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An ordinary differential equation based solution path algorithm

Wu, Yichao
Fonte: PubMed Publicador: PubMed
Tipo: Artigo de Revista Científica
Publicado em //2011 EN
Relevância na Pesquisa
819.95375%
Efron, Hastie, Johnstone and Tibshirani (2004) proposed Least Angle Regression (LAR), a solution path algorithm for the least squares regression. They pointed out that a slight modification of the LAR gives the LASSO (Tibshirani, 1996) solution path. However it is largely unknown how to extend this solution path algorithm to models beyond the least squares regression. In this work, we propose an extension of the LAR for generalized linear models and the quasi-likelihood model by showing that the corresponding solution path is piecewise given by solutions of ordinary differential equation systems. Our contribution is twofold. First, we provide a theoretical understanding on how the corresponding solution path propagates. Second, we propose an ordinary differential equation based algorithm to obtain the whole solution path.

The systems biology simulation core algorithm

Keller, Roland; Dörr, Alexander; Tabira, Akito; Funahashi, Akira; Ziller, Michael J; Adams, Richard; Rodriguez, Nicolas; Novère, Nicolas Le; Hiroi, Noriko; Planatscher, Hannes; Zell, Andreas; Dräger, Andreas
Fonte: BioMed Central Publicador: BioMed Central
Tipo: Artigo de Revista Científica
EN_US
Relevância na Pesquisa
923.1119%
Background: With the increasing availability of high dimensional time course data for metabolites, genes, and fluxes, the mathematical description of dynamical systems has become an essential aspect of research in systems biology. Models are often encoded in formats such as SBML, whose structure is very complex and difficult to evaluate due to many special cases. Results: This article describes an efficient algorithm to solve SBML models that are interpreted in terms of ordinary differential equations. We begin our consideration with a formal representation of the mathematical form of the models and explain all parts of the algorithm in detail, including several preprocessing steps. We provide a flexible reference implementation as part of the Systems Biology Simulation Core Library, a community-driven project providing a large collection of numerical solvers and a sophisticated interface hierarchy for the definition of custom differential equation systems. To demonstrate the capabilities of the new algorithm, it has been tested with the entire SBML Test Suite and all models of BioModels Database. Conclusions: The formal description of the mathematics behind the SBML format facilitates the implementation of the algorithm within specifically tailored programs. The reference implementation can be used as a simulation backend for Java™-based programs. Source code...

An ordinary differential equation model for the multistep transformation to cancer

Spencer, S.; Berryman, M.; Garcia, J.; Abbott, D.
Fonte: Academic Press Ltd Publicador: Academic Press Ltd
Tipo: Artigo de Revista Científica
Publicado em //2004 EN
Relevância na Pesquisa
812.13875%
Cancer is viewed as a multistep process whereby a normal cell is transformed into a cancer cell through the acquisition of mutations. We reduce the complexities of cancer progression to a simple set of underlying rules that govern the transformation of normal cells to malignant cells. In doing so, we derive an ordinary differential equation model that explores how the balance of angiogenesis, cell death rates, genetic instability, and replication rates give rise to different kinetics in the development of cancer. The key predictions of the model are that cancer develops fastest through a particular ordering of mutations and that mutations in genes that maintain genomic integrity would be the most deleterious type of mutations to inherit. In addition, we perform a sensitivity analysis on the parameters included in the model to determine the probable contribution of each. This paper presents a novel approach to viewing the genetic basis of cancer from a systems biology perspective and provides the groundwork for other models that can be directly tied to clinical and molecular data.; http://www.elsevier.com/wps/find/journaldescription.cws_home/622904/description#description; Sabrina L. Spencer, Matthew J. Berryman, José A. García and Derek Abbott; Copyright © 2004 Elsevier Ltd. All rights reserved.

Unnecessary Exact Solutions of Nonlinear Ordinary Differential Equations

Kudryashov, Nikolay A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/11/2010
Relevância na Pesquisa
722.26164%
We analyze the paper by Wazwaz and Mehanna [Wazwaz A.M., Mehanna M.S., A variety of exact travelling wave solutions for the (2+1) -- dimensional Boiti -- Leon -- Pempinelli equation, Appl. Math. Comp. 217 (2010) 1484 -- 1490]. Using the tanh -- coth method and the Exp -- function method the authors claim that they have found exact solutions of the (2+1) -- dimensional Boiti -- Leon -- Pempinelli equation. We demonstrate that the authors have obtained the exact solutions of the well known nonlinear ordinary differential equation. We illustrate that all solutions presented by the authors can be reduced to the well-known solutions. Wazwaz and Mehanna made a number of typical mistakes in finding exact solutions of nonlinear differential equations. Taking the results of this paper we introduce the definition of unnecessary exact solutions for the nonlinear ordinary differential equations.; Comment: 13 pages

