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The aim of this note is to correct an error in the formulation of Theorem 1 by Savin and Terwiesch [Savin, S., C. Terwiesch. 2005. Optimal product launch times in a duopoly: Balancing life-cycle revenues with product cost. Oper. Res. 53(1) 26–47].

Some time ago has been studied mathematical models for biogas production due to its importance in the use of control and optimization of re\-new\-able resources and clean energy. In this paper we combine two algebraic methods to obtain solutions of Abel equation of first kind that arise from a mathematical model to biogas production formulated in France on 2001. The aim of this paper is obtain Liouvillian solutions of Abel's equations through Hamiltonian Algebrization. As an illustration, we present graphics of solutions for Abel equations and solutions for algebrized Abel equations., Comment: 12 pages, 3 figures

N-path filters exploiting switched-series-R-C networks can realize high-Q blocking-tolerant band-pass filters. Moreover, their center frequency is flexibly programmable by a digital clock. Unfortunately, the time variant nature of these circuits also results in unwanted signal folding. This paper proves analytically that folding can be reduced and band pass filtering can be improved by adding an inductance in series with the switched-R-C network. In contrast, a shunt capacitor degrades band-pass filter performance. The interaction between the reactive series impedance and the switched capacitors of an N-path filter complicates analysis due to memory effects associated with reactive components. Assuming $N$ identical signal paths with 1/ $N$ duty cycle, we show it is possible to solve the set of differential equations, by assuming that the signals in each path only differ in delay. Analytical equations are verified versus simulations, and the benefits in filter properties and reduction in signal folding are demonstrated.

Is it possible to predict the future? How accurate is the prediction for the future? These questions are fascinating and intriguing ones in particular for young generations who look at their future with curiosity. For a long time, many have tried to quantitatively predict future behavior of systems more accurately with techniques such as time series analysis and derived dynamical models based on observed data. The paper proposes a lecture structure in which elements of chaos, which greatly impacts the predictive capabilities of dynamical models, are introduced through two classical examples of nonlinear dynamical systems, namely Lorenz attractor and Lotka-Volterra equations. In a possible lecture, these two structures are introduced in a basic and intuitive way, followed by equilibria analyses and Lyapunov control approaches. The paper intends to give a possible structure of an interdisciplinary lecture in chaotic systems, for all students in general and non-engineering students in particular, to kindle students' interest in challenging ideas and models. By presenting an intuitive learning-based approach and the results of the implementation, the paper contributes to the discourse on interdisciplinary education. The lecture is a part of a course within a Complementary Study at Leuphana Unversity of Lüneburg. The material which inspired the proposed lecture structure is taken from the scripts of the Master Complementary Course titled Modelling and Control of Dynamical Systems using Linear and Nonlinear Differential Equations held at Leuphana University of Lüneburg. © 2022 SEFI 2022 - 50th Annual Conference of the European Society for Engineering Education, Proceedings. All rights reserved.

[SPA]En este trabajo se analizará la evolución de la pandemia COVID-19 mediante un modelo SDE (en ecuaciones diferenciales estocásticas) con un sistema de tipo SIRD (Susceptibles-Infectados-ResistentesMuertos). Simularemos este modelo usando Matlab y estudiaremos los resultados obtenidos, justificándolos a partir de la teoría y analizaremos los resultados obtenidos. [ENG]Stochastic modelling has come to play a very important role in economics and virtually any other branches of science where differential equations cannot relate accurately to the reality of certain events. In this paper, we are going to study the evolution of the COVID-19 pandemic using a SDE (stochastic differential equation) model to portray a SIRD (Susceptible-Infected-Resistant-Dead) system. The aim of this work is to introduce the reader to SDE theory at an introductory level and then to carry that knowledge onto Matlab in order to correlate theory and experimental results. First, we will introduce the theory needed to understand the simulations and the models that will be used on the next chapter. Then, the results will be analyzed and discussed on the bases of the previous explained theory. This methodology will be followed throughout the whole text, excluding the last chapter, where personal opinions will be given regarding the whole paper and the most significant results. We are going to simulate a specific model using Matlab to obtain simulation results. These are analyzed on the bases of the SDE theory that was explained in the previous chapter. Finally, the conclusions will be drawn from our results to prove whether the practical use of stochastic differential equations are useful in a population system model or not. Escuela Técnica Superior de Ingeniería Industrial Universidad Politécnica de Cartagena

Let FolR(2, d) be the space of real algebraic foliations of degree d in RP(2). For fixed d, let IntR(2, d) = {F 2 FolR(2, d) | F has a non-constant rational first integral}. Given F 2 IntR(2, d), with primitive first integral G, set O(F) = number of real ovals of the generic level (G = c). Let O(d) = sup{O(F) | F 2 IntR(2, d)}. The main purpose of this paper is to prove that O(d) = +1 for all d _ 5., Let FolR(2, d) be the space of real algebraic foliations of degree d in RP(2). For fixed d, let IntR(2, d) = {F € FolR (2, d) | F has non-constant rational first integral}. Given F € IntR(2, d), with primitive first integral G, set O(F) = number of real ovals of thegeneric level (G = c). Let O(d) = sup{O(F) | F € IntR(2, d)}.The main purpose of this paper is to prove that O(d) = +∞ for all d ≥ 5.

