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We study the problem of seeking the source of a scalar signal with a force- and torque-controlled unicycle. The vehicle does not have the capability to measure or track its position. The only information about the vehicle's position relative to the source is given by measurements of the signal. We propose a feedback law that induces a uniform forward motion and an almost uniform rotational motion. This in turn leads to an almost circular motion of the unicycle; i.e., a motion along the boundary of a disk. An application of the divergence theorem reveals that the measurements of the signal along the boundary of the disk provide access to the gradient of an averaged signal, where the average is taken over the entire disk. Such a local average of the sensed signal can eliminate undesired local extrema, which in turn is beneficial for the goal of reaching a global extremum. We prove practical asymptotic stability for the closed-loop system with respect to extrema of the averaged signal. Our theoretical analysis takes L ∞ -measurement errors of the signal into account. We compare the performance of the proposed control law to another source seeking method in the presence of local extrema through numerical simulations. [ABSTRACT FROM AUTHOR]

A direct method (self-regularization method) for the numerical solution of a weakly singular integral equation of the first kind on a closed surface is considered. This equation is an integral formulation of the internal and external three-dimensional Dirichlet problems for the Laplace equation if their solutions are sought in the form of a single-layer potential. It is approximated by a system of linear algebraic equations, which is solved numerically. In this case, a new method of averaging the kernel of the integral operator is used. It preserves the conditional correctness of the discretized problem and significantly increases the rate of convergence of its solution to the exact solution of the integral equation. [ABSTRACT FROM AUTHOR]

We investigate how the temporal evolution of the rotation axis of a hypothetical exo-Earth is affected by the presence of a satellite, an exo-Moon. Namely, we study analytically and numerically how the range of the nutation angle of an exo-Earth changes if an exo-Moon is added to a system comprised of an exo-Sun, the exo-Earth and exoplanets. We say that the impact of an exo-Moon is stabilising if upon including the exo-Moon the range of the nutation angle decreases, and destabilising otherwise. The problem is considered in a general set-up. The exo-Earth is supposed to be rigid, axially symmetric and almost spherical, the difference between the largest and the smallest principal moments of inertia being a small parameter of the problem. Assuming the orbits of the celestial bodies to be quasi-periodic, we apply time averaging over fast variables associated with order one frequencies to study rotation of the exo-Earth at times large relative the respective periods. Non-resonant frequencies are assumed. For a system comprised of the exo-Sun and exoplanets in the absence of small orbital frequencies, the system is integrable, which allows to calculate the range of the nutation angle as a function of initial conditions. Using these expressions, we identify a class of systems for which we prove analytically that the impact of the exo-Moon is stabilising and a class where it is destabilising. Namely, if the orbits of the planets are circular and their orbital planes coincide then the impact is destabilising. The impact is stabilising if the angle between orbital planes of the exo-Moon and the exo-Earth vanishes. We also investigate numerically how the impact of the exo-Moon in a particular system comprised of a star and two planets varies on modifying parameters of the orbits of the exo-Moon and the second planet, and the initial nutation angle. [ABSTRACT FROM AUTHOR]

The paper deals with the transformation of a weakly nonlinear system of differential equations in a special form into a simplified form and its relation to the normal form and averaging. An original method of simplification is proposed, that is, a way to determine the coefficients of a given nonlinear system in order to simplify it. We call this established method the degree equalization method, it does not require integration and is simpler and more efficient than the classical Krylov-Bogolyubov method of normalization. The method is illustrated with several examples and provides an application to the analysis of cardiac activity modelled using van der Pol equation. [ABSTRACT FROM AUTHOR]

A viscous liquid flow along a semi-infinite plate with small periodic irregularities on the surface is considered for large Reynolds numbers. The flow near the plate is described by the Prandtl equations with induced pressure, which are non-classical PDEs because they contain a limiting term. The goal is to construct a numerical algorithm for solving these equations with periodic boundary conditions. The results of numerical modeling of the flow are presented. [ABSTRACT FROM AUTHOR]

