A class of fourth-order parabolic equation with logarithmic nonlinearity.
Ontology highlight
ABSTRACT:
SUBMITTER: Li P
PROVIDER: S-EPMC6267419 | biostudies-other | 2018
REPOSITORIES: biostudies-other
Similar Datasets
Project description:Second order linear differential equations can be used as models for regulation since under a range of parameter values they can account for return to equilibrium as well as potential oscillations in regulated variables. One method that can estimate parameters of these equations from intensive time series data is the method of Latent Differential Equations (LDE). However, the LDE method can exhibit bias in its parameters if the dimension of the time delay embedding and thus the width of the convolution kernel is not chosen wisely. This article presents a simulation study showing that a constrained fourth order Latent Differential Equation (FOLDE) model for the second order system almost completely eliminates bias as long as the width of the convolution kernel is less than two thirds the period of oscillations in the data. The FOLDE model adds two degrees of freedom over the standard LDE model but significantly improves model fit.
| S-EPMC7518521 | biostudies-literature
Project description:This paper contributes to the environmental literature by (i) demonstrating that the estimated coefficients and the statistical significance of the non-leading terms in quadratic, cubic, and quartic logarithmic environmental Kuznets curve (EKC) specifications are arbitrary and should therefore not be used to choose the preferred specification and (ii) detailing a proposed general-to-specific type methodology for choosing the appropriate specifications when attempting to estimate higher-order polynomials such as cubic and quartic logarithmic EKC relationships. Testing for the existence and shape of the well-known EKC phenomenon is a hot topic in the environmental economics literature. The conventional approach widely employs quadratic and cubic specifications and more recently also the quartic specification, where the variables are in logarithmic form. However, it is important that researchers understand whether the estimated EKC coefficients, turning points, and elasticities are statistically acceptable, economically interpretable, and comparable. In addition, it is vital that researchers have a clear structured non-arbitrary methodology for determining the preferred specification and hence shape of the estimated EKC. We therefore show mathematically and empirically the arbitrary nature of estimated non-leading coefficients in quadratic, cubic, and quartic logarithmic EKC specifications, being dependent upon the units of measurement chosen for the independent variables (e.g. dependent upon a rescaling of the variables such as moving from $m to $bn). Consequently, the practice followed in many previously papers, whereby the estimates of the non-leading terms are used in the decision to choose the preferred specification of an estimated EKC relationship, is incorrect and should not be followed since it potentially could lead to misleading conclusions. Instead, it should be based upon the sign and statistical significance of the estimated coefficients of the leading terms, the location of turning point(s), and the sign and statistical significance of the estimated elasticities. Furthermore, we suggest that researchers should follow a proposed general-to-specific type methodology for choosing the appropriate order of polynomials when attempting to estimate higher-order polynomial logarithmic EKCs.
| S-EPMC8355002 | biostudies-literature
Project description:Constrained fourth-order latent differential equation (FOLDE) models have been proposed (e.g., Boker et al. 2020) as alternative to second-order latent differential equation (SOLDE) models to estimate second-order linear differential equation systems such as the damped linear oscillator model. When, however, only a relatively small number of measurement occasions T are available (i.e., [Formula: see text]), the recommendation of which model to use is not clear (Boker et al. 2020). Based on a data set, which consists of [Formula: see text] observations of daily stress for [Formula: see text] individuals, we illustrate that FOLDE can help to choose an embedding dimension, even in the case of a small T. This is of great importance, as parameter estimates depend on the embedding dimension as well as on the latent differential equations model. Consequently, the wavelength as quantity of potential substantive interest may vary considerably. We extend the modeling approaches used in past research by including multiple subjects, by accounting for individual differences in equilibrium, and by including multiple instead of one single observed indicator.
| S-EPMC7826318 | biostudies-literature
Project description:Throughout this paper, we mainly consider the parabolic p-Laplacian equation with a weighted absorption \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{t}-\operatorname{div} (|\nabla u|^{p-2}\nabla u )=-\lambda |x|^{\alpha} {\chi}_{\{u>0\}}u^{-\beta}$\end{document}ut?div(|?u|p?2?u)=??|x|??{u>0}u?? in a bounded domain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega\subseteq\mathbb{R}^{n}$\end{document}??Rn (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq1$\end{document}n?1) with Lipschitz continuous boundary subject to homogeneous Dirichlet boundary condition. Here \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda>0$\end{document}?>0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha>-n$\end{document}?>?n are parameters, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta\in(0,1)$\end{document}??(0,1) is a given constant. Under the assumptions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{0}\in W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega)$\end{document}u0?W01,p(?)?L?(?), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{0}\geq0$\end{document}u0?0 a.e. in ?, we can establish conditions of local and global in time existence of nonnegative solutions, and show that every global solution completely quenches in finite time a.e. in ?. Moreover, we give some numerical experiments to illustrate the theoretical results.
