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The rich multifractal properties of fluid turbulence illustrated by the work of Parisi and Frisch are related explicitly to Leray's weak solutions of the three-dimensional Navier-Stokes equations. Directly from this correspondence it is found that the set on which energy dissipates, $\mathbb{F}_{m}$, has a range of dimensions $\Dim=3/m$ ($1 \leq m \leq \infty$), and a corresponding range of sub-Kolmogorov dissipation inverse length scales $L\eta_{m}^{-1} \leq Re^{3/(1+\Dim)}$ spanning $Re^{3/4}$ to $Re^{3}$. Correspondingly, the multifractal model scaling parameter $h$, must obey $h \geq h_{min}$ with $-\twothirds \leq h_{min} \leq \third$.

The Rayleigh-Taylor (RT) instability occurs at the interface between two fluids of different densities when the lighter fluid pushes against the heavier. For example, RT instability fingers are evident in the Crab Nebula. It plays a critical role in all known forms of fusion. The variable density model (VDM), which is comprised of a set of partial differential equations, best models the flow in the mixing layer. This paper shows that the VDM can be exactly transformed into the inhomogeneous, incompressible, forced Navier-Stokes equations.

Shell models have found wide application in the study of hydrodynamic turbulence because they are easily solved numerically even at very large Reynolds numbers. Although bereft of spatial variation, they accurately reproduce the main statistical properties of fully-developed homogeneous and isotropic turbulence. Moreover, they enjoy regularity properties which still remain open for the three-dimensional (3D) Navier-Stokes equations (NSEs). The goal of this study is to make a rigorous comparison between shell models and the NSEs. It turns out that only the estimate of the mean energy dissipation rate is the same in both systems. The estimates of the velocity and its higher-order derivatives display a weaker Reynolds number dependence for shell models than for the 3D NSEs. Indeed, the velocity-derivative estimates for shell models are found to be equivalent to those corresponding to a velocity gradient averaged version of the 3D Navier-Stokes equations (VGA-NSEs), while the velocity estimates are even milder. Numerical simulations over a wide range of Reynolds numbers confirm the estimates for shell models., 22 pages, 3 figures

Visual manifestations of intermittency in computations of three dimensional Navier-Stokes fluid turbulence appear as the low-dimensional or `thin' filamentary sets on which vorticity and strain accumulate as energy cascades down to small scales. In order to study this phenomenon, the first task of this paper is to investigate how weak solutions of the Navier-Stokes equations can be associated with a cascade and, as a consequence, with an infinite sequence of inverse length scales. It turns out that this sequence converges to a finite limit. The second task is to show how these results scale with integer dimension $D=1,\,2,\,3$ and, in the light of the occurrence of thin sets, to discuss the mechanism of how the fluid might find the smoothest, most dissipative class of solutions rather than the most singular., 1 figure containing 2 sub-figures

The Beale–Kato–Majda theorem contains a single criterion that controls the behaviour of solutions of the 3 D incompressible Euler equations. Versions of this theorem are discussed in terms of the regularity issues surrounding the 3 D incompressible Euler and Navier–Stokes equations together with a phase-field model for the statistical mechanics of binary mixtures called the 3 D Cahn–Hilliard–Navier–Stokes (CHNS) equations. A theorem of BKM-type is established for the CHNS equations for the full parameter range. Moreover, for this latter set, it is shown that there exists a Reynolds number and a bound on the energy-dissipation rate that, remarkably, reproduces the R e 3 ∕ 4 upper bound on the inverse Kolmogorov length normally associated with the Navier–Stokes equations alone. An alternative length-scale is introduced and discussed, together with a set of pseudo-spectral computations on a 12 8 3 grid.

Nonequilibrium information erasure below kTln2 Thin sets in three-dimensional Navier-Stokes turbulence Effects of time defects in modulated systems Subnuclear particle spectra without explicit quark masses?

