Although neural interactions are usually characterized only by their **coupling** strength and directionality, there is often a need to go beyond this by establishing the functional mechanisms of the interaction. We introduce the use of dynamical Bayesian inference for estimation of the **coupling** **functions** of neural oscillations in the presence of noise. By grouping the partial functional contributions, the **coupling** is decomposed into its functional components and its most important characteristics—strength and form—are quantified. The method is applied to characterize the δ-to-α phase-to-phase neural **coupling** **functions** from electroencephalographic (EEG) data of the human resting state, and the differences that arise when the eyes are either open (EO) or closed (EC) are evaluated. The δ-to-α phase-to-phase **coupling** **functions** were reconstructed, quantified, compared, and followed as they evolved in time. Using phase-shuffled surrogates to test for significance, we show how the strength of the direct **coupling**, and the similarity and variability of the **coupling** **functions**, characterize the EO and EC states for different regions of the brain. We confirm an earlier observation that the direct **coupling** is stronger during EC, and we show for the first time that the **coupling** function is significantly less variable. Given the current understanding of the effects of e.g., aging and dementia on δ-waves, as well as the effect of cognitive and emotional tasks on α-waves, one may expect that new insights into the neural mechanisms underlying certain diseases will be obtained from studies of **coupling** **functions**. In principle, any pair of coupled oscillations could be studied in the same way as those shown here.

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Comparing the cardiorespiratory and neural interactions shows that the former exhibits the more stable and invariant form, in that it varies less between different time windows, arguably implying greater determinism as described in the time-averaged **coupling** function. On the other hand, the neural **coupling** **functions** vary more and are individually less similar to the time- averaged **coupling** function. (These findings are consistent with previous results for the resting state in the multi-subject studies [11]). Common to both of the interactions is that they are time-varying, to a lesser or greater extent, and that the **coupling** strength and the form of the function often vary over time quite differently from each other. Hence, these characteristics can have correspondingly different effects on the outcome and the possible transitions caused by the interactions—a phenomenon worth exploring further theoretically.

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Given these properties, it is hardly surprising that cou- pling **functions** have recently attracted considerable atten- tion within the scientific community. They have mediated applications, not only in different subfields of physics, but also beyond physics, predicated by the development of powerful methods enabling the reconstruction of **coupling** **functions** from measured data. The reconstruction within these methods is based on a variety of inference techniques, e.g., least-squares and kernel smoothing fits (Rosenblum and Pikovsky, 2001; Kralemann et al., 2013), dynamical Bayesian inference (Stankovski et al., 2012), maximum likelihood (multiple-shooting) methods (Tokuda et al., 2007), stochastic modeling (Schwabedal and Pikovsky, 2010), and the phase resetting (Galán, Ermentrout, and Urban, 2005; Timme, 2007; Levnaji ć and Pikovsky, 2011). Both the connectivity between systems and the associated methods employed for revealing it are often differentiated into structural, functional, and effective connectivity (Friston, 2011; Park and Friston, 2013). Structural connectivity is defined by the existence of a physical link, such as anatomical synaptic links in the brain or a conducting wire between electronic systems. Functional connectivity refers to the statistical dependences between systems, such as, for exam- ple, correlation or coherence measures. Effective connectivity is defined as the influence one system exerts over another, under a particular model of causal dynamics. Importantly in this context, the methods used for the reconstruction of **coupling** **functions** belong to the group of effective connec- tivity techniques, i.e., they exploit a model of differential equations and allow for dynamical mechanisms — such as the **coupling** **functions** themselves — to be inferred from data.

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show the surrogate **coupling** **functions** computed to check the validity of the results presented in each row. The polar plot in the top-right corner of each figure indicates the similarity index ρ for the average form (coloured arrow) and for the individual subjects (grey arrows). Note how, with ageing, the forms lose amplitude in the central network and resemble the variability of surrogates in the peripheral network.

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Nevertheless, it appears that in this case and u nder the assum ption o f “to a given com m on variable x , it exists one and only one value x* o f the d iscip lin ary variable x that m inim izes the objective function^”, the array that uses COSMOS to approxim ate the **coupling** **functions** is a good approxim ation o f the **coupling** **functions**. This assumption comes true if the objective function is convex. However, in general case our tests indicate that if this condition is not verified, the method is not capable o f determining the correct value for approxim ation. M oreover, w e show ed that the values that are kept for approximation - the ones found at the end o f subsystem optimization - are not always the values which give the best results. The system tends to solutions that are optimum for the objective **functions** but not enough to solutions that respect the **coupling** **functions**.

