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Calibration curve plays an important role in filter paper method because it affects accuracy in soil suction values. Osmotic coefficient and relative humidity govern the osmotic suction required to develop calibration curve and equation. Most of the researchers used the readily available total (osmotic) suction values from various studies to develop the calibration curve. Very few researchers used the osmotic coefficient in calibration and studied its effect on calibration and measuring suction. In this paper, the effect of the osmotic suction proposed by various researchers on calibration of Whatman 42 Filter paper was studied. Logarithmic and exponential calibration equations were developed and compared. The NaCl solution used in the present study provides limited total suction range of 0–30,000 kPa. The total suction of clayey soil was measured to validate the calibration curves proposed in this study. The feasibility of proposed osmotic coefficients is verified by measuring total suction of soil. The total suction estimated from calibration curve using osmotic coefficients was found to be more accurate.

We investigate a modified spatial stochastic Lotka-Volterra formulation of the rock-paper-scissors model using off-lattice stochastic simulations. In this model one of the species moves preferentially in a specific direction -- the level of preference being controlled by a noise strength parameter $\eta \in [0, 1]$ ($\eta = 0$ and $\eta = 1$ corresponding to total preference and no preference, respectively) -- while the other two species have no referred direction of motion. We study the behaviour of the system starting from random initial conditions, showing that the species with asymmetric mobility has always an advantage over its predator. We also determine the optimal value of the noise strength parameter which gives the maximum advantage to that species. Finally, we find that the critical number of individuals, below which the probability of extinction becomes significant, decreases as the noise level increases, thus showing that the addition of a preferred mobility direction studied in the present paper does not favour coexistence., Comment: 5 pages, 7 figures

The report entitled "L. Liu, D. Li, F. Bai, A relative Lempel-Ziv complexity: Application to comparing biological sequences, Chem. Phys. Lett. 530 (2012) 107-112" mentions on the powerful construction of phylogenetic trees based on Lempel-Ziv algorithm. On the other hand, the method explained in the paper does not give promising result on the data set on invasive Caulerpa taxifolia in the Mediterranean Sea. The phylogenetic trees are obtained by the proposed method of the aforementioned paper in this short note. (C) 2018 Elsevier B.V. All rights reserved.

Connectivity is the safety network for biodiversity conservation because connected habitats are more effective for saving the species and ecological functions. The nature of coupling for connectivity also plays an important role in the co-existence of species in cyclic-dominance. The rock-paper-scissors game is one of the paradigmatic mathematical model in evolutionary game theory to understand the mechanism of biodiversity in cyclic-dominance. In this paper, the metapopulation model for rock-paper-scissors with mutations is presented in which the total population is divided into patches and the patches form a network of complete graph. The migration among patches is allowed through simple random walk. The replicator-mutator equations are used with the migration term. When migration is allowed then the population of the patches will synchronized and attain stable state through Hopf bifurcation. Apart form this, two phases are observed when the strategies of one of the species mutate to other two species: co-existence of all the species phase and existence of one kind of species phase. The transition from one phase to another phase is taking place due to transcritical bifurcation. The dynamics of the population of species of rock, paper, scissors is studied in the environment of homogeneous and heterogeneous mutation. Numerical simulations have been performed when mutation is allowed in all the patches (homogeneous mutation) and some of the patches (heterogeneous mutation). It has been observed that when the number of patches is increased in the case of heterogeneous mutation then the population of any of the species will not extinct and all the species will co-exist.

We realized a study of the shortest path fractal dimension in three dimensions for randomly crumpled paper balls. We took measurements between all possible combinations of pairs of points in crumpled and flat configurations, we found that a correlation between these distances exist, even more, such mean experimental value is dmin=1.2953±0.02 that coincides almost numerically with the very known 3D shortest path fractal dimension for percolation systems reported in computational simulations.

The purpose of this comment paper is to show that the results of the paper (Cui, 2021) are incorrect.

One of the simplest examples of a robust heteroclinic cycle involves three saddle equilibria: each one is unstable to the next in turn, and connections from one to the next occur within invariant subspaces. Such a situation can be described by a third-order ordinary differential equation (ODE), and typical trajectories approach each equilibrium point in turn, spending progressively longer to cycle around the three points but never stopping. This cycle has been invoked as a model of cyclic competition between populations adopting three strategies, characterised as Rock, Paper and Scissors. When spatial distribution and mobility of the populations is taken into account, waves of Rock can invade regions of Scissors, only to be invaded by Paper in turn. The dynamics is described by a set of partial differential equations (PDEs) that has travelling wave (in one dimension) and spiral (in two dimensions) solutions. In this paper, we explore how the robust heteroclinic cycle in the ODE manifests itself in the PDEs. Taking the wavespeed as a parameter, and moving into a travelling frame, the PDEs reduce to a sixth-order set of ODEs, in which travelling waves are created in a Hopf bifurcation and are destroyed in three different heteroclinic bifurcations, depending on parameters, as the travelling wave approaches the heteroclinic cycle. We explore the three different heteroclinic bifurcations, none of which have been observed in the context of robust heteroclinic cycles previously. These results are an important step towards a full understanding of the spiral patterns found in two dimensions, with possible application to travelling waves and spirals in other population dynamics models., Comment: 36 pages, 8 figures

