{M n} The partial cathode current density for metal discharge at the near cathode - i _{M f} The partial cathode current density for metal discharge at the far cathode - K A dimensionless parameter =b/2.3E _f - l Linear Ratio =l _f/l _n ori _n i _f - l _n Linear distance of the near cathode (cm) - l _f Linear distance of the far cathode (cm) - M Metal distribution ratio - m _n Weight of metal deposited on the near cathode - m _f Weight of metal deposited on the far cathode - R Secondary current distribution ratio=i _n/i_f - r A dimensionless parameter related toK andf and given byf=(1/L) ^r/K - W A dimensionless parameter related current efficiency ratioR ^W–1 =d _n/d_f - Specific resistivity of the electrolyte( cm^–2) - _n The overpotential at the near cathode (V) - _f The overpotential at the far cathode - i ₀ The exchange current density     

Zirconium tetra-n-butylate modified with different organic acids: Hydrolysis and polymerization of the products

In this work; (a) complexation reaction of zirconium tetra-n-butylate, Zr(OBu ⁿ )₄, with MAc and different organic acids. (b) the hydrolysis reaction of modified Zr species, and (c) the polymerization reaction of complex products are studied. Zr(OBu ⁿ )₄ was reacted with different mole ratios of methacrylic acid (MAc) at room temperature and the maximum combination ratio was found to be 12 [Zr(OBu ⁿ )₄MAc] by FT IR. The modification of zirconium tetra-n-butylate with the acid mixtures [methacrylic acid-acetic acid (MeCOOH), methacrylic acid-propionic acid (EtCOOH), methacrylic acidbutyric acid (PrCOOH)] was made for a combination ratio of 111 [MAcRCOOHZr(OBu ⁿ )₄RMe. Et, Pr] and the products were characterized by¹H-NMR, FT-IR, and UV-spectroscopies. Following their synthesis, hydrolysis of the complexes with various amounts of water and polymerization with benzoyl peroxide were realized. The hydrolysis and polymerization products of the complexes were studied by Karl-Fischer Coulometric titration and thermal analysis respectively. Methyl-ethyl-ketone (MEK) and chloroform were chosen as solvents.     

On the solubility parameter of polar polymers

The binary cluster integral, β, was computed from intrinsic viscosity data. Subtracting from β the polar contribution, β_e, calculated from YRCR theory,⁹ the nonpolar interaction parameter, β_n, was found. The calculations were performed for poly(vinylacetate) and poly(methyl methacrylate), each in 16 solvents. The correlation between β_n and the solvent solubility parameter, δ₁, was found to be similar to that reported^8,17 for solutions of natural rubber, cis-polybutadiene and for poly(vinyl chloride). This correlation can be crudely approximated by the formula where E and F are functions of the ill-defined symmetry of the solvent molecule and δ_m is the δ₁ value for the local maximum of the function. At δ₁ = constant, the more spherical is the molecule, the higher is the β_n value. It was shown that for most cases separation of the solvent into two classes (linear and nonlinear) is sufficient. This β_n behavior finds support in the Funk and Prausnitz⁶ report on aromatic–saturated hydrocarbon mixtures and in the theoretical calculations of Huggins.^21,22     

Influence of Hydroxyvalerate Content on the Crystallization Kinetics of Poly(hydroxybutyrate-co-hydroxyvalerate)

Influence of hydroxyvalerate (HV) content on the crystallization kinetics of hydroxybutyrate (HB) units in random poly(hydroxybutyrate-co-hydroxyvalerate) (P(HB-co-HV)) copolymers has been studied by differential scanning calorimetry (DSC) and polarized optical microscopy (POM). It was found that the crystallization kinetics of HB units was strongly affected by the presence of the ethyl side chain of HV units whether under the non-isothermal or isothermal crystallization conditions. The spherulitic growth rate (G) and overallall crystallization rate (k _n ) of HB units decreased with increasing HV content. Both G and k _n exhibited the maxima, G ^max and k _n ^max, which showed a gradual shift toward lower temperatures with increasing HV content, which may be attributed to the depression in the equilibrium melting point (T _m ⁰) and nucleation factor (K _g). The temperature corresponding k _n ^max was different from G ^max due to the depression of nucleation rate with the degree of undercooling was more susceptible than that of the growth rate. According to the Lauritzen–Hoffman theory of secondary nucleation, the crystallization of HB units in P(HB-co-HV) copolymers was similar to that of neat PHB as regime III and n=4 growth process.     