Using the scale invariance of the Navier-Stokes equations to define appropriate space-and-time-averaged inverse length scales associated with weak solutions of the $3D$ Navier-Stokes equations, an infinite `chessboard' of estimates for these inverse length scales is displayed in terms of labels $(n,\,m)$ corresponding to $n$ derivatives of the velocity field in $L^{2m}$. The $(1,\,1)$ position corresponds to the inverse Kolmogorov length $Re^{3/4}$. These estimates ultimately converge to a finite limit, $Re^3$, as $n,\,m\to \infty$, although this limit is too large to lie within the physical validity of the equations for realistically large Reynolds numbers. Moreover, all the known time-averaged estimates for weak solutions can be rolled into one single estimate, labelled by $(n,\,m)$. In contrast, those required for strong solutions to exist can be written in another single estimate, also labelled by $(n,\,m)$, the only difference being a factor of 2 in the exponent. This appears to be a generalisation of the Prodi-Serrin conditions for $n\geq 1$., Comment: 2 tables, no figures

Low-Reynolds-number polymer solutions exhibit a chaotic behaviour known as 'elastic turbulence' when the Weissenberg number exceeds a critical value. The two-dimensional Oldroyd-B model is the simplest constitutive model that reproduces this phenomenon. To make a practical estimate of the resolution scale of the dynamics requires an assumption that an attractor of the Oldroyd-B model exists : numerical simulations show that the quantities on which this assumption is based are bounded. We estimate the Lyapunov dimension of this assumed attractor as a function of the Weissenberg number by combining a mathematical analysis of the model with direct numerical simulations., 11 pages, 2 figures

The 3D incompressible Euler equations with a passive scalar $\theta$ are considered in a smooth domain $\Omega\subset \mathbb{R}^{3}$ with no-normal-flow boundary conditions $\bu\cdot\bhn|_{\partial\Omega} = 0$. It is shown that smooth solutions blow up in a finite time if a null (zero) point develops in the vector $\bB = \nabla q\times\nabla\theta$, provided $\bB$ has no null points initially\,: $\bom = \mbox{curl}\,\bu$ is the vorticity and $q = \bom\cdot\nabla\theta$ is a potential vorticity. The presence of the passive scalar concentration $\theta$ is an essential component of this criterion in detecting the formation of a singularity. The problem is discussed in the light of a kinematic result by Graham and Henyey (2000) on the non-existence of Clebsch potentials in the neighbourhood of null points., Comment: 5 pages, no figures

We study the three-dimensional, incompressible, non-hydrostatic Boussinesq fluid equations, which are applicable to the dynamics of the oceans and atmosphere. These equations describe the interplay between velocity and buoyancy in a rotating frame. A hierarchy of dynamical variables is introduced whose members $\Omega_{m}(t)$ ($1 \leq m < \infty$) are made up from the respective sum of the $L^{2m}$-norms of vorticity and the density gradient. Each $\Omega_{m}(t)$ has a lower bound in terms of the inverse Rossby number, $Ro^{-1}$, that turns out to be crucial to the argument. For convenience, the $\Omega_{m}$ are also scaled into a new set of variables $D_{m}(t)$. By assuming the existence and uniqueness of solutions, conditional upper bounds are found on the $D_{m}(t)$ in terms of $Ro^{-1}$ and the Reynolds number $Re$. These upper bounds vary across bands in the $\{D_{1},\,D_{m}\}$ phase plane. The boundaries of these bands depend subtly upon $Ro^{-1}$, $Re$, and the inverse Froude number $Fr^{-1}$. For example, solutions in the lower band conditionally live in an absorbing ball in which the maximum value of $\Omega_{1}$ deviates from $Re^{3/4}$ as a function of $Ro^{-1},\,Re$ and $Fr^{-1}$., Comment: 24 pages, 3 figures and 1 table

A series of numerical experiments is suggested for the three-dimensional Navier-Stokes and Euler equations on a periodic domain based on a set of $L^{2m}$-norms of vorticity $\Omega_{m}$ for $m\geq 1$. These are scaled to form the dimensionless sequence $D_{m}= (\varpi_{0}^{-1}\Omega_{m})^{\alpha_{m}}$ where $\varpi_{0}$ is a constant frequency and $\alpha_{m} = 2m/(4m-3)$. A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of $D_{m}$ and $D_{m+1}$ while another is furnished by imposing a critical lower bound on $\int_{0}^{t}D_{m}\,d\tau$. The behaviour of the $D_{m}$ is also important in the Euler case in suggesting a method by which possible singular behaviour might also be tested., Comment: To appear in the Procedia IUTAM volume of papers Topological Fluid Dynamics

