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[Image: see text] Accurately determining the acid dissociation constants (K(a) or their logarithmic form, pK(a)) of small molecules and large biomolecules has proven to be pivotal for the study different biological processes and developing new drugs. This Viewpoint summarizes some of the most common methodologies and recent advances described for pK(a) prediction using computational techniques when experimental values are not easily accessible such as in proteins and/or for the screening of large libraries of new compounds.

This paper contributes to the environmental literature by (i) demonstrating that the estimated coefficients and the statistical significance of the non-leading terms in quadratic, cubic, and quartic logarithmic environmental Kuznets curve (EKC) specifications are arbitrary and should therefore not be used to choose the preferred specification and (ii) detailing a proposed general-to-specific type methodology for choosing the appropriate specifications when attempting to estimate higher-order polynomials such as cubic and quartic logarithmic EKC relationships. Testing for the existence and shape of the well-known EKC phenomenon is a hot topic in the environmental economics literature. The conventional approach widely employs quadratic and cubic specifications and more recently also the quartic specification, where the variables are in logarithmic form. However, it is important that researchers understand whether the estimated EKC coefficients, turning points, and elasticities are statistically acceptable, economically interpretable, and comparable. In addition, it is vital that researchers have a clear structured non-arbitrary methodology for determining the preferred specification and hence shape of the estimated EKC. We therefore show mathematically and empirically the arbitrary nature of estimated non-leading coefficients in quadratic, cubic, and quartic logarithmic EKC specifications, being dependent upon the units of measurement chosen for the independent variables (e.g. dependent upon a rescaling of the variables such as moving from $m to $bn). Consequently, the practice followed in many previously papers, whereby the estimates of the non-leading terms are used in the decision to choose the preferred specification of an estimated EKC relationship, is incorrect and should not be followed since it potentially could lead to misleading conclusions. Instead, it should be based upon the sign and statistical significance of the estimated coefficients of the leading terms, the location of turning point(s), and the sign and statistical significance of the estimated elasticities. Furthermore, we suggest that researchers should follow a proposed general-to-specific type methodology for choosing the appropriate order of polynomials when attempting to estimate higher-order polynomial logarithmic EKCs. Supplementary Information The online version contains supplementary material available at 10.1007/s11356-021-13463-y.

As we know, the inflation models have been considered by researchers in various fields of physics such as cosmology and particle physics for explaining early Universe. According to different conditions and conjectures, there are several types of inflation models. In this article, we want to examine a new condition for the inflation models which is known as the swampland criterion. Therefore, we consider f(R) model that includes a polynomial plus a logarithmic term as $$R+\alpha R^{2}+\beta R^{n}+\gamma R^{2}\log \gamma R$$ . So, by considering the corresponding modified f(R) gravity on the brane, we obtain the modified cosmological parameters such as spectral index, tensor-to-scalar ratio and a number of e-folds. These modified parameters give us the opportunity to test the compatibility of f(R) gravity model with swampland criteria. Finally, by using some figures as well as observable data, we specify the range of modified cosmological parameters.

The article is a response to a recent opinion piece that log concentration values should not be applied in analytical chemistry. An essential aim in the development of analytical chemistry methods is to obtain more sensitive and accurate detection values. For the application of chemical analysis methods, the obtained experiment data need to fit with the mathematical functions in the first place. As influenced by different detection principles and analytical methods, data can be displayed in a coordinate system with two linear axes for linear function fitting, or the data can first be taken through a logarithmic transformation and then for function fitting. Using raw data or data after logarithmic transformation primarily depends on analytical principles, without special rules of data formats. For example, ultraviolet-visible spectrophotometric data are more suitable for direct linear fitting. However, enzyme-catalyzed reaction or electrochemical data in logarithmic form are more appropriate for function fitting. This transformation of data form will not affect the soundness of fit statistics; rather, it simplifies the complexity of function analysis and calculation, which are the essence of analytical chemistry. In this brief article, we provide justification and legitimacy of the application of logarithmic processing in various fields of quantitative analytical chemistry.

