How collectively integrated are ecological communities?

Beyond abiotic conditions, do population dynamics mostly depend on a species' direct predators, preys and conspecifics? Or can indirect feedback that ripples across the whole community be equally important? Determining where ecological communities sit on the spectrum between these two characterizations requires a metric able to capture the difference between them. Here we show that the spectral radius of a community's interaction matrix provides such a metric, thus a measure of ecological collectivity, which is accessible from imperfect knowledge of biotic interactions and related to observable signatures. This measure of collectivity integrates existing approaches to complexity, interaction structure and indirect interactions. Our work thus provides an original perspective on the question of to what degree communities are more than loose collections of species or simple interaction motifs and explains when pragmatic reductionist approaches ought to suffice or fail when applied to ecological communities.

The unreasonable effectiveness of the total quasi-steady state approximation, and its limitations.

In this note, we discuss the range of parameters for which the total quasi-steady-state approximation of the Michaelis-Menten reaction mechanism holds validity. We challenge the prevalent notion that total quasi-steady-state approximation is "roughly valid" across all parameters, showing that its validity cannot be assumed, even roughly, across the entire parameter space - when the initial substrate concentration is high. On the positive side, we show that the linearized one-dimensional equation for total substrate is a valid approximation when the initial reduced substrate concentration s0/KM is small. Moreover, we obtain a precise picture of the long-term time course of both substrate and complex.

Integration of statistical and simulation analyses for ternary hybrid nanofluid over a moving surface with melting heat transfer.

In industrial and engineering fields including lamination, melt-spinning, continuous casting, and fiber spinning, the flow caused by a continually moving surface is significant. Therefore, the problem of ternary hybrid nanofluid flow over a moving surface is studied. This study explores the stability and statistical analyses of the magnetohydrodynamics (MHD) forced flow of the ternary hybrid nanofluid with melting heat transfer phenomena. The impacts of viscous dissipation, Joule heating, and thermal radiation are also included in the flow. Different fluids including ternary hybrid nanofluid, hybrid nanofluids, and nanofluids with base fluid ethylene glycol (EG) are examined and compared, where magnetite (Fe3O4) and silica (SiO2) are taken as the magnetic nanomaterials while silver (Ag) is chosen as the nonmagnetic nanomaterial. The skin friction coefficient and the local Nusselt number are estimated through regression analysis. By employing similarity transformations, the governing partial differential equations are converted into non-linear ordinary differential equations. Then, the least square method is applied to solve the equations analytically. Dual solutions are established in a particular range of moving parameterλ. Due to this, a stability test is implemented to find the stable solution by using the bvp4c function in MATLAB software. It is found that the first solution is the stable one while the second is unstable. The use of ternary hybrid nanomaterials improves the heat transport rate. The increasing values of the Eckert number enlarge the heat passage. The fluid velocity and temperature profiles for nonmagnetic nanomaterials are higher than that of magnetic nanomaterials. The uniqueness and originality of this study stems from the fact that, to the best of the authors' knowledge, it is the first to use this combination technique.

Derivation of Bose's Entropy Spectral Density from the Multiplicity of Energy Eigenvalues.

The modern textbook analysis of the thermal state of photons inside a three-dimensional reflective cavity is based on the three quantum numbers that characterize photon's energy eigenvalues coming out when the boundary conditions are imposed. The crucial passage from the quantum numbers to the continuous frequency is operated by introducing a three-dimensional continuous version of the three discrete quantum numbers, which leads to the energy spectral density and to the entropy spectral density. This standard analysis obscures the role of the multiplicity of energy eigenvalues associated to the same eigenfrequency. In this paper we review the past derivations of Bose's entropy spectral density and present a new analysis of energy spectral density and entropy spectral density based on the multiplicity of energy eigenvalues. Our analysis explicitly defines the eigenfrequency distribution of energy and entropy and uses it as a starting point for the passage from the discrete eigenfrequencies to the continuous frequency.

Sample size considerations for stepped wedge designs with subclusters.