We have undertaken a study of the complex Lorenz equations x = −σx + σy . y = (r − z)x − ay . z = −bz + 12(x∗y + xy∗) . where x and y are complex and z is real. The complex parameters r and a are defined by r = r1 + ir2; a = 1 − ie and σ and b are real. Behaviour remarkably different from the real Lorenz model occurs. Only the origin is a fixed point except for the special case e + r2 = 0. We have been able to determine analytically two critical values of r1, namely r1c and r1c . The origin is a stable fixed point for 0 r1c, a Hopf bifurcation to a limit cycle occurs. We have an exact analytic solution for this limit cycle which is always stable if σ + 1 then this limit is only stable in the region r1c rlc, a transition to a finite amplitude oscillation about the limit cycle occurs. The nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the Stuart-Landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency. This latter bifurcation is either a sub- or super-critical Hopf-like bifurcation to a doubly periodic motion, the direction of bifurcation depending on the parameter values. The nature of the bifurcation is complicated by the existence of a zero eigenvalue.

We summarize some recently obtained results on real and complex Lorenz equations and discuss their possible significance in relation to real fluid dynamical processes.

Pedlosky (1972 a ) has derived a set of envelope equations which describe the nonlinear evolution of wave-packets in a marginally stable or unstable baroclinic shear flow using a quasi-geostrophic two-layer model on a beta-plane. We show that these envelope equations can be transformed either into the sine-Gordon equation ( ϕ ξז = sin ϕ ) for real amplitude or the so-called self-induced transparency (s. i. t.) equations of nonlinear optics for complex amplitude. The initial value problem for both equations can be solved analytically by use of the inverse scattering method and it is well known that a variety of soliton solutions exist. This two-layer system is therefore a new example of a physical system exhibiting soliton behaviour whose evolution is predictable for all time. Solutions of the sine-Gordon equation on a spatially periodic domain are more relevant for comparison with annulus experiments. Numerical integrations suggest that initial data recurs almost periodically in time thereby giving qualitative agreement with results from rotating fluid annulus experiments. The significance of such solutions to meteorology and oceanography is discussed.

Initially, this paper reviews the mathematical issues surrounding hydrostatic primitive equations (HPEs) and non-hydrostatic primitive equations (NPEs) that have been used extensively in numerical weather prediction and climate modelling. A new impetus has been provided by a recent proof of the existence and uniqueness of solutions of viscous HPEs on a cylinder with Neumann-like boundary conditions on the top and bottom. In contrast, the regularity of solutions of NPEs remains an open question. With this HPE regularity result in mind, the second issue examined in this paper is whether extreme events are allowed to arise spontaneously in their solutions. Such events could include, for example, the sudden appearance and disappearance of locally intense fronts that do not involve deep convection. Analytical methods are used to show that for viscous HPEs, the creation of small-scale structures is allowed locally in space and time at sizes that scale inversely with the Reynolds number.

The three-dimensional Euler equations have stood for a quarter of a millenium as a challenge to mathematicians and physicists. While much has been discovered, the nature of solutions is still largely a mystery. This paper surveys some of the issues, such as singularity formation, that have cost so much effort in the last 25 years. In this light we review the Beale‐Kato‐Majda theorem and its consequences and then list some of the results of numerical experiments that have been attempted. A different line of endeavour focuses on work concerning the pressure Hessian and how it may be used and modelled. The Euler equations are finally discussed in terms of their membership of a class of general Lagrangian evolution equations. Using Hamilton’s quaternions, these are reformulated in an elegant manner to describe the motion and rotation of fluid particles. c 2007 Elsevier B.V. All rights reserved.

One of the outstanding open questions in modern applied mathematics is whether solutions of the incompressible Euler equations develop a singularity in the vorticity field in a finite time. This paper briefly reviews some of the issues concerning this problem, together with some observations that may suggest that it may be more subtle than first thought.

