Lambert W-function Research Articles

Simplifying the Analysis of Enzyme Kinetics of Cytochrome <i>c</i> Oxidase by the Lambert-W Function

Conventional analysis of enzyme-catalyzed reactions uses a set of initial rates of product formation or substrate decay at a variety of substrate concentrations. Alternatively to the conventional methods, attempts have been made to use an integrated Michaelis-Menten equation to assess the values of the Michaelis-Menten KM and turnover kcat constants directly from a single time course of an enzymatic reaction. However, because of weak convergence, previous fits of the integrated Michaelis-Menten equation to a single trace of the reaction have no proven records of success. Here we propose a reliable method with fast convergence based on an explicit solution of the Michaelis-Menten equation in terms of the Lambert-W function with transformed variables. Tests of the method with stopped-flow measurements of the catalytic reaction of cytochrome c oxidase, as well as with simulated data, demonstrate applicability of the approach to de termine KM and kcat constants free of any systematic errors. This study indicates that the approach could be an alternative solution for the characterization of enzymatic reactions, saving time, sample and efforts. The single trace method can greatly assist the real time monitoring of enzymatic activity, in particular when a fast control is mandatory. It may be the only alternative when conventional analysis does not apply, e.g. because of limited amount of sample.

Inverting the MYEGA equation for viscosity

The Mauro–Yue–Ellison–Gupta–Allan (MYEGA) viscosity expression was recently suggested as an effective three-parameter representation for liquid viscosity as a function of temperature, competitive with the popular Vogel–Fulcher–Tammann (VFT) expression and arguably superior to it, especially at low temperatures. Unlike the VFT function, the MYEGA function cannot be analytically inverted to give temperature as a function of viscosity. This letter describes a simple mathematical trick for inverting the MYEGA equation to solve for temperature as a function of viscosity. It is shown that the solution requires a well-studied transcendental equation involving the Lambert W-function.

On a nonlinear viscoelastic model of Hunt-Crossley impact

We consider a nonlinear viscoelastic model of the impact of a body on a stationary Hunt-Crossley obstacle. We obtain the first integral of the equation of motion and determine the coefficient of restitution, the kinetic energy lost at the impact, and their dependence on the impact velocity. We find the solution of the equation of motion of the body in terms of integrals by using the Lambert W-function and present the results of mathematical modeling.

An exact discretization of Michaelis–Menten type population equations

Many single population models involve the inclusion of Michaelis–Menten type terms. We derive an exact discretization for such rational functions and demonstrate how these relations may be incorporated into more complex models representing the dynamics of single populations. The Lambert W-function plays an important role in this work.

Numerical determination of parasitic resistances of a solar cell using the Lambert W-function

Several methods are currently available to determine the values of series and shunt resistance of a solar cell. A new method is presented here to numerically locate these values using the popular Newton–Raphson technique at maximum power point. Equations based on the Lambert W-function and their partial derivatives are provided so that all calculations may be performed in a Matlab or similar environment. The results of this new method are presented and compared with two published methods and the analytically obtained for a blue and grey type solar cell in earlier work (Charles et al., 1981). Additionally, three modules of various type (single, poly, and amorphous crystalline) were investigated. Values determined agreed with experimentally verified results and were obtained with quadratic convergence in all instances provided initial estimates of the roots were within vicinity of the actual roots.

On the Lambert-W function for constrained resource allocation in cooperative networks

In cooperative networks, a variety of resource allocation problems can be formulated as constrained optimization with system-wide objective, e.g., maximizing the total system throughput, capacity or ergodic capacity, subject to constraints from individual users, e.g., minimum data rate, transmit power limit, and from the system, e.g., power budget, total number of subcarriers, availability of the channel state information (CSI). Most constrained resource allocation schemes for cooperative networks require rigorous optimization processes using numerical methods since closed-form solutions are rarely found. In this article, we show that the Lambert-W function can be efficiently used to obtain closed-form solutions for some constrained resource allocation problems. Simulation results are provided to compare the performance of the proposed schemes with other resource allocation schemes.

The Laplace Transform of the Cut-and-Join Equation and the Bouchard–Mariño Conjecture on Hurwitz Numbers

We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil–Petersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Mariño using topological string theory.