The stepped wedge cluster randomized trial (SW-CRT) is an increasingly popular design for evaluating health service delivery or policy interventions. An essential consideration of this design is the need to account for both within-period and between-period correlations in sample size calculations. Especially when embedded in health care delivery systems, many SW-CRTs may have subclusters nested in clusters, within which outcomes are collected longitudinally. However, existing sample size methods that account for between-period correlations have not allowed for multiple levels of clustering. We present computationally efficient sample size procedures that properly differentiate within-period and between-period intracluster correlation coefficients in SW-CRTs in the presence of subclusters. We introduce an extended block exchangeable correlation matrix to characterize the complex dependencies of outcomes within clusters. For Gaussian outcomes, we derive a closed-form sample size expression that depends on the correlation structure only through two eigenvalues of the extended block exchangeable correlation structure. For non-Gaussian outcomes, we present a generic sample size algorithm based on linearization and elucidate simplifications under canonical link functions. For example, we show that the approximate sample size formula under a logistic linear mixed model depends on three eigenvalues of the extended block exchangeable correlation matrix. We provide an extension to accommodate unequal cluster sizes and validate the proposed methods via simulations. Finally, we illustrate our methods in two real SW-CRTs with subclusters.

Adaptation in a heterogeneous environment II: to be three or not to be.

We propose a model to describe the adaptation of a phenotypically structured population in a H-patch environment connected by migration, with each patch associated with a different phenotypic optimum, and we perform a rigorous mathematical analysis of this model. We show that the large-time behaviour of the solution (persistence or extinction) depends on the sign of a principal eigenvalue, [Formula: see text], and we study the dependency of [Formula: see text] with respect to H. This analysis sheds new light on the effect of increasing the number of patches on the persistence of a population, which has implications in agroecology and for understanding zoonoses; in such cases we consider a pathogenic population and the patches correspond to different host species. The occurrence of a springboard effect, where the addition of a patch contributes to persistence, or on the contrary the emergence of a detrimental effect by increasing the number of patches on the persistence, depends in a rather complex way on the respective positions in the phenotypic space of the optimal phenotypes associated with each patch. From a mathematical point of view, an important part of the difficulty in dealing with [Formula: see text], compared to [Formula: see text] or [Formula: see text], comes from the lack of symmetry. Our results, which are based on a fixed point theorem, comparison principles, integral estimates, variational arguments, rearrangement techniques, and numerical simulations, provide a better understanding of these dependencies. In particular, we propose a precise characterisation of the situations where the addition of a third patch increases or decreases the chances of persistence, compared to a situation with only two patches.

Fitting parameters and therapies of ODE tumor models with senescence and immune system.

This work is devoted to introduce and study two quasispecies nonlinear ODE systems that model the behavior of tumor cell populations organized in different states. In the first model, replicative, senescent, extended lifespan, immortal and tumor cells are considered, while the second also includes immune cells. We fit the parameters regulating the transmission between states in order to approximate the outcomes of the models to a real progressive tumor invasion. After that, we study the identifiability of the fitted parameters, by using two sensitivity analysis methods. Then, we show that an adequate reduced fitting process (only accounting to the most identifiable parameters) gives similar results but saving computational cost. Three different therapies are introduced in the models to shrink (progressively in time) the tumor, while the replicative and senescent cells invade. Each therapy is identified to a dimensionless parameter. Then, we make a fitting process of the therapies' parameters, in various scenarios depending on the initial tumor according to the time when the therapies started. We conclude that, although the optimal combination of therapies depends on the size of initial tumor, the most efficient therapy is the reinforcement of the immune system. Finally, an identifiability analysis allows us to detect a limitation in the therapy outcomes. In fact, perturbing the optimal combination of therapies under an appropriate therapeutic vector produces virtually the same results.

Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems.

The solution of the eigenvalue problem in bounded domains with planar and cylindrical stratification is a necessary preliminary task for the construction of modal solutions to canonical problems with discontinuities. The computation of the complex eigenvalue spectrum must be very accurate since losing or misplacing one of the thereto linked modes will have an important impact on the field solution. The approach followed in a number of previous works is to construct the corresponding transcendental equation and locate its roots in the complex plane using the Newton-Raphson method or Cauchy-integral-based techniques. Nevertheless, this approach is cumbersome, and its numerical stability decreases dramatically with the number of layers. An alternative, approach consists in the numerical evaluation of the matrix eigenvalues for the weak formulation for the respective 1D Sturm-Liouville problem using linear algebra tools. An arbitrary number of layers can thus be easily and robustly treated, with continuous material gradients being a limiting case. Although this approach is often used in high frequency studies involving wave propagation, this is the first time that has been used for the induction problem arising in an eddy current inspection situation. The developed method is implemented in Matlab and is used to deal with the following problems: magnetic material with a hole, a magnetic cylinder, and a magnetic ring. In all the conducted tests, the results are obtained in a very short time, without missing a single eigenvalue.