Stability is required for real world controlled systems as it ensures that those systems can tolerate small, real world perturbations around their desired operating states. This paper shows how stability for continuous systems modeled by ordinary differential equations (ODEs) can be formally verified in differential dynamic logic (dL). The key insight is to specify ODE stability by suitably nesting the dynamic modalities of dL with first-order logic quantifiers. Elucidating the logical structure of stability properties in this way has three key benefits: i) it provides a flexible means of formally specifying various stability properties of interest, ii) it yields rigorous proofs of those stability properties from dL's axioms with dL's ODE safety and liveness proof principles, and iii) it enables formal analysis of the relationships between various stability properties which, in turn, inform proofs of those properties. These benefits are put into practice through an implementation of stability proofs for several examples in KeYmaera X, a hybrid systems theorem prover based on dL., Comment: Long version of paper at TACAS 2021 (27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, 27 Mar - 1 Apr 2021)

In 1926, Erwin Schrodinger wrote a series of papers that invented wave mechanics and set the foundation for much of the single-particle quantum mechanics that we teach today. In his first paper, he solved the Schrodinger equation using the Laplace method, which is a technique that is quite powerful, but rarely taught. This is unfortunate, because it opens the door to examining quantum mechanics from a complex-analysis perspective. Gaining this experience with complex analysis is a useful notion to consider when teaching quantum mechanics, as these techniques can be widely used outside of quantum mechanics, unlike the standard Frobenius summation method, which is normally taught, but rarely used elsewhere. The Laplace method strategy is subtle and no one has carefully gone through the arguments that Schrodinger did in this first paper, instead it is often just stated that the solution was adopted from Schlesinger's famous differential equation textbook. In this work, we show how the Laplace method can be used to solve for the quantum-mechanical energy eigenfunctions of the hydrogen atom, following Schrodinger's original solution, with all the necessary details, and illustrate how it can be taught in advanced instruction; it does require familiarity with intermediate-level complex analysis, which we also briefly review., Comment: 22 pages, 8 figures

Object and purpose of research: The purpose of this work is to develop a physical & mathematical model for the hydraulic skimmer, an innovative vehicle currently under development, intended to recover the emergency spills of oil products contained under the ice. The skimmer will be connected with the technological system of the mother ship by a flexible coaxial line. The paper studies the parameter of the water flow, laminar and turbulent, in the circular duct serving to provide hot water into the axial duct so as to heat the oil products recovered by the skimmer. Materials and methods: The work applies f-model of turbulence that allows settled flow calculation for a viscous incompressible fluid in the pipes and ducts with hydraulically smooth walls at both high and low Reynolds numbers. Main results: The first integrals (transcendental equations) have been obtained for the speed profile and the turbulence measure, which allows reducing this problem to a system of algebraic equations. The paper compares the calculation results for speed profiles and resistance coefficients versus the experimental data obtained by other authors. Conclusion: The results of the study allow making a theoretical justification for technical engineering and design solutions in developing the technological system to be installed aboard ship for recovery of the emergency spills of oil products under ice.

The relevance of this study is determined by the need to develop technologies for effective urban systems management and resolution of urban planning conflicts. The paper presents an algorithm for analyzing urban planning conflicts. The material for the study was data from social networks, microblogging, blogs, instant messaging, forums, reviews, video hosting services, thematic portals, online media, print media and TV related to the construction of the North-Eastern Chord (NEC) in Moscow (RF). To analyze the content of social media, a multimodal approach was used. The paper presents the results of research on the development of methods and approaches for constructing mathematical and neural network models for analyzing the social media users’ perceptions based on their digital footprints. Artificial neural networks, differential equations, and mathematical statistics were involved in building the models. Differential equations of dynamic systems were based on observations enabled by machine learning. Mathematical models were developed to quickly detect, prevent, and address conflicts in urban planning in order to manage urban systems efficiently. In combination with mathematical and neural network model the developed approaches, made it possible to draw a conclusion about the tense situation around the construction of the NEC, identify complaints of residents to constructors and city authorities, and propose recommendations to resolve and prevent conflicts. Research data could be of use in solving similar problems in sociology, ecology, and economics.

This paper deal with the numerical method, based on the operational matrices of the Haar wavelet orthonormal functions approach to approximate solutions to a class of coupled systems of time-fractional order partial differential equations (FPDEs.). By introducing the fractional derivative of the Caputo sense, to avoid the tedious calculations and to promote the study of wavelets to beginners, we use the integration property of this method with the aid of the aforesaid orthogonal matrices which convert the coupled system under some consideration into an easily algebraic system of Lyapunov or Sylvester equation type. The advantage of the present method, including the simple computation, computer-oriented, which requires less space to store, time-efficient, and it can be applied for solving integer (fractional) order partial differential equations. Some specific and illustrating examples have been given; figures are used to show the efficiency, as well as the accuracy of the, achieved approximated results. All numerical calculations in this paper have been carried out with MATLAB.

