Ck-Reconstruction of Surfaces from Partial Data

In this paper, we study the problem of constructing a smooth approximant of a surface defined by the equation z = f(x ₁, x ₂), the data being a finite set of patches on this surface. This problem occurs, for example, after geophysical processing such as migration of time-maps or depth-maps. The usual algorithms to solve this problem are picking points on the patches to get Lagrange's data or trying to get local junctions on patches. But the first method does not use the continuous aspect of the data and the second one does not perform well to get a global regular approximant (C ¹ or more). As an approximant of f, a discrete smoothing spline belonging to a suitable piecewise polynomial space is proposed. The originality of the method consists in the fidelity criterion used to fit the data, which takes into account their particular aspect (surface's patches): the idea is to define a function that minimizes the volume located between the data patches and the function, and which is globally C ^k. We first demonstrate the new method on a theoretical aspect and numerical results on real data are given.     

A New Method for C k -Surface Approximation from a Set of Curves, with Application to Ship Track Data in the Marianas Trench

We introduce a surface approximation technique to address the problem of fitting a surface to a given set of curves. The originality of the method lies in its ability to take into account the continuous aspect of the data, and also in the possibility to arbitrarily select the regularity (C⁰, C¹, or higher) of the approximant obtained. We demonstrate the efficiency of the approach by constructing a bathymetry map of the Marianas trench based upon a set of SONAR (SOnic Navigation And Ranging) bathymetry ship track data.     

A New Method for C k -Surface Approximation from a Set of Curves,with Application to Ship Track Data in the Marianas Trench

We introduce a surface approximation technique to address the problem of fitting a surface to a given set of curves. The originality of the method lies in its ability to take into account the continuous aspect of the data, and also in the possibility to arbitrarily select the regularity (C⁰, C¹, or higher) of the approximant obtained. We demonstrate the efficiency of the approach by constructing a bathymetry map of the Marianas trench based upon a set of SONAR (SOnic Navigation And Ranging) bathymetry ship track data.     

On the least-squares fit of small and great circles to spherically projected orientation data

Given the direction cosines a _i ^′ = (a ₁ ⁱ , a ₂ ⁱ , a ₃ ⁱ )corresponding to a set of pspherically projected fabric poles, an initial estimate x′ = (x₁, x₂, x₃, x₄)for the angular radius x₄,and direction cosines of the center of the least-squares small circle which minimizes the sum of the squares of the angular residuals $$r = \sum\limits_p {\left[ {x_4 - \cos ^{ - 1} \left( {a_1^i x_1 + a_2^i x_2 + a_3^i x_3 } \right)} \right]} ^2 $$ can be iteratively improved by taking x_j+1 = x_j + Δxwhere x_j is the value of xat the jth iteration and $$\Delta x = - H_j^{ - 1} \left[ {q_j + x_j \left( {x'_j H_j^{ - 1} x_j } \right)\left( {q_j - x'_j H_j^{ - 1} q_j } \right)} \right],$$ where As an initial approximation for xwe have found it convenient to ignore the fact that the data are constrained to lie on the surface of the reference sphere and to use the parameters of a least-squares plane through the given poles. Generalization of this approach to fitting variously constrained great and small circles is easily made. The relative merits of differently constrained fits to the same data can be tested approximately if it is assumed that the errors in the location of the poles are isotropic and normally distributed. It is thus possible to statistically assess the relative significance of conflicting structural models which predict different geometrical patterns of fabric elements.     

A method for calculating fractional s-character for bonds of tetrahedral oxyanions in crystals

A method for calculating fractional s-character, f _s , for TO bonds has been devised to apply to TO₄ tetrahedral oxyanions in crystals. These f _s -values rank bond lengths with the better correlations obtained for T atoms associated with larger bond strengths and larger electronegativities. As a simple formula, it is found that 2cot²〈?〉₃ does a good job of estimating f _s where 〈?〉₃ is the triple angle average of the three angles common to a given bond.     