In the physics literature it is common to use "in cascade" the rotating wave approximation and the adiabatic approximation for chirped pulses of two-level quantum systems driven by one external field, in particular when the resonance frequency is not known precisely. Both approximations need relatively long time and are based on averaging theory of dynamical systems. Unfortunately, the two approximations cannot be done independently since, in a sense, the two time scales interact. We study how the cascade of the two approximations can be justified, while preserving the robustness of the adiabatic strategy. Our first result, based on high-order averaging techniques, gives a precise quantification of the uncertainty interval of the resonance frequency for which the population inversion works. As a by-product, we prove controllability of an ensemble of spin systems by a single real-valued control, providing an extension of a celebrated result with two controls by Khaneja and Li. [ABSTRACT FROM AUTHOR]

This paper discusses vibrational stabilization of a class of single-input, two degree-of-freedom mechanical systems. Considering two different control formulations—position-input and force-input—and both open- and closed-loop control, we find that the sets of attainable equilibrium positions for the unactuated coordinate are identical in every case. The subset of positions that are stabilizable, however, depends on the formulation. In general, the set of equilibria that can be stabilized using open-loop force-input is larger than the set that can be stabilized using open-loop position-input. And the use of feedback expands this stabilizable set even further. As examples, this paper presents the dynamic analysis, open- and closed-loop vibrational control, and the mechanics behind the stability of two underactuated systems, the Kapitza pendulum and a one-link horizontal pendulum. [ABSTRACT FROM AUTHOR]

Let $f$ be an entire function of order $\rho\in (0,+\infty)$ with zeros on a finite system of rays $\{z: \arg z=\psi_{j}\}$, $j\in\{1,\ldots,m\}$, $0\le\psi_1

Abstract Background The three-dimensional description of rigid body kinematics is a key step in many studies in biomechanics. There are several options for describing rigid body orientation including Cardan angles, Euler angles, and quaternions; the utility of quaternions will be reviewed and elaborated. Main body of abstract The orientation of a rigid body or a joint between rigid bodies can be described by a quaternion which consists of four variables compared with Cardan or Euler angles (which require three variables). A quaternion, q = (q 0, q 1, q 2, q 3), can be considered a rotation (Ω = 2 cos−1(q 0)), about an axis defined by a unit direction vector q 1 / sin Ω 2 q 2 / sin Ω 2 q 3 / sin Ω 2 $$ \left({q}_1/\sin \left(\frac{\Omega}{2}\right),{q}_2/\sin \left(\frac{\Omega}{2}\right),{q}_3/\sin \left(\frac{\Omega}{2}\right)\right) $$ . The quaternion, compared with Cardan and Euler angles, does not suffer from singularities or Codman’s paradox. Three-dimensional angular kinematics are defined on the surface of a unit hypersphere which means numerical procedures for orientation averaging and interpolation must take account of the shape of this surface rather than assuming that Euclidean geometry based procedures are appropriate. Numerical simulations demonstrate the utility of quaternions for averaging three-dimensional orientations. In addition the use of quaternions for the interpolation of three-dimensional orientations, and for determining three-dimensional orientation derivatives is reviewed. Conclusion The unambiguous nature of defining rigid body orientation in three-dimensions using a quaternion, and its simple averaging and interpolation gives it great utility for the kinematic analysis of human movement.

