Elementary Equations Research Articles

Pseudo-potential of elastoplastic damage constitutive model and its application

Damage pseudo-potentials, from which the damage evolution equations can be acquired, are usually expressed as scalar functions of irreversible thermodynamic variables. A lower potential for an action with the dissipation property corresponds to a Helmholtz free energy function in classical mechanics; therefore, pseudo-potential acts as a cornerstone of the damage constitutive relationship. According to the analysis on state and process of a damage system, the authors study the connection between damage action, pseudo-potential and free energy function. Next a general method to define the pseudo-potential of damage process is discussed. The discrete elementary equations of continuum damage mechanics are derived from the action functional. Subsequently, the numerical implementation of the aforementioned damage model is given. Comparison between experimental evidences and numerical results verifies the soundness of the theoretical model and the numerical framework.

New approach to optimize operational conditions for the biological treatment of a high-strength thiocyanate and ammonium waste: pH as key factor

Biological treatment of coke and steel-processing wastewaters has to satisfy both industrial economic needs and environmental protection regulations. Nevertheless, as some of the pollutants contained in these waters or produced during the treatment are highly toxic, an effective and safe treatment has proved to be difficult to obtain. This paper reports the study of a biological method for the treatment of wastewaters containing free cyanide, thiocyanate and ammonium (NH 4). Laboratory-scale activated-sludge reactors were fed with a synthetic solution reproducing a steel-processing industrial wastewater and inoculated with the same industrial bacterial seeding used on-site (Ecosynergie Inc.). The results demonstrated that free cyanide and thiocyanate were efficiently degraded. Nevertheless, thiocyanate degradation and nitrification processes were actually inhibited by the free ammonia form (NH 3) in place of the ionized NH 4 form (NH 4 +) currently dosed and often unproperly named “ammonia” [IUPAC, 1997. In: McNaught, A.D., Wilkinson, A. (compilers). Compendium of Chemical Terminology. Royal Society of Chemistry, Cambridge, UK]. Optimum degradation rates were obtained for very narrow ranges of ammonia nitrogen (NH 3-N) concentrations. This result can be explained by the role of pH, which mainly controls the NH 3/NH 4 equilibrium. Pollutants and NH 3 concentrations influenced degradation rates of main pollutants. This influence was determined and expressed through elementary equations. Although the Michaelis–Menten equation could have been used to describe thiocyanate degradation, a Haldane-inhibition model was used to satisfactorily describe cyanide degradation. On the other hand, a slightly modified Haldane model was applied to describe both NH 4 oxidation using NH 3-N as substrate and thiocyanate degradation using NH 3-N as inhibitor. These findings emphasize the role of pH on degradation rates and allow one to optimize operational conditions in the biological treatment of coke and steel industrial wastewaters.

When is negativity not a problem for the ultradiscrete limit?

The “ultradiscrete limit” has provided a link between integrable difference equations and cellular automata displaying soliton-like solutions. In particular, this procedure generally turns strictly positive solutions of algebraic difference equations with positive coefficients into corresponding solutions to equations involving the “Max” operator. Although it certainly is the case that dropping these positivity conditions creates potential difficulties, it is still possible for solutions to persist under the ultradiscrete limit, even in their absence. To recognize when this will occur, one must consider whether a certain expression, involving a measure of the rates of convergence of different terms in the difference equation and their coefficients, is equal to zero. Applications discussed include the solution of elementary ordinary difference equations, a discretization of the Hirota Bilinear Difference Equation and the identification of integrals of motion for ultradiscrete equations.

Effects of classroom instruction on students’ understanding of quadratic equations

Two hundred and thirty-one students in six Grade 9 classes in two government secondary schools located near Chiang Mai, Thailand, attempted to solve the same 18 quadratic equations before and after participating in 11 lessons on quadratic equations. Data from the students’ written responses to the equations, together with data in the form of transcripts of 36 interviews with 18 interviewees (a high performer, a medium performer, and a low performer from each of the six classes), were analysed. Using a rubric for assessing students’ understanding, the analysis revealed that at the post-teaching stage students improved their performance on quadratic equations and had a better understanding of associated concepts than they had at the pre-teaching stage. However, many were still confused about the concepts of a variable and of a “solution” to a quadratic equation. After the lessons, most students had acquired neither an instrumental nor a relational understanding of the mathematics associated with solving elementary quadratic equations.

