8,260 research outputs found
Turing Pattern Formation in Reaction-Cross-Diffusion Systems with a Bilayer Geometry
Conditions for self-organisation via Turing’s mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a spatially homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further
Linear Amplification in Nonequilibrium Turbulent Boundary Layers
Resolvent analysis is applied to nonequilibrium incompressible adverse pressure gradient (APG) turbulent boundary layers (TBL) and hypersonic boundary layers with high temperature real gas effects, including chemical nonequilibrium. Resolvent analysis is an equation-based, scale-dependent decomposition of the Navier Stokes equations, linearized about a known mean flow field. The decomposition identifies the optimal response and forcing modes, ranked by their linear amplification. To treat the nonequilibrium APG TBL, a biglobal resolvent analysis approach is used to account for the streamwise and wall-normal inhomogeneities in the streamwise developing flow. For the hypersonic boundary layer in chemical nonequilibrium, the resolvent analysis is constructed using a parallel flow assumption, incorporating Nâ‚‚, Oâ‚‚, NO, N, and O as a mixture of chemically reacting gases.
Biglobal resolvent analysis is first applied to the zero pressure gradient (ZPG) TBL. Scaling relationships are determined for the spanwise wavenumber and temporal frequency that admit self-similar resolvent modes in the inner layer, mesolayer, and outer layer regions of the ZPG TBL. The APG effects on the inner scaling of the biglobal modes are shown to diminish as their self-similarity improves with increased Reynolds number. An increase in APG strength is shown to increase the linear amplification of the large-scale biglobal modes in the outer region, similar to the energization of large scale modes observed in simulation. The linear amplification of these modes grows linearly with the APG history, measured as the streamwise averaged APG strength, and relates to a novel pressure-based velocity scale.
Resolvent analysis is then used to identify the length scales most affected by the high-temperature gas effects in hypersonic TBLs. It is shown that the high-temperature gas effects primarily affect modes localized near the peak mean temperature. Due to the chemical nonequilibrium effects, the modes can be linearly amplified through changes in chemical concentration, which have non-negligible effects on the higher order modes. Correlations in the components of the small-scale resolvent modes agree qualitatively with similar correlations in simulation data.
Finally, efficient strategies for resolvent analysis are presented. These include an algorithm to autonomously sample the large amplification regions using a Bayesian Optimization-like approach and a projection-based method to approximate resolvent analysis through a reduced eigenvalue problem, derived from calculus of variations.</p
Learning Koopman eigenfunctions of stochastic diffusions with optimal importance sampling and ISOKANN
The dominant eigenfunctions of the Koopman operator characterize the metastabilities and slow-timescale dynamics of stochastic diffusion processes. In the context of molecular dynamics and Markov state modeling, they allow for a description of the location and frequencies of rare transitions, which are hard to obtain by direct simulation alone. In this article, we reformulate the eigenproblem in terms of the ISOKANN framework, an iterative algorithm that learns the eigenfunctions by alternating between short burst simulations and a mixture of machine learning and classical numerics, which naturally leads to a proof of convergence. We furthermore show how the intermediate iterates can be used to reduce the sampling variance by importance sampling and optimal control (enhanced sampling), as well as to select locations for further training (adaptive sampling). We demonstrate the usage of our proposed method in experiments, increasing the approximation accuracy by several orders of magnitude
Exploring the validity of the complete case analysis for regression models with a right-censored covariate
Despite its drawbacks, the complete case analysis is commonly used in
regression models with missing covariates. Understanding when implementing
complete cases will lead to consistent parameter estimation is vital before
use. Here, our aim is to demonstrate when a complete case analysis is
appropriate for a nuanced type of missing covariate, the randomly
right-censored covariate. Across the censored covariate literature, different
assumptions are made to ensure a complete case analysis produces a consistent
estimator, which leads to confusion in practice. We make several contributions
to dispel this confusion. First, we summarize the language surrounding the
assumptions that lead to a consistent complete case estimator. Then, we show a
unidirectional hierarchical relationship between these assumptions, which leads
us to one sufficient assumption to consider before using a complete case
analysis. Lastly, we conduct a simulation study to illustrate the performance
of a complete case analysis with a right-censored covariate under different
censoring mechanism assumptions, and we demonstrate its use with a Huntington
disease data example
Geometric Methods for Spherical Data, with Applications to Cosmology
This survey is devoted to recent developments in the statistical analysis of
spherical data, with a view to applications in Cosmology. We will start from a
brief discussion of Cosmological questions and motivations, arguing that most
Cosmological observables are spherical random fields. Then, we will introduce
some mathematical background on spherical random fields, including spectral
representations and the construction of needlet and wavelet frames. We will
then focus on some specific issues, including tools and algorithms for map
reconstruction (\textit{i.e.}, separating the different physical components
which contribute to the observed field), geometric tools for testing the
assumptions of Gaussianity and isotropy, and multiple testing methods to detect
contamination in the field due to point sources. Although these tools are
introduced in the Cosmological context, they can be applied to other situations
dealing with spherical data. Finally, we will discuss more recent and
challenging issues such as the analysis of polarization data, which can be
viewed as realizations of random fields taking values in spin fiber bundles.Comment: 25 pages, 6 figure
Semiclassical Theory and the Koopman-van Hove Equation
The phase space Koopman-van Hove (KvH) equation can be derived from the
asymptotic semiclassical analysis of partial differential equations.
