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We consider two models of the predator–prey community with prey-taxis. Both models take into account the capability of the predators to respond to prey density gradients and also to one more signal, the production of which occurs independently of the community state (such a signal can be due to the spatiotemporal inhomogeneity of the environment arising for natural or artificial reasons). We call such a signal external. The models differ to one another through the description of their responses: the first one employs the Patlak–Keller–Segel law for both responses, and the second one employs Cattaneo's model of heat transfer for both responses following to Dolak and Hillen. Assuming a short-wave external signal, we construct the complete asymptotic expansions of the short-wave solutions to both models. We use them to examine the effect of the short-wave signal on the formation of spatiotemporal patterns. We do so by comparing the stability of equilibria with no signal to that of the quasi-equilibria forced by the external signal. Such an approach refers back to Kapitza's theory for an upside-down pendulum. The overall conclusion is that the external signal is likely not capable of creating the instability domain in the parametric space from nothing but it can substantially widen the one that is non-empty with no signal. [ABSTRACT FROM AUTHOR]

The small mass limit is derived for a McKean-Vlasov equation with state-dependent friction which is a matrix-valued function. By applying the averaging approach to a non-autonomous slow-fast system with the microscopic and macroscopic scales, the convergence in distribution is obtained. [ABSTRACT FROM AUTHOR]

We study a type of perturbed polynomial Hamiltonian system in 1:1 resonance. The perturbation consists of a homogeneous quartic potential invariant by rotations of $ \pi /2 $ π / 2 radians. The existence of periodic solutions is established using reduction and averaging theories. The different types of periodic solutions, linear stability, and bifurcation curves are characterized in terms of the parameters. Finally, some choreography of bifurcations are obtained, showing in detail the evolution of the phase flow. [ABSTRACT FROM AUTHOR]

Due to their limited resolution, numerical ocean models need to be interpreted as representing filtered or averaged equations. How to interpret models in terms of formally averaged equations, however, is not always clear, particularly in the case of hybrid or generalized vertical coordinate models, which limits our ability to interpret the model results and to develop parameterizations for the unresolved eddy contributions. We here derive the averaged hydrostatic Boussinesq equations in generalized vertical coordinates for an arbitrary thickness‐weighted average. We then consider various special cases and discuss the extent to which the averaged equations are consistent with existing ocean model formulations. As previously discussed, the momentum equations in existing depth‐coordinate models are best interpreted as representing Eulerian averages (i.e., averages taken at fixed depth), while the tracer equations can be interpreted as either Eulerian or thickness‐weighted isopycnal averages. Instead we find that no averaging is fully consistent with existing formulations of the parameterizations in semi‐Lagrangian discretizations of generalized vertical coordinate ocean models such as MOM6. A coordinate‐following average would require "coordinate‐aware" parameterizations that can account for the changing nature of the eddy terms as the coordinate changes. Alternatively, the model variables can be interpreted as representing either Eulerian or (thickness‐weighted) isopycnal averages, independent of the model coordinate that is being used for the numerical discretization. Existing parameterizations in generalized vertical coordinate models, however, are not always consistent with either of these interpretations, which, respectively, would require a three‐dimensional divergence‐free eddy tracer advection or a form‐stress parameterization in the momentum equations. Plain Language Summary: Numerical ocean models represent continuous three‐dimensional physical fields using discrete data points and hence cannot adequately represent variability at all scales. Instead model variables need to be interpreted as filtered versions of the continuous physical fields. We here derive the evolution equations for horizontally filtered fields, where the horizontal filtering follows arbitrary surfaces, with the only requirement being that the surfaces do not fold over (i.e., they are iso‐surfaces of a field that is monotonic in depth). The equations for the filtered variables are formulated using an arbitrary vertical coordinate system, thus making them applicable to a wide range of different numerical ocean models. We then consider different physically motivated choices for the averaging surfaces and express the equations using different common choices for the vertical coordinate system. We conclude by discussing which equation sets are consistent with the equations solved by existing models and/or which modifications are needed to achieve consistency. The results have important implications for ocean model development, because the models need to include "parameterizations" to represent the effect of motions that have been removed by the filter. The formulation of these parameterizations needs to be consistent with the filtering operation that was assumed to obtain the model equations. Key Points: We derive and discuss the arbitrarily averaged hydrostatic Boussinesq ocean equations in generalized vertical coordinatesKnown results for Eulerian‐ and isopycnal‐mean equations in depth and isopycnal coordinates are recovered as special casesThe properties of eddy terms depend on the averaging choice, and are not always consistent with current parameterization implementations [ABSTRACT FROM AUTHOR]