Moisture Detection in Tree Trunks in Semiarid Lands Using Low-Cost Non-Invasive Capacitive Sensors with Statistical Based Anomaly Detection Approach.

This paper focuses on building a non-invasive, low-cost sensor that can be fitted over tree trunks growing in a semiarid land environment. It also proposes a new definition that characterizes tree trunks' water retention capabilities mathematically. The designed sensor measures the variations in capacitance across its probes. It uses amplification and filter stages to smooth the readings, requires little power, and is operational over a 100 kHz frequency. The sensor sends data via a Long Range (LoRa) transceiver through a gateway to a processing unit. Field experiments showed that the system provides accurate readings of the moisture content. As the sensors are non-invasive, they can be fitted to branches and trunks of various sizes without altering the structure of the wood tissue. Results show that the moisture content in tree trunks increases exponentially with respect to the measured capacitance and reflects the distinct differences between different tree types. Data of known healthy trees and unhealthy trees and defective sensor readings have been collected and analysed statistically to show how anomalies in sensor reading baseds on eigenvectors and eigenvalues of the fitted curve coefficient matrix can be detected.

On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations.

In this work, the null controllability problem for a linear system in ℓ2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with λ∈ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤- 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤- 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered ℓ∞ is not asymptotically stable if λ = - 1.

Theoretical analysis of principal components in an umbrella model of intraspecific evolution.

Principal component analysis (PCA) is one of the most frequently-used approach to describe population structure from multilocus genotype data. Regarding geographic range expansions of modern humans, interpretations of PCA have, however, been questioned, as there is uncertainty about the wave-like patterns that have been observed in principal components. It has indeed been argued that wave-like patterns are mathematical artifacts that arise generally when PCA is applied to data in which genetic differentiation increases with geographic distance. Here, we present an alternative theory for the observation of wave-like patterns in PCA. We study a coalescent model - the umbrella model - for the diffusion of genetic variants. The model is based on genetic drift without any particular geographical structure. In the umbrella model, splits from an ancestral population occur almost continuously in time, giving birth to small daughter populations at a regular pace. Our results provide detailed mathematical descriptions of eigenvalues and eigenvectors for the PCA of sampled genomic sequences under the model. When variants uniquely represented in the sample are removed, the PCA eigenvectors are defined as cosine functions of increasing periodicity, reproducing wave-like patterns observed in equilibrium isolation-by-distance models. Including singleton variants in the analysis, the eigenvectors corresponding to the largest eigenvalues exhibit complex wave shapes. The accuracy of our predictions is further investigated with coalescent simulations. Our analysis supports the hypothesis that highly structured wave-like patterns could arise from genetic drift only, and may not always be artificial outcomes of spatially structured data. Genomic data related to the peopling of the Americas are reanalyzed in the light of our new theory.

CANONICAL THRESHOLDING FOR NON-SPARSE HIGH-DIMENSIONAL LINEAR REGRESSION.

We consider a high-dimensional linear regression problem. Unlike many papers on the topic, we do not require sparsity of the regression coefficients; instead, our main structural assumption is a decay of eigenvalues of the covariance matrix of the data. We propose a new family of estimators, called the canonical thresholding estimators, which pick largest regression coefficients in the canonical form. The estimators admit an explicit form and can be linked to LASSO and Principal Component Regression (PCR). A theoretical analysis for both fixed design and random design settings is provided. Obtained bounds on the mean squared error and the prediction error of a specific estimator from the family allow to clearly state sufficient conditions on the decay of eigenvalues to ensure convergence. In addition, we promote the use of the relative errors, strongly linked with the out-of-sample R 2. The study of these relative errors leads to a new concept of joint effective dimension, which incorporates the covariance of the data and the regression coefficients simultaneously, and describes the complexity of a linear regression problem. Some minimax lower bounds are established to showcase the optimality of our procedure. Numerical simulations confirm good performance of the proposed estimators compared to the previously developed methods.

Global and local shape features of the hippocampus based on Laplace-Beltrami eigenvalues and eigenfunctions: a potential application in the lateralization of temporal lobe epilepsy.