Apparent solubilities of schwertmannite and ferrihydrite in natural stream waters polluted by mine drainage

The apparent solubilities of schwertmannite and ferrihydrite were estimated from the H⁺, OH⁻, Fe³⁺, and SO₄²⁻ activities of the natural stream waters in Korea and mine drainage in Ohio, USA. Both chemical composition of the stream waters and the mineralogy of the precipitates were determined for samples from two streams polluted by coal mine drainage. This study combines these new results with previous data from Ohio, USA to redetermine solubilities. The activities of the dissolved species necessary for the solubility determinations were calculated from the chemical compositions of the waters with the WATEQ4F computer code.Laboratory analyses of precipitates indicated that the main minerals present in Imgok and Osheep creek were schwertmannite and ferrihydrite, respectively. The schwertmannite from Imgok creek had a variable chemical formula of Fe₈O₈(OH)_8−2x(SO₄)_x· nH₂O, where 1.74 ≤ x ≤ 1.86 and 8.17 ≤ n ≤ 8.62. The chemical formula of ferrihydrite was Fe₂O₃· 1.6H₂O. With known mineralogy of the precipitates from each stream, the activities of H⁺, OH⁻, Fe³⁺, and SO₄²⁻ in the waters were plotted on logarithmic activity-activity diagrams to determine apparent solubilities of schwertmannite and ferrihydrite. The best estimate for the logarithm of the solubility product of schwertmannite, logK_s, was 10.5 ± 2.5 around 15°C. This value of logK_s constrains the logarithm of the solubility product of ferrihydrite, logK_f, to be 4.3 ± 0.5 to maintain the stability boundary with schwertmannite observed in natural waters.     

Iron control by equilibria between hydroxy-Green Rusts and solutions in hydromorphic soils

In order to verify Fe control by solution - mineral equilibria, soil solutions were sampled in hydromorphic soils on granites and shales, where the occurrence of Green Rusts had been demonstrated by Mössbauer and Raman spectroscopies. Eh and pH were measured in situ, and Fe(II) analyzed by colorimetry. Ionic Activity Products were computed from aqueous Fe(II) rather than total Fe in an attempt to avoid overestimation by including colloidal particles. Solid phases considered are Fe(II) and Fe(III) hydroxides and oxides, and the Green Rusts whose general formula is [Fe^II_1−xFe^III_x(OH)₂]^+x· [x/z A^−z]^−x, where compensating interlayer anions, A⁻, can be Cl⁻, SO₄²⁻, CO₃²⁻ or OH⁻, and where x ranges a priori from 0 to 1. In large ranges of variation of pH, pe and Fe(II) concentration, soil solutions are (i) oversaturated with respect to Fe(III) oxides; (ii) undersaturated with respect to Fe(II) oxides, chloride-, sulphate- and carbonate-Green Rusts; (iii) in equilibrium with hydroxy-Green Rusts, i.e., Fe(II)-Fe(III) mixed hydroxides. The ratios, x = Fe(III)/Fe_t, derived from the best fits for equilibrium between minerals and soil solutions are 1/3, 1/2 and 2/3, depending on the sampling site, and are in every case identical to the same ratios directly measured by Mössbauer spectroscopy. This implies reversible equilibrium between Green Rust and solution. Solubility products are proposed for the various hydroxy-Green Rusts as follows: log K_sp = 28.2 ± 0.8 for the reaction Fe₃(OH)₇ + e⁻ + 7 H⁺ = 3 Fe²⁺ + 7 H₂O; log K_sp = 25.4 ± 0.7 for the reaction Fe₂(OH)₅ + e⁻ + 5 H⁺ = 2 Fe²⁺ + 5 H₂O; log K_sp = 45.8 ± 0.9 for the reaction Fe₃(OH)₈ + 2e⁻ + 8 H⁺ = 3 Fe²⁺ + 8 H₂O at an average temperature of 9 ± 1°C, and 1 atm. pressure. Tentative values for the Gibbs free energies of formation of hydroxy-Green Rusts obtained are: Δ_fG° (Fe₃(OH)₇, cr, 282.15 K) = −1799.7 ± 6 kJ mol⁻¹, Δ_fG° (Fe₂(OH)₅, cr, 282.15 K) = −1244.1 ± 6 kJ mol⁻¹ and Δ_fG° (Fe₃(OH)₈, cr, 282.15 K) = −1944.3 ± 6 kJ mol⁻¹.     