We consider coupled slow-fast stochastic processes, where the averaged slow motion is given by a two-dimensional Hamiltonian system with multiple critical points. On a proper time scale, the evolution of the first integral converges to a diffusion process on the corresponding Reeb graph, with certain gluing conditions specified at the interior vertices, as in the case of additive white noise perturbations of Hamiltonian systems considered by M. Freidlin and A. Wentzell. The current paper provides the first result where the motion on a graph and the corresponding gluing conditions appear due to the averaging of a slow-fast system, with a Hamiltonian structure, on a large time scale. The result allows one to consider, for instance, long-time diffusion approximation for an oscillator with a potential with more than one well. [ABSTRACT FROM AUTHOR]

Time-decaying perturbations of nonlinear oscillatory systems in the plane are considered. It is assumed that the unperturbed systems are non-isochronous and the perturbations oscillate with an asymptotically constant frequency. Resonance effects and long-term asymptotic regimes for solutions are investigated. In particular, the emergence of stable states close to periodic ones is discussed. By combining the averaging technique, stability analysis, and constructing suitable Lyapunov functions, the conditions on perturbations are described that guarantee the existence and stability of the phase-locking regime with a resonant amplitude. The results obtained are applied to the perturbed Duffing oscillator. [ABSTRACT FROM AUTHOR]

At present, non-conservative perturbations of two-dimensional nonlinear Hamiltonian systems have been studied quite fully. The purpose of the study is to generalize this theory to the three-dimensional case, when the unperturbed system is nonlinear, integrable and has a region filled with closed phase trajectories. In this paper, autonomous perturbations are considered and the main attention is paid to the problem of limit cycles. Methods. The study is based on the construction of special coordinates in which the variables are divided into two slow and one fast, and in the first approximation with respect to a small parameter the equations for the slow variables are separated. Results. It is shown that hyperbolic equilibrium states of a truncated system determine closed phase trajectories, in the vicinity of which cycles appear under the perturbation. Conclusion. Thus, the problem is reduced to the study of solutions of the “generating” system of two algebraic or transcendental equations, similar to the generating Poincare–Pontryagin equation for two-dimensional systems. As examples, we considere a three-imensional van der Pol type system and the Lorentz system in the case of large Rayleigh numbers.

Abstract This paper analyzes effects of the mutual presentation of weak and strong arguments. Departing from the prevalent “the-more-the-better” heuristic, our research scrutinizes whether the inclusion of weak arguments enhances or diminishes the persuasive impact of strong arguments. Leveraging insights from judgment formation literature, we conducted four experimental studies on political and health-related topics to unravel whether the presenting weak arguments strengthens the persuasive effect of a strong argument (adding) or actually weakens this persuasive effect (averaging). The results show that providing supporting arguments of moderate strength along with a strong argument increases persuasion, representing an additive pattern. However, presenting weak supporting arguments along with a strong argument reduces the persuasive effect of the strong argument, representing an averaging pattern. Exposure to weak arguments diminishes the strength of strong ones, suggesting the omission of weak arguments. These findings underscore the vital role of strategically selecting arguments to optimize persuasion across disciplines.

