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A thermodynamic model for calculating the mass action concentrations of structural units in Fe-Al binary melts based on the atom-molecule coexistence theory, i.e., AMCT-N-i model, has been developed and verified through comparing with the reported activities of both Al and Fe in Fe-Al binary melts in a temperature from 1 573 K to 1 873 K by different researchers. The calculated mass action concentration N-Al of Al or N-Fe of Fe can be applied to ideally substitute the measured activity a(R, Al) of Al or a(R, Fe) of Fe relative to pure liquid AI(I) or Fe(I) as standard state in Fe-Al binary melts. The following equations were derived for the activity coefficient of Al in natural logarithmic form In gamma(Al) and the calculated activity coefficient of Fe in natural logarithmic form In gamma(Fe) in the temperature range from 1 573 K to 1 873 K. In gamma(Al)= -3.810+4.476x(Al) 0 < x(Al)

This paper investigates the problem of quantized feedback control for networked control systems (NCSs) with time-varying delays and time-varying sampling intervals, wherein the physical plant is a continuous-time, and the control input is a discrete-time signal. By using an input delay approach and a sector bound method, the network induced delays, the signal quantization and sampling intervals are presented in one framework in the case of the state and the control input by quantization in a logarithmic form. We exploit a novel Lyapunov functional with discontinuity, taking full advantage of the NCS characteristic information including the bounds of delays, the bounds of sampling intervals and quantization parameters. In addition, it has been shown that the Lyapunov functional is decreased at the jump instants. Furthermore, we use the Leibniz-Newton formula and free-weighting matrix method to obtain the stability analysis and stabilization conditions which are dependent on the NCS characteristic information. The proposed stability analysis and stabilizing controller design conditions can be presented in term of linear matrix inequalities, which have less conservativeness and less computational complexity. Four examples demonstrate the effectiveness of the proposed methods.

We study the multiple residue of logarithmic differential forms with poles along a reducible divisor and compute the kernel and the image of the multiple residue map. As an application we describe the weight filtration on the logarithmic de Rham complex for divisors whose irreducible components are given locally by a regular sequence of holomorphic functions. In particular, this allows us to compute the mixed Hodge structure on the cohomology of the complement of divisors of certain types without the use of theorems on resolution of singularities and the standard reduction to the case of normal crossings.

We show that in the self-consistent equations for equal-time correlation functions of velocity fluctuations obtained in the model of developed turbulence based on the maximal randomness principle, infrared divergences are absent from all orders of the perturbation theory. We analyze the additional ultraviolet (UV) divergences that appear in the two-loop approximation of the self-consistent equations. We show that in the system with conserved parity, these divergences can be eliminated using the existing ambiguity in the solution. In the case of parity violation in the system, the UV divergences have a logarithmic form, which can indicate a deviation from the Kolmogorov scaling.

We define the compound logarithmic function on simple connected region. As an application of such function, we point out a slight mistake of Hayman in his proof for logarithmic derivative theorem (Lemma 2.3, Page 36), in his book Meromorphic Functions, published in 1964. We modify his proof.

The free vibration behaviors of functionally graded rings were investigated theoretically. The material graded in the thickness direction according to the power law rule and the rings were assumed to be in plane stress and plane strain states. Based on the first-order shear deformation theory and the kinetic relation of von Kárman type, the frequency equation for free vibration of functionally graded ring was derived. The derived results were verified by those in literatures which reveals that the present theory can be appropriate to predict the free vibration characteristics for quite thick rings with the radius-to-thickness ratio from 60 down to 2.09. Comparison between the plane stress case and the plane strain case indicates a slight difference. Meanwhile, the effects of the structural dimensional parameters and the material inhomogeneous parameter are examined. It is interesting that the value of the logarithmic form of vibration frequency is inversely proportional to the logarithmic form of the radius-to-thickness ratio or the mean radius.

In the probe limit, we investigate the nonlinear electrodynamical effects of the both exponential form and the logarithmic form on the holographic paramagnetism-ferromagnetism phase transition in the background of a Schwarzschild-AdS black hole spacetime. Moreover, by comparing the exponential form of nonlinear electrodynamics with the logarithmic form of nonlinear electrodynamics and the Born-Infeld nonlinear electrodynamics which has been presented in Ref.~\cite{Wu:2016uyj}, we find that the higher nonlinear electrodynamics correction makes the critical temperature smaller and the magnetic moment harder form in the case without external field. Furthermore, the increase of nonlinear parameter b will result in extending the period of the external magnetic field. Especially, the effect of the exponential form of nonlinear electrodynamics on the periodicity of hysteresis loop is more noticeable., arXiv admin note: text overlap with arXiv:1212.6693, arXiv:1208.5902 by other authors