An analytical expression for the optimal control of an Ebola problem is obtained. The analytical solution is found as a first-order approximation to the Pontryagin Maximum Principle via the Euler-Lagrange equation. An implementation of the method is given using the computer algebra system Maple. Our analytical solutions confirm the results recently reported in the literature using numerical methods., Comment: This is a preprint of a paper whose final and definite form is in International Journal for Computational Methods in Engineering Science and Mechanics, ISSN 1550-2287 (Print), 1550-2295 (Online). Paper Submitted 14-Jul-2015; Revised 29-Oct-2015; Accepted for publication 09-Dec-2015

[EN] This paper presents a modification of a recursive method described in a previous paper of the authors, which yields necessary and sufficient conditions for the existence of solutions of a class of 𝑛�th-order linear boundary value problems, in the form of integral inequalities. Such a modification simplifies the assessment of the conditions on restricting the inequality to be verified to a single point instead of the full interval where the boundary value problem is defined. The paper also provides an error bound that needs to be considered in the integral inequalities of the previous paper when they are calculated numerically, This work has been supported by the Spanish Ministerio de Economia y Competitividad Grant MTM2013-41765-P.

In this paper we present the main results of the master thesis in applied mathematics of the second named author, which was supervised by the first named author. Such results, and for instance this paper, concerns to some differential and algebraic results involving natural matrices. The problem of solving differential equations is very ancient and is very important to get explicit solutions of differential equations to be applied in physics and other areas. In this paper, as well in the master thesis, we study the differential and algebraic structure of linear differential equations with natural matrix coefficients and generalizations. These results are original and important for researchers interested in differential algebra and applications of differential equations.

System modeling and simulation plays a crucial role in the engineering of large and complex systems from various fields, such as industrial automation or power systems. In this paper, we propose a method that can be used to easily deploy high fidelity simulations at scale, onto various target platforms. Out method is to approximate the behavior of the modeled system using a recurrent neural network. We use artificial neural networks as they easily lend themselves to high performance execution, thus avoiding the need to (manually) translate system models (typically a system of differential equations) to specialized hardware architectures. Moreover, this approach is generic in the sense that it is decoupled from typical modeling and simulation tools, such as Matlab Simulink or Dymola. This paper presents a proof-of-concept neural network architecture including the methodology for training that we used to approximate the behavior of different example systems originating from the electrical power systems domain. We present our evaluation results mainly regarding accuracy and to a certain extent performance on a GPU-based testbed. Furthermore, we detail limitations of the used approach and outline potential directions for research regarding the general applicability of our method.

The purpose of the paper is to present a model allowing the retailer to determine the optimal price of three kinds of items in a situation where the supplier provides the retailer with an interest-free loan for a contractually agreed period. The scientific aim is to verify whether an optimization problem is solvable, and determine the maximum length of the interval over which the goods can be sold with a profit in a situation where the model features two kinds of deteriorating items and one non deteriorating item. The economic theory is explained in the introductory section and serves as a basis for the drawing up of the model. Methods of analysis, synthesis, dynamic modeling and differential calculus of multivariate functions are also used. The situation where the dealer sells all his goods in time and the situation where this period is not observed are analyzed. Thanks to the exact expression of the model it is possible to assess the effect of any changes in external factors. Authors’ findings are that the used variables form compact set and analysed function is continues on this space, we can use Weistrass Theorem for additional calculation with success. Thanks to the exact expression of the model it is possible to assess the effect of any changes in external factors. The model proposed in the paper may be expanded in the future. One possibility is to consider a generalization of the model allowing for shortage of items, quantity discounts, inflation, etc.DOI: http://dx.doi.org/10.5755/j01.ee.29.3.14204

In this paper we investigate the dynamics of the classic synchronous reference frame phase-locked-loop (PLL) from a non-linear perspective. First, we demonstrate the nonlinear differential equations that describe the PLL under balanced conditions can be represented as a dissipative Hamiltonian system (DHS). After that, we find the equilibrium points of this system and their stability properties. Additional properties are investigated such as the attraction region, the conditions for exponential stability and the performance for small unbalances an d/or transients in the grid. Simulations results complement the theoretical analysis. We do not propose a new type of PLL, instead, we propose a non-linear analysis for the classic synchronous reference frame PLL. This analysis is useful for theoretical and practical studies since this PLL is widely used in industrial applications. In addition, it can give insights for better understanding of the dynamics of the phase-locked-loop1 1The presentation of this paper in the COMPEL2018 was partially supported by the Maestrfa en Ingeniería Eléctrica Universidad Tecnologica de Pereira. © 2018 IEEE. 1The presentation of this paper in the COMPEL2018 was partially supported by the Maestría en Ingeniería Eléctrica Universidad Tecnologica de Pereira

In the paper a problem of stabilizing a HIV infection dynamics mathematical model is considered. The model is described by a system of functional differential equations. A stabilizing control is constructed basing on the method of explicit solutions of Generalized Riccati Equations of the theory of analytical constructing regulator for systems with delays. For construct a feedback control we use the first variant of explicit solutions of the generalized Riccati equations (the study of control stabilizing properties based on other variants discussed in other authors articles). Stabilizing control for the system of differential equations with delay supports HIV-infection model spread at a certain sufficiently small non-zero level. Results of the research can be applied to analysis of some aspects of HIV dynamics. © 2015, IFAC Hosting by Elsevier Ltd.