In this paper, we investigate the asymptotic behavior of inertial dynamics with dry friction within the context of a Hilbert framework for convex differentiable optimization. Our study focuses on a doubly nonlinear first-order evolution inclusion that encompasses two potentials. In our analysis, we specifically focus on two main components: the differentiable function f that needs to be minimized, which influences the system's state through its gradient, and the nonsmooth dry friction potential denoted as φ = r ‖ · ‖ . It's important to note that the dry friction term acts on a linear combination of the velocity vector and the gradient of f. Consequently, any stationary point in our system corresponds to a critical point of f, unlike the case where only the velocity vector is involved in the dry friction term, resulting in an approximate critical point of f. To emphasize the crucial role of ∇ f (x) , we also explore the dual formulation of this dynamic, which possesses a Riemannian gradient structure. To address these dynamics, we employ the recently developed generic acceleration approach by Attouch, Bot, and Nguyen. This approach involves the time scaling of a continuous first-order differential equation, followed by the application of the method of averaging. By applying this methodology, we derive fast convergence results for second-order time-evolution systems with dry friction, asymptotically vanishing viscous damping, and implicit Hessian-driven damping. [ABSTRACT FROM AUTHOR]

This paper analyzes effects of the mutual presentation of weak and strong arguments. Departing from the prevalent “the-more-the-better” heuristic, our research scrutinizes whether the inclusion of weak arguments enhances or diminishes the persuasive impact of strong arguments. Leveraging insights from judgment formation literature, we conducted four experimental studies on political and health-related topics to unravel whether the presenting weak arguments strengthens the persuasive effect of a strong argument (adding) or actually weakens this persuasive effect (averaging). The results show that providing supporting arguments of moderate strength along with a strong argument increases persuasion, representing an additive pattern. However, presenting weak supporting arguments along with a strong argument reduces the persuasive effect of the strong argument, representing an averaging pattern. Exposure to weak arguments diminishes the strength of strong ones, suggesting the omission of weak arguments. These findings underscore the vital role of strategically selecting arguments to optimize persuasion across disciplines. [ABSTRACT FROM AUTHOR]

We establish global-in-time averaging for the L 2 -critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in  L 2 . [ABSTRACT FROM AUTHOR]

The present study is an extension of the work done by Peddle, Haut, and Wingate [SIAM J. Sci. Comput., 41 (2019), pp. A3476--A3497] and Haut and Wingate [SIAM J. Sci. Comput., 36 (2014), pp. A693--A713], where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples. [ABSTRACT FROM AUTHOR]

G-equations are popular level set Hamilton–Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly nonconvex and nonlinear. In the absence of coercivity and convexity, the PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection. [ABSTRACT FROM AUTHOR]

A typical model for a gyrating engine consists of an inertial wheel powered by an energy source that generates an angle-dependent torque. Examples of such engines include a pendulum with an externally applied torque, Stirling engines, and the Brownian gyrating engine. Variations in the torque are averaged out by the inertia of the system to produce limit cycle oscillations. While torque generating mechanisms are also ubiquitous in the biological world, where they typically feed on chemical gradients, inertia is not a property that one naturally associates with such processes. In the present work, seeking ways to dispense of the need for inertial effects, we study an inertia-less concept where the combined effect of coupled torque-producing components averages out variations in the ambient potential and helps overcome dissipative forces to allow sustained operation for vanishingly small inertia. We exemplify this inertia-less concept through analysis of two of the aforementioned engines, the Stirling engine, and the Brownian gyrating engine. An analogous principle may be sought in biomolecular processes as well as in modern-day technological engines, where for the latter, the coupled torque-producing components reduce vibrations that stem from the variability of the generated torque.

Summary: In this paper, we investigate the stabilizing effect of an extremum seeking control law on a class of underactuated Lagrangian control systems with symmetry. Our study is motivated by the problem of attitude optimization for satellites with reaction wheels. The goal is to minimize the value of an analytically unknown configuration‐dependent objective function without being reliant on measurements of the current configuration and velocity. This work extends our previous work on extremum seeking control for fully actuated mechanical systems on Lie groups in the absence of dissipation. We show that the earlier method for fully actuated systems can also be successfully applied to a certain class of underactuated systems, which are controlled by internal momentum exchange devices (reaction wheels) so that the total momentum is conserved. Conservation of momentum allows us to reduce the control system to a system on a smaller state manifold. The reduced control system is not of Lagrangian form but fully actuated. For the reduced control system we prove that the extremum seeking method leads to practical uniform asymptotic stability. We illustrate our findings by the example of a rigid body with internal rotors. [ABSTRACT FROM AUTHOR]