In this work, we investigate numerically a diffusion equation with nonlinear reaction, defined spatially over a closed and bounded interval of the real line. The partial differential equation is expressed in an equivalent logarithmic form, and initial and Dirichlet boundary data are imposed upon the problem. An implicit finite-difference discretization of this logarithmic model is proposed then. We show that the numerical scheme is capable of preserving the constant solutions of the continuous model. Moreover, we establish the existence of positive and bounded numerical solutions using analytical arguments. The theoretical analysis of the numerical model is carried out also. In particular, we establish that the method is a consistent technique, we provide a priori bounds for the numerical solutions, and we prove the stability and the convergence of the scheme by applying a suitable Gronwall-type inequality. As one of the consequences of stability, we show not only that the solutions of the numerical model exist, but also that they are unique.

Focusing on the weight function ω (x,t)=xαe-{1}{3}x{3}+tx, x∈[0,∞), α >-1, t>0, we state its asymptotic orthogonal polynomials. Through Toda evolution, differential equations of αn(t) and βn(t) have been worked. Consequently, we also talk about the approximate value of αn(t). Basing on the asymptotic value of αn(t), the asymptotic of second order differential equation of Pn(z) and expansion of the logarithmic form of Hankel determinant are confirmed.

A collaboration network model with improved triad formation (ITF) step is proposed and denominated as one-mode collaboration network with tunable clustering. We systematically study its structural features and find this network model can much more accurately reflect cooperative relationship in real society. We acquire expression of nodes degree distribution P(k) that possesses a power-law form with exponent γ whose range is (1,3]. Expression of clustering coefficient C(k) for nodes with degree k is also obtained and composes of a power-law section and a constant section. We verify clustering coefficient of individual nodes increases monotonically with the increase of parameter p that is the probability of ITF step occurring. When specific forms of P(k) and C(k) are given, the model’s average clustering coefficient C can be derived approximately. Finally, we analytically testify average path length L(N) increases with network size N in accordance with logarithmic form. When N is fixed, average path length of entire network decreases with the increase of p. All theoretically predicted results are consistent with simulative experimental results.

Element and mineral associations are fundamental parameters for palaeoceanographical reconstructions but laboratory methodologies are expensive, time-consuming and need a lot of material. Here, we investigate the quality and reliability of XRF measurements of major elements (Fe, Ti and Ca) using BTX II Benchtop, by comparing them with previous ICP-OES elemental analysis for a set of Late Pleistocene marine sediments from Campos Basin. Although the numerical values of the logarithmic form of the elementary ratios were different, the lnTi/Ca and lnFe/Ca ratios measured by both techniques (XRF and ICP-OES) presented similar downcore results. To correct the XRF intensity data, a linear regression model was calculated and, based on the linear equation generated, the logarithmic values of the elementary XRF ratios were corrected. After the correction, One-Sample t-test and Bland–Altman plot show that both techniques obtained similar results. In addition, a brief paleoceanographic interpretation, during the MIS 5 and MIS 4 periods, was conducted by comparing mineralogical and elementary analysis aiming to reconstruct the variations of the terrigenous input to the studied area. As a conclusion, the results from XRF measurements (BTX II) presented to confirm the viability of such a technique, showing that analysis using BTX II is a reliable, cheap, rapid and non-destructive option for obtaining elementary ratios and mineralogical downcore results at high resolution, allowing stratigraphic and paleoceanographic interpretations.

Code construction is a critical issue for polar coding. The expected construction method is of an accurate estimate of the reliability of bit-channels, but the current methods usually require high computational complexity. In this paper, we concern the input-output weight distribution of each bit-channel and derive its recursive calculation algorithm for systematic coding. The union bound and union-Bhattacharyya bound on the bit error probability are also derived to evaluate the reliability of bit-channels. Furthermore, by calculating the logarithmic form of the union-Bhattacharyya bound, we also propose two novel construction methods named the union-Bhattacharyya bound weight of the bit error probability (UBWB) and the simplified UBWB (SUBWB). Numerical results show that the proposed UBWB/SUBWB construction methods can achieve comparable performance to current methods under successive cancellation (SC) decoding and obtain obvious performance gain under SC list (SCL) decoding.