Using magnetic resonance (MR) images to evaluate changes in the shape of the hippocampus has been an active research topic. This paper presents a new shape analysis approach to quantify and visualize deformations of the hippocampus in epilepsy. The proposed method is based on Laplace-Beltrami (LB) eigenvalues and eigenfunctions as isometric invariant shape features, and thus, the procedure does not require any image registration. In addition to the LB-based shape features, total hippocampal volume and surface area are calculated using manually segmented images. Theses shape and volumetric descriptors are used to distinguish the patients with temporal lobe epilepsy (TLE) (N = 55) from healthy control subjects (N = 12, age = 32.2 ± 9.1, sex (M/F) = 6/6) and patients with right TLE (N = 26, age = 45.1 ± 11.0, sex (M/F) = 9/17) from left TLE (N = 29, age = 45.4 ± 11.9, sex (M/F) = 10/19). Experimental results illustrate the usefulness of the proposed approach for the diagnosis and lateralization of TLE with 93.0% and 86.4% of the cases, respectively. Moreover, the proposed method outperforms the volumetric analysis in terms of both sensitivity (94.9% vs. 88.1%) and specificity (83.3% vs. 50.0%) of the lateralization. The analysis of local hippocampal thickness variations suggests significant deformation in both ipsilateral and contralateral hippocampi of epileptic patients, while there were no differences between right and left hippocampi in controls. It is anticipated that the proposed method could be advantageous in the presurgical evaluation of patients with drug-resistant epilepsy; however, further validation of the method using a larger dataset is required.

Stability-instability transition in tripartite merged ecological networks.

Although ecological networks are typically constructed based on a single type of interaction, e.g. trophic interactions in a food web, a more complete picture of ecosystem composition and functioning arises from merging networks of multiple interaction types. In this work, we consider tripartite networks constructed by merging two bipartite networks, one mutualistic and one antagonistic. Taking the interactions within each sub-network to be distributed randomly, we consider the stability of the dynamics of the network based on the spectrum of its community matrix. In the asymptotic limit of a large number of species, we show that the spectrum undergoes an eigenvalue phase transition, which leads to an abrupt destabilisation of the network as the ratio of mutualists to antagonists is increased. We also derive results that show how this transition is manifest in networks of finite size, as well as when disorder is introduced in the segregation of the two interaction types. Our random-matrix results will serve as a baseline for understanding the behaviour of merged networks with more realistic structures and/or more detailed dynamics.

Power considerations for generalized estimating equations analyses of four-level cluster randomized trials.

In this article, we develop methods for sample size and power calculations in four-level intervention studies when intervention assignment is carried out at any level, with a particular focus on cluster randomized trials (CRTs). CRTs involving four levels are becoming popular in healthcare research, where the effects are measured, for example, from evaluations (level 1) within participants (level 2) in divisions (level 3) that are nested in clusters (level 4). In such multilevel CRTs, we consider three types of intraclass correlations between different evaluations to account for such clustering: that of the same participant, that of different participants from the same division, and that of different participants from different divisions in the same cluster. Assuming arbitrary link and variance functions, with the proposed correlation structure as the true correlation structure, closed-form sample size formulas for randomization carried out at any level (including individually randomized trials within a four-level clustered structure) are derived based on the generalized estimating equations approach using the model-based variance and using the sandwich variance with an independence working correlation matrix. We demonstrate that empirical power corresponds well with that predicted by the proposed method for as few as eight clusters, when data are analyzed using the matrix-adjusted estimating equations for the correlation parameters with a bias-corrected sandwich variance estimator, under both balanced and unbalanced designs.

Extreme Eigenvalues and the Emerging Outlier in Rank-One Non-Hermitian Deformations of the Gaussian Unitary Ensemble.

Complex eigenvalues of random matrices J=GUE+iγdiag(1,0,…,0) provide the simplest model for studying resonances in wave scattering from a quantum chaotic system via a single open channel. It is known that in the limit of large matrix dimensions N≫1 the eigenvalue density of J undergoes an abrupt restructuring at γ=1, the critical threshold beyond which a single eigenvalue outlier ("broad resonance") appears. We provide a detailed description of this restructuring transition, including the scaling with N of the width of the critical region about the outlier threshold γ=1 and the associated scaling for the real parts ("resonance positions") and imaginary parts ("resonance widths") of the eigenvalues which are farthest away from the real axis. In the critical regime we determine the density of such extreme eigenvalues, and show how the outlier gradually separates itself from the rest of the extreme eigenvalues. Finally, we describe the fluctuations in the height of the eigenvalue outlier for large but finite N in terms of the associated large deviation function.

