Ordinary Differential Equations

This paper develops a nonlinear macroeconomic model to analyze how unconventional monetary policy (QE, negative interest rates) exacerbates income variability through asset price inflation and wage stagnation. Using differential equations, bifurcation analysis, and empirical calibration to U.S./EU data (2008-2023), I demonstrate that central bank interventions create fractal instability in the wealth distribution. Results reveal a critical threshold beyond which fiscal inequality becomes irreversible without fiscal redistribution. The study integrates Minsky's financial instability hypothesis with Keen's debtdriven collapse frameworks, offering policy prescriptions for mitigating systemic risk.

Consider an infinitely long massless rod with a movable mass m attached, assuming a frictionless interaction between the rod and the mass. The rod undergoes rotational motion with a constant angular velocity α about a fixed pivot point O, causing the mass to move outward along its length. The objective is to determine the trajectory of the mass, specifically deriving the functional relationship between the radial coordinate r and the angular coordinate θ.

We apply a new global dynamical systems formulation to flat Robertson-Walker cosmologies with a massless and massive Yang-Mills field and a perfect-fluid with linear equation of state as the matter sources. This allows us to give proofs concerning the global dynamics of the models including asymptotic source-dominance towards the past and future time directions. For the pure massless Yang-Mills field, we also contextualize well-known explicit solutions in a global (compact) state space picture.

We consider a random matching model where heterogeneous agents endogenously choose to invest time and real resources in education. Generically, there is a steady state equilibrium, where some agents, but not all of them, choose to invest. Regular steady state equilibria are constrained Pareto inefficient in a strong sense. The Hosios (1990) condition is neither necessary, nor sufficient, for constrained Pareto optimality. We also provide restrictions on the fundamentals sufficient to guarantee that equilibria are characterized by overeducation (or undereducation), and present some results on their comparative statics properties.

A system of ordinary differential equations describes the populuation dynamics of a rabies epidemic in raccoons. The model accounts for the dynamics of vaccine, including loss of vaccine due to animal consumption and loss from factors other than racoon uptake. A control method to reduce the spread of disease is introduced through temporal distribution of vaccine packets. This work incorporates the effect of the seasonal birth pulse in the racoon population and the attendant increase in new-borns which are susceptible to the diseases, analysing the impact of the timing and length of this pulse on the optimal distribution of vaccine packets. The optimization criterion is to minimize the number of infected raccoons while minimizing the cost of distributing the vaccine. Using an optimal control setting, numerical results illustrate strategies for distributing vaccine depending on the timing of the infection outbreak with respect to the birth pulse.

A system of ordinary differential equations describes the population dynamics of a rabies epidemic in raccoons. The model accounts for the dynamics of a vaccine, including loss of vaccine due to animal consumption and loss from factors other than racoon uptake. A control method to reduce the spread of disease is introduced through temporal distribution of vaccine packets. This work incorporates the effect of the seasonal birth pulse in the racoon population and the attendant increase in new-borns which are susceptible to the diseases, analysing the impact of the timing and length of this pulse on the optimal distribution of vaccine packets. The optimization criterion is to minimize the number of infected raccoons while minimizing the cost of distributing the vaccine. Using an optimal control setting, numerical results illustrate strategies for distributing the vaccine depending on the timing of the infection outbreak with respect to the birth pulse.

In Neuroscience, mathematical modelling involving multiple spatial and temporal scales can unveil complex oscillatory activity such as excitability, subthreshold oscillations, spiking or bursting. While the number of slow and fast variables and the geometry of the system determine the type of the complex oscillations, canard structures define boundaries between them. In this study, we use geometric singular perturbation theory to identify and characterise boundaries between different dynamical regimes of multiple-timescale firing rate models of the developing spinal cord. These rate models are either three or four dimensional with state variables chosen within an overall group of two slow and two fast variables. The fast subsystem corresponds to a recurrent excitatory network with fast activity-dependent synaptic depression, and the slow variables represent the cell firing threshold and slow activity-dependent synaptic depression, respectively. We start by studying canard-induced bursting and mixed-mode oscillations in two different three-dimensional rate models. Then, in the full four-dimensional model we show that a canard-mediated slow passage creates dynamics that combine these complex oscillations and