Models for postseismic transient displacements can be formulated using logarithmic and exponential decay formulas with single or multiple timescales. The logarithmic form is associated with rate and state friction theory and afterslip, while the exponential form is associated with bulk viscoelastic relaxation of coseismic stresses. It is now quite widely understood that one can model GPS/GNSS time series manifesting postseismic transient displacements almost equally well using trajectory models constructed using logarithmic or exponential transients. This is consistent with the consensus established by the late 1970s that it is difficult to use geodetic observations to distinguish between deep aseismic afterslip and more diffuse viscoelastic relaxation as the primary mechanism of postseismic deformation. In this paper, we assess the relative explanatory value of both logarithmic and exponential forms by focusing on GNSS time series measured with better than typical signal-to-noise ratios. We find that the double logarithmic transient typically provides much better fits than double exponential transient, when both models are given equal degrees of freedom. We also discuss the relative utility of the hybrid transient formulas in which the logarithmic component is assigned shorter decay timescale parameters than the exponential component. These models fit observed postseismic displacements almost as well as double logarithmic transients.

In this paper, a piecewise-defined function is proposed to estimate traffic noise in urban areas. The proposed approach allows the use of the model even in the case of very low or zero flows for which the classical logarithmic form is not suitable. A model based on the proposed approach is calibrated for a real case and compared with the results obtained with a model based only on the logarithmic form. The results obtained show how the proposed piecewise-defined function, linear for low traffic flows and logarithmic for medium-high volumes, is able to better represent real noise pollution levels in all conditions. The proposed approach is particularly useful when comparing two plan scenarios from the point of view of noise effects.

Anisotropic interparticle correlations in the simple cubic lattice of single-domain ferroparticles (SCLF) are studied using both theory and computer simulation. The theory is based on the Helmholtz free energy expansion like classical virial series up to the second virial coefficient. The analytical formula for the Helmholtz free energy is incorporated in a logarithmic form to minimize the effects of series truncation. The new theoretical approach, including discrete summation over lattice nodes coordinates, is compared critically against the classical virial expansion of the Helmholtz free energy for the dipolar hard sphere fluid; the main differences between the Helmholtz free energy of SCLF and dipolar hard sphere fluid are discussed. The theoretical results for the Helmholtz free energy, the magnetization, and the initial magnetic susceptibility of the SCLF are compared against Molecular Dynamic simulation data. In all cases, theoretical predictions using logarithmic form of the Helmholtz free energy are seen to be superior, but they only have an applicability range of the effective dipolar coupling constant $\lambda_e < 1.5$. For highest values of $\lambda_e$, the structural transition of the magnetic dipoles in SCLF is observed in Molecular Dynamic simulation. It has been shown that for $\lambda_e \gtrsim 2$, an antiferromagnetic order appears in the system., Comment: 13 pages, 8 figures, published in Physica A

In the field of adiabatic correction for complex reactions, a simple one-stage kinetic model was used to estimate the real reaction kinetics. However, this assumption simplified the real process, inevitably generated inaccurate or even unsafe results. Therefore, it was necessary to find a new correction method for complex reactions. In this work, esterification of acetic anhydride by methanol was chosen as an object reaction of study. The reaction was studied under different conditions by Reaction Calorimeter (RC1). Then, Thermal Safety Software (TSS) was used to establish the kinetic model and estimate the parameters, where, activation energies for three stages were 67.09, 81.02, 73.77 kJˑmol− 1 respectively, and corresponding frequency factors in logarithmic form were 16.05, 19.59, 15.72 s− 1. In addition, two adiabatic tests were performed by Vent Sizing Package2 (VSP2). For accurate correction of VSP2 tests, a new correction method based on Enhanced Fisher method was proposed. Combined with kinetics, adiabatic correction of esterification reaction was achieved. Through this research, accurate kinetic parameters for a three-step kinetic model of the esterification reaction were acquired. Furthermore, the correlation coefficients between simulated curves and corrected curves were 0.976 and 0.968, which proved the accuracy of proposed new adiabatic correction method. Based on this new method, conservative corrected results were able to be acquired and be applied in safety assessment.

The results of predicting thermodynamic properties in the full composition range of Ti-Al binary melts in a temperature range from 1973 to 2273 K were obtained by coupling with the developed thermodynamic model for calculating mass action concentration Ni of structural units in Ti-Al system based on the atom and molecule coexistence theory (AMCT). Temperature dependence of the activity coefficients of Ti and Al in natural logarithmic form in the infinitely dilute solution (0>xTi

The thermodynamic temperature of an opaque material with unknown emissivity was determined from the recorded distribution of spectral intensities of the emitted radiation. The initial system of equations was derived in the logarithmic form in accordance with the Planck formula. At the first stage, the spectral dependence of the material emissivity was investigated with a special function—relative emissivity. Based on the analysis of spectral dependences of this function at different reference temperatures, a parametric model for the material emissivity was chosen. At the second stage, the desired parameters of the chosen model of spectral emissivity and the thermodynamic temperature of the material were determined. An example of the determination of a temperature of a tungsten sample from the spectral distribution of emitted radiation intensities is given. A method for narrowing the range, which includes the thermodynamic temperature of the material, is presented.