This paper deals with the fuzzy initial value problem and how to solve a linear fuzzy differential equation of first order when the initial condition is a triangle fuzzy number. This problem is then developed to the case when the initial condition is a trapezoidal fuzzy number. The paper includes also the issue of the representation of a system of linear fuzzy differential equations, a more general system of linear fuzzy differential equations is then proposed and the solution of this system is also given. An illustrative examples are given in order to consolidate the raised ideals.

Decentralized optimization of distributed stochastic dynamical systems with two or more controls of the decision makers (DMs) has been an active area of research for over half a century. Although, such decentralized optimization problems are often formulated utilizing static team and person-by-person (PbP) optimality criteria, the corresponding static team theory results have not been extended to dynamical systems. In this first part of the two-part paper, we derive team and PbP optimality conditions for distributed stochastic differential systems, when the controls of the DMs generate actions based on different information structures. The necessary conditions are given by a Hamiltonian System described by coupled backward and forward stochastic differential equations (SDEs) and a conditional Hamiltonian, conditioned on the information structures available to the controls of the DMs. The sufficient conditions state that PbP optimality implies team optimality, if the Hamiltonian is convex in the state and/or actions spaces of the controls of the DMs. We show existence of relaxed team optimal strategies, when the information structures are not affected by the controls of the DMs. Throughout the paper we discuss similarities to analogous optimality conditions of centralized decision or control of stochastic systems, and we note a connections to mean field stochastic optimal control problems. © 1963-2012 IEEE. 62 3 1194 1209

Although the explicit commutativitiy conditions for second-order linear time-varying systems have been appeared in some literature, these are all for initially relaxed systems. This paper presents explicit necessary and sufficient commutativity conditions for commutativity of second-order linear time-varying systems with non-zero initial conditions. It has appeared interesting that the second requirement for the commutativity of non-relaxed systems plays an important role on the commutativity conditions when non-zero initial conditions exist. Another highlight is that the commutativity of switched systems is considered and spoiling of commutativity at the switching instants is illustrated for the first time. The simulation results support the theory developed in the paper., Comment: 20 PAGES, 7 F\.IGURES

This paper provides an innovative practical realization of a memristor based chaotic circuit. The first part discusses the mathematical analysis of the proposed system, including calculation of an eigenvalues, bifurcation diagram and largest Lyapunov exponents. Another parts deal with circuitry realization and the influence of parasitic properties of active elements. The circuit simulations obtained by PSpice environment and the practical measurement results on a breadboard are presented in the last part of this paper. The main aim of this work is an innovative realization of the memristor based chaotic circuit with one type of energy-storage element (linear passive capacitor) and with simpler construction in comparison to other circuits. The next contribution consists in verification of designed circuit with respect to influence of parasitic properties of active elements to chaos destruction. Tento článek poskytuje inovativní realizaci chaotického systému na základě vlastností memristoru. První část pojednává o matematické analýze navrhovaného systému, včetně výpočtu vlastních čísel, bifurkačního diagramu a největších Ljapunových exponentů. Další části se zabývají obvodovou realizací a vlivem parazitních vlastností aktivních prvků. Simulace obvodů získaná v simulačním systému PSpice a výsledky praktického měření jsou uvedeny v poslední části. Hlavním přínosem této práce je inovativní realizace chaotického systému s jedním akumulačním prvkem (lineární pasivní kondenzátor) a jednoduchá konstrukce ve srovnání s dosud publikovanými zapojeními. Další přínos spočívá v ověření funkčnosti navrženého zapojení s uvažováním kritických hodnot parazitních vlastností použitých aktivních prvků.