Estimates for the three -models known as the LANS-, Leray- and Bardina models are found in terms a Reynolds number associated with a Navier-Stokes velocity field. They are tabulated for comparative purposes and show clearly that all estimates for the Leray- model are smaller than those for the LANS- and Bardina models.

We consider the time dependence of a hierarchy of scaled $L^{2m}$-norms $D_{m,\unicode[STIX]{x1D714}}$ and $D_{m,\unicode[STIX]{x1D703}}$ of the vorticity $\unicode[STIX]{x1D74E}=\unicode[STIX]{x1D735}\times \boldsymbol{u}$ and the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$, where $\unicode[STIX]{x1D703}=\log (\unicode[STIX]{x1D70C}^{\ast }/\unicode[STIX]{x1D70C}_{0}^{\ast })$, in a buoyancy-driven turbulent flow as simulated by Livescu & Ristorcelli (J. Fluid Mech., vol. 591, 2007, pp. 43–71). Here, $\unicode[STIX]{x1D70C}^{\ast }(\boldsymbol{x},t)$ is the composition density of a mixture of two incompressible miscible fluids with fluid densities $\unicode[STIX]{x1D70C}_{2}^{\ast }>\unicode[STIX]{x1D70C}_{1}^{\ast }$, and $\unicode[STIX]{x1D70C}_{0}^{\ast }$ is a reference normalization density. Using data from the publicly available Johns Hopkins turbulence database, we present evidence that the $L^{2}$-spatial average of the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ can reach extremely large values at intermediate times, even in flows with low Atwood number $At=(\unicode[STIX]{x1D70C}_{2}^{\ast }-\unicode[STIX]{x1D70C}_{1}^{\ast })/(\unicode[STIX]{x1D70C}_{2}^{\ast }+\unicode[STIX]{x1D70C}_{1}^{\ast })=0.05$, implying that very strong mixing of the density field at small scales can arise in buoyancy-driven turbulence. This large growth raises the possibility that the density gradient $\unicode[STIX]{x1D735}\unicode[STIX]{x1D703}$ might blow up in a finite time.

We derive and study the hyperbolic reduced Maxwell-Bloch (HRMB) equations, a simplified model for the dynamics of strongly correlated Bose-Einstein condensates (BECs), and in particular for the interaction between the BEC atoms and its evaporated atoms under the strong interactions. This equation is one among four, which are proven to be integrable via the existence of a Lax pair, and thus the method of inverse scattering transform. Another equation is the reduced Maxwell-Bloch equation of quantum optics and the two others do not have physical applications yet. By studying the linear stability of the constant solutions of these four equations we observe various regimes, from stable, to modulational unstable, and unstable at all frequencies. The finite-dimensional reduction of the RMB equations is also used to give more insight into the constant solutions of these equations. From this study, we find that the HRMB equation arising from strongly correlated BECs is stable under the particular condition that the transition rate of evaporation is not too large compared to the number of evaporated atoms. We then derive explicit soliton solutions of the RMB equations and use numerical simulations to show collisions of solitons and kink solitons.

We build on recent developments in the study of fluid turbulence [Gibbon \textit{et al.} Nonlinearity 27, 2605 (2014)] to define suitably scaled, order-$m$ moments, $D_m^{\pm}$, of $\omega^\pm= \omega \pm j$, where $\omega$ and $j$ are, respectively, the vorticity and current density in three-dimensional magnetohydrodynamics (MHD). We show by mathematical analysis, for unit magnetic Prandtl number $P_M$, how these moments can be used to identify three possible regimes for solutions of the MHD equations; these regimes are specified by inequalities for $D_m^{\pm}$ and $D_1^{\pm}$. We then compare our mathematical results with those from our direct numerical simulations (DNSs) and thus demonstrate that 3D MHD turbulence is like its fluid-turbulence counterpart insofar as all solutions, which we have investigated, remain in \textit{only one of these regimes}; this regime has depleted nonlinearity. We examine the implications of our results for the exponents $q^{\pm}$ that characterize the power-law dependences of the energy spectra $\mathcal{E}^{\pm}(k)$ on the wave number $k$, in the inertial range of scales. We also comment on (a) the generalization of our results to the case $P_M \neq 1$ and (b) the relation between $D_m^{\pm}$ and the order-$m$ moments of gradients of hydrodynamic fields, which are used in characterizing intermittency in turbulent flows., Comment: 14 pages, 3 figures