This paper studies the effect of temperature on the resistance of asphalt mixtures to fatigue cracking by calculating the parameters of the modified Paris’ Law more accurately at multiple temperatures in the basis of separating the DPSE components. Firstly, the repeated direct tensile (RDT) tests with two strain levels were performed on four types of asphalt mixtures to obtain the mechanical properties in the non-destructive and destructive stage under 5 sets of temperatures including 0 °C, 10 °C, 20 °C, 30 °C and 40 °C. The results show that energy ratio to develop fatigue cracking (DPSEF) increases first and then decreases with the temperature increasing, while energy ratio to develop permanent deformation (DPSEP) decreased continuously. If only considering for the relative proportion of the two destructive effects, it was found that the dominant damage mode is fatigue cracking in the 0 °C to 20 °C interval. The dominant damage mode at 40 °C is permanent deformation, and the dominant damage mode at 30 °C depends on material types. This finding helps to understand the development and distribution of fatigue cracking and permanent deformation at different temperatures. Then the DPSEF is further introduced to the modified Paris’ Law to derive the coefficients A J R and n J R . The logarithmic form of A J R and n J R at different temperatures follow the same linear relationship while remaining consistent model fitting coefficients. The damage density follows a good linear relationship with the logarithmic form of the parameter n J R which is independent of temperature. Parameter n J R can reflect the level of cracking resistance performance which is not limited by material type and temperature conditions.

A two-dimensional forward and backward algorithm for the controlled-source audio-frequency magnetotelluric (CSAMT) method is developed to invert data in the entire region (near, transition, and far) and deal with the effects of artificial sources. First, a regularization factor is introduced in the 2D magnetic inversion, and the magnetic susceptibility is updated in logarithmic form so that the inversion magnetic susceptibility is always positive. Second, the joint inversion of the CSAMT and magnetic methods is completed with the introduction of the cross gradient. By searching for the weight of the cross-gradient term in the objective function, the mutual influence between two different physical properties at different locations are avoided. Model tests show that the joint inversion based on cross-gradient theory offers better results than the single-method inversion. The 2D forward and inverse algorithm for CSAMT with source can effectively deal with artificial sources and ensures the reliability of the final joint inversion algorithm.

A new analytical description for the bainitic transformations in steel on isothermal holding is proposed, on the basis of a logistical function of the holding time (in logarithmic form). The model is characterized by the parameters a and b, which are constant in the given experimental conditions (austenitization temperature, isothermal holding temperature and time, chemical composition of the steel, etc.). The calculated temperature dependence of a and b is in good agreement with experimental data. In further research, the physical significance of coefficients a and b in the logistical function should be refined.

All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.

Transform means change in form. For example, in finding the derivative of the product of two functions, we use the product rule and transform the problem into an easier one. Multiplication operation is more difficult than addition operation. By expressing the numbers in logarithmic form, we reduce the multiplication operation into an addition.

La evaluación precisa de la biomasa de árboles es necesaria para estimar los almacenes de carbono y entender la contribución de los ecosistemas de bosques en la regulación de las emisiones de gases de efecto invernadero. El objetivo de este estudio fue desarrollar una ecuación alométrica para predecir la biomasa del fuste (Bf) para Pseudotsuga menziesii (Mirb.) Franco, en la región de Arteaga Coahuila, en el noreste de México. El diámetro normal (Dn), la altura al punto directriz (Hp) y la altura total (Ht) fueron medidos con el Criterion RD1000®. El volumen del fuste (Sv) fue obtenido con la ecuación de Pressler y después fue trasformado a biomasa mediante la densidad básica de la madera. Los datos se ajustaron con el modelo potencial Ŷ = aXb en su forma logarítmica, donde Ŷ es Sb y X el Dn evaluando los supuestos de normalidad, homocedasticidad y no autocorrelación. Un total de 110 árboles de esta especie fueron muestreados. La ecuación ln(Bf) = −2.8732 (± 0.238) + 2.4237 (± 0.066) × ln(Dn) cumplió todos los supuestos de los modelos lineales (valor de P > 0.05), lo que asegura que la predicción de biomasa de fuste sea confiable. El diámetro normal explicó el 98 % de la biomasa del fuste de P. menziesii. El sesgo en la predicción de biomasa del fuste debido a la trasformación logarítmica debe corregirse multiplicándola por un factor de corrección de 1.018. La estimación indirecta de volumen a través del método de Pressler es una forma eficiente, de bajo costo y no destructiva para generar un modelo alométrico de biomasa de fuste.