Heat capacity and thermodynamic properties of andradite garnet,Ca3Fe2Si3O12, between 10 and 1000 K and revised values for ΔfGom (298.15 K) of hedenbergite and wollastonite

The heat capacity of synthetic andradite garnet (Ca₃Fe₂Si₃O₁₂) was measured between 9.6 and 365.5 K by cryogenic adiabatic calorimetry and from 340 to 990 K by differential scanning calorimetry. At 298.15 K C^o_p,m and S^o_m are 351.9 ± 0.7 and 316.4 ± 2.0 J/(mol·K), respectively.Andradite has a λ-peak in C^o_p,m with a maximum at 11.7 ± 0.2 K which is presumably associated with the antiferromagnetic ordering of the magnetic moments of the Fe³⁺ ions. The Gibbs free energy of formation, Δ_fG^o_m (298.15 K) of andradite is −5414.8 ± 5.5 kJ/mol and was obtained by combining our entropy and heat capacity data with the known breakdown of andradite to pseudowollastonite and hematite at ≈ 1410 to 1438 K. From a reexamination of the calcite + quartz = wollastonite equilibrium data we obtained Δ_fH^o_m (298.15 K) = − 1634.5 ± 1.8 kJ/mol for wollastonite.Between 300 and 1000 K the molar heat capacity of andradite can be represented by the equation C^o_p,m = 809.24 - 7.025 × 10−2T− 7.403 × 10³T−0.5 − 6.789 × 10⁵T−2. We have also used our thermochemical data for andradite to estimate the Gibbs free energy of formation of hedenbergite (CaFeSi₂O₆) for which we obtained Δ_fG^o_m (298.15 K) = −2674.3 ± 5.8 kJ/mol.     

Scaling properties of corner frequencies of Kamchatka earthquakes

Scaling properties of earthquake populations bear the major information on the physics of the source process of an earthquake. To determine scaling properties, source spectra of more than 400 earthquakes of Kamchatka were determined in a frequency range 0.1–30 Hz using materials of digital registration of PET station, and characteristic frequencies of earthquakes were estimated. The range of magnitudes is 4–6.5, the range of distances is 80–220 km. To enable reduction of a spectrum to the source, attenuation properties of the medium around PET were determined beforehand. It is revealed that source spectra show several corner (characteristic) frequencies: f _c1, f _c2 and f _c3; where the spectral trend changes: from f ⁰ to f ^?1, from f ^?1 to f ^?2, and from f ^?2 to f ^?3, respectively. Although in some cases f _c1 ≈ f _c2 in agreement with the usual ω^?2 spectral model, the main part of spectra has more complicated character. For a large part of the studied earthquakes a source-controlled upper cutoff of acceleration spectrum, or corner frequency f _c3, is observed. This is an important fact, as the existence of f _c3 (source-controlled f _max) is not recognized in the bulk of the seismological literature. For f _c1, the observed scaling agrees with the usual hypothesis of similarity of the earthquake sources of different size (magnitude); it is close to f _c1 ～ M ₀ ^?1/3 , where M ₀ is seismic moment. For f _c2, scaling is close to f _c2 ～ M ₀ ^?0.17 ～ f _c1 ^0.5 , that indicates an expressed violation of similarity. For f _c3, scaling is close to f _c2 ～ M ₀ ^?0.08 ～ f _c1 ^0.25 , so that similarity is broken even sharper in this case. Hypotheses about possible causes of the observed scaling are discussed.     