Acoustic color is a descriptor of the dual dependences of backscattering cross section or its logarithmic form, target strength, of an underwater object on sonar frequency and angle of incidence. Use of ‘color’ refers, in one sense, to the coding of backscattering cross section by color on a figure whose axes are frequency and angle. In a larger sense, ‘color’ refers to the frequency dependence of backscattering, in analogy with optics, recognizing that the spectral dependence can convey significantly more information about the scattering object than the angle dependence alone. Here, acoustic color is computed for 13 pollack (Pollachius pollachius) swimbladders for which high-resolution morphometric data are available. The represented total fish length range is 31.5–44.5 cm. Investigated frequencies span the range 35–125 kHz, and investigated angles for vertically downward ensonification span the range of tilt angles from 45 deg head down to 45 deg head up. Since high-frequency acoustic scattering by fish is dominated by the swimbladder when present, the backscattering cross section of the swimbladder is a very good approximation to that of the entire fish. The assumption of high frequencies also supports use of the Kirchhoff-approximation scattering model applied to the digital representation of the swimbladder wall as a pressure-release surface. Computational results are shown for a particular specimen, and for the entire set of specimens when averaged with respect to the backscattering cross section. Differences of individual dependences with respect to the mean are computed. Cross-correlation coefficients of the difference functions are then computed and further quantified in a covariance matrix.

In this paper, time-dependent nonlinear numerical and analytical computational models suitable for rheological analysis of prestressed cable nets subjected to creep effects are presented. The principal aim of this contribution is to describe a developed design computational tool for a theoretical simulation of the time-dependent behavior of cable nets and predict their rheological characteristics. For this purpose, a modified finite-element method for the time-dependent analysis of materially linear and geometrically nonlinear cable nets is proposed. Creep constitutive equations in the logarithmic form are incorporated into the finite-element model to express the current lengths of cables as the functions of creep strain increments. As an extension of the Irvine approach, a geometrically nonlinear time-dependent analytical static solution of taut flat cable nets over an orthogonal square plan is presented. The cable net is replaced by an equivalent membrane. Creep strain is incorporated into the c...

The mean velocity profile (MVP) in thermally stratified atmospheric surface layers (ASLs) deviates from the classic logarithmic form. A theoretical framework was recently proposed (Katulet al.Phys. Rev. Lett., vol. 107, 2011, 268502) to link the MVP to the spectrum of turbulence and was found to successfully predict the MVP for unstable stratification. However, the theory failed to reproduce the MVP in stable conditions (Saleskyet al.Phys. Fluids, vol. 25, 2013, 105101), especially when${\it\zeta}>0.2$(where${\it\zeta}$is the atmospheric stability parameter). In the present study, it is demonstrated that this shortcoming is due to the failure to identify the appropriate length scale that characterizes the size of momentum transporting eddies in the stable ASL. Beyond${\it\zeta}\approx 0.2$(near where the original theory fails), the Ozmidov length scale becomes smaller than the distance from the wall$z$and hence is a more stringent constraint for characterizing the size of turbulent eddies. An expression is derived to connect the Ozmidov length scale to the normalized MVP (${\it\phi}_{m}$), allowing${\it\phi}_{m}$to be solved numerically. It is found that the revised theory produces a prediction of${\it\phi}_{m}$in good agreement with the widely used empirical Businger–Dyer relation and two experimental datasets in the stable ASL. The results here demonstrate that the behaviour of${\it\phi}_{m}$in the stable ASL is closely linked to the size of momentum transporting eddies, which can be characterized by the Ozmidov scale under mildly to moderately stable conditions ($0.2).

International audience; Starting from the general expression for the ground state correlation energy in the adiabatic connection fluctuation dissipation theorem (ACFDT) framework, it is shown that the dielectric matrix formulation, which is usually applied to calculate the direct random phase approximation (dRPA) correlation energy, can be used for alternative RPA expressions including exchange effects. Within this famework, the ACFDT analog of the second order screened exchange (SOSEX) approximation leads to a logarithmic formula for the correlation energy similar to the direct RPA expression. Alternatively, the contribution of the exchange can be included in the kernel used to evaluate the response functions. In this case the use of an approximate kernel is crucial to simplify the formalism and to obtain a correlation energy in logarithmic form. Technical details of the implementation of these methods are discussed and it is shown that one can take advantage of density fitting or Cholesky decomposition techniques to improve the computational efficiency; a discussion on the numerical quadrature made on the frequency variable is also provided. A series of test calculations on atomic correlation energies and molecular reaction energies shows that exchange effects are instrumental to improve over direct RPA results.

We study an element of the ring of algebraic integers having the special form 1 − zλ. We obtain a formula for calculating its logarithmic functions. Thus, we verify the conjecture that logarithmic functions of the element 1 − zλ are polynomials in z such that z appears only in powers that are divisors of the number of the logarithmic function.