Kinetic investigation of the chlorine reduction reaction on electrochemically oxidised ruthenium

The rate and mechanism of the electroreduction of chlorine on electrooxidised ruthenium has been investigated with focus on the effect of solution pH. Current/potential curves for the reduction process in solutions with constant chloride concentration of 1.0 mol dm −3 and varying H + concentration have been obtained with the use of the rotating disk electrode technique (RDE). It was found that the chlorine reduction rate is highly inhibited in solutions with high H + concentrations and that it can be satisfactorily described by the Erenburg mechanism, previously suggested for the chlorine evolution on RuO 2 and RTO. The expression of the kinetic current as a function of chlorine and H + concentration was obtained by solving the elementary rate equations of the kinetic mechanism. The kinetic constants obtained from the correlation of the kinetic current expression to the experimental data were used to simulate the dependence of the surface coverages and elementary reaction rates on overpotential.

Derivation of macroscopic equations for individual cell‐based models: a formal approach

AbstractIn this paper we review the theory of cells (particles) that evolve according to a dynamics determined by friction and that interact between themselves by means of suitable potentials. We derive by means of elementary arguments several macroscopic equations that describe the evolution of cell density. Some new results are also obtained—a formal derivation of a limit equation in the case of attractive potential as well as in the case of repulsive potential with a hard‐core part are presented. Finally we discuss the possible relevance of those results within the framework of individual cell‐based models. Several classes of potentials, including hard‐core, repulsive and potentials with attractive parts are discussed. The effect of noise terms in the equation is also considered. Copyright © 2005 John Wiley & Sons, Ltd.

Solution of elementary equations in the Minkowski geometric algebra of complex sets

The solution of elementary equations in the Minkowski geometric algebra of complex sets is addressed. For given circular disks $\mathcal{A}$ and ℬ with radii a and b, a solution of the linear equation $\mathcal{A}\otimes \mathcal{X}=\mathcal{B}$ in an unknown set $\mathcal{X}$ exists if and only if a≤b. When it exists, the solution $\mathcal{X}$ is generically the region bounded by the inner loop of a Cartesian oval (which may specialize to a limacon of Pascal, an ellipse, a line segment, or a single point in certain degenerate cases). Furthermore, when a<b<1, the solution of the nonlinear monomial equation $\mathcal{A}\otimes(\otimes^{n}\mathcal{X})=\mathcal{B}$ is shown to be the region that is bounded by a single loop of a generalized form of the ovals of Cassini. The latter result is obtained by considering the nth Minkowski root of the region bounded by the inner loop of a Cartesian oval. Preliminary consideration is also given to the problems of solving univariate polynomial equations and multivariate linear equations with complex disk coefficients.

Optimization of the Freeze-Drying Cycle: A New Model for Pressure Rise Analysis

The principal aim of this study was to evaluate the Pressure Rise Analysis (PRA) method as a nonintrusive method for monitoring the product temperature during primary drying of the freeze-drying process of model pharmaceutical formulations. The principle of this method, based on the MTM method initially published by Milton et al.1 consisted in interrupting rapidly the water vapor flow from the sublimation chamber to the condenser chamber and by analyzing the resulting dynamics of the chamber total pressure increase. A new physical model, named PRA model, based on elementary heat and mass balance equations and on constitutive equations expressing the elementary fluxes, was proposed and validated in this study for interpreting the experimental pressure rise data. It was possible to identify very precisely the values of some key parameters of the freeze-drying process such as the ice sublimation interface temperature, the mass transfer resistance of the dried layer and the overall heat transfer coefficient of the vial. The identified ice front temperatures were compared with experimental data obtained from vial bottom temperatures measured by thin thermocouples during freeze-drying runs of 5% w/v mannitol solutions. These two sets of data were found consistent with a maximum difference of no more than 2°C. The dried layer mass transfer resistance increased linearly as a function of its thickness, and the values were coherent with the few literature data published for this system. The method also led to reliable values of the vial overall heat transfer coefficient of approximately 20 Wm−2 K−1 in accordance with the published data for this type of vials and these experimental freeze-drying conditions.

Physico-chemical aspects of polyethylene processing in an open mixer. Part 9: complements to the experimental kinetics of oxidation product formation

Polyethylene oxidation on processing in air has been discussed several times. In spite of substantial amounts of work, there remains considerable uncertainty with respect to the mechanisms and kinetics involved. The aim of this work is presenting additional data and highlighting similarities and differences between the rates of formation of various oxidation products. The data available point to significant correlation between the rates of formation of alcohol, ketone and trans-vinylene groups. Hence, these groups should be formed in the same or, at least, in similar processes. There is no general correlation between the formation of carboxylic acids and γ-lactones. There is also no direct relation between the two last oxidation products and the other products detected in the system. The Arrhenius-plots of the experimental rates deduced from fitting the data to two elementary rate equations yield many values for the activation energy. However, usually, these values cannot be related to well-defined chemical reactions. Even if the values look reasonable, there remains considerable doubt concerning the correct attribution to some reaction. In fact, further work shows that the apparent agreement is mainly fortuitous. It seems mandatory to take a new approach at data interpretation.