Abstract: We derive a new analytical solution for the first-order, short-periodic perturbations due to planetary oblateness and systematically compare our results to the classical Brouwer–Lyddane transformation. Our approach is based on the Milankovitch vectorial elements and is free of all the mathematical singularities. Being a non-canonical set, our derivation follows the scheme used by Kozai in his oblateness solution. We adopt the mean longitude as the fast variable and present a compact power-series solution in eccentricity for its short-periodic perturbations that relies on Hansen’s coefficients. We also use a numerical averaging algorithm based on the fast-Fourier transform to further validate our new mean-to-osculating and inverse transformations. This technique constitutes a new approach for deriving short-periodic corrections and exhibits performance that are comparable to other existing and well-established theories, with the advantage that it can be potentially extended to modeling non-conservative orbit perturbations.

Abstract Background This study aimed to develop a deep learning (DL) algorithm that enhances the quality of a single-frame enface OCTA scan to make it comparable to 4-frame averaged scan without the need for the repeated acquisitions required for averaging. Methods Each of the healthy eyes and eyes from diabetic subjects that were prospectively enrolled in this cross-sectional study underwent four repeated 6 × 6 mm macular scans (PLEX Elite 9000 SS-OCT), and the repeated scans of each eye were co-registered to produce 4-frame averages. This prospective dataset of original (single-frame) enface scans and their corresponding averaged scans was divided into a training dataset and a validation dataset. In the training dataset, a DL algorithm (named pseudoaveraging) was trained using original scans as input and 4-frame averages as target. In the validation dataset, the pseudoaveraging algorithm was applied to single-frame scans to produce pseudoaveraged scans, and the single-frame and its corresponding averaged and pseudoaveraged scans were all qualitatively compared. In a separate retrospectively collected dataset of single-frame scans from eyes of diabetic subjects, the DL algorithm was applied, and the produced pseudoaveraged scan was qualitatively compared against its corresponding original. Results This study included 39 eyes that comprised the prospective dataset (split into 5 eyes for training and 34 eyes for validating the DL algorithm), and 105 eyes that comprised the retrospective test dataset. Of the total 144 study eyes, 58% had any level of diabetic retinopathy (with and without diabetic macular edema), and the rest were from healthy eyes or eyes of diabetic subjects but without diabetic retinopathy and without macular edema. Grading results in the validation dataset showed that the pseudoaveraged enface scan ranked best in overall scan quality, background noise reduction, and visibility of microaneurysms (p

Introduction: Understanding the neural code has been one of the central aims of neuroscience research for decades. Spikes are commonly referred to as the units of information transfer, but multi-unit activity (MUA) recordings are routinely analyzed in aggregate forms such as binned spike counts, peristimulus time histograms, firing rates, or population codes. Various forms of averaging also occur in the brain, from the spatial averaging of spikes within dendritic trees to their temporal averaging through synaptic dynamics. However, how these forms of averaging are related to each other or to the spatial and temporal units of information representation within the neural code has remained poorly understood. Materials and methods: In this work we developed NeuroPixelHD, a symbolic hyperdimensional model of MUA, and used it to decode the spatial location and identity of static images shown to n = 9 mice in the Allen Institute Visual Coding—NeuroPixels dataset from large-scale MUA recordings. We parametrically varied the spatial and temporal resolutions of the MUA data provided to the model, and compared its resulting decoding accuracy. Results: For almost all subjects, we found 125ms temporal resolution to maximize decoding accuracy for both the spatial location of Gabor patches (81 classes for patches presented over a 9×9 grid) as well as the identity of natural images (118 classes corresponding to 118 images) across the whole brain. This optimal temporal resolution nevertheless varied greatly between dierent regions, followed a sensory-associate hierarchy, and was significantly modulated by the central frequency of theta-band oscillations across dierent regions. Spatially, the optimal resolution was at either of two mesoscale levels for almost all mice: the area level, where the spiking activity of all neurons within each brain area are combined, and the population level, where neuronal spikes within each area are combined across fast spiking (putatively inhibitory) and regular spiking (putatively excitatory) neurons, respectively. We also observed an expected interplay between optimal spatial and temporal resolutions, whereby increasing the amount of averaging across one dimension (space or time) decreases the amount of averaging that is optimal across the other dimension, and vice versa. Discussion: Our findings corroborate existing empirical practices of spatiotemporal binning and averaging in MUA data analysis, and provide a rigorous computational framework for optimizing the level of such aggregations. Our findings can also synthesize these empirical practices. [ABSTRACT FROM AUTHOR]