Este análisis de la película "La Ola" (2008), del director alemán Dennis Gansel, invita a la revisión de los roles del educador y el educando en la sociedad contemporánea. Partiendo de una activa y di­recta interacción entre docente y discente, en el filme, el profesor Rainer Wenger toma como punto de partida de su cla­se la discusión surgida por un grupo de estudiantes en torno al tema de la 'Autocracia", evidenciando la con­secuente huella que el régimen nazi dejó en la sociedad alemana durante el siglo XX. Este filme se inspira en el experimento social la "Tercera Ola" realizado por Ron Jones (1967).
El artículo, propone una reflexión en torno a la docencia y el cine desde el análisis iconológico, insistiendo en la relación relato y cultura y recordando que "el autor es un lugar en el que habla la cultura".

In this research, implicit discrete schemes which form our block integrators were developed for solving Initial Valued Problems of Ordinary Differential Equations of the form 𝑦 ′′ = 𝑓(𝑥, 𝑦, 𝑦 ′). The equivalent second order Runge-Kutta type Methods (RKTM) were also constructed for the same purpose. Both methods were demonstrated on linear and nonlinear problems of Ordinary Differential Equations. Numerical results obtained from RKTM show that the method is competitive with the existing one.

Los contenidos y metodos de los sistemas dinamicos son de gran interes en la matematica aplicada actual; tanto por el soporte cientifico como por la necesidad de interactuar con los modelos matematicos de fenomenos de diversas disciplinas. En este trabajo, esta preocupacion se centra en estudiar el origen, la interpretacion dinamica, el caos y la sincronizacion del modelo de Lorenz. La motivacion surge del curso de ecuaciones diferenciales, orientado por el profesor Mauro Montealegre Cardenas, a quien agradecemos su colaboracion durante la elaboracion de esta tesis.

This brief set of notes for a lecture on mathematical meteorology generalizes existing estimates of tornadic intensity as a function of CAPE and time-dependent Carnot Efficiency. Approximate Angular Momentum conservation on small time intervals yields generalized state equations for variable mass vortices subject to temperature-dependent variations in the Carnot term.

We tackle the two-body problem associated to Hénon-Heiles' renowned potential, using the qualitative analysis methods, for the limit situation of escape. The study of the infinity flow reveals many "exotic" features as regards behaviour of escape solutions.

In this paper, we propose a new form of the fuzzy number of right hand side vector and then solve fuzzy linear system. We show that the solution of fuzzy linear system is always fuzzy vector. We compare the result of proposed idea with other methods by some exam-ples.

Symmetries in modern physics are a fundamental theme under study that allows to appreciate the Particle Physics. In addition, gauge theories are tools that allow to do a description more complete. With this paper, we analyze a gauge symmetry that appears in complex holomorphic systems. A complex system can be reduced to different real systems, using different gauge conditions and several real systems are connected by gauge transformations in the complex space. We prove that the space of solutions of one system is related to another one using a gauge transformation. Gauge transformations are in some cases canonical transformations. However, in other cases are more general transformations that change the symplectic structure, but there is still a map between systems. We establish a construction extend of gauge transformations and show how to extend the analysis to the quantum case using path integrals by means of the Batalin-Fradkin-Vilkovisky theorem and with in the canonical formalism, where we show explicitly that solutions of the Schrödinger equation are gauge related.

In this paper, Rayleigh wave equation has been solved numerically for finding an approximate solution by Successive approximation method and Finite difference method. Example showed that Successive approximation method is much faster and effective for this kind of problems than Finite difference method.