Effect of 2-D Random Field Discretization on Failure Probability and Failure Mechanism in Probabilistic Slope Stability

This paper investigates, using the random field theory and Monte Carlo simulation, the effects of random field discretization on failure probability, p _f, and failure mechanism of cohesive soil slope stability. The spatial sizes of the discretized elements in random field Δx, Δy in horizontal and vertical directions, respectively, are assigned a series of combinational values in order to model the discretization accuracy. The p _f of deterministic critical slip surface (DCSS) and that of the slope system both are analyzed. The numerical simulation results have demonstrated that both the ratios of Δy/λ _y (λ _y  = scale of fluctuation in vertical direction) and Δx/λ _x (λ _x  = scale of fluctuation in horizontal direction) contribute in a similar manner to the accuracy of p _f of DCSS. The effect of random field discretization on the p _f can be negligible if both the ratios of Δx/λ _x and Δy/λ _y are no greater than 0.1. The normalized discrepancy tends to increase at a linear rate with Δy/λ _y when Δx/λ _x is larger than 0.1, and vice versa for p _f of DCSS. The random field discretization tends to have more considerable influence on the p _f of DCSS than on that of the slope system. The variation of p _f versus λ _x and λ _y may exhibit opposite trends for the cases where the limit state functions of slope failure are defined on DCSS and on the slope system as well. Finally, the p _f of slope system converges in a more rapid manner to that of DCSS than the failure mechanism does to DCSS as the spatial variability of soil property grows from significant to negligible.     

High temperature calorimetry of sulfide systems

Enthalpies of solution of synthetic pentlandite Fe_4.5Ni_4.5S₈, natural violarite (Fe_0.2941Ni_0.7059)₃S₄ from Vermillion mine, Sudbury, Ontario, synthetic pyrrhotite, FeS, synthetic high temperature NiS, synthetic vaesite, NiS₂, synthetic pyrite, FeS₂, Ni and Fe have been measured in a Ni_0.6S_0.4 melt at 1,100 K. Using these data and the standard enthalpies of formation of binary sulfides, given in literature, standard enthalpies of formation of pentlandite and violarite were calculated. The following values are reported: ΔH _f ^{o, Pent} =?837.37±14.59 kJ mol^?1 and ΔH _f ^{o, Viol} =?378.02±11.81 kJ mol^?1. While there are no thermo-chemical data for pentlandite with which our new value can be compared, an equilibrium investigation of stoichiometric violarite by Craig (1971) gives a significantly less negative enthalpy of formation. It is suggested that the difference may be due to the higher degree of order in the natural sample.     

Activity-composition relationship in Tschermak's substituted Fe biotites at 700°C, 2 kbar

The reaction-displacement technique was applied to the end-member reaction annite = sanidine + magnetite + H₂ in order to determine the activity of the annite component (a _Ann) in iron biotites with variable degrees of the Tschermak's substitution (^[6]Fe + ^[4]Si = ^[6]Al + ^[4]Al). Based on the simplified relation a _Ann = f _{H 2}/foH₂ (foH₂ = hydrogen fugacity of the end-member reaction at P, T), two types of experiments were performed at 700°C / 2 kbar: Type I used Fe-Al biotites of known starting composition together with sanidine + magnetite + H₂O. This assemblage was exposed to various f _{H 2} conditions (f _{H 2} < foH₂) produced in the pressure vessel either by using different ratios of water/oil as pressure medium (f _{H 2} in this case was measured by the hydrogen sensor technique), or by the Ni′NiO buffer. The composition of the Fe-Al biotites changed through incorporation or release of the annite component in response to the externally imposed f _{H 2}. By using opposite biotite starting compositions, the equilibrium composition as a function of f _H2 was bracketed. For type II, f _{H 2} in equilibrium with a specific combination of fine-grained Fe-Al biotite (+ sanidine + magnetite + H₂O) was measured internally by application of the hydrogen sensor technique. Both type I and type II experiments yield consistent results demonstrating that a fine-grained assemblage of Fe-Al biotite (+ sanidine + magnetite + H₂O) is able to act as a sliding-scale buffer. The final chemical composition of the Fe-Al biotite after the experiments was determined by electron microprobe and Mössbauer spectroscopy. The ^[4]Al and ^[6]Al in the biotites are coupled according to the Tschermak's substitution. In the tetrahedral sheet 0.1 Al-atoms per formula unit are present in excess to the amount required to balance ^[6]Al, and all Fe-Al biotites contain 8–10% Fe³⁺. Therefore, they are not members of the pure annite - siderophyllite join, but have an almost constant amount (15 Mol%) of two additional Fe³⁺-bearing components (ferri-siderophyllite and a vacancy end-member). The volume - composition relationship obtained does not indicate excess molar volumes of mixing for the annite (Ann) - siderophyllite (Sid) binary. The data are consistent with a molar volume of annite of 15.46 ± 0.02 Jbar^–1 and of 15.06 ± 0.02 Jbar^–1 for siderophyllite. The experimentally determined activity - composition relation shows that biotites on the join annite - siderophyllite deviate negatively from ideality. A symmetric interaction parameter W_AnnSid is sufficient to represent the data within error. This was constrained as: W _AnnSid = –29 ± 4 kJmol^–1. This is in contradiction to empirical interaction parameters derived from natural assemblages for this binary that predict positive deviation from ideality. Reasons for this discrepancy are discussed.     