A recent paper by Karin et al. introduced a mathematical notion called dynamical compensation (DC) of biological circuits. DC was shown to play an important role in glucose homeostasis as well as other key physiological regulatory mechanisms. Karin et al. went on to provide a sufficient condition to test whether a given system has the DC property. Here, we show how DC can be formulated in terms of a well-known concept in systems biology, statistics, and control theory—that of parameter structural non-identifiability. Viewing DC as a parameter identification problem enables one to take advantage of powerful theoretical and computational tools to test a system for DC. We obtain as a special case the sufficient criterion discussed by Karin et al. We also draw connections to system equivalence and to the fold-change detection property., Author summary A recently introduced mathematical notion called dynamical compensation of biological circuits was shown to play an important role in glucose homeostasis and other key physiological regulatory mechanisms. This paper explains how dynamical compensation can be formulated in terms of a well-known concept in systems biology, statistics, and control theory—that of parameter structural non-identifiability. Viewing dynamical compensation as a parameter identification problem enables one to take advantage of powerful theoretical and computational tools to test a system for dynamical compensation. As a special case, one obtains the sufficient criterion for dynamical compensation. The paper also draws connections to system equivalence and to the fold-change detection property. The non-identifiability characterization brings up an interesting contrast in the way in which one thinks of these properties in the two fields. From the point of view of robustness of behavior, one wishes that parameters do not influence much the response of a system. On the other hand, from the systems and parameter identification point of view, the more that a parameter affects behavior, the easier it is to estimate it, and poor sensitivity is taken as an indication of a poorly parametrized model.

Cloud computing has paved the way to the flexible deployment of software applications. This flexibility offers service providers a number of options to tailor their deployments to the observed and foreseen customer workloads, without incurring in large capital costs. However, cloud deployments pose novel challenges regarding application reliability and performance. Examples include managing the reliability of deployments that make use of spot instances, or coping with the performance variability caused by multiple tenants in a virtualized environment. In this paper, we introduce Line, a tool for performance and reliability analysis of software applications. Line solves layered queueing network (LQN) models, a popular class of stochastic models in software performance engineering, by setting up and solving an associated system of ordinary differential equations. A key differentiator of Line compared to existing solvers for LQNs is that Line incorporates a model of the environment the application operates in. This enables the modeling of reliability and performance issues such as resource failures, server breakdowns and repairs, slow start-up times, resource interference due to multitenancy, among others. This paper describes the Line tool, its support for performance and reliability modeling, and illustrates its potential by comparing Line predictions against data obtained from a cloud deployment. We also illustrate the applicability of Line with a case study on reliability-aware resource provisioning. © 1963-2012 IEEE.

The consideration of uncertainty in differential equations leads to the emergent area of random differential equations. Under this approach, inputs become random variables and/or stochastic processes. Often one assumes that inputs are independent, a hypothesis that simplifies the mathematical treatment although it could not bemet in applications. In this paper,we analyse, through the Airy equation, the influence of statistical dependence of inputs on the output, computing its expectation and standard deviation by Fröbenius and Polynomial Chaos methods.The results are compared with Monte Carlo sampling. The analysis is conducted by the Airy equation since, as in the deterministic scenario its solutions are highly oscillatory, it is expected that differences will be better highlighted. To illustrate our study, and motivated by the ubiquity of Gaussian random variables in numerous practical problems, we assume that inputs follow a multivariate Gaussian distribution throughout the paper. The application of Fröbenius method to solve Airy equation is based on an extension of the method to the case where inputs are dependent. The numerical results show that the existence of statistical dependence among the inputs and its magnitude entails changes on the variability of the output., This work has been partially supported by the Ministerio de Economia y Competitividad Grants MTM2009-08587 and DPI2010-20891-C02-01 and Universitat Politecnica de Valencia Grant PAID06-11-2070.

Este trabajo versa sobre la utilidad de las ecuaciones diferenciales y las ecuaciones en diferencias finitas para la resolución de distintos problemas en el ámbito de la economía y la empresa.En Economía es frecuente estudiar la evolución de los valores de una misma variable en distintos instantes temporales. Si la variable "tiempo" se considera como algo continuo, la evolución se estudia mediante ecuaciones diferenciales. Sin embargo, si el "tiempo" es tratado de manera discreta, se utilizan entonces ecuaciones en diferencias finitas.Concretamente, nuestro objetivo no solo es exponer la evolución que han sufrido las nociones de ecuaciones diferenciales y ecuaciones en diferencias finitas sino también dar una visión (no exhaustiva) de sus múltiples aplicaciones a cuestiones relativas a fenómenos económicos y financieros.------------------------------------ This paper deals with the use of differential equations and finite difference methods for solving several problems in the field of Economics and Business Administration.Economics usually needs to study the evolution of the values which are taken by a given variable in different moments. If the time variable works in a continuous way, its evolution is studied by differential equations. Otherwise, time is a discrete variable and finite difference methods must be used.In addition, to expound the evolution of the notions of differential and difference equations, the goal of this paper is to show a general view (but not comprehensive) of their many applications for explaining economical and financial phenomena., Artículo revisado por pares

We prove existence of solution to a local fractional nonlinear differential equation with initial condition. For that we introduce the notion of tube solution., Comment: This is a preprint of a paper whose final and definite form will be published in Journal of Computational and Applied Mathematics, ISSN: 0377-0427. Paper Submitted 04/Jul/2015; Revised 14/Dec/2015 and 03/Jan/2016; Accepted for publication 08/Jan/2016

The simplest epidemiologic model composed by mutually exclusive compartments SIR (susceptible-infected-susceptible) is presented to describe a reality. From health concerns to situations related with marketing, informatics or even sociology, several are the fields that are using this epidemiological model as a first approach to better understand a situation. In this paper, the basic transmission model is analyzed, as well as simple tools that allows us to extract a great deal of information about possible solutions. A set of applications - traditional and new ones - is described to show the importance of this model., Comment: Please cite this paper as: Rodrigues, Helena Sofia (2016). Application of SIR epidemiological model: new trends, International Journal of Applied Mathematics and Informatics, 10: 92--97

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0 or 1 , by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m=5,7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1, m as sum of row terms of corresponding triangle. This version might be out of date, proceed by the link to see actual version., 16 pages, 8 figures, 2 tables, typos revised, added missing references, results generalized and shortened, derivations detailed.