The author, David Aranovsky, is the sole originator and rightful author of all the mathematical discoveries, derivations, formulations, and conceptual frameworks presented herein. The insights, proofs, and conclusions articulated in this work-including the derivation and identification of fundamental mathematical constants-are entirely original and result from independent thought, research, and analysis. Structured Constants Declaration ℵ (Aleph)-The geometric necessity ratio: The fundamental scaling constant governing structured reality, bridging geometric recursion with deterministic expansion. ‫ש‬ (Shin)-The universal growth constant (aka cosmological constant): The structured refinement factor underpinning deterministic wave scaling, replacing Euler's recursive approximationism. ‫ח‬ (Het)-The fundamental circle constant: The algebraic exact solution governing circular and wave-based structures, eliminating reliance on π-based approximations.

The Mulholland equation is a third-order ordinary differential equation characterized by its nonlinearity. It particularly represents a central restoring force. Mulholland equations have many real applications in various fields, such as modern control theory, phase plane analysis, stability analysis, bifurcation analysis, etc. This paper employed the modified direct iteration method to solve the Mulholland equation, analytically incorporating all its terms (sometimes in approximated forms) during each iterative step. The analytical solutions are compared with existing results. The analytical solutions also showed remarkable precision when compared with numerical outcomes. It also becomes clear that compared to other approaches already in use, the modified direct iteration method is substantially easier to use, more accurate, efficient, and uncomplicated. Moreover, the fourth approximated frequency exhibited only a 0.022 percentage error. The suggested method can be widely applied to different engineering issues, while it is primarily demonstrated in nonlinear models with strong nonlinear factors.

Declaration of Authorship, Originality, and Data Ownership I, David Aranovsky, am the sole originator and rightful author of all the mathematical discoveries, derivations, formulations, and conceptual frameworks presented herein. The insights, proofs, and conclusions articulated in this work-including the identification of as the fundamental circle constant-are entirely my own.

In looking for qualitative differences between quaternionic and complex formulations of quantum physical theories, we provide a detailed discussion of the behavior of a wave packet in presence of a quaternionic time-independent potential step. In this paper, we restrict our attention to diffusion phenomena. For the group velocity of the wave packet moving in the potential region and for the reflection and transmission times, the study shows a striking difference between the complex and quaternionic formulations which could be matter of further theoretical discussions and could represent the starting point for a possible experimental investigation.