An Explicit Approximation of Colebrook's Equation for Fluid Flow Friction Factor

The Colebrook equation for determination of hydraulic resistances is implicit in fluid flow friction factors. Hence, it has to be approximately solved using an iterative procedure or using some of the approximate explicit formulas that were developed by many authors. The author shows one approximation of the Colebrook equation based on the Lambert W-function.

Extraction of equivalent circuit parameters of solar cell: influence of temperature

A new method to evaluate the five parameters of illuminated solar cells is described. The method is based on nonlinear least squares approach. On calculating the lsqcurvefit-function with constraints, between the experimental current-voltage I(V) characteristic and a theoretical arbitrary characteristic based on Lambert W-function. The used model is implemented as a MATLAB® script which yields the I(V) characteristics of the LILT under test. The model has been validated against by applying it to experimental I(V) characteristics. Some parameters of the model have been measured directly whereas others have been evaluated by means of direct computation on the data sheet or by means of best-fit on the measured data. The results have been compared and an analysis of the errors is presented.

A simple approach to determine the solar cell diode ideality factor under illumination

A simple approach, which can estimate the diode ideality factor of a high efficiency pn junction solar cell under illumination by using its current–voltage data, is explained. We have proposed that an analytical method based on Lambert W-function is sufficient for the extraction of the diode ideality factor of a solar cell modeled by double junction behavior with considerable compliance. Various illumination intensities are also considered in order to specify the reliable limit of the method. The dependence of the ideality factor and the reverse saturation current with light intensity has also been investigated in order to provide insight into the alteration of electrical conduction at junction interface at room temperature.

Solving Share Equations in Logit Models Using the LambertW Function

Though individual demand and supply equations can readily be expressed in logit models, closed-form solutions for equilibrium shares and prices are intractable due to the presence of products of polynomial and exponential terms. This hinders the employment of logit models in theoretical studies, and also makes it difficult to develop reduced-form expressions for share and price as a function of exogenous variables for use in empirical studies. In this paper we propose that a mathematical function called the ‘LambertW’ be employed in solving logit models for equilibrium shares and prices. We derive closed form solutions for price and share in both the monopoly case as well as in the presence of competition. In the competitive case, the prices of the focal firm and the competitor are dependent on each other; hence the equilibrium prices are endogenous and need to be determined simultaneously. To solve this issue, we provide a simple technique that researchers can employ to derive the optimal prices for both the focal firm and the competitor simultaneously.

One-point pseudospectral collocation for the one-dimensional Bratu equation

The one-dimensional planar Bratu problem is uxx+λ exp(u)=0 subject to u(±1)=0. Because there is an analytical solution, this problem has been widely used to test numerical and perturbative schemes. We show that over the entire lower branch, and most of the upper branch, the solution is well approximated by a parabola, u(x)≈u0 (1−x2) where u0 is determined by collocation at a single point x=ξ. The collocation equation can be solved explicitly in terms of the Lambert W-function as u(0)≈−W(−λ(1−ξ2)/2)/(1−ξ2) where both real-valued branches of the W-function yield good approximations to the two branches of the Bratu function. We carefully analyze the consequences of the choice of ξ. We also analyze the rate of convergence of a series of even Chebyshev polynomials which extends the one-point approximation to arbitrary accuracy. The Bratu function is so smooth that it is actually poor for comparing methods because even a bad, inefficient algorithm is successful. It is, however, a solution so smooth that a numerical scheme (the collocation or pseudospectral method) yields an explicit, analytical approximation. We also fill some gaps in theory of the Bratu equation. We prove that the general solution can be written in terms of a single, parameter-free β(x) without knowledge of the explicit solution. The analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly. We give three overlapping perturbative approximations to the eigenrelation, allowing the analytical solution to be easily evaluated throughout the entire parameter space.

On the quantum critical behaviour of a model of structural phase transitions with long-range interaction

The pure quantum limit of an exactly solvable lattice model describing structural phase transitions in an anharmonic crystal with long-range interaction is considered. At the upper quantum critical dimension the free energy density at T = 0 in the neighbourhood of the quantum critical point is exactly calculated in terms of the Lambert W-function. For the three real physical dimensions, the exact results, obtained here, and the asymptotic ones are compared. It is pointed out that the Lambert W-function turns out to be a very effective tool for an exact computation of non-universal characteristics in the upper critical dimensions, especially in a broader neighbourhood of the critical region.