The connection of Monge-Bateman equations with ordinary differential equations and their generalisation

Leznov, A. N.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/08/1999
Relevância na Pesquisa
719.95375%
It is shown that the Monge equation is equivalent to the ordinary differential equation $\ddot X=0$ of free motion. Equations of Monge type (with their general solutions) are connected with each ordinary differential equation of second order $\ddot X=F(\dot X,X; t)$, integrable by quadratures. The result is generalised to a system of equations of the second order, which is in one to one correspondence with the multidimensional Monge-Bateman system.; Comment: LaTeX, 6 pages

Linearisable third order ordinary differential equations and generalised Sundman transformations

Euler, N.; Wolf, T.; Leach, P. G. L.; Euler, M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 14/03/2002
Relevância na Pesquisa
699.2141%
We calculate in detail the conditions which allow the most general third order ordinary differential equation to be linearised in X'''(T)=0 under the transformation X(T)=F(x,t), dT=G(x,t)dt. Further generalisations are considered.; Comment: 33 pages

Formal Paths, Iterated Integrals and the Center Problem for Ordinary Differential Equations

Brudnyi, Alexander
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 26/02/2007
Relevância na Pesquisa
699.2141%
We continue the study of the center problem for the ordinary differential equation $v'=\sum_{i=1}^{\infty}a_{i}(x)v^{i+1}$ started in our earlier papers. In this paper we present the highlights of the algebraic theory of centers.; Comment: 35 pages

Linearization of Second-Order Ordinary Differential Equations by Generalized Sundman Transformations

Nakpim, Warisa; Meleshko, Sergey V.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/06/2010
Relevância na Pesquisa
716.0041%
The linearization problem of a second-order ordinary differential equation by the generalized Sundman transformation was considered earlier by Duarte, Moreira and Santos using the Laguerre form. The results obtained in the present paper demonstrate that their solution of the linearization problem for a second-order ordinary differential equation via the generalized Sundman transformation is not complete. We also give examples which show that the Laguerre form is not sufficient for the linearization problem via the generalized Sundman transformation.

Fourth-order ordinary differential equation obtained by similarity reduction of the modifed Sawada-Kotera equation

Sasano, Yusuke
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
713.1346%
We study a one-parameter family of the fourth-order ordinary differential equations obtained by similarity reduction of the modifed Sawada-Kotera equation. We show that the birational transformations take this equation to the polynomial Hamiltonian system in dimension four. We make this polynomial Hamiltonian from the viewpoint of accessible singularity and local index. We also give its symmetry and holomorphy conditions. These properties are new. Moreover, we introduce a symmetric form in dimension five for this Hamiltonian system by taking the two invariant divisors as the dependent variables. Thanks to the symmetric form, we show that this system admits the affine Weyl group symmetry of type $A_2^{(2)}$ as the group of its B{\"a}cklund transformations.; Comment: 19 pages

An introduction to the NMPC-Graph as general schema for causal modelling of nonlinear, multivariate, dynamic, and recursive systems with focus on time-series prediction

Jahnz, Christoph
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
714.9591%
While the disciplines of physics and engineering sciences in many cases have taken advantage from accurate time-series prediction of system behaviour by applying ordinary differential equation systems upon precise basic physical laws such approach hardly could be adopted by other scientific disciplines where precise mathematical basic laws are unknown. A new modelling schema, the NMPC-graph, opens the possibility of interdisciplinary and generic nonlinear, multivariate, dynamic, and recursive causal modelling in domains where basic laws are only known as qualitative relationships among parameters while their precise mathematical nature remains undisclosed at modelling time. The symbolism of NMPC-graph is kept simple and suited for analysts without advanced mathematical skills. This article presents the definition of the NMPC-graph modelling method and its six component types. Further, it shows how to solve the inverse problem of deriving a nonlinear ordinary differential equation system from any NMPC-graph in conjunction with historic calibration data by means of machine learning. This article further discusses how such a derived NMPC-model can be used for hypothesis testing and time-series prediction with the expectation of gaining prediction accuracy in comparison to conventional prediction methods.; Comment: 29 pages...