An autonomous system of ordinary differential equations describing nonlinear oscillations on the plane is considered. The influence of non-autonomous perturbations decaying at infinity in time is investigated. Such systems are usually called asymptotically autonomous and arise, in particular, as a result of the reduction of multidimensional autonomous and non-autonomous systems. In this paper, we consider a special class of oscillatory perturbations that satisfy the non-resonance condition and do not vanish at the equilibrium of the unperturbed system. We construct a near-identity transformation that averages the system in the first asymptotic terms at infinity in time, and study the structure of the simplified equations. Under some natural assumptions, we describe possible long-term asymptotic regimes for solutions and analyse their stability by constructing Lyapunov functions. In particular, we show how oscillatory terms of perturbations can break the stability of the system and discuss conditions under which the perturbed system behaves like the corresponding unperturbed autonomous system or has new stable regimes. The results obtained are applied to some examples of oscillatory systems with time-decaying oscillatory perturbations. [ABSTRACT FROM AUTHOR]

This paper develops an averaging approach on macroscopic scales to derive Smoluchowski–Kramers approximation for a Langevin equation with state dependent friction in d-dimensional space. In this approach we couple the microscopic dynamics to the macroscopic scales. The weak convergence rate is also presented. [ABSTRACT FROM AUTHOR]

Recent advances in protein structure prediction, driven by AlphaFold 2 and machine learning, demonstrate proficiency in static structures but encounter challenges in capturing essential dynamic features crucial for understanding biological function. In this context, homology-based modeling emerges as a cost-effective and computationally efficient alternative. The MODELLER (version 10.5, accessed on 30 November 2023) algorithm can be harnessed for this purpose since it computes intermediate models during simulated annealing, enabling the exploration of attainable configurational states and energies while minimizing its objective function. There have been a few attempts to date to improve the models generated by its algorithm, and in particular, there is no literature regarding the implementation of an averaging procedure involving the intermediate models in the MODELLER algorithm. In this study, we examined MODELLER's output using 225 target-template pairs, extracting the best representatives of intermediate models. Applying an averaging procedure to the selected intermediate structures based on statistical potentials, we aimed to determine: (1) whether averaging improves the quality of structural models during the building phase; (2) if ranking by statistical potentials reliably selects the best models, leading to improved final model quality; (3) whether using a single template versus multiple templates affects the averaging approach; (4) whether the "ensemble" nature of the MODELLER building phase can be harnessed to capture low-energy conformations in holo structures modeling. Our findings indicate that while improvements typically fall short of a few decimal points in the model evaluation metric, a notable fraction of configurations exhibit slightly higher similarity to the native structure than MODELLER's proposed final model. The averaging-building procedure proves particularly beneficial in (1) regions of low sequence identity between the target and template(s), the most challenging aspect of homology modeling; (2) holo protein conformations generation, an area in which MODELLER and related tools usually fall short of the expected performance. [ABSTRACT FROM AUTHOR]

We consider populations with time-varying growth rates living in sinks. Each population, when isolated, would become extinct. Dispersal-induced growth (DIG) occurs when the populations are able to persist and grow exponentially when dispersal among the populations is present. We provide a mathematical analysis of this surprising phenomenon, in the context of a deterministic model with periodic variation of growth rates and non-symmetric migration which are assumed to be piecewise continuous. We also consider a stochastic model with random variation of growth rates and migration. This work extends existing results of the literature on the DIG effects obtained for periodic continuous growth rates and time independent symmetric migration. [ABSTRACT FROM AUTHOR]