Solution behavior of C-O-H volatiles in FeO-Na2O-Al2O3-SiO2 melts in equilibrium with liquid iron alloy and graphite at 4 GPa and 1550°C

In order to elucidate the solution behavior of carbon and hydrogen in iron-bearing magmatic melts in equilibrium with a metallic iron phase and graphite at oxygen fugacity (fO₂) values 2–5 orders of magnitude below the iron-wustite buffer equilibrium, fO₂ (IW), experiments were carried out at 4 GPa and 1550°C with melts of FeO-Na₂O-SiO₂-Al₂O₃ compositions. Melt reduction in response to an fO₂ decrease was accompanied by a decrease in FeO content. The values of fO₂ in the experiments were determined on the basis of equilibrium between Fe-C-Si alloy and silicate liquid. Infrared and Raman spectroscopy showed that carbon compounds are formed in FeO-Na₂O-SiO₂-Al₂O₃ melts: CH₄ molecules, CH₃ complexes (Si-O-CH₃), and complexes with double C=O bonds. The content of CO₂ molecules and carbonate ions (CO ₃ ^2? ) is very low. In addition to carbon-bearing compounds, dissolved hydrogen occurs in melt as H₂ and H₂O molecules and OH^? groups. The spectral characteristics of FeO-Na₂O-SiO₂-Al₂O₃ glasses indicate the occurrence of redox reactions in the melt, which are accompanied at decreasing fO₂ by a significant decrease in H₂O and OH^?, a slight decrease in H₂, and a significant concomitant increase in CH₄ content. The content of species with the double C=O bond increases considerably at decreasing fO₂ and reaches a maximum at ΔlogfO₂(IW) = ?3. According to the obtained IR spectra, the total water content (OH^? + H₂O) in the glasses is 1.2–5.8 wt % and decreases with decreasing fO₂. The high H₂O contents are due largely to oxygen release related to FeO reduction in the melt. The total carbon content at high H₂O (4.9–5.8 wt %) is approximately 0.4 wt %. The carbon content in liquid iron alloys depends on silicon content and, probably, oxygen solubility and ranges from 0.3 to 3.65 wt %. Low carbon contents were observed at a significant increase in Si content in liquid iron alloy, which may be as high as ～13 wt % at fO₂ values 4–5 orders of magnitude below fO₂(IW).     

Crystal chemistry of perovskite-type compounds in the tausonite-loparite series, (Sr1−2 x Na x La x )TiO3

Perovskite-type compounds in the series tausonite-loparite, (Sr_{1?2 x} Na _x La _x )TiO₃, were synthesized by solid-state reaction (final heating at 1200–1300?°C), and studied using “conventional” and synchrotron X-ray powder diffractometry. The structures of intermediate compositions were determined using the Rietveld profile refinement method. In the compositional range 0?≤x?≤ 0.1, the series comprises perovskites characterized by an undistorted cubic structure (space group Pm3¯m, a?≈ 3.905–3.902?Å, Z?=?1). Intermediate compounds in the range 0.15?≤?x?≤?0.35 crystallize with tetragonal symmetry (I4/mcm, a?≈? , c?≈? , Z?=?4) derived from the cubic aristotype by antiphase rotation of the TiO₆ octahedra about a fourfold axis. The angle of rotation estimated from the positional parameters of oxygen atoms ranges from 2.5(7)° to 5.5(4)°. The cubic-to-tetragonal transition arises from substitution of Sr²⁺ by the comparatively smaller Na¹⁺ and La³⁺ cations. A further transition from the tetragonal to rhombohedral symmetry (R3¯c, a?≈? , c?≈?2 , Z?=?6) occurs between x?=?0.35 and 0.40, and apparently does not involve formation of perovskite with an intermediate two-tilt structure (Imma). The rhombohedral structure is characterized by a multicomponent octahedral tilt about a threefold axis ranging in magnitude from 6.5(2)° to 7.7(2)°. In the series (Sr_{1?2 x} Na _x La _x )TiO₃, the unit-cell dimensions decrease, and the degree of structural distortion increases with x.     