Semiclassical theory yields the Hamilton-Jacobi equation for the complex phase
factor and the transport equation for the amplitude. These two equations can be
combined to form a nonlinear semiclassical version of the KvH equation in
configuration space. Every solution of the configuration space KvH equation
satisfies both the semiclassical phase space KvH equation and the
Hamilton-Jacobi constraint. For configuration space solutions, this constraint
resolves the paradox that there are two different conserved densities in phase
space. For integrable systems, the KvH spectrum is the Cartesian product of a
classical and a semiclassical spectrum. If the classical spectrum is
eliminated, then, with the correct choice of Jeffreys-Wentzel-Kramers-Brillouin
(JWKB) matching conditions, the semiclassical spectrum satisfies the
Einstein-Brillouin-Keller quantization conditions which include the correction
due to the Maslov index. However, semiclassical analysis uses different choices
for boundary conditions, continuity requirements, and the domain of definition.
For example, use of the complex JWKB method allows for the treatment of
tunneling through the complexification of phase space. Finally, although KvH
wavefunctions include the possibility of interference effects, interference is
not observable when all observables are approximated as local operators on
phase space. Observing interference effects requires consideration of nonlocal
operations, e.g. through higher orders in the asymptotic theory.Comment: 49 pages, 10 figure
Phenomenological analysis of simple ion channel block in large populations of uncoupled cardiomyocytes
Current understanding of arrhythmia mechanisms and design of anti-arrhythmic drug therapies hinges on the assumption that myocytes from the same region of a single heart have similar, if not identical, action potential waveforms and drug responses. On the contrary, recent experiments reveal significant heterogeneity in uncoupled healthy myocytes both from different hearts as well as from identical regions within a single heart. In this work, a methodology is developed for quantifying the individual electrophysiological properties of large numbers of uncoupled cardiomyocytes under ion channel block in terms of the parameters values of a conceptual fast-slow model of electrical excitability. The approach is applied to a population of nearly 500 rabbit ventricular myocytes for which action potential duration (APD) before and after the application of the drug nifedipine was experimentally measured (Lachaud et al., 2022, Cardiovasc. Res.). To this end, drug action is represented by a multiplicative factor to an effective ion conductance, a closed form asymptotic expression for APD is derived and inverted to determine model parameters as functions of APD and ΔAPD (drug-induced change in APD) for each myocyte. Two free protocol-related quantities are calibrated to experiment using an adaptive-domain procedure based on an original assumption of optimal excitability. The explicit APD expression and the resulting set of model parameter values allow (a) direct evaluation of conditions necessary to maintain fixed APD or ΔAPD, (b) predictions of the proportion of cells remaining excitable after drug application, (c) predictions of stimulus period dependency and (d) predictions of dose-response curves, the latter being in agreement with additional experimental data
Critical growth of cerebral tissue in organoids: theory and experiments
We develop a Fokker-Planck theory of tissue growth with three types of cells
(symmetrically dividing, asymmetrically dividing and non-dividing) as main
agents to study the growth dynamics of human cerebral organoids. Fitting the
theory to lineage tracing data obtained in next generation sequencing
experiments, we show that the growth of cerebral organoids is a critical
process. We derive analytical expressions describing the time evolution of
clonal lineage sizes and show how power-law distributions arise in the limit of
long times due to the vanishing of a characteristic growth scale
Whitham modulation theory and two-phase instabilities for generalized nonlinear Schr\"{o}dinger equations with full dispersion
The generalized nonlinear Schr\"odinger equation with full dispersion (FDNLS)
is considered in the semiclassical regime. The Whitham modulation equations are
obtained for the FDNLS equation with general linear dispersion and a
generalized, local nonlinearity. Assuming the existence of a four-parameter
family of two-phase solutions, a multiple-scales approach yields a system of
four independent, first order, quasi-linear conservation laws of hydrodynamic
type that correspond to the slow evolution of the two wavenumbers, mass, and
momentum of modulated periodic traveling waves. The modulation equations are
further analyzed in the dispersionless and weakly nonlinear regimes. The
ill-posedness of the dispersionless equations corresponds to the classical
criterion for modulational instability (MI). For modulations of linear waves,
ill-posedness coincides with the generalized MI criterion, recently identified
by Amiranashvili and Tobisch (New J. Phys. 21 (2019)). A new instability index
is identified by the transition from real to complex characteristics for the
weakly nonlinear modulation equations. This instability is associated with
long-wavelength modulations of nonlinear two-phase wavetrains and can exist
even when the corresponding one-phase wavetrain is stable according to the
generalized MI criterion. Another interpretation is that, while infinitesimal
perturbations of a periodic wave may not grow, small but finite amplitude
perturbations may grow, hence this index identifies a nonlinear instability
mechanism for one-phase waves. Classifications of instability indices for
multiple FDNLS equations with higher order dispersion, including applications
to finite depth water waves and the discrete NLS equation are presented and
compared with direct numerical simulations.Comment: 26 pages, 7 figure
Cohomological Arithmetic Statistics for Principally Polarized Abelian Varieties over Finite Fields
There is a natural probability measure on the set of isomorphism classes of
principally polarized Abelian varieties of dimension over ,
weighted by the number of automorphisms. The distributions of the number of
-rational points are related to the cohomology of fiber powers of
the universal family of principally polarized Abelian varieties. To that end we
compute the cohomology for
using results of Eichler-Shimura and for using results of
Lee-Weintraub and Petersen, and we compute the compactly supported Euler
characteristics for
using results of Hain and conjectures of Bergstr\"om-Faber-van der Geer.
In each of these cases we identify the range in which the point counts
are polynomial in . Using results
of Borel and Grushevsky-Hulek-Tommasi on cohomological stability, we adapt
arguments of Achter-Erman-Kedlaya-Wood-Zureick-Brown to pose a conjecture about
the asymptotics of the point counts
in the limit .Comment: 29 pages, comments welcome
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