SUMMARY The discrete optimal orientation design of the composite laminate can be treated as a material selection problem dealt with by using the concept of continuous topology optimization method. In this work, a new bi-value coding parameterization (BCP) scheme of closed form is proposed to this aim. The basic idea of the BCP scheme is to ‘code’ each material phase using integer values of +1 and –1 so that each available material phase has one unique ‘code’ consisting of +1 and/or –1 assigned to design variables. Theoretical and numerical comparisons between the proposed BCP scheme and existing schemes show that the BCP has the advantage of an evident reduction of the number of design variables in logarithmic form. The benefit is particularly remarkable when the number of candidate materials becomes important in large-scale problems. Numerical tests with up to 36 candidate material orientations are illustrated for the first time to indicate the reliability and efficiency of the BCP scheme in solving this kind of problem. It proves that the BCP is an interesting and valuable scheme to achieve the optimal orientations for large-scale design problems. Besides, a four-layer laminate example is tested to demonstrate that the proposed BCP scheme can easily be extended to multilayer problems. Copyright © 2012 John Wiley & Sons, Ltd.

Combustion kinetic parameters (i.e., activation energy and frequency factor) of coal have been proven to relate closely to coal properties; however, the quantitative relationship between them still requires further study. This paper adopts a support vector regression machine (SVR) to generate the models of the non-linear relationship between combustion kinetic parameters and coal quality. Kinetic analyses on the thermo-gravimetry (TG) data of 80 coal samples were performed to prepare training data and testing data for the SVR. The models developed were used in the estimation of the combustion kinetic parameters of ten testing samples. The predicted results showed that the root mean square errors (RMSEs) were 2.571 for the activation energy and 0.565 for the frequency factor in logarithmic form, respectively. TG curves defined by predicted kinetic parameters were fitted to the experimental data with a high degree of precision.

Existing analytical models for mean wind profiles within canopies are applicable only in dense canopy scenarios, where all momentum is absorbed by canopy elements and, hence, the effect of the ground on turbulent mixing is not important. Here, we propose a new analytical model that can simulate mean wind profiles within sparse canopies under neutral conditions. The model adopts a linearized canopy-drag parametrization and a first-order turbulence closure scheme taking into account the effects of both the ground and canopy elements on turbulent mixing. The resulting wind profile within a sparser canopy appears to be more like a logarithmic form, with the no-slip condition at the ground being satisfied. The analytical solution converges exactly to the standard surface-layer logarithmic wind profile in the case of zero canopy density (i.e., no-canopy scenario) and tends to be an exponential wind profile for a dense canopy; this feature is unique compared with existing analytical models for canopy wind profiles. Results from the new model are in good agreement with those from laboratory experiments and numerical simulations.

Nonlinear time series analysis aims at understanding the dynamics of stochastic or chaotic processes. In recent years, quite a few methods have been proposed to transform a single time series to a complex network so that the dynamics of the process can be understood by investigating the topological properties of the network. We study the topological properties of horizontal visibility graphs constructed from fractional Brownian motions with different Hurst index $H\in(0,1)$. Special attention has been paid to the impact of Hurst index on the topological properties. It is found that the clustering coefficient $C$ decreases when $H$ increases. We also found that the mean length $L$ of the shortest paths increases exponentially with $H$ for fixed length $N$ of the original time series. In addition, $L$ increases linearly with respect to $N$ when $H$ is close to 1 and in a logarithmic form when $H$ is close to 0. Although the occurrence of different motifs changes with $H$, the motif rank pattern remains unchanged for different $H$. Adopting the node-covering box-counting method, the horizontal visibility graphs are found to be fractals and the fractal dimension $d_B$ decreases with $H$. Furthermore, the Pearson coefficients of the networks are positive and the degree-degree correlations increase with the degree, which indicate that the horizontal visibility graphs are assortative. With the increase of $H$, the Pearson coefficient decreases first and then increases, in which the turning point is around $H=0.6$. The presence of both fractality and assortativity in the horizontal visibility graphs converted from fractional Brownian motions is different from many cases where fractal networks are usually disassortative., 12 pages, 8 figures

The results obtained in determining the creep strain of mineral wool slabs under compressive stress, used for insulating flat roofs and facades, cast-in-place floors, curtain and external basement walls, as well as for sound insulation of floors, are presented. The creep strain tests were conducted under a compressive stress of σ c =0.35σ 10%. Interval forecasting of creep strain was made by extrapolating the creep behaviour and approximated in accordance with EN 1606 by a power equation and reduced to a linear form using logarithms. This was performed for a lead time of 10 years. The extension of the range of the confidence interval due to discount of the prediction data, i.e. a decrease in their informativity was allowed for by an additional coefficient. Analysis of the experimental data obtained from the tests having 65 and 122 days duration showed that the prediction of creep strains for 10 years can be made based on data obtained in experiments with durations shorter than the 122 days as specified by EN 13162. Interval prediction of creep strains (with a confidence probability of 90%) was based on using the mean square deviation of the actual direct observations of creep strains in logarithmic form to have the linear trend in a retrospective area.