Solubility of CO2 in Glassy PMMA and PS over a Wide Pressure Range: The Effect of Carbonyl Groups

The equilibrium sorption isotherms of CO₂ in glassy PMMA and PS at 32, 42, and 52C over a wide pressure range from 20 to 340 atm were investigated. The normalized sorption concentration (C) isotherms for the polymers in terms of pressure (P) can be described fairly well by two power laws C=kP ⁿ for below and above critical pressure (P _c). The exponent n is a measure of sorption intensity and is closely associated with the interaction between CO₂ and the polymer. From the temperature variation of n values, a negligible interaction between CO₂ and the polymer is found in the sorption process below P _c whereas the interaction is significant above P _c. For a given temperature, the n value for PMMA is 12 times higher than that for PS as a result of the specific interaction of CO₂ and the carbonyl groups in PMMA. The pre-exponential constant k is a measure of sorption capacity and is closely associated with the heat of sorption. In sorption above P _c, k is found to decrease with increasing temperature due to the exothermic sorption process that leads to a decrease in sorption capacity with temperature. From the sorption isobars of PMMA and PS, we observe that the temperature necessary for chemisorption to occur is lower for PMMA than for PS; this results from the specific interaction of CO₂ and the carbonyl groups of PMMA.     

Comparison of the Composition and Structural Parameters of W/O Microemulsions Containing Gemini Imidazoliums with Those Containing Monomeric Analogues

The composition and structural parameters of W/O microemulsions containing the gemini surfactant 1,4‐bis(3‐alkylimidazolium‐1‐yl) butane bromide [(C_n‐4‐C_n)Br₂, n = 12, 14, 16] + pentan‐1‐ol + octane + water and W/O microemulsions containing the ionic liquid surfactant 1‐alkyl‐3‐methylimidazolium (C_nmimBr, n = 12, 14, 16) + pentan‐1‐ol + octane + water were studied and compared. The mole fractions of the n‐alkyl alcohol at the interfacial layer in (C_n‐4‐C_n)Br₂ based microemulsion systems are always larger than those in C_nmimBr based microemulsion systems. However, the mole fractions of the n‐alkyl alcohol in the oil phase are nearly the same for both the microemulsion systems. The (C_n‐4‐C_n)Br₂ based microemulsion systems have greater absolute values of the free enthalpy values than that for C_nmimBr based systems. In the (C_n‐4‐C_n)Br₂ based microemulsion systems, a large number of cosurfactants at the interfacial layer is conducive to the formation of a smaller droplet W/O microemulsion. The effects of n‐alkyl alcohols, alkanes, salinity and temperature on the composition and structural parameters of the (C_n‐4‐C_n)Br₂ based and C_nmimBr based microemulsion systems were also investigated and discussed.     

The voidage problem in gas-electrolyte dispersions

Assessing the ohmic interelectrode resistance of electrochemical reactors with gas evolution requires data for the gas void fraction of gas-electrolyte dispersions. A voidage equation is derived taking account of the internal liquid flow in stationary electrolytes and at small liquid superficial velocities. The equation is a general form of available voidage equations.Nomenclature C non-dimensional constant, Equation 8 - n exponent, Equation 5 - S cross-sectional area (m²) - v _G gas velocity (m s^–1) - v _L liquid velocity (m s^–1) - v _s rising velocity of a bubble swarm (m s^–1) - v _l terminal rising velocity of a single bubble (m s^–1) - V_G volume flow rate of gas (m³ s^–1) - V_L volume flow rate of liquid (m³ s^–1) - fraction of cross-sectional area - volume (void) fraction of gas - _m geometric maximum of void fraction - maximum of void fraction in infinite gas flow Indices i internal - t total     