Achieving synchronization in arrays of coupled differential systems with time-varying couplings

Yi, Xinlei; Lu, Wenlian; Chen, Tianping
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 09/11/2013
Relevância na Pesquisa
700.67%
In this paper, we study complete synchronization of the complex dynamical networks described by linearly coupled ordinary differential equation systems (LCODEs). The coupling considered here is time-varying in both the network structure and the reaction dynamics. Inspired by our previous paper [6], the extended Hajnal diameter is introduced and used to measure the synchronization in a general differential system. Then we find that the Hajnal diameter of the linear system induced by the time-varying coupling matrix and the largest Lyapunov exponent of the synchronized system play the key roles in synchronization analysis of LCODEs with the identity inner coupling matrix. As an application, we obtain a general sufficient condition guaranteeing directed time-varying graph to reach consensus. Example with numerical simulation is provided to show the effectiveness the theoretical results.; Comment: 22 pages, 4 figures

On the complete integrability and linearization of nonlinear ordinary differential equations - Part II: Third order equations

Chandrasekar, V. K.; Senthilvelan, M.; Lakshmanan, M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 17/10/2005
Relevância na Pesquisa
714.4542%
We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle-Singer method. We describe a procedure to deduce all the integrals of motion associated with the given equation so that the general solution follows straightforwardly from these integrals. The method is illustrated with several examples. Further, we propose a powerful method of identifying linearizing transformations. The proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation (GLT). In addition to the above, we provide an algorithm to invert the nonlocal linearizing transformation. Through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation.; Comment: Submitted to Proceedings of the Royal Society London Series A, 21 pages

The functional formulation of second-order ordinary differential equations

Chladek, P.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 28/10/2003
Relevância na Pesquisa
716.0041%
In this paper, the necessary and sufficient conditions in order that a smooth mapping F be a dependence of a complete solution of some second-order ordinary differential equation on Neumann conditions are deduced. These necessary and sufficient conditions consist of functional equations for F and of a smooth extensibility condition. Illustrative examples are presented to demonstrate this result. In these examples, the mentioned functional equations for F are related to the functional equations for geodesics, to Jensen's equation, to the functional equations for conic sections and to Neuman's result for linear ordinary differential equations.; Comment: 8 pages, Latex

Discrete gradient methods for preserving a first integral of an ordinary differential equation

Norton, Richard A.; Quispel, G. R. W.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/01/2013
Relevância na Pesquisa
699.2141%
In this paper we consider discrete gradient methods for approximating the solution and preserving a first integral (also called a constant of motion) of autonomous ordinary differential equations. We prove under mild conditions for a large class of discrete gradient methods that the numerical solution exists and is locally unique, and that for arbitrary $p \in \mathbb{N}$ we may construct a method that is of order $p$. In the proofs of these results we also show that the constants in the time step constraint and the error bounds may be chosen independently from the distance to critical points of the first integral. In the case when the first integral is quadratic, for arbitrary $p \in \mathbb{N}$, we have devised a new method that is linearly implicit at each time step and of order $p$. This new method has significant advantages in terms of efficiency. We illustrate our theory with a numerical example.; Comment: 22 pages, 4 figures

Convexity of reachable sets of nonlinear ordinary differential equations

Reißig, Gunther
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 26/11/2012
Relevância na Pesquisa
710.5046%
We present a necessary and sufficient condition for the reachable set, i.e., the set of states reachable from a ball of initial states at some time, of an ordinary differential equation to be convex. In particular, convexity is guaranteed if the ball of initial states is sufficiently small, and we provide an upper bound on the radius of that ball, which can be directly obtained from the right hand side of the differential equation. In finite dimensions, our results cover the case of ellipsoids of initial states. A potential application of our results is inner and outer polyhedral approximation of reachable sets, which becomes extremely simple and almost universally applicable if these sets are known to be convex. We demonstrate by means of an example that the balls of initial states for which the latter property follows from our results are large enough to be used in actual computations.; Comment: Accepted version

Invariant manifolds for a singular ordinary differential equation

Bianchini, Stefano; Spinolo, Laura V.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
713.1346%
We study the singular ordinary differential equation $$ \frac{d U}{d t} = f (U) / z (U) + g (U), $$ where $U \in R^N$, the functions $f \in R^N $ and $g \in R^N $ are of class $C^2$ and $z $ is a real valued $C^2$ function. The equation is singular in the sense that $z (U)$ can attain the value 0. We focus on the solutions of the singular ODE that belong to a small neighborhood of a point $\bar U$ such that $f (\bar U) = g (\bar U) = \vec 0$, $z (\bar U) =0$. We investigate the existence of manifolds that are locally invariant for the singular ODE and that contain orbits with a suitable prescribed asymptotic behaviour. Under suitable hypotheses on the set $\{U: z (U) = 0 \}$, we extend to the case of the singular ODE the definitions of center manifold, center stable manifold and of uniformly stable manifold. An application of our analysis concerns the study of the viscous profiles with small total variation for a class of mixed hyperbolic-parabolic systems in one space variable. Such a class includes the compressible Navier Stokes equation.; Comment: 35 pages, more general case considered