In this paper, traversable wormholes have been studied in [Formula: see text] gravity, where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are constant. A simplest form of shape function and a logarithmic form of redshift function is considered to construct wormhole solutions. The range of parameters providing the wormhole solutions free from the matter violating the energy conditions is explored. Further, the effect of charge is analyzed on wormhole solutions.

This paper is concerned with the Cauchy problem for the one-dimensional compressible Euler equations with critical time-dependent overdamping. In this critical case, the damped Euler system is proved to admit a unique global solution that converges time-asymptotically to the corresponding nonlinear diffusion wave. Particularly, the optimal convergence rate in logarithmic form is first derived by the technical time-weighted method.

Spectral theory of differential operators is a newly emerging interdisciplinary theory in twentieth century. Based on quantum physics, it provides a unified theoretical framework and solution for problems of differential equations. The present paper studies of a class of symmetric differential operators with logarithmic coefficients. Since logarithmic function discrete slower than exponential function and power and integrals of some logarithmic functions cannot be expressed by elementary functions, the paper uses the method of operator decomposition and quadratic comparison. By using several inequalities and convergence of and functions to the corresponding lower bound estimate of quadratic form, it is concluded that under certain conditions, the sufficient condition and necessary condition of discreteness of this kind of differential operators is the spectrum of all self-adjoint extensions.

In this work, two coupled Artificial Neural Networks (ANNs) were proposed to determine the first two higher-order resonant frequencies in radial vibrational modes and their associate effective electromechanical coupling coefficient (keff) of the ring-shaped piezoelectric transformer. In the first network, the inputs to the ANN were composed of ring width-to-thickness ratio (w/t), ring thickness (t) and the modes of the first two higher-order resonances (R1 and R2). Then, the output from the first network, i.e. the universal resonant frequency (the resonant frequency (fr) multiplied by ring thickness (t)) in logarithmic form (ln(frt)), was later supplied to the second network along with w/t and the order of the vibrational modes (R1 and R2) to predict ln(keff). The data for this coupled ANNs modeling was derived from Finite Element calculation analysis on the ring-shaped piezoelectric transformer. From the results, while it was unlikely to obtain reliable predicted results with just one ANN, the us...

Twenty-eight research questions on meromorphic functions and complex differential equations are listed and discussed. The main purpose of this paper is to make this collection of problems available to everyone.

Studies on allometric models are essential for quantifying the potential of a stand, especially when the objective is to know the wood stock through estimates. This study aimed to select height, volume and artificial form factor equations for a teak stand with data from a sample with 81 trees at 96 months in different diameter classes. The correlation analysis was performed between the variables total height, diameter with and without bark at 1.30 m soil height (dbhcc and dbhsc), total volume with and without bark (VTcc and VTsc) and artificial form factor with and without bark (ffcc and ffsc), using the Pearson coefficient. Six models were evaluated for each category: height, volume and artificial form factor, which were selected by the following criteria: analysis of variance, adjusted coefficient of determination, standard error of estimate, Akaike information criterion and graphical analysis of residues. The selected equations were validated by applying the L & O test. The variables dbhcc, dbhsc, HT, VTcc and VTsc presented significant correlations, ranging from moderate to very strong, the others presented weak correlation. The model that presented the best relation of total height to diameter was the model of Azevedo and contributors. For the total volume with and without bark, Schumacher and Hall model in its logarithmic form was the one that presented the best results. The adjusted models for artificial form factor did not present satisfactory results and, therefore, the use of average form factor is recommended. The mean form factor values by diameter class ranged from 0.4498 to 0.7139 and 0.4480 to 0.7075 for diameter with and without bark, respectively.

In this paper, we establish regularity conditions of weak solutions for the 3D incompressible viscous magnetohydrodynamics (MHD) equations, and show that weak solutions to this system are actually regular if the pressure or the gradient of velocity and magnetic fields satisfies some integrable conditions with the logarithmic form in the critical Besov space B ∞ , ∞ − 1 .