The volatile component of some pumice-forming alkaline magmas from the Azores and Canary Islands

The abundances of pre-eruptive magmatic volatile species in the system H-O-S may be determined by application of thermodynamic methods to phenocryst assemblages commonly found in volcanic rocks, as demonstrated by Rutherford and Heming (1978). These methods are applied to alkaline pumice deposits, of airfall and ignimbrite type, from Tenerife (Canary Islands), Sao Miguel and Faial (Azores). It is argued that reliable temperature and fO₂ buffering mechanisms found in rhyolitic magmas appear not to operate in more alkaline liquids. fH₂O is estimated using biotite; the high values found are shown to be compatible with the violently explosive nature of the magmas concerned. fS₂ is estimated from pyrrhotite composition. fH₂, fH₂S, fSO₂, fSO₃ are calculated from gas equilibria. Water fugacity may be very roughly estimated for non-biotite bearing samples from data on the sulphur species. Abundances of these species are similar in alkaline and calc-alkaline salic magmas. Volcanological implications, relating to the release of volatiles during explosive eruptions, are considered.     

Steady-state kinetics and equilibrium between ground water and granitic rock

A hypothesis is presented that the dissolution of albite includes the exchange of sodium for hydrogen ion in a surface layer of the mineral and the structural collapse of the residual anionic lattice of the layer. The ion exchange is described by the first law of diffusion (D_25°C = 3 × 10^?22 and 1.5 × 10^?20 cm²sec^?1 at P_CO2 = 0 and 26.2 atm, respectively). The surface residual layer reaches a steady-state thickness ranging from n × 10^?8 to n × 10^?5 cm according to the temperature and P_CO2. The increase in aqueous sodium with time in a continuous ground-water system is described by a simple exponential equation. The equation is used to estimate the percolation time of ground water from the data on the chemical composition of a water sample. The probable times range from 14 to 3840 days for various ground-water systems and are compared to the times of percolation calculated from the geothermal and hydraulic data. Both estimates are found to be in general agreement. The concentrations of Al and Si in cold water from granitic rocks are shown to be controlled by the chemical equilibrium with respect to an aged aluminosilicate. The aluminosilicate precipitates from ground water as an amorphous isoelectric solid. Its chemical composition is represented by a simplified stoichiometric formula [Al(OH)₃](_1?x)[SiO₂]_x and varies linearly with pH of the solution. The atoms of Al, O and H tend to occupy a fixed position in the solid given by the gibbsite structure upon aging in the field. The solubility product of the solid is estimated from the published data on experimental and field research into the dissolution of feldspars: logK = (1 ? x) × log [Al³⁺] + xlog [H₄SiO₄] ? (3 ? 3x) log [H⁺] = 8.56 ? 11.26x, where x is the molar fraction of silica in the aluminosilicate.     