In this paper, we study quantum corrections to the temperature and entropy of a regular Ay\'{o}n-Beato-Garc\'{\i}a-Bronnikov black hole solution by using tunneling approach beyond semiclassical approximation. We use the first law of black hole thermodynamics as a differential of entropy with two parameters, mass and charge. It is found that the leading order correction to the entropy is of logarithmic form. In the absence of the charge, i.e., $e=0$, these corrections approximate the corresponding corrections for the Schwarzschild black hole., Comment: 15 pages, accepted for publication in Astrophysics and Space Science

In this article, a theoretical procedure has been developed for the kinetic study of the glass-crystal transformations under continuous heating regime in materials involving formation and growth of nuclei, obtaining the corresponding parameters: kinetic exponent, activation energy, and pre-exponential frequency. It is an integral procedure, which is based on a transformation rate independent of the thermal history of the material and assumes that the quoted rate depends only on the transformed fraction, x, across the f(x) function, and temperature, considering that these variables are independent ones. Therefore, the transformation rate is expressed as the product of two separable functions of absolute temperature and the transformed fraction. The quoted f(x) function corresponds to a theoretical method that we have developed and recently published, whose details are given in the Sect. “Basic theory” of this study. The above-mentioned integral procedure considers the same pair of temperatures for the different heating rates and obtains a constant value for temperature integral and, therefore, a plot of a function of the transformed fraction versus the reciprocal of the heating rate leads to a straight line with an intercept of zero, if the value of kinetic exponent is correctly chosen. The process may be repeated for other pairs of temperatures and, consequently, other straight lines will be obtained. By using different values of kinetic exponent for each of the quoted lines, it is taken the best correlation coefficient to choose the quoted exponent more suitable. On the other hand, by using the first mean value theorem to approach the temperature integral, one obtains a relationship between a function of the temperature and other function of the transformed fraction. The logarithmic form of the quoted relationship leads to a straight line, whose slope and intercept allow the obtaining of the activation energy and of the pre-exponential frequency. In addition, this study applies the quoted procedure to the analysis of the crystallization kinetics of the Ge0.08Sb0.15Se0.77 glassy alloy resulting in ranges of variation both of the kinetic exponent, n, with the temperature and of the activation energy, E, with the crystallized fraction, which contain the values of n and E, obtained according to the already quoted theoretical method, that we have developed and recently published. This fact shows the reliability of the theoretical procedure described in this article to analyze the glass-crystal transformation kinetics of glassy alloy.

The Cu solubility was measured in the CaO-B2O3 and BaO-B2O3 slag systems to understand the dissolution mechanism of Cu in the slags. The Cu solubility had a linear relationship with oxygen partial pressure in the CaO-B2O3 slag system, which corresponds with previous studies. Also, the Cu solubilities in slag decreased with increasing the slag basicity, which value of slope was close to –0.5 in logarithmic form. From the results of experiment, the Cu dissolution mechanism established as follows: $$ {\text{Cu}} + \frac{1}{4}{\text{O}}_{2} = {\text{Cu}}^{ + } + \frac{1}{2}{\text{O}}^{2 - } $$ To compensate the effects of oxygen partial pressure and chemical potential of Cu in alloy on the Cu solubility of slag, the concept of cuprous capacity was suggested as follows: $$ C_{{\rm Cu^{ + } }} = \frac{K}{{f_{{\rm Cu^{ + } }} \cdot a_{{\rm O^{2 - } }}^{\frac{1}{2}} }} = \frac{{({\text{mass pct Cu}})}}{{a{}_{\rm Cu} \cdot p_{{\rm O_{2} }}^{\frac{1}{4}} }} $$ The cuprous capacity showed decreasing tendency as the slag basicity increased in the B2O3 bearing slags as well as in CaO-SiO2-Al2O3(Satd.) and FeO-Fe2O3-SiO2(Satd.) slag systems. It is suggested that the proposed dissolution mechanism could be available in various slag systems.

This paper focuses on the long-term creep characterization of Gr. 91 steel using creep constitutive equations. The models of three such equations, a combination of power-law form and omega model (CPO), a combination of exponential form and omega model (CEO), and a combination of logarithmic form and omega model (CLO), which are described as sum decaying primary creep and accelerating tertiary creep, are proposed. A series of creep rupture data was obtained through creep tests with various applied loads at 600 °C. On the basis of the creep data, a nonlinear least-square fitting (NLSF) analysis was carried out to provide the best fit with the experimental data in optimizing the parameter constants of an individual equation. The results of the NLSF analysis showed that in the lower stress regions of 160 MPa (σ/σys 0.65), the CPO model showed better agreement than the other two models. It was found that the CEO model was superior to the CPO and CLO models in the modeling of long-term creep curves. Using the CEO model, the long-term creep curves of Gr. 91 steel were numerically characterized, and its creep life was predicted accurately.