This thesis paper is mainly analytic and comparative among various numerical methods for solving differential equations but Chapter-4 contains two proposed numerical methods based on (i) Predictor-Corrector formula for solving ordinary differential equation of first order and first degree (ii) Finite-difference approximation formula for solving partial differential equation of elliptic type. Two types of problems are discussed in detail in this thesis work, namely the ordinary differential equation in Chapters-2 & Chapter-3 and partial differential equations in Chapter-4. Also, Chapter-5 highlights the boundary value problems. The chapters of this thesis paper are organized as follows: Chapter-1 of the thesis is an overview of differential equations and a brief discussion of their solutions by numerical methods. Chapter-2 deals with the solution of ordinary differential equations by Taylor’s series method, Picard’s method of successive approximation, and Euler’s method. Derivation of Taylor’s series method with truncation error and application are discussed here. The solution of ordinary differential equations by Picard’s method of successive approximations and its application is discussed in detail. The definition of Euler’s method is mentioned, the simple pendulum problem is solved to demonstrate Euler’s method. Error estimations and geometrical representation of Euler’s method and the improved Euler’s method are mentioned as a Predictor-Corrector form, which forms being discussed in Chapter-3 next. Also in it, the comparison between Taylor’s series method and Picard’s method of successive approximation has given. Moreover, the advantages and disadvantages of these three methods are narrated in it.  Chapter-3 provides a complete idea of the Predictor-Corrector method. Derivation of Milne’s predictor-corrector formula and Adams-Moulton Predictor-Corrector formula with their local truncation errors and applications are discussed here. Solutions of ordinary differential equations by the Runge-Kutta method with error estimation are studied in this chapter. Some improved extensions of the Runge-Kutta method are explained. Also, the general form of the Runge-Kutta method has been given here. The law of the rate of nuclear decay is solved in this chapter by means of standard fourth-order Runge-Kutta method and then the obtained solution is compared with the exact solution, which is an application of the numerical method to the nuclear physics. Comparison between the Predictor-Corrector method and the Runge-Kutta method discussed in detail. Also, the advantages and disadvantages of these two methods are discussed in it. Chapter-4 gives a review of the solution of partial differential equations. Three types of partial differential equations such as elliptic equations, parabolic equations, and hyperbolic equations with methods of their solutions are discussed at length. To solve the method of the elliptic equation of iterations and relaxation are discussed. Schmidt's method and the Crank-Nicholson method are discussed to solve parabolic equations. The solution of vibrations of a stretched string is mentioned as a method of solution of hyperbolic equations. The solution of vibrations of the rectangular membrane by the Rayleigh-Ritz method has been given here. A comparison between the iterative method and relaxation method has highlighted and then a total discussion of Rayleigh-Ritz with methods of iteration and relaxation reviewed in this chapter. Chapter-5 deals with the solution of the boundary value problems in both ordinary differential equations and partial differential equations. It provides a brief discussion of the finite-difference approximation method and shooting method with their applications. Also, the applications of Green’s function to solve boundary value problems are discussed in detail with the application. Moreover, the B-Spline method for solving two-point boundary value problems of order Four is introduced in this chapter at length. Derivations of cubic B-splines have been represented. Cubic B-spline solutions of the special linear fourth-order boundary value problems, the general case of the boundary value problem, treatment of non-linear problems, and singular problems have been discussed here.  Chapter-6 contains the proposal for the modification of two numerical methods. One of which proposed a modification of Milne’s predictor-corrector formula for solving ordinary differential equations of the first order and first degree, namely Milne’s (modified) Predictor-Corrector formula. One more step-length and one more term in Newton’s interpolation formula being calculated for deriving the predictor and corrector formulae of Milne’s (modified) Predictor-Corrector formula. Also, a modified formula for solving the elliptic equation by finite-difference approximation is proposed, namely surrounding 9-point formula. This formula is obtained by combining standard 5-point formula and diagonal 5-point formula, which gives a more contribution to find mesh points of a given domain in a certain region. Moreover, the advantages of the proposed methods over previous methods are mentioned at the end of this chapter. Chapter-7 provides us the conclusions of this thesis paper. In this chapter, we have chosen the better methods in every chapter by comparing them. Also, the advantages and limitations of Milne’s (modified) predictor-corrector formulae and surrounding 9-point formula are given here. Finally, recommendations for future research and a list of few further works have been mentioned.

This letter presents an extended analysis and a novel upper bound of the subclass of Linear Quadratic Near Potential Differential Games (LQ NPDG). LQ NPDGs are a subclass of potential differential games, for which there is a distance between an LQ exact potential differential game and the LQ NPDG. LQ NPDGs exhibit a unique characteristic: The smaller the distance from an LQ exact potential differential game, the more closer their dynamic trajectories. This letter introduces a novel upper bound for this distance. Moreover, a linear relation between this distance and the resulting trajectory errors is established, opening the possibility for further application of LQ NPDGs.

The main purpose of this research is to find out the best method through iterative methods for solving the nonlinear equation. In this study, the four iterative methods are examined and emphasized to solve the nonlinear equations. From this method explained, the rate of convergence is demonstrated among the 1st degree based iterative methods. After that, the graphical development is established here with the help of the four iterative methods and these results are tested with various functions. An example of the algebraic equation is taken to exhibit the comparison of the approximate error among the methods. Moreover, two examples of the algebraic and transcendental equation are applied to verify the best method, as well as the level of errors, are shown graphically.

Network analysis is an essential component of systems biology approaches toward understanding the molecular and cellular interactions underlying biological systems functionalities and their perturbations in disease. Regulatory and signalling pathways involve DNA, RNA, proteins and metabolites as key elements to coordinate most aspects of cellular functioning. Cellular processes depend on the structure and dynamics of gene regulatory networks and can be studied by employing a network representation of molecular interactions. This chapter describes several types of biological networks, how combination of different analytic approaches can be used to study diseases, and provides a list of selected tools for network visualization and analysis. It also introduces proteinprotein interaction networks, gene regulatory networks, signalling networks and metabolic networks to illustrate concepts underlying network representation of cellular processes and molecular interactions. It finally discusses how the level of An erratum to this chapter is available at 10.