Nanostructured devices and materials, such as carbon nanotubes, Atomic Force Microscope, MEMS, etc. attract increasing attention in the scientific world. It has been realized that the classical elasticity is not capable to capture the mechanical behavior of them precisely. There is a wide consensus among the scientists that nonlocal elasticity is more capable than the classical counterpart to model the mechanical behavior of nanostructured materials and devices. In this paper a method which is useful for solving problems in nonlocal is introduced. Airy's stress functions for plane stress problems in nonlocal elasticity are studied. The nonlocal constitutive equations in integral form are discussed and a method is suggested to invert the constitutive equation which allows expressing strains in terms of stresses. A qualitative discussion is given on this inversion process. For the nonlocality kernel of exponential form, the differential equation for Airy's functions in nonlocal elasticity is obtained by introducing the strains into the compatibility condition. Appropriate polynomial forms for the Airy's function are considered and are applied to solve beam bending problems. The solutions are compared with their classical counterparts. The results are given in a series of figures and tables and are discussed in detail. This paper is concluded by indicating the implications of the presented study in nanomechanics and nanotechnology. Copyright © 2011 American Scientific Publishers.

The functions of computable analysis are defined by enhancing normal Turing machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as real recursive functions. Since there are numerous incompatible models of computation over the reals, it is interesting to find that the two very different models we consider can be set up to yield exactly the same functions. Bournez and Hainry used a function algebra to characterize computable analysis, restricted to the twice continuously differentiable functions with compact domains. In our earlier paper, we found a different (and apparently more natural) function algebra that also yields computable analysis, with the same restriction. In this paper we improve earlier work, finding three function algebras characterizing computable analysis, removing the restriction to twice continuously differentiable functions and allowing unbounded domains. One of these function algebras is built upon the widely studied real primitive recursive functions. Furthermore, the proof of this paper uses our previously developed method of approximation, whose applicability is further evidenced by this paper.

PurposeThe purpose of this paper is to solve both the prey and predator problem and the problem of the spread of a non‐fatal disease in a population which is assumed to have constant size over the period of the epidemic.Design/methodology/approachThe differential transform method (DTM) is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations of these problems.FindingsResults obtained using the scheme presented here agree well with the results obtained by the Adomian decomposition and power series methods. Some plots are presented to show the reliability and simplicity of the method.Originality/valueThis paper is believed to represent a new application for DTM on solving systems of nonlinear ordinary differential equations.

As a part of the international DECOVALEX III project, and the European BENCHPAR project, the impact of thermal-hydrological-mechanical (THM) couplings on the performance of a bentonite-back-filled nuclear waste repository in near-field crystalline rocks is evaluated in a Bench-Mark Test problem (BMT1) and the results are presented in a series of three companion papers in this issue. This is the third paper with focuses on the effects of THM processes at a repository located in a sparsely fractured rock. Several independent coupled THM analyses presented in this paper show that THM couplings have the most significant impact on the mechanical stress evolution, which is important for repository design, construction and post-closure monitoring considerations. The results show that the stress evolution in the bentonite-back-filled excavations and the surrounding rock depends on the post-closure evolution of both fields of temperature and fluid pressure. It is further shown that the time required to full resaturation may play an important role for the mechanical integrity of the repository drifts. In this sense, the presence of hydraulically conducting fractures in the near-field rock might actually improve the mechanical performance of the repository. Hydraulically conducting fractures in the near-field rocks enhances the water supply to the buffers/back-fills, which promotes a more timely process of resaturation and development of swelling pressures in the back-fill, thus provides timely confining stress and support to the rock walls. In one particular case simulated in this study, it was shown that failure in the drift walls could be prevented if the compressive stresses in back-fill were fully developed within 50 yr, which is when thermally induced rock strain begins to create high differential (failure-prone) stresses in the near-field rocks.

The main object of this paper is to investigate several geometric properties of the solutions of the following second-order linear differential equation: w″(z)+a(z)w′(z)+b(z)w(z)=0, where the functions a(z) and b(z) are analytic in the open unit disk U. Relevant connections of the results presented in this paper with those given earlier by, for example, Robertson, Miller and Saitoh are also considered.