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be able to withstand, not only man-made attacks, but also inter- ruptions arising from the technical infrastructure and the real- ization of the communication links themselves. The technical perturbations often result in increased noise and interference, which tend to alter and reduce the quality of communications and information content, which in turn can affect the infor- mation forensic procedures [3]–[7]. Many different types of communications protocol have been designed, including the use of logical and mathematical procedures, signal processing manipulations, dynamical chaotic systems, and quantum infor- mation approaches [8]–[20]. The focus in the present paper is on a secure communications protocol based on the **coupling** **functions** between dynamical systems. The protocol itself is introduced in [21]; here, we present a new experimental realization designed to test its robustness to noise, as discussed below in Secs. III and IV.

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Typically, the first change seen in Dst at the beginning of a storm is a positive swing, due to an increase in the dayside magnetopause current from an enhanced solar wind dynamic pressure. The injection of energetic particles into the ring current, resulting in prolonged negative Dst values, occurs primarily on the nightside. It takes time, of the order of an hour, for newly merged IMF and geomagnetic field lines to convect to the nightside of the planet and diffuse through the magnetotail, at which point the open geomagnetic field reconnects (closes), and undergoes dipolarization. Many for- mula are available that describe the negative excursions of Dst in terms of the solar wind parameters, some of these are listed in section 5. These “**coupling** **functions**” are often (incorrectly) assumed to directly represent the rate of par- ticle injection into the ring current, but it is no coincidence that the **functions** appear to describe the rate of magnetic field merging on the dayside. Following the explanation of Siscoe and Petschek [1997], the merging of magnetic flux on the dayside results in a negative swing in Dst firstly due to the deformation of the magnetotail, and the enhanced cross- tail current. Around an hour later, when this merged flux reconnects on the nightside, the injection of particles into the ring current offsets the loss of Dst from the restored geomagnetic field. At this point the negative Dst contribu- tion is transferred from the magnetic field to an enhanced ring current. Recently Vasyli¯ unas [2006] points out that the deformation of the geomagnetic tail can be represented by the amount of open (merged) magnetic flux, which is largely piled up in the magnetotail. If during tail reconnec- tion the gain in -Dst from the ring current exactly cancels the loss from the reduced magnetotail contribution, then the **coupling** **functions**, which describe the enhancement of -Dst over the course of a storm, will be identical to the rate of dayside magnetic field merging. However, there is no rea- son to believe that exactly half of the magnetotail energy is transferred to the ring current, so that its contribution to Dst exactly replaces that of the deformed magnetotail.

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Typically, the ﬁrst change seen in Dst at the beginning of a storm is a positive swing, due to an increase in the dayside magnetopause current from an enhanced solar wind dynamic pressure. The injection of energetic particles into the ring current, resulting in prolonged negative Dst values, occurs primarily on the nightside. It takes time, of the order of an hour, for newly merged IMF and geomagnetic ﬁeld lines to convect to the nightside of the planet and diﬀuse through the magnetotail, at which point the open geomagnetic ﬁeld reconnects (closes) and undergoes dipolarization. Many formulas are available that describe the negative excursions of Dst in terms of the solar wind parameters; some of these are listed in section 5. These “**coupling** **functions**” are often (incorrectly) assumed to directly represent the rate of particle injection into the ring current, but it is no coincidence that the **functions** appear to describe the rate of magnetic ﬁeld merging on the dayside. Following the explanation of Siscoe and Petschek [1997], the merging of magnetic ﬂux on the dayside results in a negative swing in Dst ﬁrst due to the deformation of the magnetotail and the enhanced cross-tail current. Around an hour later, when this merged ﬂux reconnects on the nightside, the injection of particles into the ring current oﬀsets the loss of Dst from the restored geomagnetic ﬁeld. At this point the neg- ative Dst contribution is transferred from the magnetic ﬁeld to an enhanced ring current. Recently, Vasyli¯unas [2006] points out that the deformation of the geomagnetic tail can be represented by the amount of open (merged) magnetic ﬂux, which is largely piled up in the magnetotail. If during tail reconnection the gain in −Dst from the ring current exactly cancels the loss from the reduced magnetotail contribution, then the cou- pling **functions**, which describe the enhancement of −Dst over the course of a storm, will be identical to the rate of dayside magnetic ﬁeld merging. However, there is no reason to believe that exactly half of the mag- netotail energy is transferred to the ring current, so that its contribution to Dst exactly replaces that of the deformed magnetotail.