Q1. Optical density is: A. The ratio of incident light to transmitted light in logarithmic form B. Measured using densitometer C. A measure of the degree of film darkening D. Referred as radiographic density E. 1.0 if 100% of the incident light is transmitted

Based on the factorization in perturbative QCD, a jet cross sections in heavy-ion collisions can be expressed as a convolution of the jet cross section in $p+p$ collisions and a jet energy loss distribution. Using this simple expression and the Markov Chain Monte Carlo method, we carry out Bayesian analyses of experimental data on jet spectra to extract energy loss distributions for both single inclusive and $\gamma$-triggered jets in $Pb+Pb$ collisions with different centralities at two colliding energies at the Large Hadron Collider. The average jet energy loss has a dependence on the initial jet energy that is slightly stronger than a logarithmic form and decreases from central to peripheral collisions. The extracted jet energy loss distributions with a scaling behavior in $x=\Delta p_T /\langle \Delta p_T\rangle$ have a large width. These are consistent with the linear Boltzmann transport model simulations, in which the observed jet quenching is caused on the average by only a few out-of-cone scatterings., Comment: 5 pages in RevTex, 3 figures, final version to appear in Phys. Rev. Letters

Taking random information systems as study subjects, a new attribute reduction method is proposed by using correlation between two subsets. Information entropy about logarithmic form is used to measure uncertainty of knowledge, and the correlation connection between information entropy, conditional information entropy, joint information entropy and mutual information entropy are analyzed in a random information system. Correlation coefficients are introduced to describe the correlation between two attribute subsets. Attribute correlations are introduced to depict how important the attributes. A new algorithm based on attribute correlation is given in random information systems, and the experimental result shows that the algorithms is effective.

The purpose of this paper is to study singular holomorphic foliations of arbitrary codimension defined by logarithmic forms on projective spaces., 45 pages

Many crystalline materials are known to exhibit creep at low temperatures (T

In this paper, we deal with the growth and the oscillation of solutions of the linear difference equation an (z) f (z + n) + an-1 (z) f (z + n - 1) + ··· + a1 (z) f (z + 1) + a0 (z) f (z) = 0; where an(z),···, a0(z) are meromorphic functions of finite logarithmic order such that an(z)a0(z) 6≢ 0.

The prediction of delayed deformations of concrete at very long term is particularly important for prestressed structures such as nuclear power plant containments. The basic creep, which corresponds to the deformation of the concrete under load, without exchange of water with the environment, is a major component of the delayed strains. The analysis of several experimental results indicates that this component can be expressed in a logarithmic form. However, Brooks very long term tests (more than 30 years) seem to indicate a deviation from this trend and an acceleration of creep after 3 years of testing. This could be due to a non-linear behavior but these deviations can also be explained by experimental artefacts. In the case of Brooks tests, because the sample were stored in renewed water, calcium leaching could explain the evolution of the behavior on the very long term. Assuming a simple model for leaching and that the leached part of the sample does not sustain stress, it is possible to estimate the evolution of the stress within the core of the sample. Using the creep model fitted on the first part of the tests (before 1000 days) and in the framework of the superposition principle, it is possible to show that leaching indeed could explain the very long term behavior.

The empirical Colebrook equation from 1939 is still accepted as an informal standard way to calculate the friction factor of turbulent flows (4000 8) through pipes with roughness between negligible relative roughness (ε/D ⟶ 0) to very rough (up to ε/D = 0.05). The Colebrook equation includes the flow friction factor λ in an implicit logarithmic form, λ being a function of the Reynolds number Re and the relative roughness of inner pipe surface ε/D: λ = f(λ, Re, ε/D). To evaluate the error introduced by the many available explicit approximations to the Colebrook equation, λ ≈ f(Re, ε/D), it is necessary to determinate the value of the friction factor λ from the Colebrook equation as accurately as possible. The most accurate way to achieve that is by using some kind of the iterative method. The most used iterative approach is the simple fixed-point method, which requires up to 10 iterations to achieve a good level of accuracy. The simple fixed-point method does not require derivatives of the Colebrook function, while the most of the other presented methods in this paper do require. The methods based on the accelerated Householder’s approach (3rd order, 2nd order: Halley’s and Schröder’s method, and 1st order: Newton–Raphson) require few iterations less, while the three-point iterative methods require only 1 to 3 iterations to achieve the same level of accuracy. The paper also discusses strategies for finding the derivatives of the Colebrook function in symbolic form, for avoiding the use of the derivatives (secant method), and for choosing an optimal starting point for the iterative procedure. The Householder approach to the Colebrook’ equations expressed through the Lambert W-function is also analyzed. Finally, it is presented one approximation to the Colebrook equation with an error of no more than 0.0617%.