Development of a correlation between the non-polar solubility parameter,refractive index and molecular structure. I. Aliphatic hydrocarbons

For non-polar liquids (e.g. the alkanes) the cohesion energy density (λ²) can be shown to be a function of the refractive index (nD ), molal volume (V) and molecular structure according to: where Δ is the non-polar solubility parameter and the increments g_ij are determined from molecular structure. The difference between values of Δ calculated by this formula and experimental data is <0.1% for the C₅–C₁₆ n-alkanes and <0.8% for the C₅–C₈ branched isomers. The main object of this correlation was to provide a method for estimating the London dispersion force contribution to the cohesive energy of branched polar liquids.     

Optimal Rate of Convergence for Empirical Quantiles and Distribution Functions for Time Series

Given a stationary sequence , we are interested in the rate of convergence in the central limit theorem of the empirical quantiles and the empirical distribution function. Under a general notion of weak dependence, we show a Berry–Esseen result with optimal rate n^?1/2. The setup includes many prominent time series models, such as functions of ARMA or (augmented) GARCH processes. In this context, optimal Berry–Esseen rates for empirical quantiles appear to be novel.     

Maximum likelihood estimation for nearly non‐stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near‐unit root autoregressive process Y_t = ρ_nY_t?1 + ?_t with α‐stable noise {?_t} is studied in this paper. Herein ρ_n = 1 ? γ/n, γ ≥ 0 is a constant, Y₀ is a fixed random variable and ε_t is an α‐stable random variable with characteristic function φ(t,θ) for some parameter θ. It is shown that when 0 < α < 1 or α > 1 and E?₁ = 0, the limit distribution of the MLE of ρ_n and θ are mixtures of a stable process and Gaussian processes. On the other hand, when α > 1 and E?₁ ≠ 0, the limit distribution of the MLE of ρ_n and θ are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index α is less than 1.     

Differential scanning calorimetry analysis of thermoset cure kinetics: Phenolic resole resin

The Differential Scanning Calorimetry (DSC) trace for a commercial phenolic resole resin shows two distinct peaks. Assuming that these represent two independent cure reactions results in a kinetic model of the form: with κ_i = κ_io exp(-B_i/T). The Arrhenius parameters were estimated from a plot of ln(β/T) versus 1/T_p. The parameters, p, n₁, and n₂ were obtained by writing the DSC response predicted by the equation above in terms of a function which contains temperature as the only variable. with $ \theta _i = \left({1/\beta} \right)\int_{T_0}^T {\kappa _i dT \le r_i} $ dT ? r_i and r_i = 1/(1-n_i). Fitting this equation to the DSC response measured at a scan rate of 4°C/min obtains p ≈ 0.66; n₁ ≈ 0.55; n₂ ≈ 2.2; B₁ ≈ 8285; B₂ ≈ 7480; κ₁ ≈ 1. 12 × 10⁸ s^?1; κ₂ ≈ 0.99 × 10⁶ S^?1.     

Polymer Complexes XLV. Spectral Studies on Metal-Ligand Bonding in Novel Poly-Schiff Base Complexes

Polymer complexes derived from cinnamaldehyde and 2-substituted aniline with Cu(II), Pd(II), Pt(II), UO₂(II), Rh(III), Ru(III), and Pd(IV) have been synthesized and characterized by IR, electronic, EPR, ¹H-NMR, and ¹³C-NMR spectra, as well as by elemental analysis, thermogravimetry, and magnetic susceptibility measurements. The important bands in the IR spectra and the main ¹H- and ¹³C-NMR signals are assigned and discussed in relation to molecular structure. The wavelengths of the principal electronic absorption peaks have been accounted for quantitatively in terms of crystal field theory and various parameters were evaluated. The complexes, [Ru(HL _n )Cl₃] _n , are penta-coordinate; and, a trigonal-bipyramidal environment is commensurate for the Ru(III) ion. The presence of a coordinated water molecule in complex the Cu(II) complex (20) was demonstrated by thermogravimetry. The compounds, [N-(3-phenylacrylidene)-2-mercaptoaniline] (HL₁) and cinnamaldehyde-2-aminophenol (HL₂), act as monobasic and neutral bidentate ligands. The B-value suggests a strong covalency in the metal-ligand -bond. Tentative structures of the polymer complexes are proposed.     