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The heavy quarks are valence quarks which propagate through the thermal medium ac- cording to the solution of the initial value problem in NRQCD. Their propagators do not satisfy anti-periodic thermal boundary conditions so the heavy quarks are not in thermal equilibrium with the medium. This is illustrated in the representation of the correlation function in eq. (2.7) which is manifestly not symmetric about τ = 1/2T, while the kernel, K(τ, ω), is independent of the temperature. The thermal modification of the correlators can therefore be directly attributed to the modification of the associated spectral function. The asymmetry of the hadronic correlation **functions** can be seen explicitly in figure 2. These simplifications result from replacing ω → 2m b + ω and taking the m b /T → ∞ limit

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Holographic Schwinger-Keldysh formulation Let us now turn to the gravity dual de- scription of the Schwinger-Keldysh formalism reviewed in the previous paragraph. For the asymptotically AdS spaces containing a black hole which we consider here, there exists an analog of Kruskal coordinates. Kruskal coordinates in general relativity cover the entire space- time manifold of the maximally extended Schwarzschild solution and they are well-behaved everywhere outside the physical singularity, i.e. they show no coordinate singularities as other coordinates do, e.g. at the horizon. The identity (3.1) suggests, that one has to know the explicit form of the classical action including the solution of the equation of motion for the field φ in terms of boundary values for the field in order to take derivatives of the expression on the left hand side as shown in (3.2) and get an explicit expression for the correlation func- tion. Now the main idea is to use this standard AdS/CFT prescription to get the correlation **functions** but to carefully impose boundary conditions on the gravity fields in the analog of the Kruskal time coordinate and not in the ordinary Minkowski time. These boundary con- ditions on the gravity fields will be subject of a detailed discussion on the level of two point correlators in the next section 3.1.2. Let us note here only that these boundary conditions are the point where the naive Minkowski formulation of the AdS/CFT correlator prescription fails. The reason for this is the fact that in ordinary coordinates the boundary conditions in Euclidean space-time are completely fixed by the requirement of regularity but this is not the case in the Minkowski version. For example a scalar gravity field has to fulfill a second order equation of motion and therefore one needs to fix two boundary conditions. One of these is fixed by the boundary data φ | bdy = φ bdy . The other condition is imposed at the horizon where

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Therefore, hybrid method is widely adopted for optimization problems with many local minima [9, 10]. In this work two cost **functions** were formulated to avoid a trap of local optimum, where the entries of **coupling** matrix were used as independent variables in the optimization process. The gradient-based hybrid method combines fmincon with solvopt algorithm, which performs a local search within only a limited number of iterations with good ﬁtness values. This hybrid method can provide good accuracy to ﬁnd the ﬁnal solution, while maintaining the speed of search. Compared with conventional techniques, the initial **coupling** matrix with random values rather than with guess has the potential to be useful in the synthesis of **coupling** matrices when local optimization methods, which rely upon on the provision of a good initial guess at the solution, fail. The proposed method was veriﬁed and illustrated by numerical tests which show the eﬀectiveness of the procedure developed in the paper. 2. POLYNOMIAL DEFINITIONS AND CIRCUIT MODEL OF FILTERS

Abstract. Agricultural management practices influence soil structure, but the characterization of these modifications and consequences are still not completely understood. In this study, we combine X-ray microtomography with retention and hydraulic conductivity measurements in the context of tillage simplification. First, this association is used to validate microtomography information with a quick scan method. Secondly, X-ray microtomography is used to increase our knowledge of soil structural differences. Notably, we show a good match for retention and conductivity **functions** be- tween macroscopic measurements and microtomographic in- formation. Microtomography refines the shape of the reten- tion function, highlighting the presence of a secondary pore system in our soils. Analysis of structural parameters for these pores appears to be of interest and offers additional clues for soil structure differentiation, through – among oth- ers – connectivity and tortuosity parameters. These elements make microtomography a highly competitive instrument for routine soil characterization.

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Myosins are actin-activated ATPases that convert the chemical energy stored in ATP into the mechanical swing of its lever-arm. The members of the myosin family exhibit a wide range of cellular **functions**. Myosin Ib (myo1b) is single-headed and may link the cell membrane to the actin network acting as a tension sensor; while myosin V (myoV) is double-headed and can act as a cargo transporter in cells. The very different **functions** of myo1b and myoV arise from differences in their chemical and mechanical activities. We examined the chemomechanical properties of myo1b using stopped flow and optical trap experiments, from which were determined mechanical step sizes and kinetic rates associated with the chemomechanical steps of the myo1b crossbridge cycle. Most importantly, we found that the rates are slow and the rate associated with ADP release during actin attachment is greatly decreased by force, which could allow it to act as a tension sensor. These kinetic rates and force sensitivity of myo1b are strongly regulated by the signaling molecule, calcium. MyoV steps along actin in a complex and dense cellular environment; how this is done can be understood from its intrinsic stepping behavior as measured from the changes in lever-arm conformation as it steps along actin using single molecule techniques and a novel analytic tool I developed. From this we find that myoV mainly walks straight along actin, but can take steps around the long axis of actin. The frequency of these azimuthal steps depends on the length of the myoV lever-arm.