Universal boundary reflection amplitudes

For all affine Toda field theories we propose a new type of generic boundary bootstrap equations, which can be viewed as a very specific combination of elementary boundary bootstrap equations. These equations allow to construct general solutions for the boundary reflection amplitudes, which are valid for theories related to all simple Lie algebras, that is simply laced and non-simply laced. We provide a detailed study of these solutions for concrete Lie algebras in various representations. The boundary bootstrap equations relating different types of exited boundary states are not automatically solved by our expressions.

The dynamics of a bungee rocket

In this paper we concentrate on some aspects of the modelling of the so-calledbungee rocket, mainly from the perspective of undergraduate mechanics. Abungee rocket consists of a rigid spherical capsule into which two people arestrapped. The capsule is then catapulted vertically upwards by very strong rubbercords between two high towers to provide the ‘thrill of a life-time’. The motion issimple rectilinear translation of a particle, yet is surprisingly rich in problemswhich are interesting from a teaching perspective at different levels, ranging fromvery elementary static and dynamic equations to a slightly more advancedapplication of a conservation principle and some reasonably advanced aspects inconnection with integrating the equations with respect to time and including airresistance.

Solution techniques for elementary partial differential equations

Ordinary Differential Equations: Brief Revision First-Order Equations Homogeneous Linear Equations with Constant Coefficients Nonhomogeneous Linear Equations with Constant Coefficients Cauchy-Euler Equations Functions and Operators Fourier Series The Full Fourier Series Fourier Sine Series Fourier Cosine Series Convergence and Differentiation Sturm-Liouville Problems Regular Sturm-Liouville Problems Other Problems Bessel Functions Legendre Polynomials Spherical Harmonics Some Fundamental Equations of Mathematical Physics The Heat Equation The Laplace Equation The Wave Equation Other Equations The Method of Separation of Variables The Heat Equation The Wave Equation The Laplace Equation Other Equations Equations with More than Two Variables Linear Nonhomogeneous Problems Equilibrium Solutions Nonhomogeneous Problems The Method of Eigenfunction Expansion The Heat Equation The Wave Equation The Laplace Equation Other Equations The Fourier Transformations The Full Fourier Transformation The Fourier Sine and Cosine Transformations Other Applications The Laplace Transformation Definition and Properties Applications The Method of Green's Functions The Heat Equation The Laplace Equation The Wave Equation General Second-Order Linear Partial Differential Equations with Two Independent Variables The Canonical Form Hyperbolic Equations Parabolic Equations Elliptic Equations The Method of Characteristics First-Order Linear Equations First-Order Quasilinear Equations The One-Dimensional Wave Equation Other Hyperbolic Equations Perturbation and Asymptotic Methods Asymptotic Series Regular Perturbation Problems Singular Perturbation Problems Complex Variable Methods Elliptic Equations Systems of Equations Answers to Odd-Numbered Exercises Appendix Bibliography Index Exercises appear at the end of each chapter.

Information obtained in cladistic analysis

The application of elementary equations from information theory to the elements involved in cladistic analysis is formalized mathematically. An equation is derived that quantifies the amount of information obtained by constructing a cladogram from a cladistic data matrix. Given particular conditions, the amount of information obtained increases monotonically with increases of the number of taxa involved and, so, may be used directly as a comparative measure of species richness for sister groups; in general, however, the amount of information obtained is related to the distribution of character states on the cladogram(s) deduced. An example is presented in which clades representing 11 phyla in the animal kingdom are compared in terms of information yielded. The amount of information obtained is consistent for different numbers of taxa and characters used in classifications. Speculative evolutionary explanations are presented for differences of information yielded among the phyla analyzed.

Hemodynamics for medical students.

The flow of blood through the cardiovascular system depends on basic principles of liquid flow in tubes elucidated by Bernoulli and Poiseuille. The elementary equations are described involving pressures related to velocity, acceleration/deceleration, gravity, and viscous resistance to flow (Bernoulli-Poiseuille equation). The roles of vascular diameter and number of branches are emphasized. In the closed vascular system, the importance of gravity is deemphasized, and the occurrence of turbulence in large vessels is pointed out.

Theory of Frequency-Dependent Polarization of General Planar Electrodes with Zeta Potentials of Arbitrary Magnitude in Ionic Media: 1. Theoretical Foundations and General Results

Electrode polarization effects have long aggravated the efforts of low frequency analysis, particularly those investigations carried out on biological material or in highly conductive media. Beginning from elementary equations of electrostatics and hydrodynamics, a comprehensive model is devised to account for the screening of a general planar electrode by an ionic double layer. The surface geometry of the planar electrode is left unspecified to include any type of micromachined array. Building on the previous work by DeLacey and White (1982, J. Chem. Soc. Faraday Trans. 278, 457) using a variational theorem, we extend their numerical results with compact analytic solutions, analogous to the Debye–Hückel potential for dc systems, but applicable now to dynamic ac experiments. The variational approach generates functions that are not restricted by perturbation expansions or numerical convergence, representing optimal approximations to the exact solutions.