The old idea that an infinite dimensional dynamical system may have its high modes or frequencies slaved to low modes or frequencies is re-visited in the context of the $3D$ Navier-Stokes equations. A set of dimensionless frequencies $\{\tilde{\Omega}_{m}(t)\}$ are used which are based on $L^{2m}$-norms of the vorticity. To avoid using derivatives a closure is assumed that suggests that the $\tilde{\Omega}_{m}$ ($m>1$) are slaved to $\tilde{\Omega}_{1}$ (the global enstrophy) in the form $\tilde{\Omega}_{m} = \tilde{\Omega}_{1}\mathcal{F}_{m}(\tilde{\Omega}_{1})$. This is shaped by the constraint of two H\"older inequalities and a time average from which emerges a form for $\mathcal{F}_{m}$ which has been observed in previous numerical Navier-Stokes and MHD simulations. When written as a phase plane in a scaled form, this relation is parametrized by a set of functions $1 \leq \lambda_{m}(\tau) \leq 4$, where curves of constant $\lambda_{m}$ form the boundaries between tongue-shaped regions. In regions where $2.5 \leq \lambda_{m} \leq 4$ and $1 \leq \lambda_{m} \leq 2$ the Navier-Stokes equations are shown to be regular\,: numerical simulations appear to lie in the latter region. Only in the central region $2 < \lambda_{m} < 2.5$ has no proof of regularity been found., Comment: 10 pages; 2 figures

An upper bound on the mixing efficiency is derived for a passive scalar under the influence of advection and diffusion with a body source. For a given stirring velocity field, the mixing efficiency is measured in terms of an equivalent diffusivity, which is the molecular diffusivity that would be required to achieve the same level of fluctuations in the scalar concentration in the absence of stirring, for the same source distribution. The bound on the equivalent diffusivity depends only on the functional "shape" of both the source and the advecting field. Direct numerical simulations performed for a simple advecting flow to test the bounds are reported., Comment: 10 pages, 2 figures, JFM format (included)

For the class of cylindrically symmetric velocity fields U (r, z, t) = {u(r, t), v(r, t), zγ(r, t)}, two infinite energy exact solutions of the three-dimensional incompressible Euler equations are exhibited that blow up at every point in space in finite time. The first solution is embedded within the second as a special case and in both cases v = 0. Both solutions represent three-dimensional vortices which take the form of hollow cylinders for which the vorticity vector is ω = (0, ωθ, 0). An analysis on characteristics shows how more general solutions can be constructed and analysed.

By considering the three-dimensional incompressible Euler equations, a 4-vector ζ is constructed out of a combination of scalar and vector products of the vorticity ω and the vortex stretching vector ω ·∇ u = Sω. The evolution equation for ζ can then be cast naturally into a quaternionic Riccati equation. This is easily transformed into a quaternionic zero-eigenvalue Schrodinger equation whose potential depends on the Hessian matrix of the pressure. With minor modifications, this system can alternatively be written in complex notation. An infinite set of solutions of scalar zero-eigenvalue Schrodinger equations has been found by Adler and Moser, which are discussed in the context of the present problem. Similarly, when the equations for ideal magneto-hydrodynamics (MHD) are written in Elsasser variables, a pair of 4-vectors ζ ± are governed by coupled

Repetitive tapping is used to investigate temporal perception, memory, and reproduction. Intertap intervals and their variability, arise from cognitive and motor processes during the task. We used a measure of readiness potential onset to determine motor component latency during the timed interval. Subjects performed a paced, two-handed tapping task at four target intervals (1.5-2.75 s). Overall latency of production increased with increasing ISI, as did variability across target interval, conforming to a generalized Weber's law. In contrast, average motor latency was roughly 0.5 s across ISI. This constant motor latency may also indicate constant variability attributable to motor processing.