Dyslexic individuals have been reported to have reduced global motion sensitivity, which could be attributed to various causes including atypical magnocellular or dorsal stream function, impaired spatial integration, increased internal noise and/or reduced external noise exclusion. Here, we applied an equivalent noise experimental paradigm alongside a traditional motion-coherence task to determine what limits global motion processing in dyslexia. We also presented static analogues of the motion tasks (orientation tasks) to investigate whether perceptual differences in dyslexia were restricted to motion processing. We compared the performance of 48 dyslexic and 48 typically developing children aged 8 to 14 years in these tasks and used equivalent noise modelling to estimate levels of internal noise (the precision associated with estimating each element's direction/orientation) and sampling (the effective number of samples integrated to judge the overall direction/orientation). While group differences were subtle, dyslexic children had significantly higher internal noise estimates for motion discrimination, and higher orientation-coherence thresholds, than typical children. Thus, while perceptual differences in dyslexia do not appear to be restricted to motion tasks, motion and orientation processing seem to be affected differently. The pattern of results also differs from that previously reported in autistic children, suggesting perceptual processing differences are condition-specific.

The paper deals with constructive semiclassical asymptotics of eigenfunctions of the Dirac operator describing graphene in a constant magnetic field. Two cases are considered: (a) a strong magnetic field, and (b) a radially symmetric electric field and small mass. Using standard semiclassical methods, we reduce the problem to a pencil of magnetic Schrödinger operators with a correction term. In both cases, the classical system defined by the principal symbol turns out to be integrable, but the correction term destroys the integrability. In case (a), where the correction removes the frequency degeneracy (resonance), we use the averaging method to reduce the problem to an integrable system not only in the leading approximation but also with the correction taken into account. The tori of the resulting system generate a series of asymptotic eigenfunctions of the original operator. In case (b), the system defined by the principal symbol is nondegenerate. Fixing an invariant torus with Diophantine frequencies for this system and looking for a solution of the transport equation for it, we obtain a series of asymptotic eigenfunctions that are in one-to-one correspondence with tori that satisfy the Bohr–Sommerfeld quantization rule and lie in a small neighborhood of the chosen Diophantine torus. In both cases, the construction of the asymptotic eigenfunctions is based on the global representation of the Maslov canonical operator on a two-dimensional torus projected onto the configuration space into an annular domain with two simple caustics via the Airy function and its derivative. A numerical implementation of our formulas in examples shows their efficiency. [ABSTRACT FROM AUTHOR]

New boundary conditions in the regions of vessel junctions for a one-dimensional network model of hemodynamics are proposed. It is shown that these conditions ensure the continuity of the solution and its derivatives at the points of vessel junctions. In the asymptotic limit, they give solutions that coincide with the solution in one continuous vessel. Nonreflecting boundary conditions at the endpoints of the terminal vessels are proposed. Results of numerical experiments that confirm the results of theoretical analysis are presented. [ABSTRACT FROM AUTHOR]

The effect of decaying oscillatory perturbations on autonomous Hamiltonian systems in the plane with a stable equilibrium is investigated. It is assumed that perturbations preserve the equilibrium and satisfy a resonance condition. The behaviour of the perturbed trajectories in the vicinity of the equilibrium is investigated. Depending on the structure of the perturbations, various asymptotic regimes at infinity in time are possible. In particular, a phase locking and a phase drifting can occur in the systems. The paper investigates the bifurcations associated with a change of Lyapunov stability of the equilibrium in both regimes. The proposed stability analysis is based on a combination of the averaging method and the construction of Lyapunov functions. [ABSTRACT FROM AUTHOR]

This paper derives a weak convergence theorem of the time discretization of the slow component for a two-time-scale stochastic evolutionary equations on interval [0, 1]. Here the drift coefficient of the slow component is cubic with linear coupling between slow and fast components. [ABSTRACT FROM AUTHOR]