| S-EPMC6154082 | biostudies-other
Project description:We consider a Markov process in continuous time with a finite number of discrete states. The time-dependent probabilities of being in any state of the Markov chain are governed by a set of ordinary differential equations, whose dimension might be large even for trivial systems. Here, we derive a reduced ODE set that accurately approximates the probabilities of subspaces of interest with a known error bound. Our methodology is based on model reduction by balanced truncation and can be considerably more computationally efficient than solving the chemical master equation directly. We show the applicability of our method by analysing stochastic chemical reactions. First, we obtain a reduced order model for the infinitesimal generator of a Markov chain that models a reversible, monomolecular reaction. Later, we obtain a reduced order model for a catalytic conversion of substrate to a product (a so-called Michaelis-Menten mechanism), and compare its dynamics with a rapid equilibrium approximation method. For this example, we highlight the savings on the computational load obtained by means of the reduced-order model. Furthermore, we revisit the substrate catalytic conversion by obtaining a lower-order model that approximates the probability of having predefined ranges of product molecules. In such an example, we obtain an approximation of the output of a model with 5151 states by a reduced model with 16 states. Finally, we obtain a reduced-order model of the Brusselator.
| S-EPMC4133211 | biostudies-literature
Project description:We investigate the positive solutions of the semilinear parabolic system with coupled nonlinear nonlocal sources subject to weighted nonlocal Dirichlet boundary conditions. The blow-up and global existence criteria are obtained.
| S-EPMC3947733 | biostudies-other
Project description:An intriguing consequence of ongoing riboswitch discovery efforts is the occasional identification of metabolic or toxicity response pathways for unusual ligands. Recently, we reported the experimental validation of three distinct bacterial riboswitch classes that regulate gene expression in response to the selective binding of a guanidinium ion. These riboswitch classes, called guanidine-I, -II, and -III, regulate numerous genes whose protein products include previously misannotated guanidine exporters and enzymes that degrade guanidine via an initial carboxylation reaction. Guanidine is now recognized as the primal substrate of many multidrug efflux pumps that are important for bacterial resistance to certain antibiotics. Guanidine carboxylase enzymes had long been annotated as urea carboxylase enzymes but are now understood to participate in guanidine degradation. Herein, we report the existence of a fourth riboswitch class for this ligand, called guanidine-IV. Members of this class use a novel aptamer to selectively bind guanidine and use an unusual expression platform arrangement that is predicted to activate gene expression when ligand is present. The wide distribution of this abundant riboswitch class, coupled with the striking diversity of other guanidine-sensing RNAs, demonstrates that many bacterial species maintain sophisticated sensory and genetic mechanisms to avoid guanidine toxicity. This finding further highlights the mystery regarding the natural source of this nitrogen-rich chemical moiety.
| S-EPMC7919255 | biostudies-literature
Project description:Riboswitches are RNAs that specifically sense a small molecule and regulate genes accordingly. The recent discovery of guanidine-binding riboswitches revealed the biological significance of this compound, and uncovered genes related to its biology. For example, certain sugE genes encode guanidine exporters and are activated by the riboswitches to reduce toxic levels of guanidine in the cell. In order to study guanidine biology and riboswitches, we applied a bioinformatics strategy for discovering additional guanidine riboswitches by searching for new candidate motifs associated with sugE genes. Based on in vitro and in vivo experiments, we determined that one of our six best candidates is a new structural class of guanidine riboswitches. The expression of a genetic reporter was induced 80-fold in response to addition of 5 mM guanidine in Staphylococcus aureus. This new class, called the guanidine-IV riboswitch, reveals additional guanidine-associated protein domains that are extremely rarely or never associated with previously established guanidine riboswitches. Among these protein domains are two transporter families that are structurally distinct from SugE, and could represent novel types of guanidine exporters. These results establish a new metabolite-binding RNA, further validate a bioinformatics method for finding riboswitches and suggest substrate specificities for as-yet uncharacterized transporter proteins.
| S-EPMC7736828 | biostudies-literature
Project description:Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.
| S-EPMC3869418 | biostudies-literature
Project description:It is now possible to calculate the ab initio quantum mechanics of very large biological molecules. Two things lead to this perspective, namely, (i) the advances of parallel supercomputers, and (ii) the discovery of a quantum formalism called quantum crystallography and the use of quantum kernels, a method that is well suited for parallel computation. The kernel energy method (KEM) carried to second order has been used to calculate the quantum mechanical ab initio molecular energy of peptides, protein (insulin and collagen), DNA, and RNA and the interaction of drugs with their biochemical molecular targets. The results were found to have good accuracy. In this article, the accuracy of the KEM is investigated up to an approximation including fourth-order interactions among kernels. Remarkable accuracy is achieved in the calculation of the energy of the ground state of the important biological molecule Leu1-zervamicin, whose crystal structure is known and used in the calculations.
| S-EPMC2538851 | biostudies-other