This paper focuses on optimizing an unknown cost function through extremum seeking (ES) control in the presence of a slow nonlinear dynamic sensor responsible for measuring the cost. In contrast to traditional perturbation-based ES control, which often suffers from sluggish convergence, the proposed method eliminates the time-scale separation between sensor dynamics and ES control by using the relative degree of the nonlinear sensor system. To improve the convergence rate, the authors incorporate high-frequency dither signals and a differentiator. To enhance the robustness with the existence of rapid disturbances, an off-the-shelf linear high-gain differentiator is applied. The first result demonstrates that, for any desired convergence rate, with properly tuned parameters for the proposed ES algorithm, the input of the cost function can converge to an arbitrarily small neighborhood of the optimal solution, starting from any initial condition within any given compact set. Furthermore, the second result shows the robustness of the proposed ES control in the presence of sufficiently fast, zero-mean periodic disturbances. Simulation results substantiate these theoretical findings. [ABSTRACT FROM AUTHOR]

We study nonconservative quasi-periodic (with frequencies) perturbations of two-dimensional Hamiltonian systems with nonmonotonic rotation. It is assumed that the perturbation contains the so-called parametric terms. The behavior of solutions in the vicinity of degenerate resonances is described. Conditions for the existence of resonance -dimensional invariant tori for which there are no generating ones in the unperturbed system are found. The class of perturbations for which such tori can exist is indicated. The results are applied to the asymmetric Duffing equation under a parametric quasi-periodic perturbation. [ABSTRACT FROM AUTHOR]

The objects we encounter in everyday settings are composed of multicolored surfaces. To utilize the color information of these surfaces, it is necessary to efficiently summarize the color distribution of object surfaces. Studies examining such summarization (averaging) processes have shown that the three attributes of color -- hue, saturation, and brightness -- are averaged in different manners. Hue can be averaged accurately and approximated by the arithmetic mean on the uniform chromaticity scale diagram. Averaging of saturation exhibits a bias, with the average tending to shift towards a more saturated direction than the colorimetric mean. Furthermore, this bias becomes more pronounced when there is greater variability (noise) in the color distribution. While there is also a bias observed in the averaging of brightness, the underlying processing strategy seems to be adaptively adjusted depending on the task at hand. These summarizing strategies are believed to be acquired to enhance the reliability and certainty of sensory information in noisy natural environments. Identifying the factors that determine these processing strategies remains a challenge for future research. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to constructing multistep combination forecasts in a nonstationary framework with stochastic and deterministic trends. Existing forecast combination approaches in the stationary setup typically target the in-sample asymptotic mean squared error (AMSE), relying on its approximate equivalence with the asymptotic forecast risk (AFR). Such equivalence, however, breaks down in a nonstationary setup. This paper develops combination forecasts based on minimizing an accumulated prediction errors (APE) criterion that directly targets the AFR and remains valid whether the time series is stationary or not. We show that the performance of APE-weighted forecasts is close to that of the optimal, infeasible combination forecasts. Simulation experiments are used to demonstrate the finite sample efficacy of the proposed procedure relative to Mallows/Cross-Validation weighting that target the AMSE as well as underscore the importance of accounting for both persistence and lag order uncertainty. An application to forecasting US macroeconomic time series confirms the simulation findings and illustrates the benefits of employing the APE criterion for real as well as nominal variables at both short and long horizons. A practical implication of our analysis is that the degree of persistence can play an important role in the choice of combination weights. [ABSTRACT FROM AUTHOR]