The radar on the Global Precipitation Measurement (GPM) mission observes precipitation at 13.6 GHz (Ku-band) and 35.6 GHz (Ka-band) and also receives echoes from the earth's surface. Statistics of surface measurements for non-raining conditions are saved in a database for later use in estimating the precipitation path-integrated attenuation. Previous work by Meneghini and Jones (2011) showed that while averaging over larger latitude/longitude bins increase the number of samples, it can also increase sample variance due to spatial inhomogeneity in the data. As a result, Meneghini and Kim (2017) proposed a new, adaptive method of database construction, in which the number of measurements averaged depends on the spatial homogeneity. The purpose of this work is to re-visit previous, single-frequency results using dual-frequency data and optimal interpolation (kriging). Results include that (1) temporal inhomogeneity can create similar results as spatial, (2) Ka-band behavior is similar to Ku-band, (3) the Ku-/Ka-band difference has less spatial inhomogeneity than either band by itself, and (4) kriging and the adaptive method can reduce the sample variance. The author concludes that finer spatial and temporal resolution is necessary in constructing the database for single frequencies but less so for the Ku-/Ka-band difference. The adaptive approach reduces sample standard deviation with a relatively modest computational increase. [ABSTRACT FROM AUTHOR]

The seasonal temperature and humidity regime of the road foundation in the active layer of soil largely determines the reliability and safety of the automobile road use. The aim of this article is a quantitative assessment of the influence of variation in ice content in the active layer of the road foundation soil in the permafrost area on the depth of seasonal soil thawing. Analytical solutions of one-phase flat Stefan problem at varying ice content along the coordinate were obtained. The solution is in the dimensionless form as a Fourier function based on Stefan number and dimensionless simplexes. The dependence of ice content change on the distance was assumed to be linear. Calculations using the obtained formulas were conducted, comparing against formulas assuming that ice content is a constant. It was determined at which ice content gradient averaging the ice content does not lead to an error greater than permitted in engineering practice in determination of the thawing depth. The results of variant calculations are presented as 2D and 3D charts to assess the influence of the initial ice content in the soil and the degree of its variation over the period of road use on the thawing depth of road foundation.

We examine the ability of observers to extract summary statistics (such as the mean and the relative-variance) from rapid numerical sequences of two digit numbers presented at a rate of 4/s. In four experiments (total N = 100), we find that the participants show a remarkable ability to extract such summary statistics and that their precision in the estimation of the sequence-mean improves with the sequence-length (subject to individual differences). Using model selection for individual participants we find that, when only the sequence-average is estimated, most participants rely on a holistic process of frequency based estimation with a minority who rely on a (rule-based and capacity limited) mid-range strategy. When both the sequence-average and the relative variance are estimated, about half of the participants rely on these two strategies. Importantly, the holistic strategy appears more efficient in terms of its precision. We discuss implications for the domains of two pathways numerical processing and decision-making.

A heterogeneous material obtained by means of cold gas dynamic spraying with subsequent laser processing is studied. The reasons for reduction of the particle size of boron carbide B4C under the action of a laser beam and the chemical composition of the material containing the "lost" material of B4C particles are examined. A mathematical model, which takes into account the dissolution of B4C particles and the presence of a chemical reaction, is developed. The elastic properties of the deposited heterogeneous material are analyzed with the use of continuum approaches that describe heterogeneous media and are based on averaging of the original stochastic equations. [ABSTRACT FROM AUTHOR]

In this paper, we generalize our previous results of Newton-based extremum seeking of higher derivatives via predictors with averaging-based estimates for uncertain delays. We show necessary conditions for the robustness to small delay uncertainties of the proposed extremum seeking controller with predictor feedback. Local stability and exponential convergence to a small neighborhood of the unknown extremum can still be guaranteed. A numerical simulation example is presented to illustrate the performance of the predictor-based control scheme for uncertain time-delay compensation. [ABSTRACT FROM AUTHOR]

The slow dynamics of unbalanced rotors with a passive self-balancing system are investigated considering the interaction of the mechanical system with a limited power engine. The slow dynamics equations are obtained using the averaging technique for partially strongly damped systems. Stationary system configurations, different types of nonstationary solutions while passing through resonance, and areas of stability and attraction are investigated. [ABSTRACT FROM AUTHOR]

It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase in the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of complexification-averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period-doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes), and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors' knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations. [ABSTRACT FROM AUTHOR]