The type of mathematical model of the production function is very important for an adequate prediction of optimal fertilizer rates and their effect on yield. In this case, easily calibrated regression models can’t be used for extrapolation, and the models proposed by Mitscherlich, Spillman, Baule, and von Boguslawski are difficult to calibrate even in the one-dimensional case. An explanation of the good approximating capacity of logarithmic transformation of the argument of the linearizing ascending branch of the production function is found on the basis of model investigations. An algorithm for simplified calibration of the von Boguslawski mathematical model is developed: (1) transform fertilizer rates into a logarithmic form after introducing a correction for the initial content of the nutrient element in soil; (2) estimate the maximum yield under experimental conditions; convert the yield data into a relative form; (3) discriminate the data based on the interval permissible for approximation; (4) find the parameters of the linear model by the least-squares method and on their basis calculate the parameters of the von Boguslawski model, thereby correcting the form of the approximation and obtaining the possibility of its extrapolation.

The rheology of solutions of randomly methylated (1.8) β-cyclodextrin (β-CD) in 1-ethyl-3-methyl imidazolium acetate ([EMIM][Ac]) was studied in detail by rotational and oscillatory shear measurements. It was found that a gel structure was formed even at a relatively low β-CD concentration, which induced intriguing results in oscillation experiments. The solutions showed abrupt changes of the structurally dependent rheological moduli and reproducible transitions from gel to sol state at specific shear stress. For all β-CD solutions, a non-Newtonian flow behavior including shear thinning was obtained. The influence of temperature and β-CD concentration on the flow behavior was studied for all solutions, and the flow activation energies were calculated from the logarithmic form of the Arrhenius equation for nonassociating electrolytes.

We propose some nonlinear Schrodinger equations by adding some higher order terms to the Lagrangian density of Schrodinger field, and obtain the Gross-Pitaevskii (GP) equation and the logarithmic form equation naturally. In addition, we prove the coefficient of nonlinear term is very small, i.e., the nonlinearity of Schrodinger equation is weak.

A Galton model, in logarithmic form is used to explain the convergence of countries' productivities over time, including β and σ convergence. The ‘regression fallacy’ does not arise. When comparing growths and levels of productivity, countries should be classified by their initial levels of productivity, preferably after logarithmic transformation. Errors in the data may justify using an error-in-variables model, or the geometric mean of the original and reverse regressions, but this is not certain.

An estimate is obtained for the growth of a meromorphic function near to a logarithmic singularity of the derivative. This estimate is applied to show that if f is meromorphic of finite lower order in the plane, such that the second derivative f″ has finitely many zeros and the multiplicities of the poles z of f grow at most polynomially in ¦z¦, then f has finitely many poles. Subsequent results then consider the zeros of linear differential polynomials F = f (k)+a k−1 f (k−1)+…+ a 0 f, where f is transcendental and meromorphic of finite order in the plane, and the coefficients a j are constants.

Two generalized polynomial expressions, one based on a logarithmic form and the other on an exponential form have been set up that give pressure–volume ( P – V ) relationship and higher derivative properties mimicking the Stacey reciprocal K-primed equation of state. The results have been obtained for pressure P , bulk modulus K and its pressure derivative K ′ for six metals viz. Ag, Al, Au, Cu, Mo and W at different values of compression down to V / V 0 =0.5. The zero-pressure values of input parameters K 0 and K 0 ′ have been taken from the literature, whereas K ∞ ′ and K 0 K 0 ″ have been fixed to match the Stacey reciprocal K-primed equation. The polynomial equations thus formulated can be used as a substitute for the Stacey equations of state (EOS) for determining P – V relationship and higher derivative properties such as K and K ′.

The logarithmic form for turbulent flow analysis guarantees the positivity of the turbulence variables as k and ? of the k-? model by using the natural logarithm of these variables. In the present study, the logarithmic form is incorporated into the finite element solution procedure for the unsteady turbulent flow analysis. A backward facing step flow using the standard k-? model and a flow around a 2D square cylinder using the modified k-? model (the Kato-Launder model) are simulated. These results show that the logarithmic form effectively keeps adequate balance of turbulence variables and makes the analysis stable during transient or unsteady processes.

We present some types of estimates for logarithmic derivatives of meromorphic functions, which include an improvement of Gundersen’s estimates in another situation.

