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## An ordinary differential equation based solution path algorithm

Fonte: PubMed
Publicador: PubMed

Tipo: Artigo de Revista Científica

Publicado em //2011
EN

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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.

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## The systems biology simulation core algorithm

Fonte: BioMed Central
Publicador: BioMed Central

Tipo: Artigo de Revista Científica

EN_US

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#Systems biology#Biological networks#Mathematical modeling#Simulation#Algorithms#Ordinary differential equation systems#Numerical integration#Software engineering

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...

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## An ordinary differential equation model for the multistep transformation to cancer

Fonte: Academic Press Ltd
Publicador: Academic Press Ltd

Tipo: Artigo de Revista Científica

Publicado em //2004
EN

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#Humans#Cell Transformation, Neoplastic#Mutation#Cell Division#Systems Biology#Neoplasms#Neovascularization, Pathologic

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.

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## Unnecessary Exact Solutions of Nonlinear Ordinary Differential Equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/11/2010

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/08/1999

Relevância na Pesquisa

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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

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## Linearisable third order ordinary differential equations and generalised Sundman transformations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/03/2002

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#Nonlinear Sciences - Exactly Solvable and Integrable Systems#Mathematics - Classical Analysis and ODEs

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/02/2007

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/06/2010

Relevância na Pesquisa

716.0041%

#Mathematics - Classical Analysis and ODEs#Nonlinear Sciences - Exactly Solvable and Integrable Systems

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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Algebraic Geometry#Mathematical Physics#Mathematics - Classical Analysis and ODEs#Mathematics - Dynamical Systems#34M55, 34M45, 58F05, 32S65

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

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## 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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/11/2013

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/10/2005

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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

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## The functional formulation of second-order ordinary differential equations

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/01/2013

Relevância na Pesquisa

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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

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## Convexity of reachable sets of nonlinear ordinary differential equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/11/2012

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#Mathematics - Optimization and Control#Computer Science - Systems and Control#93B03 (Primary), 52A05, 93C10, 93C15 (Secondary)

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

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## Invariant manifolds for a singular ordinary differential equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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## An ordinary differential equation model for the multistep transformation to cancer

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

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

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

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

Fonte: Universidade de Cambridge
Publicador: Universidade de Cambridge

Tipo: Conferência ou Objeto de Conferência

EN

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#Ordinary Differential Equations#State discretisation#Effective continuous time solution of ODE's#First order sensitivity equations#Discrete events#Stiffness resolution#one-variable-at-a-time scheme#High-performance computing#discrete event location#Sensitivity Equations#ODE

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...

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

Fonte: John Wiley & Sons Inc
Publicador: John Wiley & Sons Inc

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Keywords: Computer simulation#Convergence of numerical methods#Discrete time control systems#Linear control systems#Observability#Online systems#Ordinary differential equations#State estimation#State space methods#Stochastic control systems#Adaptive estim Adaptive estimation

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.

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