Generalized Solution of Inverse Problem for Ising Connection Matrix on d-Dimensional Hypercubic Lattice.

We analyze a connection matrix of a d-dimensional Ising system and solve the inverse problem, restoring the constants of interaction between spins, based on the known spectrum of its eigenvalues. When the boundary conditions are periodic, we can account for interactions between spins that are arbitrarily far. In the case of the free boundary conditions, we have to restrict ourselves with interactions between the given spin and the spins of the first d coordination spheres.

On Positive Definite Kernels of Integral Operators Corresponding to the Boundary Value Problems for Fractional Differential Equations.

In the spectral analysis of operators associated with Sturm-Liouville-type boundary value problems for fractional differential equations, the problem of positive definiteness or the problem of Hermitian nonnegativity of the corresponding kernels plays an important role. The present paper is mainly devoted to this problem. It should be noted that the operators under study are non-self-adjoint, their spectral structure is not well investigated. In this paper we use various methods to prove the Hermitian non-negativity of the studied kernels; in particular, a study of matrices that approximate the Green's function of the boundary value problem for a differential equation of fractional order is carried out. Using the well-known Livshits theorem, it is shown that the system of eigenfunctions of considered operator is complete in the space L2(0,1). Generally speaking, it should be noted that this very important problem turned out to be very difficult.

SEIR order parameters and eigenvectors of the three stages of completed COVID-19 epidemics: with an illustration for Thailand January to May 2020.

By end of October 2020, the COVID-19 pandemic has taken a tragic toll of 1150 000 lives and this number is expected to increase. Despite the pandemic is raging in most parts of the world, in a few countries COVID-19 epidemics subsided due to successful implementations of intervention measures. A unifying perspective of the beginnings, middle stages, and endings of such completed COVID-19 epidemics is developed based on the order parameter and eigenvalue concepts of nonlinear physics, in general, and synergetics, in particular. To this end, a standard susceptible-exposed-infected-recovered (SEIR) epidemiological model is used. It is shown that COVID-19 epidemic outbreaks follow a suitably defined SEIR order parameter. Intervention measures switch the eigenvalue of the order parameter from a positive to a negative value, and in doing so, stabilize the COVID-19 disease-free state. The subsiding of COVID-19 epidemics eventually follows the remnant of the order parameter of the infection dynamical system. These considerations are illustrated for the COVID-19 epidemic in Thailand from January to May 2020. The decay of effective contact rates throughout the three epidemic stages is demonstrated. Evidence for the sign-switching of the dominant eigenvalue is given and the order parameter and its stage-3 remnant are identified. The presumed impacts of interventions measures implemented in Thailand are discussed in this context.

Robust Coarse-to-Fine Registration Scheme for Mobile Laser Scanner Point Clouds Using Multiscale Eigenvalue Statistic-Based Descriptor.

To overcome the drawbacks of pairwise registration for mobile laser scanner (MLS) point clouds, such as difficulty in searching the corresponding points and inaccuracy registration matrix, a robust coarse-to-fine registration method is proposed to align different frames of MLS point clouds into a common coordinate system. The method identifies the correct corresponding point pairs from the source and target point clouds, and then calculates the transform matrix. First, the performance of a multiscale eigenvalue statistic-based descriptor with different combinations of parameters is evaluated to identify the optimal combination. Second, based on the geometric distribution of points in the neighborhood of the keypoint, a weighted covariance matrix is constructed, by which the multiscale eigenvalues are calculated as the feature description language. Third, the corresponding points between the source and target point clouds are estimated in the feature space, and the incorrect ones are eliminated via a geometric consistency constraint. Finally, the estimated corresponding point pairs are used for coarse registration. The value of coarse registration is regarded as the initial value for the iterative closest point algorithm. Subsequently, the final fine registration result is obtained. The results of the registration experiments with Autonomous Systems Lab (ASL) Datasets show that the proposed method can accurately align MLS point clouds in different frames and outperform the comparative methods.