Analysis of reinforced concrete beams strengthened in flexure with composite laminates: Discussion

Discussions 835 The authors are to be commended for attempting to pro- vide a refined, yet simple, flexural analysis tool for rein - forced concrete beams strengthened with composite laminates. Although there is much merit in the paper, there are also some aspects of the work that need careful examina- tion before anyone begins to apply it to any real structure. The purpose of this discussion is to point out the danger that is inherent in analyses that (i) assume perfect bond between the FRP laminate and the concrete and (ii) assume the shear and normal stresses at the interface of the laminate and the substrate to be governed by classical beam theory. The writer is confident that the authors know about these dan- gers, but have forgotten to mention them in the paper. Both experiments (Tumialan et al. 1999) and theory (e.g., Rabinovitch and Frostig 1999) have shown that the shear and normal stresses at the interface of the laminate-epoxy and epoxy-concrete have a complex distribution which can- not be found by using the simple beam theory. In fact, beam theory erroneously gives the normal stresses perpendicular to the interface to be zero, but more refined analyses based on theory of elasticity indicate significant tensile stresses de - veloping at the interface. The beam theory also leads one to believe that the interfacial longitudinal shear stresses can be calculated using the elementary shear stress equations given in strength of material books. In reality these stresses have a much more complex distribution. Furthermore, when a con- crete beam cracks, the stresses at the interface in the vicinity of the crack greatly deviate from the distribution predicted by simple beam theory or by any other theory that ignores the presence of cracks. The experimentally observed delamination of either the substrate concrete or the FRP laminate is caused by the combined effect of these high in- terfacial stresses. Given that premature delamination will cause a strength- ened reinforced concrete beam to fail at a load lower than predicted by the program described in the paper, the results of the program could lead to unsafe design. This is the dan- ger that users of the program must be aware of. Figure 6 of the paper illustrates this point where the predicted and the actual load differ significantly. Note that the measures taken to prevent delamination, namely, the use of the U-shaped composite anchors, cannot be modelled by the program as described. The program is unable to quantify the effect of the end anchors and thus unless it can be shown by some other method that delamination cannot occur, it may overes- timate the actual capacity of a member with or without an- chors.

Un problema planteado como actividad de investigación: estudio de las posibles trayectorias para el lanzamiento efectivo de un tiro libre de baloncesto

Although students learn kinematics from their first contact with physics, for a series of reasons preconceptions and conceptual and methodological mistakes accumulate, hindering progress.We propose a simple and motivating exercise that will not only improve the understanding of elementary equations of kinematics, but also detect and facilitate the correction of such mistakes.

Superradiance in semiconductors

The dependences of the characteristic superradiance time in quantum well InGaAs/GaAs laser heterostructures on the pump current, temperature, and cross sections of the active region are studied by analyzing electroluminescence spectra. The number of dipoles involved in forming a superradiant pulse is estimated using elementary equations from the theory of superradiance in two-level systems. A mechanism is proposed for superradiance in semiconductors.

Introduction to fuzzy partial differential equations

This paper considers solutions to elementary fuzzy partial differential equations. We follow the same strategy as in Buckley and Feuring (Fuzzy Sets and Systems, to appear) which is: (1) first check to see if the Buckley-Feuring method produces a solution; and (2) if the Buckley-Feuring method does not give a solution, then see if the Seikkala procedure generates a solution. Examples are presented showing the Buckley-Feuring solution and the Seikkala solution.

Current conveyor chaos generators

New RC chaos generators that utilize the second-generation current conveyor (CCII) as the active building block are presented. The very small input resistance associated with the current conveyor low-impedance input terminal is shown to be responsible for stimulating the circuits' chaotic nature. The proposed chaotic oscillators originate from two sinusoidal oscillator circuits which are modified for chaos by replacing one of the linear resistors with a nonlinear resistor of antisymmetric current-voltage characteristics and adding a single capacitor. The elementary linear design equations of the sinusoidal oscillators are used as a start point for chaos modification to estimate all component values and identify tunable elements. Mathematical models of the proposed generators are derived and further modified to demonstrate chaotic behavior with odd symmetrical nonlinearities. Suitability for VLSI integration is discussed. PSpice circuit simulations, numerical simulations of the derived models, and experimental results are included.