Exact Analytical Solution for the Critical Layer Thickness of a Lattice-Mismatched Heteroepitaxial Layer

Based on the Lambert W-function, an exact analytical solution for the critical thickness of a lattice-mismatched heteroepitaxial layer is presented. The new expression in exact and algebraic closed form eliminates the need for complex iterative computation. Its high accuracy is proved by comparison of the calculated critical thickness versus fractional atomic content of an alloy epilayer with the respective numerical solution.

A note on Chrystal’s equation

The exact, explicit form of the transcendental solution of Chrystal’s equation, a first order nonlinear ordinary differential equation (ODE) of degree two, is derived in terms of the Lambert W-function. It is shown that this case of the general solution is dual-valued over a finite interval and that, for a special case of the coefficients, its zeros involve the Golden ratio. Additionally, a number of applications involving special cases of this ODE are noted and the main properties of the Lambert W-function are briefly reviewed.

Model of CW pumped passively Q-switched laser

An analytical model of a CW pumped passively Q-switched laser in plane wave approximation is presented. The model was based on possibility to express the analytical solution of simplified rate equations with the help of LambertW function. The LambertW function is now commonly available in most of mathematical software and became to be easy to use. The significant simplification of saturable absorber dynamics is made and its presence is described only by its initial and saturated transmission. Using this model it is possible to estimate giant pulse parameters, like pulse length and energy density, the pulse repetition rate, and the laser input-output power characteristic. The validity limits of this model were verified using numerical solution of more complicated rate equations model which included nonlinear response of the modulator. The model was compared with experimental results obtained for passively Q-switched diode pumped Nd:YAG/V:YAG microchip laser.

Comment on “Existence and stability of periodic solution of a Lotka–Volterra predator–prey model with state dependent impulsive effects” [J. Comput. Appl. Math. 224 (2009) 544–555

In this paper, according to integrated pest management principles, a class of Lotka–Volterra predator–prey model with state dependent impulsive effects is presented. In this model, the control strategies by releasing natural enemies and spraying pesticide at different thresholds are considered. The sufficient conditions for the existence and stability of the positive order-1 periodic solution are given by the Poincaré map and the properties of the LambertW function.

A copper electrolysis cell model including effects of the ohmic potential loss in the cell

A computational model of a Cu electrolysis cell is presented. The model includes the microscopic mass transfer and electric field effects, as well as the effects of the ohmic loss of potential in the cell bulk electrolyte. A lumped (0D) estimate for the cell ohmic loss is created by utilizing a 3D electric field model of the cell and the Lambert W-function is applied to include the effects of ohmic loss to the cell model. The model structure enables computing the cell current, the electrode overpotentials and the cell ohmic loss, with the only model input variable being the cell voltage. The cell model is implemented as a coupled 0D + 2D computational model, which is solved with the finite elements method. Cyclic voltammetry (CV) is conducted in an electrolysis cell, whose dimensions can be accurately adjusted and measured. The electrode equation parameters are found by fitting the model against the data of one CV experiment, after which the model is evaluated against three other measurements.

Efficient Resolution of the Colebrook Equation

A robust, fast, and accurate method for solving the Colebrook-like equations is presented. The algorithm is efficient for the whole range of parameters involved in the Colebrook equation. The computations are not more demanding than simplified approximations, but they are much more accurate. The algorithm is also faster and more robust than the Colebrook solution expressed in term of the Lambert W-function. Matlab and FORTRAN codes are provided.

A CAD-compatible closed form approximation for the inversion charge areal density in double-gate MOSFETs

In developing the drain current model of a symmetrically driven, undoped (or lightly doped) symmetric double-gate MOSFET (SDGFET), one encounters a transcendental equation relating the value of an intermediate variable β (which is related to the inversion charge areal density and also surface-potential) to the gate and drain voltages; as a result, it doesn’t have a closed form solution. From a compact modeling perspective, it is desirable to have closed form expressions in order to implement them in a circuit simulator. In this paper, we present an accurate closed form approximation for the inversion charge areal density, based on the Lambert-W function. We benchmark our approximation against other existing approximations and show that our approximation is computationally the most efficient and numerically the most robust, at a reduced but acceptable accuracy. Hence, it is suitable for use in implementing inversion charge based compact models.