An ordinary differential equation model for the multistep transformation to cancer

Spencer, Sabrina L.; Berryman, Matthew J.; Garcia, Jose A.; Abbott, Derek
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/03/2004
Relevância na Pesquisa
706.7544%
Cancer is viewed as a multistep process whereby a normal cell is transformed into a cancer cell through the acquisition of mutations. We reduce the complexities of cancer progression to a simple set of underlying rules that govern the transformation of normal cells to malignant cells. In doing so, we derive an ordinary differential equation model that explores how the balance of angiogenesis, cell death rates, genetic instability, and replication rates give rise to different kinetics in the development of cancer. The key predictions of the model are that cancer develops fastest through a particular ordering of mutations and that mutations in genes that maintain genomic integrity would be the most deleterious type of mutations to inherit. In addition, we perform a sensitivity analysis on the parameters included in the model to determine the probable contribution of each. This paper presents a novel approach to viewing the genetic basis of cancer from a systems biology perspective and provides the groundwork for other models that can be directly tied to clinical and molecular data.; Comment: 12 pages, submitted to Journal of Theoretical Biology

Symmetries of nonlinear ordinary differential equations: the modified Emden equation as a case study

Senthilvelan, M.; Chandrasekar, V. K.; Mohanasubha, R.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 13/02/2015
Relevância na Pesquisa
727.76875%
Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries, contact symmetries, hidden symmetries, nonlocal symmetries, $\lambda$-symmetries, adjoint symmetries and telescopic vector fields of a second-order ordinary differential equation. We also illustrate the algorithm involved in each method by considering a nonlinear oscillator equation as an example. The connections between (i) symmetries and integrating factors and (ii) symmetries and integrals are also discussed and illustrated through the same example. The interconnections between some of the above symmetries, that is (i) Lie point symmetries and $\lambda$-symmetries and (ii) exponential nonlocal symmetries and $\lambda$-symmetries are also discussed. The order reduction procedure is invoked to derive the general solution of the second-order equation.; Comment: 31 pages, To appear in the proceedings of NMI workshop on nonlinear integrable systems and their applications which was held at Centre for Nonlinear Dynamics, Tiruchirappalli, India

Quantised State Simulation (QSS): Advances in the Numerical Solution of Ordinary Differential Equations

Vassiliadis, Vassilios S.; Fiorelli, Fabio
Fonte: Universidade de Cambridge Publicador: Universidade de Cambridge
Tipo: Conferência ou Objeto de Conferência
EN
Relevância na Pesquisa
938.2533%
This presentation handout presents the idea of discretising the state variables (quantising them) instead of time, to effect the numerical integration/simulation of ordinary differential equation systems. The aim is to provide the computational technology to address huge scale systems at extremely high efficiency, at unprecedented speeds over existing methods. The methodology results in a matrix free algorithm, which can also be very readily be used for sensitivity evaluations and is highly parallelisable (in fact it is completely scalable). A multitude of potential uses is outlined, i.e. replacing totally the need for stochastic simulation algorithms for dynamical systems as the method is completely rigorous and robust, and nonrandom. In other words it comprises an innovative standard type numerical integration scheme. The potential for applications in Molecular Dynamic Simulation, Polymerisation reaction simulations, population dynamic balances, and of course in combustion reactions is tremendous. The importance also and its particular incomparable strength over stochastic simulation methods is that it can produce rigorous high precision values for sensitivity equations. As such it can be integrated within a parameter estimation scheme robustly...

On-line almost-sure parameter estimation for partially observed discrete-time linear systems with known noise charateristics

Elliott, Robert J; Ford, Jason; Moore, John
Fonte: John Wiley & Sons Inc Publicador: John Wiley & Sons Inc
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
711.1794%
In this paper we discuss parameter estimators for fully and partially observed discrete-time linear stochastic systems (in state-space form) with known noise characteristics. We propose finite-dimensional parameter estimators that are based on estimates of summed functions of the state, rather than of the states themselves. We limit our investigation to estimation of the state transition matrix and the observation matrix. We establish almost-sure convergence results for our proposed parameter estimators using standard martingale convergence results, the Kronecker lemma and an ordinary differential equation approach. We also provide simulation studies which illustrate the performance of these estimators.