Competitive mechanisms ofn-butane isomerization on sulfated zirconia catalysts

Isotopic analysis ofn-butane isomerization over sulfated zirconium oxide, using double¹³C-labelledn-butane, shows that at low temperature this isomerization is an inter-molecular process. Probably, a C₈ surface intermediate is formed which isomerizes and undergoes -fission; the iso-C₄ fragments are desorbed asi-butane. Previously, the same mechanism was indicated for Fe, Mn promoted catalysts. The isomerization rate at 130°C is drastically lowered by gaseous H₂, because the concentration of the unsaturated species, required for the formation of the C₈ intermediate, is low under such conditions. Whereas₁₃C-scrambledi-butane is a true primary product, isotopic scrambling ofn-butane continues after chemical equilibrium betweenn-butane andi-butane has almost been reached; i.e.¹³C scrambledn-butane is a secondary product. Intra-molecular rearrangement of carbon atoms inn-butane precedes intermolecular scrambling. The similarity of the isomerization mechanism over unpromoted sulfated zirconia and Fe, Mn promoted sulfated zirconia is paralleled by an equal strength of the acid sites in both catalysts. The shift in the FTIR band of CO adsorbed on the Lewis sites indicates that these sites, presumably surface Zr⁴⁺ ions, are weaker acids than A1³⁺ in dehydrated alumina.     

Simulation of nonequimolecular polycondensation in continuous flow-stirred tank reactors

Conversion and molecular weight distribution are computed and compared for uncatalyzed and catalyzed nonequimolecular polycondensation in continuous flow-stirred tank reactors (CSTRs) using two different kinetic schemes proposed by Flory and Lin, respectively. The contrast between the two schemes is also remarkable as found in the batch reactors. The polydispersity indices in the CSTRs are substantially larger than those obtained in the batch reactors. Also, the molecular weight distribution splits into two curves for odd and even homologues regardless of the two different schemes. An extremely long rersidence time is needed to obtain the higher conversion accompanying a large polydispersity index as compared to the batch reactors. The polydispersity can be expressed in CSTR as x?_w/x?_n = 2 Σn²N_n(R+1?2P)/(1+R)²[A]₀.     

Electrochemical reduction of silver thiosulphate complexes Part II Mechanism and kinetics

In this paper the thermodynamic data for complex formation between Ag⁺ and S₂O_{3 2–} ions, determined previously, are applied to kinetic investigation of the reduction of silver thiosulphate complexes. Both electrochemical (linear sweep voltammetry on a rotating disc electrode) and surface analytical (Auger electron spectroscopy) techniques are used. The deposits resulting from the electrodeposition of silver thiosulphate complexes are shown to be composed of silver and to be polycrystalline. The reduction follows a mechanism involving mass and charge transfer and chemical reaction steps. The relevant kinetic parameters are calculated and a rate equation describing the kinetics of the reduction is given.List of symbols a activity (M) - c concentration (M) - j current density (A m^–2) - j _c current density of charge transfer (A m^–2) - j _m current density of mass transfer (A m^–2) - k rate constant (m s^–1) - y activity coefficient (molarity scale) - D diffusion coefficient against gradient of concentration (m² s^–1) - D diffusion coefficient against gradient of electrochemical potential (m² s^–1) - E electrode potential vs NHE (V) - I ionic strength (M) - T temperature (K) Greek symbols a transfer coefficient - _1n stability constant of Ag(S₂O₃) _n (^2n–1)- - kinematic viscosity (m² s^–1) - rotation speed of the electrode (rad s^–1) Indices b bulk of the solution - f free (= uncomplexed) - 1,n related to complex Ag(S₂O₃)_n ^(n–1) - t total Constants F Faraday constant (96486 A s mol^–1) - R universal gas constant (8.3145 Jmol^–1 K^–1)