We provide some analytical account of the influence of the internal boundaries of a heterogeneous medium on the propagation of an elastic stress field through it. The generalized derivative serves as the mathematical concept displaying the microstructure of a heterogeneous system. Using the generalized derivative we modify the operator in the initial model of linear elasticity. The Green's function (built on the operator) displays the microstructural features of the system. We use the method of conditional moments to obtain the effective coefficients of elasticity which are included in the averaged equations and describing the elastic properties of a heterogeneous medium. This approach leads to the integrals containing the modified averaged Green's function and the correlation function of the structure geometry. Using these terms, we integrally take into account the microstructure of the system in the final effective elasticity coefficients. [ABSTRACT FROM AUTHOR]

We consider a motion about the center of mass of a spheroid with a cavity filled with a viscous fluid. It is assumed that the velocity of the fluid is sufficiently high, so the corresponding Reynolds number is small. The torque of forces acting on the body by the viscous fluid in the cavity is determined by the method developed in the works of F.L. Chernousko. Asymptotic approach permits to obtain some qualitative results and to describe nonlinear evolution of angular motion using simplified averaged equations and numerical solution. [ABSTRACT FROM AUTHOR]

We consider a nonlinear nonautonomous system of two ordinary differential equations with a stable fixed point and assume that the non-Hamiltonian part of the system tends to zero at infinity. We examine the asymptotic behavior of a two-parameter family of solutions that start from a neighborhood of the stable equilibrium. The proposed construction of asymptotic solutions is based on the averaging method and the transition in the original system to new dependent variables, one of which is the angle of the limit Hamiltonian system, and the other is the Lyapunov function for the complete system. [ABSTRACT FROM AUTHOR]

We examine the ability of observers to extract summary statistics (such as the mean and the relative-variance) from rapid numerical sequences of two digit numbers presented at a rate of 4/s. In four experiments (total N = 100), we find that the participants show a remarkable ability to extract such summary statistics and that their precision in the estimation of the sequence-mean improves with the sequence-length (subject to individual differences). Using model selection for individual participants we find that, when only the sequence-average is estimated, most participants rely on a holistic process of frequency based estimation with a minority who rely on a (rule-based and capacity limited) mid-range strategy. When both the sequence-average and the relative variance are estimated, about half of the participants rely on these two strategies. Importantly, the holistic strategy appears more efficient in terms of its precision. We discuss implications for the domains of two pathways numerical processing and decision-making. [ABSTRACT FROM AUTHOR]

An autonomous nonconservative mechanical system with two degrees of freedom is studied. The system is subject to a feedback control with two control impact gain factors. It is required to select the values of these factors in such a way as to ensure existence of asynchronous self-sustained oscillations with prescribed properties. An iterative method is proposed to search for the corresponding values of control impact gain factors. This approach is based on constructing auxiliary second-order systems and generating limit cycles in these systems. The algorithm that is used for this purpose represents a modification of the Andronov–Pontryagin method, but does not require the presence of a small parameter in the system. The efficiency of this approach is illustrated on a problem of generating asynchronous self-sustained oscillations/rotations in a model of an aerodynamic pendulum. The applicability conditions of the algorithm and the possible modifications are discussed. [ABSTRACT FROM AUTHOR]

Conventional tuned mass dampers are designed either for a local frequency response or for broadband applications. In either case, optimization of both the resonance amplitudes and the tuned frequency amplitude is not possible. This paper considers a novel friction damper, the tuned wedge damper, which utilizes piecewise-defined contact geometries to decouple vibration absorption from vibration damping. Consequently, a simultaneous optimization of the tuned frequency and the amplitude in the resonance regimes is possible. The damper is investigated via numerical and analytical methods. Experimental investigations validate the theoretical insights and offer the first proof of concept for the implementation of the tuned wedge damper. • Damper based on dry friction and piecewise-defined contact surfaces. • The decoupling of damping and absorption was achieved. • Damper optimization leads to low amplitude at resonance and at absorption frequency. • Numerical, analytical, and experimental analysis. • Theoretical behavior was verified by experiments. [ABSTRACT FROM AUTHOR]