A class of stretched solutions of the equations for three-dimensional, incompressible, ideal magneto-hydrodynamics (MHD) is studied. In Elsasser variables, V ± = U ± B, these solutions have the form V ± = v ± ,v ± 3 where v ± = v ± (x, y, t)and v ± 3 (x, y, z, t)= zγ ± (x, y, t)+ β ± (x, y, t) .T wodimensional partial differential equations for γ ± , v ± and β ± are obtained that are valid in a tubular domain which is infinite in the z-direction with periodic cross section. Pseudo-spectral computations of these equations provide evidence for a blow-up in finite time in the above variables. This apparent blow-up is an infinite energy process that gives rise to certain subtleties; while all the variables appear to blow-up simultaneously, the two-dimensional part of the magnetic field b = 1 v + − v − blows up at a very late stage. This singularity in b is hard to detect numerically but supporting analytical evidence of a Lagrangian nature is provided for its existence. In three dimensions these solutions correspond to magnetic vortices developing along the axis of the tube prior to breakdown.

In associative models of simple conditioning, conductive connections (associations, Hebbian synapses) are strengthened by the repetitive temporal pairing of stimuli. The associations cause the animal to behave more adaptively, but they do not encode information about objectively specifiable properties of the conditioning experience. In information processing (computational) models, the temporal intervals in that experience are timed and the results recorded in memory for later use in computations that determine the decisions whether and when to respond to the conditioned stimulus. The predictions of these latter models depend on the ratios of remembered and currently experienced temporal intervals; hence, they are time-scale invariant. Two examples of empirical time-scale invariance are described: Neither the delay of reinforcement nor the ratio of reinforced to unreinforced presentations of the conditioned stimulus affects rates of acquisition and extinction. Time-scale invariance has far-reaching implications for models of the processes that underlie conditioning, for example, models of Hebbian synapses.

When humans and animals compare two numbers, responding is faster and more accurate with increasing numerical disparity and decreasing numerical size. Researchers explaining these distance and size effects often assume that the subjective number continuum is logarithmically compressed. An alternative hypothesis is that the subjective number continuum is linear, but positions farther along it are proportionately fuzzier, that is, less precisely located. These two hypotheses have been treated as functionally equivalent because of their similar empirical predictions. The current experiment sought to resolve this issue with a paradigm originally developed to address the subjective representation of time (time left). In our adaptation, pigeons were required to compare a constant number with the number remaining after a numerical subtraction. Our results indicate that subjective number is linearly, not logarithmically, related to objective number.

We study numerically a class of stretched solutions of the three-dimensional Euler and Navier–Stokes equations identified by Gibbon, Fokas, and Doering (1999). Pseudo-spectral computations of a Euler flow starting from a simple smooth initial condition suggests a breakdown in finite time. Moreover, this singularity apparently persists in the Navier–Stokes case. Independent evidence for the existence of a singularity is given by a Taylor series expansion in time. The mechanism underlying the formation of this singularity is the two-dimensionalization of the vorticity vector under strong compression; that is, the intensification of the azimuthal components associated with the diminishing of the axial component. It is suggested that the hollowing of the vortex accompanying this phenomenon may have some relevance to studies in vortex breakdown.

The effects of signal modality on duration classification in college students were studied with the duration bisection task. When auditory and visual signals were presented in the same test session and shared common anchor durations, visual signals were classified as shorter than equivalent duration auditory signals. This occurred when auditory and visual signals were presented sequentially in the same test session and when presented simultaneously but asynchronously. Presentation of a single modality signal within a test session, or both modalities but with different anchor durations did not result in classification differences. The authors posit a model in which auditory and visual signals drive an internal clock at different rates. The clock rate difference is due to an attentional effect on the mode switch and is revealed only when the memories for the short and long anchor durations consist of a mix of contributions from accumulations generated by both the fast auditory and slower visual clock rates. When this occurs auditory signals seem longer than visual signals relative to the composite memory representation.