A theoretical method has been considered to determine the suitable form of the glass–crystal transformation function, and to calculate the kinetic parameters by using differential scanning calorimetry data, obtained from experiments carried out under non-isothermal regime. It is an integral method, which is based on a transformation rate independent of the thermal history and expressed as the product of two separable functions of absolute temperature and the volume fraction transformed. Considering the same temperatures for the different heating rates, one obtains a constant value for temperature integral, and therefore a plot of a function of the volume fraction transformed versus the reciprocal of the heating rate leads to a straight line with an intercept of zero, if the reaction mechanism is correctly chosen. Besides, by using the first mean value theorem to approach the temperature integral, one obtains a relationship between a function of the temperature and other function of the volume fraction transformed. The logarithmic form of the quoted relationships leads to a straight line, whose slope and intercept allow to obtain the activation energy and the frequency factor, respectively. The theoretical method has been applied to the crystallization kinetics of the Ge 0.13 Sb 0.23 Se 0.64 glassy alloy and it has been found that the kinetic model of normal grain growth is most suitable to describe the crystallization of the quoted alloy. The values obtained for the activation energy, E , and the logarithm of the frequency factor, K 0 , have been 188.3 kJ mol −1 and 36.7 ( K 0 in s −1 ), respectively. The phases at which the alloy crystallizes after the thermal process have been identified by X-ray diffraction. The diffractogram of the transformed material suggests the presence of microcrystallites of Sb 2 Se 3 and GeSe, remaining in a residual amorphous matrix.

In Laser Beam (L.B.) and Electron Beam (E.B.) drilling, the energy distribution significantly affects both the penetrating efficiency and working performance, both of which are usually estimated by numerical skill or experimental measure. Through the application of a stimulation model, an unstable solution with the finite difference method will result near the solidliquid interface unless much finer grid sizes are set up. To improve on the above defect, nonuniform grids are therefore utilized; this will complicate the built-up of the program and also easily causes the simulated energy distribution to be divergent in the iteration process. In this study, an estimated small Peclet number and observed narrow-deep cavity made the convective and radial diffusion terms small enough to be neglected in the governing equation. From these assumptions, the model was then used to investigate the drilling efficiency where two-phase flow convection could be simplified further into one dimension and thus the analytical solution becomes possible by transferring the penetration velocity into the logarithmic form. When compared with the experiment made by Allmen [1] , the present model shows good agreement in higher energy density and relative errors are no more than 10%.

We empirically study the genealogical trees of ten families for about five centuries in Korea. Although each family tree contains only the paternal part, the family names of women married to the family have been recorded, which allows us to estimate roughly the family name distributions for the past five hundred years. Revealed is the fact that the unique Korean family name distribution, characterized by a logarithmic form of the cumulative distribution and an exponentially decaying rank-size plot often called the Zipf plot, has remained unchanged for a long time. We discuss the implications of our results within a recently suggested theoretical model and compare them with observations in other countries in which power-law forms are abundantly found., Comment: 5 pages, 3 figures

Details are given of the apparatus used for measuring adsorption isotherms of krypton at −196°, which allowed a rapid and precise measurement of pressure. Surface areas have been calculated, by the B.E.T. method, from the krypton adsorption isotherms on samples of ground quartz and a silica gel. The surface areas of the quartz samples determined by other workers, using B.E.T. nitrogen adsorption and electron microscopy, agreed well with the krypton values, provided that the atomic cross-sectional area of krypton was taken as 19·5Ǎ2. It was not possible to obtain a ‘standard’ curve from the krypton adsorption isotherms on the non-porous silicas, but when these isotherms were plotted in logarithmic form, all gave a substantial linear region with the same slope. The logarithmic plot of the isotherm on the silica gel had a different slope; it would seem likely, therefore, that capillary condensation occurred on this sample at low pressure and that the B.E.T. area is invalid here.

A procedure has been developed to determine the reliable form of the glass-crystal transformation function, and to deduce the kinetic parameters by using differential scanning calorimetry data, obtained from experiments performed under non-isothermal conditions. It is an integral method, which is based on a transformation rate independent of the thermal history and expressed as the product of two separable functions of absolute temperature and the fraction transformed. Considering the same temperatures for the different heating rates, one obtains a constant value for temperature integral, and, therefore, a plot of a function of the volume fraction transformed versus the reciprocal of the heating rate leads to a straight line with an intercept of zero, if the reaction mechanism is correctly chosen. Besides, by using the first mean value theorem to approach the temperature integral, one obtains a relationship between a function of the temperature and other function of the volume fraction transformed. The logarithmic form of the quoted relationship leads to a straight line, whose slope and intercept allow to obtain the activation energy and the frequency factor. The method developed has been applied to the crystallization kinetics of the Sb0.12As0.36Se0.52 glassy alloy and it has been found that the kinetic model of normal grain growth is the most suitable to describe the crystallization of the quoted alloy. The mean values obtained for the activation energy and the frequency factor have been 27.36 kcal mol−1 and 1.5×109 s−1, respectively.