The math model for a mass-spring-damper given here is where the mass is oscillating in a vertical position. This math model is different than the one which is oscillating in a horizontal position provided in Chapter 10 on Ship Dynamics. This may be more suitable for Naval Architecture and Marine (Ship) Engineering. Also, in electrical engineering circuits course on RLC there is a similar math model where this derivation can be helpful. Some textbooks omit the derivation because its lengthy or if it is useful it can be found in another relevant textbook.

The textbook used for this math model is 'Introduction to Numerical and Analytical Methods with Matlab for Engineers and Scientist'. Author/Engineer: William Bober.

The Naval Architecture textbook: Applied Naval Architecture by Robert Zubaly; first edition sixth printing. Both SI and English unit problems. Prerequisite: Completion of high school math or year one college math. The textbook has some of the relevant math required explained very clearly, which is the basic introduction of an integral in calculus, that is the area under a curve. 2nd order differential equation could require further courses in mathematics. Here, the purpose was to provide some exposure on this math's side of things to gain an understanding on what could be involved in ship dynamics. Aotearoa - New Zealand.

Apologies for any errors and omissions.

Regards,
Karl Bogha.

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully

In this paper, we discuss the asymptotic properties and efficiency of several a posteriori estimates for the global error of collocation methods. Proofs of the asymptotic correctness are given for regular problems and for problems with a singularity of the first kind. We were also strongly interested in finding out which of our error estimates can be applied for the efficient solution of boundary value problems in ordinary differential equations with an essential singularity. Particularly, we compare estimates based on the defect correction principle with a strategy based on mesh halving.

This paper deals with an initial boundary value problem with an integral condition for the two-dimensional diffusion equation. Thanks to an appropriate transformation, the study of the given problem is reduced to that of a one-dimensional problem. Existence, uniqueness, and continuous dependence upon data of a weak solution of this latter are proved by means of the Rothe method. Besides, convergence and an error estimate for a semidiscrete approximation are obtained.

Euler’s Method, Taylor’s Method are the most fundamental and easiest methods to solve first order ordinary differential equations (ODEs). Many other methods like Runge-Kutta Method have been developed on the basis of these method. In this paper, the basic ideas behind Euler's Method, Taylor's Method, and Runge-Kutta Method, as well as the geometrical interpretation have been discussed. The main focus is confined to the mathematical interpretation and graphical representation of these method and to find a way to reduce the errors. In order to verify the accuracy of these methods, we compare numerical solutions to exact solutions. Numerical experiment and graphical representation of a specific problem have been discussed in this paper. MATLAB programs have been used for graphical representation and FORTRAN programs have been used for computational efficiency.

Un texte d'Euler sur les fonctions continues et les fonctions discontinues, véritable programme d'organisation de l'analyse au 18e siècle Cahiers du séminaire d'histoire des mathématiques 1 re série, tome 9 (1988), p. 23-68 <http © Cahiers du séminaire d'histoire des mathématiques, 1988, tous droits réservés. L'accès aux archives de la revue « Cahiers du séminaire d'histoire des mathématiques » implique l'accord avec les conditions générales d'utilisation (). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques Un texte d'Euler sur les fonctions continues et les fonctions discontinues, véritable programme d'organisation de l'analyse au 18e siècle. Jean Dhombres * 1. La crise fonctionnelle au XVIIIème siècle 1 . 1 Cadre général l 1.2 Exemple des cordes vibrantes 1.3 Exemple de la composition des forces 1.4 Exemple de la formule du binôme de Newton 2. Les fonctions continues selon Euler 2.1 1 Définition 2.2 Validité de la définition , 2.3 L'hésitation entre deux approches des fonctions 2.4 L'Analyse considérée dans son extension légitime 3. L'architecture du texte d'Euler 3.1 1 Ordre des arguments 3.2 Une classification des fonctions: solution générale 3.3 Structure de l'analyse différentielle et intégrale 4. L'exemple géométrique choisi par Euler 4. 1 Reconstitution possible de la démarche d'Euler 4.2 Solution géométrique 4.3 Solution analytique: les trois premières étapes 4.4 Construction géométrique déduite de la solution analytique 4.5 Podaire d'une courbe par rapport à un point Conclusion: la postérité du texte d'Euler Traduction française du texte d'Euler et notes sur la traduction * Conférence donnée le 11 février 1987 au Séminaire d'histoire des mathématiques. Un teHte d'Euler sur les fonctions continues et les fonctions discontinues, uéritable programme d'organisation de l'analyse au 18e siècle. Jean Dhombres Centre d'histoire des sciences et des techniques