In this paper, we present a detailed analysis of first-order perturbations of the Kerr metric in the slow-rotation limit. We perform the calculation by perturbing the Schwarzschild metric plus up to second-order corrections in the spin in the Regge-Wheeler gauge. The apparent coupling between different angular momentum axial-led and polar-led modes can be removed by suitably combining the perturbation equations and projecting them onto spin-weighted spherical harmonics. In this way, we derive the corrections to the Regge-Wheeler and the Zerilli equations up to second-order in the spin. We show that the two potentials remain isospectral as in the non-rotating limit. However, it is easy to demonstrate it only for a precise choice of the tortoise coordinate. The isospectrality with slow-rotating Teukolsky equation is also verified. We discuss the main implication of this result for the problem of vacuum metric reconstruction, providing the transformation rule between slow-spinning Teukolsky variables and metric perturbations. The existence of this relation leaves us with the conjecture that a resummation of the expansion in the spin is possible, leading to two decoupled differential equations for perturbations of the Kerr metric., Comment: 21 pages (12 + appendix and bibliography). Minor changes and additions to the bibliography

International audience; This paper proposes a new approach for set-membership state estimation of switched discrete-time linear systems subject to bounded disturbances and noises. A zonotopic outer approximation of the state estimation domain is computed and a new criterion is proposed to reduce the size of the zonotope at each sample time. The zonotopic set-membership estimator design for switched systems is provided within the LMI framework. The extension of the proposed scheme to deal with unknown inputs is also presented. An application to vehicle lateral dynamics state estimation is provided. Simulation results demonstrate the effectiveness of the proposed algorithm and highlight its advantages over the existing methods.

В работе исследуются условия, которые нужно наложить на правую часть системы для того, чтобы при достаточно малом диаметре разбиения ломаные Эйлера сходились к пучку решений системы, в частности, чтобы из всякой последовательности ломаных Эйлера можно было выделить сходящуюся на всем рассматриваемом промежутке времени к решению подпоследовательность. Найдено условие (для заданной, выписываемой явно, константы, для любой липшицевого с этой константой отображения в фазовую плоскость, множество точек разрыва функции динамики имеет нулевую по Лебегу меру на графиках таких отображений), которое гарантирует сходимость ломаных Эйлера к пучку решений системы, если только диаметр соответствующих ломаным разбиений стремится к нулю. Рядом примеров показано, что данное условие не может быть ослаблено; в частности, сходимости может не быть даже если для всякой порожденной в рамках системы траектории сужение функции динамики на этот график интегрируемо по Риману, константа в указанном выше условии также не может быть уменьшена. В работе ломаные Эйлера погружаются в семейство решений интегрального уравнения с запаздыванием специального вида, для которых в свою очередь, и проводится доказательство основного результата. Вследствие этого, результаты статьи имеют место и в более широком классе численных методов, например для ломаных со счетным числом звеньев. We study the conditions on right-hand side of a system that guarantee the convergence of Euler's broken lines to the funnel of solutions of the system for sufficiently small diameter of partition; in particular, the condition that lets us select a subsequence from any sequence of Euler's broken lines that would converge to the solution on a given time interval. We obtain the condition that guarantees the convergence of Euler's broken lines to the funnel of solutions of the system as the diameter of partitions corresponding to the broken lines tends to zero. The condition is specified for a given explicit constant such that for any mapping that is Liepshitz continuous with this constant and maps onto the phase plane, the set of points of discontinuity has the zero Lebesgue measure (on the graph of this mapping). Several examples are given to demonstrate that this condition cannot be relaxed; specifically, there may be no convergence even if, for each trajectory generated by the system, the restriction of the dynamics function to that graph is Riemann integrable; the constant from the condition above can never be decreased either. In the paper, Euler's broken lines are embedded into the family of solutions of delay integral equations of the special form, for which, in its own turn, the main result of the paper is proved. It is due to this fact that the results of the paper hold for a broader class of numerical methods, for example, for broken lines with countable number of segments. Дмитрий Валерьевич Хлопин, кандидат физико-математических наук, заведующий отделом, Институт математики и механики им. Н.Н. Красовского УрО РАН, (г. Екатеринбург, Российская Федерация), khlopin@imm.uran.ru. D. V. Khlopin, Krasovskii Institute of Mathematics and Mechanics, UrB RAS; Yekaterinburg, Russian Federation, khlopin@imm.uran.ru

Due to its simplicity, the theory of KUBELKA-MUNK [1] has found a wide acceptance for modeling the optical properties of light scattering materials. However, the concept is not explicitly adapted to predict halftone prints on paper. In this respect, a recent improvement was given by BERG. Our approach is an extension of BERG'S model in order to reduce the gap between the mathematical description of the paper's point spread function and the experimental results of simple reflectance measurements.

The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

Mathematics constitutes a key component in engineering education. Engineering students are traditionally offered a number of mathematics courses which provide the knowledge needed at the workplace. Unfortunately, many students perceive mathematics as a discipline that teaches mostly procedures not relevant to their future careers and often view it as one of the main obstacles on their way to an engineering degree. In this paper, we discuss how introducing university students in a standard Differential Equations course to mathematical modelling (MM), a powerful strategy for solving real-life problems, contributes to the development of their mathematical competencies, motivates their interest to mathematics, promotes the use of advanced mathematical thinking, methods of applied mathematics, and digital computational tools.