Numerical modeling of unidirectional flow in self-affine fractures using the lubrication approximation requires averaging of the transmissivity between the nodes. Seven averaging techniques are reviewed: arithmetic averaging of transmissivity; harmonic averaging of transmissivity; two averaging techniques derived by cell-based collocation method; global reconstruction of profile by means of multiquadrics; arithmetic averaging of aperture; harmonic averaging of aperture. In order to evaluate the performance of the seven techniques in terms of pressure errors and hydraulic aperture errors, self-affine profiles of 1024 nodes with different Hurst exponents (0.4 to 0.8) are generated. Every second node is then removed, resulting in 512-node profiles. Apertures at removed nodes are used in reference flow simulations on the 512-node profiles. Then, simulations with the seven averaging techniques are performed on 512-node profiles. Errors are computed with regard to the results obtained in the reference simulations. Reconstruction with multiquadrics is found to provide superior accuracy on self-affine profiles, followed by harmonic averaging of transmissivity or harmonic averaging of the aperture (Some of the errors analyzed in this study are minimized with the two last mentioned schemes.). Multiquadrics reconstruction is found to provide the best accuracy also on a smooth periodic profile. [ABSTRACT FROM AUTHOR]

IntroductionUnderstanding the neural code has been one of the central aims of neuroscience research for decades. Spikes are commonly referred to as the units of information transfer, but multi-unit activity (MUA) recordings are routinely analyzed in aggregate forms such as binned spike counts, peri-stimulus time histograms, firing rates, or population codes. Various forms of averaging also occur in the brain, from the spatial averaging of spikes within dendritic trees to their temporal averaging through synaptic dynamics. However, how these forms of averaging are related to each other or to the spatial and temporal units of information representation within the neural code has remained poorly understood.Materials and methodsIn this work we developed NeuroPixelHD, a symbolic hyperdimensional model of MUA, and used it to decode the spatial location and identity of static images shown to n = 9 mice in the Allen Institute Visual Coding—NeuroPixels dataset from large-scale MUA recordings. We parametrically varied the spatial and temporal resolutions of the MUA data provided to the model, and compared its resulting decoding accuracy.ResultsFor almost all subjects, we found 125ms temporal resolution to maximize decoding accuracy for both the spatial location of Gabor patches (81 classes for patches presented over a 9×9 grid) as well as the identity of natural images (118 classes corresponding to 118 images) across the whole brain. This optimal temporal resolution nevertheless varied greatly between different regions, followed a sensory-associate hierarchy, and was significantly modulated by the central frequency of theta-band oscillations across different regions. Spatially, the optimal resolution was at either of two mesoscale levels for almost all mice: the area level, where the spiking activity of all neurons within each brain area are combined, and the population level, where neuronal spikes within each area are combined across fast spiking (putatively inhibitory) and regular spiking (putatively excitatory) neurons, respectively. We also observed an expected interplay between optimal spatial and temporal resolutions, whereby increasing the amount of averaging across one dimension (space or time) decreases the amount of averaging that is optimal across the other dimension, and vice versa.DiscussionOur findings corroborate existing empirical practices of spatiotemporal binning and averaging in MUA data analysis, and provide a rigorous computational framework for optimizing the level of such aggregations. Our findings can also synthesize these empirical practices with existing knowledge of the various sources of biological averaging in the brain into a new theory of neural information processing in which the unit of information varies dynamically based on neuronal signal and noise correlations across space and time.

We consider an optimal control problem for a differential inclusion of the Carathéodory type affine with respect to the control with a coercive cost functional on a semiaxis and with fast oscillating time-dependent coefficients. We prove that, when the small parameter converges to zero, the solution to this problem tends to some solution of the optimal control problem with averaged coefficients, where the averaging we understand in the sense of the Kuratowski upper limit. [ABSTRACT FROM AUTHOR]

Motivated by applications to mathematical biology, we study the averaging problem for slow–fast systems, in the case in which the fast dynamics is a stochastic process with multiple invariant measures. We consider both the case in which the fast process is decoupled from the slow process and the case in which the two components are fully coupled. We work in the setting in which the slow process evolves according to an ordinary differential equation (ODE) and the fast process is a continuous time Markov process with finite state space and show that, in this setting, the limiting (averaged) dynamics can be described as a random ODE (i.e. an ODE with random coefficients). [ABSTRACT FROM AUTHOR]