In diesem Beitrag wurde ein neues Verfahren zur Berechnung des eingekoppelten Stromes in einen Volumenleiter vorgestellt, das auf der asymptotischen Annäherungsmethode und der Methode der kleinsten Quadrate beruht. Als Anregung wurde eine ebene Welle angenommen. Um solche Problemstellungen zu untersuchen, existierte bisher nur eine verallgemeinerte Leitungstheorie mit modalen Parametern oder die Möglichkeit der numerischen Simulation. Der Vorteil der vorgestellten asymptotischen Annäherungsmethode ist offensichtlich, da die Streuungskoeffizienten unabhängig von der Leitungslänge sind. Daher ist es relativ einfach, die Berechnung der Streuungskoeffizienten für eine relativ lange Leitung, basierend auf den Ergebnissen von kurzen Leitungen, durchzuführen. Die beschreibende Integralgleichung mit gemischten Potentialen braucht nur über eine verminderte Konfiguration gelöst werden, wodurch man eine kurze Rechenzeit und kleine Speicheranforderungen erzielt [Farhad Rachidi, Sergey Tkachenko...

In this paper, we derive a new composition law obtained by substituting a B-series into the vector field appearing in another B-series. We derive explicit formulas for the computation of this law and study its algebraic properties. We then focus on the specific case of ...

We use the branching rules on SU (3) to show that if two Aloff-Wallach spaces M k,l and M k ,l are isospectral for the Laplacian acting on smooth functions, they are isometric. We also show that 1 is the non-zero smallest eigenvalue among all Aloff-Wallach spaces and compute the multiplicities.

The purpose of this to produce efficient numerical methods with the same order of accuracy as that of the main starting values for exact solutions of fourth order differential equation without reducing it to a system of first order differential equations. The methods of the differential systems arising from the approximate solution to the problem are adopted using the Runge-Kutta method and stages. The methods were compared and contrasted based on the results obtained. The comparison shows that Euler method gives accurate approximate result than Runge-Kutta method. After the derivation of the formulae of O(h2), the comparison was done in regards to identify the formula with higher accuracy.

In this article, we extend the classic Banach contraction principle to a complete metric space equipped with a partial order. We accomplish this by establishing some coincidence point for h-non decreasing self mapping satisfying certain rational type contractions in the framework of a metric space equipped with partial order. These contributions extend the existing literature on metric spaces and fixed point theory. Through illustrative examples, we showcase the practical applicability of our proposed notions and results. In addition, we present applications of our main results for a self mapping involving the integral type contractions.

This paper explains the developments on factorization and decomposition of linear differential equations in the last two decades. The results are applied for developing solution procedures for these differential equations. Although the subject is more than 100 years old, it has been rediscovered as recently as about 20 years ago. A fundamental ingredient has been the easy availability of symbolic computation systems to accomplish the extensive calculations usually involved in applications; to this end the interactive website has been provided. Although originally only developed for ordinary equations, it has been extended to large classes of partial equations as well. In the first part Loewy's results for ordinary equations are outlined. Thereafter those results of differential algebra are summarized that are required for extending Loewy's theory to partial equations. In the remaining part a fairly complete discussion of second-and some third-order partial differential equations in the plane is given; it is shown that Loewy's result remains essentially true for these equations. Finally, several open problems and possible extensions are discussed.