A greedy average block sparse Kaczmarz method is developed for the sparse solution of the linear system of equations. The convergence theory of this method is established and the upper bound of its convergence rate with adaptive stepsize is derived. Numerical experiments are presented to verify the efficiency of the proposed method, which outperforms the existing sparse Kaczmarz methods in terms of the number of iteration steps and computational costs. [ABSTRACT FROM AUTHOR]

In this work, the transport equations of ionic species in concrete are studied. First, the equations at the porescale are considered, which are then averaged over a representative elementary volume. The so obtained transport equations at the macroscopic scale are thoroughly examined and each term is interpreted. Furthermore, it is shown that the tortuosity-connectivity does not slow the average speed of the ionic species down. The transport equations in the representative elementary volume are then compared with the equations obtained in an equivalent pore. Lastly, comparing Darcy's law and the Hagen–Poiseuille equation in a cylindrical equivalent pore, the tortuosity-connectivity parameter is obtained for four different concretes. The proposed model provides very good results when compared with the experimentally obtained chloride profiles for two additional concretes. [ABSTRACT FROM AUTHOR]

Rodents can be trained to associate different durations with different stimuli (e.g., light/sound). When the associated stimuli are presented together, maximal responding is observed around the average of individual durations (akin to averaging). The current study investigated whether mice can also average independently trained numerosities. Mice were initially trained to make 10 or 20 lever presses on a single (run) lever to obtain a reward and each fixed-ratio schedule was signaled either with an auditory or visual stimulus. Then, mice were trained to press another lever to obtain the reward after they responded on the run lever for the minimum number of presses [Fixed Consecutive Number (FCN)-10 or -20 trials] signaled by the corresponding discriminative stimulus. Following this training, FCN trials with the compound stimulus were introduced to test the counting behavior of mice when they encountered conflicting information regarding the number of responses required to obtain the reward. Our results showed that the numbers of responses on these compound test trials were around the average of the number of responses in FCN-10 and FCN-20 trials particularly when the auditory stimulus was associated with a fewer number of required responses. The counting strategy explained the behavior of the majority of the mice in the FCN-Compound test trials (as opposed to the timing strategy). The number of responses in FCN-Compound trials was accounted for equally well by the arithmetic, geometric, and Bayesian averages of the number of responses observed in FCN-10 and FCN-20 trials. [ABSTRACT FROM AUTHOR]

This paper presents dynamic analysis and optimal control of vibratory systems with asymmetric drag and asymmetric added mass, moving in a resisting medium. The paper considers the general equation of motion of drag-based vibratory systems and discusses the effects of drag asymmetry, mass asymmetry, and input waveform on the response of the system. The paper also investigates the effects of drag law on the response of the vibratory systems in different flow regimes. Using averaging methods, the averaged dynamics of the general vibratory system is determined and used for the dynamic analysis. The averaged dynamics are also used for the optimal control of the system with symmetric drag and symmetric mass. The optimal control results using the averaged dynamics are compared with the existing results determined using the conventional optimal control theories. It is shown that, for high enough frequencies, in the presented optimal control method using averaging techniques and constrained optimization, simple algebraic equations are used and the results match those obtained using the conventional optimal control theories. [ABSTRACT FROM AUTHOR]

Data-driven prediction is becoming increasingly widespread as the volume of data available grows and as algorithmic development matches this growth. The nature of the predictions made and the manner in which they should be interpreted depend crucially on the extent to which the variables chosen for prediction are Markovian or approximately Markovian. Multiscale systems provide a framework in which this issue can be analyzed. In this work kernel analog forecasting methods are studied from the perspective of data generated by multiscale dynamical systems. The problems chosen exhibit a variety of different Markovian closures, using both averaging and homogenization; furthermore, settings where scale separation is not present and the predicted variables are non-Markovian are also considered. The studies provide guidance for the interpretation of data-driven prediction methods when used in practice. [ABSTRACT FROM AUTHOR]