Background: This study aimed to develop a deep learning (DL) algorithm that enhances the quality of a single-frame enface OCTA scan to make it comparable to 4-frame averaged scan without the need for the repeated acquisitions required for averaging. Methods: Each of the healthy eyes and eyes from diabetic subjects that were prospectively enrolled in this cross-sectional study underwent four repeated 6 × 6 mm macular scans (PLEX Elite 9000 SS-OCT), and the repeated scans of each eye were co-registered to produce 4-frame averages. This prospective dataset of original (single-frame) enface scans and their corresponding averaged scans was divided into a training dataset and a validation dataset. In the training dataset, a DL algorithm (named pseudoaveraging) was trained using original scans as input and 4-frame averages as target. In the validation dataset, the pseudoaveraging algorithm was applied to single-frame scans to produce pseudoaveraged scans, and the single-frame and its corresponding averaged and pseudoaveraged scans were all qualitatively compared. In a separate retrospectively collected dataset of single-frame scans from eyes of diabetic subjects, the DL algorithm was applied, and the produced pseudoaveraged scan was qualitatively compared against its corresponding original. Results: This study included 39 eyes that comprised the prospective dataset (split into 5 eyes for training and 34 eyes for validating the DL algorithm), and 105 eyes that comprised the retrospective test dataset. Of the total 144 study eyes, 58% had any level of diabetic retinopathy (with and without diabetic macular edema), and the rest were from healthy eyes or eyes of diabetic subjects but without diabetic retinopathy and without macular edema. Grading results in the validation dataset showed that the pseudoaveraged enface scan ranked best in overall scan quality, background noise reduction, and visibility of microaneurysms (p < 0.05). Averaged scan ranked best for motion artifact reduction (p < 0.05). Grading results in the test dataset showed that pseudoaveraging resulted in enhanced small vessels, reduction of background noise, and motion artifact in 100%, 82%, and 98% of scans, respectively. Rates of false-positive/-negative perfusion were zero. Conclusion: Pseudoaveraging is a feasible DL approach to more efficiently improve enface OCTA scan quality without introducing notable image artifacts. [ABSTRACT FROM AUTHOR]

The atmosphere and ocean are described by highly oscillatory PDEs that challenge both our understanding of their dynamics and their numerical approximation. This paper presents a preliminary numerical study of one type of phase averaging applied to mean flows in the 2D Boussinesq equations that also has application to numerical methods. The phase averaging technique, well-known in dynamical systems theory, relies on a mapping using the exponential operator, and then an averaging over the phase. The exponential operator has connections to the Craya–Herring basis pioneered by Jack Herring to study the fluid dynamics of oscillatory, nonlinear fluid dynamics. In this paper, we perform numerical experiments to study the effect of this averaging technique on the time evolution of the solution. We explore its potential as a definition for mean flows. We also show that, as expected from theory, the phase-averaging method can reduce the magnitude of the time rate of change in the PDEs, making them potentially suitable for time stepping methods. [ABSTRACT FROM AUTHOR]

The outbreak of Corona Virus 2019(Covid-19) is a great threat to the whole world. It is crucial to early detect patients infected with covid-19 and treat them to mitigate the rapid spread of this disease. It is an immediate priority to overcome the traditional screening and develop an accurate as well as speedy covid-19 automatic diagnosis system. Computer Tomography (CT) and Chest X-Ray imaging coupled with deep learning models to develop and test Computer Aided Screening (CAS) of covid-19 images from the normal images. In this paper classification and screening of covid-19 disease are performed by using pre-trained convolutional neural networks and a proposed hybrid model on an available standard dataset of chest X-Ray images. The proposed hybrid model employs the pre-trained Convolutional Neural Network models and Transfer Learning models. Our proposed model consists of three stages where extraction of features is performed in first stage by using pre-trained machine learning model. Deep features are extracted by using the infusion of the Transfer Learning Technique in the second stage of the model. The third stage uses Flatten and Classification layers to diagnose of Covid-19 patients. In order to assure the consistency of the proposed model, by considering standard dataset X-Ray images. Simulation results of performance metrics of Accuracy, F1 Score, Precision, Recall, ROC, and AUC curve, and training and testing loss are used to evaluate and compare the proposed model with existing models. Experimental result demonstrates that the hybrid model improves the screening process for Covid-19 disease by achieving higher accuracy. [ABSTRACT FROM AUTHOR]