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with basal body positioning defects, suggesting that the precise spatial pattern of PAK activation might also be important. Together with our previous findings, these results reveal multiple crucial **functions** of PAK signaling in hair bundle morphogenesis. During the early phase of hair bundle formation, cortical PAK activity serves to position the basal body and direct hair bundle orientation. Subsequently, PAK signaling is required for the cohesion of the nascent hair bundles (Grimsley-Myers et al., 2009). As a motor molecule, Kif3a may regulate PAK activity through direct or indirect mechanisms. As our in vitro results indicate that both Rac and PAK signaling are required for basal body positioning, we speculate that Kif3a may transport a cargo that in turn regulates activation of Rac GTPases at cortical locations. Alternatively, Kif3a may transport a cargo that mediates the interaction between microtubules and cortical actin, thereby regulating cortical PAK activity.

The main objective of this paper is to couple the project domain decomposition method with asymmetric collocation method based on radial basis **functions** to solve Poisson problem. The paper is organized as follows. In Section 2, we use a general elliptic problem to analyze the **coupling** method of PDM and RBF-based meshless collocation method. In Section 3, several numerical examples, consisting of pure mathematical test and computation of the ﬁelds of an inﬁnite square grounding metal slot, are given to validate the proposed method. Conclusions are drawn in the ﬁnal section, Section 4.

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shifted to upper frequencies. Specially, when sl = 10 mm, that is the ground plane is truncation, however, the mutual **coupling** is more than −15 dB in WLAN 5.2/5.8 band. Note that the case of 8 mm is the proposed optimal design, which is approximately 0.25λ. Effects of the slot separation (sd) on the isolation are studied in Figure 3(b). Results for three different slot separation of sd = 1.5, 4, 6.5 mm with sl = 8 mm are presented. Obvious variations in mutual **coupling** are also seen when sd varies. With a increase in sd, the frequency of lowest mutual **coupling** is shifted to lower frequencies. It can be observed that sd = 4 mm is the optimal design.

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If the problem is solved by direct methods, for example, by the Galerkin method, the solution convergence can be better if the basis **functions** correctly describe special features of E-ﬁeld distribution in a slot. So, it is very important to choose proper **functions** for approximation of the E -ﬁeld distribution inside the slot. A trigonometric basis, Chebyshev or Gegenbauer polynomials are often used for slots homogeneously ﬁlled with dielectric or for unﬁlled slots. The distribution of the slot magnetic current in the ﬁrst approximation of an asymptotic method was presented in [20] as a sum of symmetric and asymmetric **functions**, which take into account the structure of the exciting ﬁeld.

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by a continuous local martingale. It has arisen in statistical studies: see [2] for an application to polynomial regression. There are potential applications (for example, to model relativistic diffusion of photons [3], also to model the motion of a tracer in fluid flow [12]); however its main interest is as a simple model for a non-elliptic diffusion. **Coupling** properties have been studied in [5] and numerically in [12], also (when supplemented by further iterated time integrals) in [17]; this problem provides the simplest non-trivial example of a diffusion with nilpotent group symmetries which admits a Markovian or immersed **coupling**. Most interest to date has focussed on the classical two-component Kolmogorov diffusion. Here we follow [17] in considering **coupling** for (in the most part) the case of index k (k iterated time integrals), since the general structure adds clarity to the arguments.

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In [12] a numerical solution of the full Hamiltonian describing the donor electron is proposed, obtained through exact diagonalization in the basis of the undoped crystal Bloch **functions**—the band minima basis method (BMB). Predictions made using this method are not limited by any of the EMT approximations, but only by the convergence and numerical accuracy of the computation, and by the validity of the pseudopotential used [25]. Such detailed and numerically intensive microscopic calculations predict that the strength of the **coupling** is reduced, and its oscillations have their amplitude decreased as compared to the calculations performed with the multivalley wave function involving Kohn and Luttinger envelopes [14]. This happens since the KL approach includes the correct valley structure without taking into account its consequences on the donor Hamiltonian, i.e., central cell corrections. However, all calculations so far still fail to get a reliable description of the electronic density in the region close to the donor nucleus.

second- and third-order expressions in sections 2 and 3 have been obtained by fitting the coefficients not written as fractions in the non-distribution parts to the exact coefficient **functions** at x ≥ 10 −6 . Where useful, the coefficients of δ (1−x) have been adjusted (even from zero) to fine-tune the accuracy of Mellin moments and convolutions. The resulting accuracy of Eqs. (2.6) – (2.11) and (3.3) – (3.8) and their convolutions with typical quark distributions of hadrons is 0.1% or better except where the **functions** are very small. Towards smaller x the accuracy deteriorates, but the results are still accurate to about 1% and 3% at x = 10 −8 and x = 10 −10 , respectively.