International audience; The structural properties of model two-dimensional (2D) ionic liquids are examined, with a particular focus on the charge ordering process, with the use of computer simulation and integral equation theories. The influence of the logarithmic form of the Coulomb interaction, versus that of a 3D screened interaction form, is analysed. Charge order is found to hold and to be analogous for both interaction models, despite their very different form. The influence of charge ordering in the low density regime is discussed in relation to well known properties of 2D Coulomb fluids, such as the Kosterlitz-Thouless transition and criticality. The present study suggests the existence of a stable thermodynamic labile cluster phase, implying the existence of a liquid-liquid " transition " above the liquid-gas binodal. The liquid-gas and Kosterlitz-Thouless transitions would then take place inside the predicted cluster phase.

We consider a heteroclinic connection in a planar system, between two symmetric hyperbolic saddles of which the eigenvalues are resonant. Starting with a C-infinity normal form, defined globally near the connection, we normally linearize the vector field by using finitely smooth tags of logarithmic form. We furthermore define an asymptotic entry-exit relation, and we discuss on two examples how to deal with counting limit cycles near a limit periodic set involving such a connection. (C) 2017 Elsevier Inc. All rights reserved.

In this note we consider M-theory compactified on a warped Calabi-Yau fourfold including the eight-derivative terms in the eleven-dimensional action known in the literature. We dimensionally reduce this theory on geometries with one Kahler modulus and determine the resulting three-dimensional Kahler potential and complex coordinate. The logarithmic form of the corrections suggests that they might admit a physical interpretation in terms of one-loop corrections to the effective action. Including only the known terms the no-scale condition in three dimensions is broken, but we discuss caveats to this conclusion. In particular, we consider additional new eight-derivative terms in eleven dimensions and show that they are strongly constrained by compatibility with the Calabi-Yau threefold reduction. We examine their impact on the Calabi-Yau fourfold reduction and the restoration of the no-scale property., 24 pages

This article deals with the irreducible components of the space of codi- mension one foliations in a projective space defined by logarithmic forms of a certain degree. We study the geometry of the natural parametrization of the logarithmic com- ponents and we give a new proof of the stability of logarithmic foliations, obtaining also that these irreducible components are reduced. Fil: Cukierman, Fernando Miguel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; Argentina Fil: Gargiulo Acea, Javier Nicolás. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; Argentina Fil: Massri, Cesar Dario. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; Argentina

The concept of logarithmic order in the unit disc forms a bridge between meromorphic functions of unbounded Nevanlinna characteristic and meromorphic functions of (usual) zero order of growth. A collection of fundamental results for meromorphic functions of finite logarithmic order is given. Some of these results are reminiscent from the finite order case. Part I of this paper culminates in solving the inverse problem related to the famous defect relation in the case of finite logarithmic order. Part II deals with the analytic case.

The problem of guaranteed cost control for the networked control systems (NCSs) with time-varying delays, time-varying sampling intervals and signals quantization was investigated, wherein the physical plant was continuous-time one, and the control input was discrete-time one. By using an input delay approach and a sector bound method, the network induced delays, quantization parameter and sampling intervals were presented in one framework in the case of the state and the control input by quantized in a logarithmic form. A novel Lyapunov function with discontinuity, which took full advantages of the NCS characteristic information, was exploited. In addition, it was shown that Lyapunov function decreased at the jump instants. Furthermore, the Leibniz-Newton formula and free-weighting matrix methods were used to obtain the guaranteed cost controller design conditions which were dependent on the NCS characteristic information. A numerical example was used to illustrate the effectiveness of the proposed methods.

We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered D \mathcal {D} -modules. We discuss in particular the logarithmic degeneration behavior at the large radius limit point and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauß-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D \mathcal {D} -module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined by this filtration and show the existence of a primitive form, which yields the construction of Frobenius structures with logarithmic poles associated to the mirror Laurent polynomial. As a final application, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski open subset of the complexified Kähler moduli space of the variety.