A well known limitation with stretched vortex solutions of the 3D Navier–Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which is stretched by a strain field that is decoupled from them. It is shown here that these drawbacks can be partially circumvented by considering a class of velocity fields of the type u=(u1(x,y,t),u2(x,y,t),γ(x,y,t)z+W(x,y,t)) where u1,u2,γ and W are functions of x,y and t but not z. It turns out that the equations for the third component of vorticity ω3 and W decouple. More specifically, solutions of Burgers type can be constructed by introducing a strain field into u such that u = −(γ/2)x−(γ/2)y,γz + −ψ y ,ψ x ,W . The strain rate, γ(t), is solely a function of time and is related to the pressure via a Riccati equation γ +γ 2 +p zz (t)=0 . A constraint on pzz(t) is that it must be spatially uniform. The decoupling of ω3 and W allows the equation for ω3 to be mapped to the usual general 2D problem through the use of Lundgren’s transformation, while that for W can be mapped to the equation of a 2D passive scalar. When ω3 stretches then W compresses and vice versa. Various solutions for W are discussed and some 2π-periodic θ-dependent solutions for W are presented which take the form of a convergent power series in a similarity variable. Hence the vorticity ω = r −1 W θ ,−W r ,ω 3 has nonzero components in the azimuthal and radial as well as the axial directions. For the Euler problem, the equation for W can sustain a vortex sheet type of solution where jumps in W occur when θ passes through multiples of 2π.

Timing of conditioned keypecking was investigated using a delay autoshaping procedure. In two experiments, the CS-US interval (ISI) was varied and the temporal pattern of responding during unreinforced probe trials extending beyond the reinforced ISI was recorded. Both experiments showed temporal control of keypecking at ISIs of 4, 8, and 16 sec, regardless of whether the probe duration was held constant (Experiment 1) or covaried with the ISI (Experiment 2). Analyses showed that the timing of keypecking was scalar. Increased responding near the end of the probe was due possibly to independent timing of the probe duration. In a third experiment, a group of subjects trained at an ISI of 8 sec were extinguished by presentation of only probe trials. Although the maximal response rate declined progressively, timing of keypecking did not change.

Dysfunction of the basal ganglia and the brain nuclei interconnected with them leads to disturbances of movement and cognition, including disordered timing of movement and perceptual timing deflcits. Patients with Parkinson's disease (PD) were studied in temporal reproduction tasks. We examined PD patients when brain dopamine (DA) transmission was impaired (OFF state) and when DA transmission was reestablished, at the time of maximal clinical beneflt following administration of levodopa + apomorphine (ON state). Patients reproduced target times of 8 and 21 sec trained in blocked trials with the peak interval procedure, which were veridical in the ON state, comparable to normative performance by healthy young and aged controls (Experiment 1). In the OFF state, temporal reproduction was impaired in both accuracy and precision (variance). The 8-sec signal was reproduced as longer and the 21-sec signal was reproduced as shorter than they actually were (Experiment 1). This fimigrationfl effect was dependent upon training of two different durations. When PD patients were trained on 21 sec only (Experiment 2), they showed a reproduction error in the long direction, opposite to the error produced under the dual training condition of Experiment 1. The results are discussed as a mutual attraction between temporal processing systems, in memory and clock stages, when dopaminergic regulation in the striatum is dysfunctional.

The properties of the internal clock, temporal memory, and decision processes used to time short durations were investigated. The peak-interval procedure was used to evaluate the timing of 8-, 12-, and 21-s intervals, and analyses were conducted on the mean response functions and on individual trials. A distractor task prevented counting, and visual feedback on accuracy and precision was provided after each trial. Mean response distributions were (a) centered at the appropriate real-time criteria, (b) highly symmetrical, and (c) scalar in their variability. Analysis of individual trials indicated more memory variability relative to response threshold variability. Taken together, these results demonstrate that humans show the same qualitative timing properties that other animals do, but with some quantitative differences.

It is shown that a rigorous estimate for the fractal dimension of the global attractor of the three-dimensional incompressible Navier - Stokes equations on periodic boundary conditions is given by where L is the box length and is a Kolmogorov length defined by with the energy dissipation rate given by . By interpreting the dimension of the attractor as the number of degrees of freedom of the system, we obtain an estimate for an average natural length scale for the flow which is given by