Crystallographic phase transition and valence fluctuation in synthetic mn-bearing ilvaite CaFe2+ 2-x Mn2+ xFe3+ [Si2O7/O/(OH)]

The dependence of the electronic and the crystallographic structure on temperature of synthetic Mnbearing ilvaites CaFe²⁺ _2-xMn²⁺ _xFe³⁺ [Si₂O₇/O/OH] with 0≤x≤0.19 has been investigated. The change of the electronic structure was studied by ⁵⁷Fe Mössbauer spectroscopy. The spectra show an increasing valence fluctuation rate between Fe²⁺ and Fe³⁺ in the double chain of edge-sharing octahedra with increasing temperature resulting in a mixed valent state of iron. The valence fluctuation rate is distinctly increased by the Mnsubstitution. The temperature of the crystallographic phase transition T _x as studied by a high temperature Guinier method is distinctly lowered by the Mn-substitution (x = 0.0, T _x=390K; x = 0.12, T _x =370K; x = 0.19, T _x=295K). The reasons for this behaviour are discussed in terms of Fe^{2 +}, Fe^{3 +} cation order-disorder, electronic relaxation rate, and relaxation of the lattice. In the monoclinic phase there is electron hopping between Fe^{2 +}, Fe^{3 +} pairs whereas in the orthorhombic phase there is extended electron delocalization via a narrow, d-band mechanism.     

Capillary Pressure Curves from Centrifuge Data: A Semi-Iterative Approach

The problem of determining capillary pressure functions from centrifuge data leads to an integral equation of the form _a ^x K(x,t)f(t)dt=g(x),x[a,b],(1)where the kernel K is known exactly and given by the underlying mathematical model. g is only known with a limited degree of accuracy in a finite and discrete set of points x ₁,...,x _M . However, the sought function f(t) is continuous. By the nature of the right-hand side, g(x), equation (1) is a discrete inverse problem which is ill-posed in the sense of Hadamard [9]. By a parameterization of the sought function, equation (1) reduces to a system of linear equations of the form Ac=b+ ,where b is the observation vector and A arises from discretization of the forward problem. is the error vector associated with b, and c contains the model parameters. The matrix A is usually ill-conditioned. The ill-conditioning is closely connected to the parameterization of the problem [23].In this paper a semi-iterative regularization method for solving the Volterra integral equation in the ₂-norm, namely, Brakhage's -method [2], is investigated. The iterative method is tested on synthetically generated, and on experimental data.     

Exoelectron spectroscopy of traps in surface layers of phenakite and quartz

The possibilities of exoelectron spectroscopy to investigate defects in dielectrics are demonstrated for phenakite Be₂SiO₄, its structural analogs Zn₂ SiO₄, Be₂GeO₄, solid solutions Be₂Si_1?x Ge _x O₄ (x=0÷1) and α-quartz. Emission maxima at 330 and 670 K in phenakite have been found to be due to [GeO₄]^5? andE' centers, respectively. Structural disturbances in the silicon and oxygen positions have been shown to control the exoemission activity of the crystals. Radiation induced decrease of exoemission activity connected with generation ofE' centers by neutron irradiation has been discovered. The energy level scheme of active centers in the subsurface region of Be₂SiO₄ has been established.     

Standard Gibbs free energy of formation for Cu2O,NiO, CoO,and FexO: High resolution electrochemical measurements using zirconia solid electrolytes from 900–1400 K

Galvanic cells with oxygen-specific solid electrolytes made of calcia-stabilized zirconia have been used to make equilibrium measurements of the standard Gibbs free energy of formation, Δ_fG⁰_m,(T), for copper (I) oxide (Cu₂O), nickel (II) oxide (NiO), cobalt (II) oxide (CoO), and wüstite (Fe_xO) over the temperature range from 900–1400 K. The measured values of Δ_fG⁰_m at 1300 K are −73950, −123555, −142150, and −179459 J · mol⁻¹ for Cu₂O, NiO, CoO, and Fe_0.947O, respectively. The precision of these measurements is ± 30–60 J · mol⁻¹, and their absolute accuracy is estimated to be ± 100–200 J·mol⁻¹. Using values of –76.557, −94.895, −79.551, and −71.291 J · K⁻¹ · mol⁻¹ for the entropies of formation, Δ_fS_m⁰, (298.15 K), the calculated enthalpies of formation, Δ_fH_m⁰, (298.15 K), are −170508, −240110, −237390, and −266458 J · mol⁻¹ for Cu₂O, NiO, CoO, and Fe_0.947O, respectively. These values of Δ_fS_m⁰ (298.15 K) and Δ_fH_m⁰ (298.15 K) are in good agreement with the best available calorimetric measurements.