Ensemble statistics of a visual scene can be estimated to provide a gist of the scene without detailed analysis of all individual items. The simplest and most widely studied ensemble statistic is mean estimation, which requires averaging an ensemble of elements. Averaging is useful to estimate the mean of an ensemble and discard the variance. The source of variance can be external (i.e., variance across the physical elements) or internal (i.e., imprecisions in the estimates of the elements by the visual system). The equivalent noise paradigm is often used to measure the impact of the internal variance (i.e., the equivalent input noise). This paradigm relies on the assumption that the averaging process is equally effective independently of the main source of variance, internal or external, so any difference between the processing when the main source of variance is internal or external must be assumed not to affect the averaging efficiency. The current fMRI study compared the neural activity when the main variance is caused by the stimulus (i.e., high variance) and when it is caused by imprecisions in the estimates of the elements by the visual system (i.e., low variance). The results showed that the right superior frontal and left middle frontal gyri can be significantly more activated when the variance in the orientation of the Gabors was high than when it was low. Consequently, the use of the equivalent noise paradigm requires the assumption that such additional neural activity in high variance does not affect the averaging efficiency. [ABSTRACT FROM AUTHOR]

Ensemble perception is efficient because it summarizes redundant and complex information. However, it loses the fine details of individual items during the averaging process. Such characteristics of ensemble perception are similar to those of coarse processing. Here, we tested whether extracting an average of a set was similar to coarse processing. To manipulate coarse processing, we used the fast flicker adaptation known as suppressing coarse information processed by the magnocellular pathway. We hypothesized that if computing the average of a set relied on coarse processing, the precision of an averaging task should decrease after adaptation compared to baseline (no-adaptation). Across experiments with various features (orientation in Experiment 1, size in Experiment 2, and facial expression in Experiment 3), we found that suppressing coarse information did not disrupt the performance of the averaging tasks. Rather, adaptation increased the precision of mean representation. The precision of mean representation might have increased because fine information was relatively enhanced after adaptation. Our results suggest that the quality of ensemble representation relies on that of individual items. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by divergence-free vector fields. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians (below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian. [ABSTRACT FROM AUTHOR]

We apply Reeb's theorem to prove the existence of periodic orbits in the rotating Hénon–Heiles system. To this end, a sort of detuned normal form is calculated that yields a reduced system with at most four non degenerate equilibrium points. Linear stability and bifurcations of equilibrium solutions mimic those for periodic solutions of the original system. We also determine heteroclinic connections that can account for transport phenomena. [ABSTRACT FROM AUTHOR]

Due to the stochastic nature of pore space geometry, particle velocities in a colloidal‐suspension flux are also stochastically distributed. This phenomenon is captured by Boltzmann's kinetic equation. We formulate a BGK‐form of Boltzmann's equation for particulate flow with a particle capture rate proportional to the particle speed and derive an exact method for the model's averaging. The averaged equation is of the form of a reaction‐advection‐diffusion equation with the average particle speed lower than the carrier fluid velocity. This delay, reported in numerous laboratory studies, is explained by preferential capture of fast particles. The large‐scale model coefficients of delay, dispersion, and filtration are explicitly expressed via the microscale mixing length, equilibrium velocity distribution, and filtration coefficient. The properties of the averaged coefficients are discussed, and emerging dependencies between them are presented. The derived large‐scale equation resolves two paradoxes of the traditional model for suspension‐colloidal transport in porous media. The large‐scale equation closely matches the laboratory breakthrough curves. Key Points: Exact upscaling of Boltzmann's equation for colloidal transport yields advective‐reaction‐diffusion equation with particle arrival delayThe averaged large‐scale model closely matches laboratory coreflooding data [ABSTRACT FROM AUTHOR]