The issue of intermittency in numerical solutions of the 3D Navier–Stokes equations on a periodic box ${[0, L] }^{3} $ is addressed through four sets of numerical simulations that calculate a new set of variables defined by ${D}_{m} (t)= {({ \varpi }_{0}^{- 1} {\Omega }_{m} )}^{{\alpha }_{m} } $ for $1\leq m\leq \infty $ where ${\alpha }_{m} = 2m/ (4m- 3)$ and ${[{\Omega }_{m} (t)] }^{2m} = {L}^{- 3} \int \nolimits _{\mathscr{V}} {\vert \boldsymbol{\omega} \vert }^{2m} \hspace{0.167em} \mathrm{d} V$ with ${\varpi }_{0} = \nu {L}^{- 2} $. All four simulations unexpectedly show that the ${D}_{m} $ are ordered for $m= 1, \ldots , 9$ such that ${D}_{m+ 1} \lt {D}_{m} $. Moreover, the ${D}_{m} $ squeeze together such that ${D}_{m+ 1} / {D}_{m} \nearrow 1$ as $m$ increases. The values of ${D}_{1} $ lie far above the values of the rest of the ${D}_{m} $, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of $409{6}^{3} $.

In 1988, Constantin, Foias and Temam found a sharp (to within logarithms) upper bound on the Lyapunov dimension of the attractor of the periodic 2d Navier-Stokes equations which took the form cG 2 3 (1 + log G) 1 3 , where G is the Grashof number. This result is derived by a method independent of that used to estimate the attractor dimension. It involves defining a set of time averaged squared ‘wavenumbers’ 〈κn,r2〉 (r cG(1 + log G) 1 2 for large n. When applied to the 3d Navier-Stokes equations this method also shows that for weak solutions there is an a priori upper bound on the time averaged infinite set of wavenumbers 〈κn,1〉 which takes the form L〈κ n,1 〉 ≤ c n ( L Λ K ) 4 , n ≥ 2, where ΛK is the Kolmogorov length, which is bounded from below. Other 3d Navier-Stokes results can be derived from this in a simple fashion.

An experiment with starlings was conducted to investigate the effect of variability in inter-reinforcement intervals on foraging decisions. The experimental design simulated an environment in which food was distributed in patches. Patches contained zero to four food items which could be collected by pecking at a key. All patches ended with sudden depletion. The time elapsed since the last reinforcement was the only way to detect the depletion of the patch. Once a patch was depleted, a new patch could be reached by completion of a travel requirement of 20 flights between two perches. Key pecks within a patch and the time of the last response in a patch (giving-in time) were recorded. The level of variability in the inter-reinforcement intervals was varied between different conditions. An increase in inter-reinforcement interval variability resulted in a flattening of response rate functions and giving-in time distributions, and in more asymmetry of the response functions, but not of the giving-in time distributions. Two theoretical models of decision making are presented, which differ in the assumptions about memory constraints. In one case, all inter-reinforcement intervals are remembered but in the other, only the intervals with extreme values are remembered. Both models accommodate response rates as a function of trial time, but only the second is compatible with the observed departure decision. Our results are compatible with net rate maximization.

We generalise and in certain cases improve on previous a priori estimates of Sobolev norms of solutions to the generalised complex Ginzburg-Landau equation. A set of dynamic length scales based on ratios of these norms is defined. We are able to derive lower bounds for time averages and long-time limits of these length scales. The bounds scale like the inverses of our L∞ bounds.

A comparison was made of the effects of long-term exposure to fixed-interval reinforcement with unsignaled and with signaled reinforcement omissions. When successive fixed-interval cycles ended in reinforcement, subjects showed a clear “scalloped” response pattern. When reinforcement was omitted, but a brief signal was given in lieu of reinforcement, responding in the next cycle started earlier than it did after reinforcement. When reinforcement was omitted without exteroceptive cues, response rates peaked near the time of reinforcement and then declined to a flat but substantial level. The classicreinforcement-omission effect was observed, in that total responding was greatest after uncued omission, somewhat less after cued omission, and least after reinforcement. However, response rate plotted as a function of time showed that the uncued-omission condition had a very different function from that of the cued-omission or reinforcement conditions. A failed-discrimination account of the omission effect might accommodate the three functions if the discrimination is considered to be a temporal one. The temporal-discrimination account argues that high-rate responding reflects the accumulation of subjective time proximal to the memory of the time of reinforcement. The accumulation resets completely with food reinforcement, incompletely with cues in lieu of reinforcement, and not at all in the absence of cues.