Optically active propiothiolactones I (R = OMe, Me, H, Cl, and NO2) were prepared by debenzylation of the corresponding 2-para-substituted-benzenesulfonamido-3-benzyl-mercaptopropionyl chlorides in the presence of 3 moles of aluminum bromide under the conditions of Friedel-Crafts synthesis. The para-substituted-benzenesulfonamido-propio-thiolactones were polymerized in bulk and in DMF solution in the presence of benzyl mercaptan as initiator. The rate of polymerization was determined by measuring the change of specific rotation with time. The relative reactivity, expressed in logarithmic form, was plotted against Hammett's σ funtions showing a linear relationship, with reaction constant σ = –0.36. The temperature dependence of rate constants was determined and the activation energy in the polymerization of I (R = OMe, Me, and Cl) was determined: EA = 9.9 kcal/mole for I (R = OMe); 12.3 kcal/mole for I (R = Me), and 15.2 kcal/mole for I (R = Cl).

In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2, any Brody curve is algebraically degenerate. In the case of logarithmic Kodaira dimension 1, we still get the same result under a very mild condition on the Stein factorization map of the quasi-Albanese map of the log surface, but we show by giving a counter-example that the result is not true any more in general. Finally we prove that a logarithmic surface having logarithmic irregularity 2 admits certain types of algebraically non degenerate entire curves if and only if its logarithmic Kodaira dimension is zero, and we also give a characterization of this case in terms of the quasi-Albanese map.

We develop a renormalization group approach for cyclizing polymers for the case when chain ends are initially close together (ring initial conditions). We analyze the behavior at times much shorter than the longest polymer relaxation time. In agreement with our previous work (Europhys. Lett. 73, 621 (2006)) we find that the leading time dependence of the reaction rate k(t) for ring initial conditions and equilibrium initial conditions are related, namely k ring(t) ∝ t -δ and k eq(t) ∝ t 1-δ for times less than the longest polymer relaxation time. Here δ is an effective exponent which approaches δ = 5/4 for very long Rouse chains. Our present analysis also suggests a “sub-leading” term proportional to (ln t)/t which should be particularly significant for smaller values of the renormalized reaction rate and early times. For Zimm dynamics, our RG analysis indicates that the leading time dependence for the reaction rate is k(t) ∼ 1/t for very long chains. The leading term is again consistent with the expected relation between ring and equilibrium initial conditions. We also find a logarithmic correction term which we “exponentiate” to a logarithmic form with a Landau pole. The presence of the logarithm is particularly important for smaller chains and, in the Zimm case, large values of the reaction rate.

We introduce the notion of logarithmic index of a vector field on a hypersurface and prove that the homological index can be expressed via the logarithmic index. Then both invariants are described in terms of logarithmic differential forms for Saito free divisors, which are hypersurfaces with nonisolated singularities, and all contracting homology groups of the complex of regular holomorphic forms on such a hypersurface are computed. In conclusion, we consider the case of normal hypersurfaces, including the case of an isolated singularity, and describe the contracting homology of the complex of regular meromorphic forms with the help of the residue of logarithmic forms.

By using a slow growth scale, the logarithmic order, with which to measure the growth of functions, we obtain basic results on the value distribution of a class of meromorphic functions of zero order.

This article deals with the theory of logarithmic differential forms and with some of its less known applications, developed by the author in the past few years, in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. At first, some relations between properties of logarithmic differential forms and torsion differentials on singular hypersurfaces are studied. Then a natural extension of the classical Poincare residue in the case of singular hypersurfaces, and a variant of the Grothendieck local duality for non-isolated non-normal reduced hypersurface singularities are described in detail. In conclusion a new method for computing the index of vector fields on hypersurfaces with arbitrary singularities, and some applications to the theory of Pfaffian systems of differential equations of Fuchsian type are discussed.

Let $f:\Cal C\to S$ be a flat family of curves over a smooth curve $S$ such that $f$ is smooth over $S_0=S\ssm\{s_0\}$ and $f^{-1}(s_0)=\Cal C_0$ is irreducible with one node. We have an associated family $\Cal M_{S_0}\to S_0$ of moduli spaces of semistable vector bundles and the relative theta line bundle $\Theta_{S_0}$. We are interested in the problem: to find suitable degeneration $\Cal M_S$ of moduli spaces and extension $\Theta_S$ of theta line bundles such that the direct image of $\Theta_S$ is a vector bundle on $S$ with a logarithmic projective connection. In this paper, we figured out the conditions of existence of the connection and solved the problem for rank one., Comment: 26 pages, amstex

The adsorption of copper(II) to Shirasu, a pyroclastic flow deposit, was examined by a batch-and-column method. The effect of the grain size was investigated in the range