Primarily this work intends to investigate the existence of best proximity points (pairs) for new classes of cyclic (noncyclic) mappings via simulation functions and measure of noncompactness. Use of different classes of additional functions make it possible to generalize the contractive inequalities in this work. As an application of the main conclusions, a survey for the existence of optimal solutions of a system of integro-differential equations under some new conditions is presented. As an application of our existence results, we establish the existence of a solution for the following system of integro-differential equations in the space of all bounded and continuous real functions on [0, +∞[ under suitable assumptions on F 1 , F 2 .

Cancer is a burning problem in the modern health field. In this research mainly we focused on how
we can treat for a cancer by using chemotherapy and immune therapy as individual monotherapies and as a
combined therapy. In previously, researchers have constructed mathematical models to analyse the combined
therapy treatment with saturation effects for immune therapy treatment but here we introduce mass action kinetics
for immune therapy and this model reflects as a continues attacking process to tumour cells by immune cells with
the help of chemotherapy drug. We introduce some threshold levels which we can remove cancer completely and
control the cancer in a constant level.

The Carrier envelope offset phase or the so called &quot;absolute phase&quot; governs the Spatio- temporal evaluation of the electric field of a short laser pulse and this field in turn governs the interaction of matter with such pulses. The precise measurement and stabilization of this quantity is of great importance for attosecond physics and virtually every other application that benefits from ultra short laser pulses. The aim of this project is to build a Spectrometer, which we call the Stereo ATI phase-meter, for measuring the absolute phase of short laser pulses. It involves the stereo measurement of photo electrons of xenon gas ejected after ATI (Above Threshold Ionization). This approach to phase measurement is based on the belief that the spatial symmetry of photo-electron emission from a gaseous medium should be governed by the symmetry.

Network analysis is an essential component of systems biology approaches toward understanding the molecular and cellular interactions underlying biological systems functionalities and their perturbations in disease. Regulatory and signalling pathways involve DNA, RNA, proteins and metabolites as key elements to coordinate most aspects of cellular functioning. Cellular processes depend on the structure and dynamics of gene regulatory networks and can be studied by employing a network representation of molecular interactions. This chapter describes several types of biological networks, how combination of different analytic approaches can be used to study diseases, and provides a list of selected tools for network visualization and analysis. It also introduces proteinprotein interaction networks, gene regulatory networks, signalling networks and metabolic networks to illustrate concepts underlying network representation of cellular processes and molecular interactions. It finally discusses how the level of An erratum to this chapter is available at 10.

This note deals with an example of class of ordinary differential equations which are singular near a codimension 2 set, with an homogeneous singularity, of degree 0. Under some structural assumptions, we prove that for almost all initial data there exists a unique global solution and study the evolution of the Lebesgue measure of a transported set of initial data.

In this paper, a seventh order numerical method is presented for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point. The two-point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a seventh order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two-point boundary value problem is obtained from the theory of singular perturbations. It is used in the seventh order compact difference scheme to get a two term recurrence relation and is solved. Several linear and nonlinear singular perturbation problems have been solved and the numerical results are presented to support the theory.

Apresenta-se um material referente à Transformada de Laplace, com o objetivo de ser utilizado na disciplina de Equações Diferenciais na Pontifícia Universidade Católica de Goiás como material de aula para o professor e como material de apoio para os alunos. Inicialmente é apresentada a história de vida do físico, matemático e astrônomo francês Pierre-Simon Laplace assim como o contexto histórico no qual a Transformada de Laplace foi desenvolvida. Em sequência, é feita uma breve revisão de conteúdos importantes ao estudo da Transformada de Laplace, é apresentada a transformada em si, sua definição, as propriedades de linearidade, condições suficientes para existência da Transformada de Laplace, as Transformadas de Laplace de derivadas e integrais e a apresentação da Transformada de Laplace inversa. Por fim, apresenta-se duas aplicações da Transformada de Laplace em Equações Diferenciais: Os circuitos Resistor-Indutor-Capacitor e o sistema Massa-Mola. O conteúdo foi construído através de cuidadosa análise bibliográfica, buscando na medida do possível adaptar o material a uma linguagem mais acessível que outros materiais de mesmo tema.