Recently, a class of adaptive control schemes called L1 Adaptive Control (L1-AC) has been proposed and widely advertised in aerospace control for achieving fast and robust adaptation and better performance than the existing Model Reference Adaptive Control (MRAC) Schemes. The L1-AC scheme is designed mainly for plants with full state measurement even though the name L1-AC has been used as an umbrella name for more general classes of plants. In this paper, we show that the L1-AC for plants with measured states is simply a standard MRAC with a low pass filter inserted in front of the control input. The analysis of the scheme is almost identical to that of MRAC as the same Lyapunov function is used to establish stability. The motivation for using the filter is the fact that for this class of adaptive schemes i.e. MRAC for plants with full state measurement the tracking error can be made arbitrarily small during transient by increasing the adaptive gain. A high adaptive gain however makes the differential equation of the adaptive law or estimator very stiff and leads to numerical problems that cause high oscillations in the estimated parameters leading to loss of adaptivity and deviations from what the theoretical properties dictate. The L1-AC approach mistook these numerical oscillations as properties of the adaptive scheme and inserted an input low pass filter in order to filter them out. While the filter helps reduce the frequency of these oscillations in the control law the price paid is high. First the numerical instability does not go away and the estimated parameters continue to oscillate without converging to the true parameters even in the presence of suffciently rich signals. Second, due to the filter the tracking error is no longer guaranteed to converge to zero and the transient bounds for the tracking error also depend on the filter. As a result, the tracking properties of the L1-AC scheme are worse than what a simple MRAC scheme can generate with adaptive gains that could be high but away from the region of numerical instability. In addition, the presence of the filter reduces the robust stability margins in the presence of unmodeled dynamics and provides literally no advantage as simple robust MRAC techniques can solve the same problem achieving much better properties. The authors of L1-AC often compare the properties of a numerically unstable MRAC due to extremely high adaptive gains something that is prohibited by robust adaptive control with those of the filtered MRAC aka L1-AC to show that L1-AC performs better. Such comparisons are not only misleading but do not reveal what causes what giving the reader a false impression of a new theory that results to new performance and robustness improvements. The use of filters in adaptive control is not new and they are used to improve the performance and robustness of certain adaptive control schemes with-out destroying their ideal tracking properties. In this paper we present such a scheme that guarantees good transient performance and robustness and reveals the trade off between transient performance and robust stability. Sponsors: Draper Laboratory Conference code: 99256 Cited By :2

In this paper, we consider nonconvex control problems of stochastic differential equations driven by relaxed controls adapted, in the weak star sense, to a current of sigma algebras generated by observable processes. We cover in a unified way both continuous diffusion and jump processes. We present existence of optimal controls before we construct the necessary conditions of optimality (unlike some papers in this area) using only functional analysis. We develop a stochastic Hamiltonian system of equations on a rigorous basis using the semimartingale representation theory and the Riesz representation theorem, leading naturally to the existence of the adjoint process which satisfies a backward stochastic differential equation. In other words, our approach predicts the existence of the adjoint process as a natural consequence of Riesz representation theory ensuring at the same time the (weak star) measurability. This is unlike other papers, where the adjoint process is introduced before its existence is proved. We believe this is one of our major contributions in this paper. We also discuss the realizability of relaxed controls by regular controls using the Krein- Millman theorem. We believe this is another major contribution of this paper. We also believe that our approach is direct and easy to understand following simply the precise logic of functional analysis. © 2013 Society for Industrial and Applied Mathematics. 51 4 3235 3257

This paper presents a concise and orderly methodology to obtain universal solutions to different problems in science and engineering using the nondimensionalization of the governing equations of the physical–chemical problem posed. For its application, a deep knowledge of the problem is necessary since it will facilitate the adequate choice of the references necessary for its resolution. In addition, the application of the methodology to examples of coupled ordinary differential equations is shown, resulting in an interesting tool to teach postgraduate students in the branches of physics, mathematics, and engineering. The first example used for a system of coupled ordinary differential equations is a model of a continuous flow chemical reactor, where it is worth noting; on the one hand, the methodology used to choose the reference (characteristic) time and, on the other, the equivalence between the characteristic times obtained for each one of the species. The following universal curves are obtained, which are validated by comparing them with the results obtained by numerical simulation, where it stands out that the universal solution includes an unknown that must be previously obtained. The resolution of this unknown implies having a deep knowledge of the problem, a common characteristic when using the methodology proposed in this work for different engineering or physicochemical problems. Finally, the second example is a coupled oscillator, where it is worth noting that the appearance of characteristic periods that implicitly or explicitly affect the particles’ movement is striking.

In this work, we study the oscillation of the fourth order neutral differential equations with delay argument. By means of generalized Riccati transformation technique, we obtain new oscillation criteria for oscillation of this equation. An example is given to clarify the main results in this paper.