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues in a specific Banach algebra, namely the one generated by the compact operators and the identity in the case when the underlying Banach space is infinite dimensional. The situation is more complex than encountered in previous investigations concerning logarithmic residues. Logarithmic derivatives may have essential singularities and the geometric properties of the Banach space play a role. Topological properties of the set of logarithmic residues are considered too. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A critical examination is made of the mathematical principles that are used for the kinetic interpretation of nonisothermal thermoanalytical rate measurement by a representative approximate method, the widely applied Coats–Redfern (CR) equation. It is concluded that the dominant feature in the identification of that isothermal kinetic equation, which most satisfactorily expresses the rate characteristics, is the (effective) exponent in the set of equations comparatively considered. Consequently, the form of the CR equation possesses only limited ability to distinguish between ‘fits’ to alternative kinetic models. This is entirely consistent with literature conclusions expressing the view that data from a single nonisothermal experiment is insufficient to identify the three kinetic parameters: kinetic model, g ( α )= kt , and both Arrhenius terms, A and E . An usual first step in kinetic analysis by the CR (and other related) equations is to incorporate the kinetic model ( g ( α )= kt ) into the expression used to calculate A and E . Because the rate equation (effective) exponent is a dominant and, without supporting observations, is an unknown parameter, this introduces the well-known ambiguity that alternative kinetic parameters are obtained by the uncritical use of this method. Accordingly, the following replacement calculation sequence is recommended as being more trustworthy. At least two, preferably several, nonisothermal experiments are undertaken, each at a different (usually constant) rate of temperature increase. For a comprehensive range of constant increments of reaction, Δ α i , the different rates (d α /d t ) at different reaction temperatures, T i , are determined from the several experiments and the activation energy, E i , for each successive reaction interval can be calculated. The constancy, or otherwise, of E i with α i throughout the reaction is thus established. From these data, the rate constant for each α i value can be extrapolated to a selected representative temperature T R , perhaps at the mid-point of a reaction ( α =0.5). From these data a pseudoisothermal α – t ( T R constant) curve can be constructed, suitable for analysis by the usual methods. Various advantages from this approach are perceived. The unfortunate role of the (effective) kinetic model exponent in combination with the logarithmic form of the CR equation in analyzing the data, which inhibits recognition of the kinetic model, is avoided. The laborious calculations will not be a problem for modern high-speed computers, unlike the situation existing when the CR equation was first introduced in 1964. Suitable programs may be used to maximize the accuracy of the method, through the use of several nonisothermal experiments and small Δ α increments in the analysis. Kinetic comparisons can be extended to a wider range of kinetic models than the limited selection that often restrict the possibilities for nonisothermal rate data interpretation by the most widely used approximate methods.

We prepared the indium doped MgB 2 bulk sample by conventional solid state reaction. Systematic measurements of resistance versus temperature were performed on the sample with different transport current and different applied magnetic field. The curves of resistance in opposition to temperature are broadened with different transport current in transmitting temperature range. We present a method for the first time to analyze the experimental data and explore the pinning mechanisms of MgB 2 for such kind of experiments. By analyzing on the experimental data, we found that the effective potential U eff ( T , H , J ) is of logarithmic form in our experimental temperature and applied magnetic field regime. The curve U 0 ( T , J ) is linear dependent on temperature but the slope coefficient is changing lower near onset critical temperature.

Summary Length–weight and length–length relationships are reported for 32 fish species belonging to seven families and 26 genera from the Tian-e-zhou Oxbow in the middle reach of Yangtze River, near Shishou City, China. The weight (W) - length (L) relationships were estimated using the logarithmic form of the equation W = aLb. The exponent b varied from 2.60 to 3.65 (mean ± SE: 3.12 ± 0.25). The 14 weight–length relationships, 13 length–length relationships, and seven maximum lengths are presented to FishBase for the first time.

The Generalized Minimal Massive Gravity (GMMG) theory is realized by adding the CS deformation term, the higher derivative deformation term, and an extra term to pure Einstein gravity with a negative cosmological constant. In the present paper we obtain exact solutions to the GMMG field equations in the non-linear regime of the model. GMMG model about $AdS_3$ space is conjectured to be dual to a 2-dimensional CFT. We study the theory in critical points corresponding to the central charges $c_-=0$ or $c_+=0$, in the non-linear regime. We show that $AdS_3$ wave solutions are present, and have logarithmic form in critical points. Then we study the $AdS_3$ non-linear deformation solution. Furthermore we obtain logarithmic deformation of extremal BTZ black hole. After that using Abbott-Deser-Tekin method we calculate the energy and angular momentum of these types of black hole solutions., Comment: 16 pages, no figure, some Refs added, minor changes, as published in Gen Relativ Gravit

We develop the Newton-Kantorovich method to solve the system of2×2nonlinear Volterra integral equations where the unknown function is in logarithmic form. A new majorant function is introduced which leads to the increment of the convergence interval. The existence and uniqueness of approximate solution are proved and a numerical example is provided to show the validation of the method.

Using contemporary accounts, we describe the scope of the Henderson approximation, and its relationship to the mass action law of Guldberg & Waage. The Henderson approximation, apparently written in logarithmic form by Hasselbalch, has the same form as the mass action law, but has a completely different meaning and a much more restricted range of applicability. The Guldberg-Waage law is the fundamental mass action relationship valid for all chemical equilibria, whereas the Henderson approximation is useful only within a limited range of a sufficiently concentrated two-component buffer mixture.