Classic frameworks of rock-paper-scissors game have been assumed in a closed community that a density of each group is only affected by internal factors such as competition interplay among groups and reproduction itself. In real systems in ecological and social sciences, however, the survival and a change of a density of a group can be also affected by various external factors. One of common features in real population systems in ecological and social sciences is population flow that is characterized by population inflow and outflow in a group or a society, which has been usually overlooked in previous works on models of rock-paper-scissors game. In this paper, we suggest the rock-paper-scissors system by implementing population flow and investigate its effect on biodiversity. For two scenarios of either balanced or imbalanced population flow, we found that the population flow can strongly affect group diversity by exhibiting rich phenomena. In particular, while the balanced flow can only lead the persistent coexistence of all groups which accompanies a phase transition through supercritical Hopf bifurcation on different carrying simplices, the imbalanced flow strongly facilitates rich dynamics such as alternative stable survival states by exhibiting various group survival states and multistability of sole group survivals by showing not fully covered but spirally entangled basins of initial densities due to local stabilities of associated fixed points. In addition, we found that, the system can exhibit oscillatory dynamics for coexistence by relativistic interplay of population flows which can capture the robustness of the coexistence state. Applying population flow in the rock-paper-scissors system can ultimately change a community paradigm from closed to open one, and our foundation can eventually reveal that population flow can be also a significant factor on a group density which is independent to fundamental interactions among groups.

In this letter, we investigate the population dynamics in a May-Leonard formulation of the rock-paper-scissors game in which one or two species, which we shall refer to as "weak", have a reduced predation or reproduction probability. We show that in a nonspatial model the stationary solution where all three species coexist is always unstable, while in a spatial stochastic model coexistence is possible for a wide parameter space. We find, that a reduced predation probability results in a significantly higher abundance of "weak" species, in models with either one or two "weak" species, as long as the simulation lattices are sufficiently large for coexistence to prevail. On the other hand, we show that a reduced reproduction probability has a smaller impact on the abundance of "weak" species, generally leading to a slight decrease of its population size -- the increase of the population size of one of the "weak" species being more than compensated by the reduction of the other, in the two species case. We further show that the species abundances in models where both predation and reproduction probabilities are simultaneously reduced may be accurately estimated from the results obtained considering only a reduction of either the predation or the reproduction probability., 6 pages, 7 figures, to appear in EPL

In this note, we make two revisions of the paper [2]. The first one is the asymptotic behavior of the energy functional as $$t\rightarrow T$$ (see [2, Theorem 1.6]), where T is the blow-up time. The second one is the equivalent conditions for the solutions blowing up in finite time or existing globally (see [2, Theorem 1.8]).

Translation of the seminal 1877 paper by Ludwig Boltzmann which for the first time established the probabilistic basis of entropy. Includes a scientific commentary.

The aim of this paper is to discuss the main result in the paper by D.Y. Gao and X. Lu [On the extrema of a nonconvex functional with double-well potential in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a detailed study of the problem considered in that paper, pointing out the importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$ for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective function has not local extrema with the mentioned constraints for $p\in [1,4)$, and has (up to an additive constant) only a local maximizer for $p=\infty$, unlike the conclusion of the main result of the discussed paper where it is mentioned that there are (up to additive constants) two local minimizers and a local maximizer. We also show that the same conclusions are valid for the similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema of a nonconvex functional with double-well potential in higher dimensions, arXiv:1607.03995]., 12 pages; in this version we added the forgotten condition $F(x) \ne 0$ for $x\in (a,b)$ on page 3

We briefly review a history of the impact of the famous 1963 paper by E Lorenz on hydrodynamics, physics and mathematics communities on both sides of the iron curtain. This paper was an attempt to apply the ideas and methods of dynamical systems theory to the problem of weather forecast. Its major discovery was the phenomenon of chaos in dissipative dynamical systems which makes such forecasts rather problematic, if at all possible. In this connection we present some recent results which demonstrate that both a short-term and a long-term forecast are actually possible for the most chaotic dynamical (as well as for the most random, like IID and Markov chain) systems. Moreover, there is a sharp transition between the time interval where one may use a short-term forecast and the times where a long-term forecast is applicable. Finally we discuss how these findings could be incorporated into the forecast strategy outlined in the Lorenz's paper.

Guesstimation 2.0 reveals the simple and effective techniques needed to estimate virtually anything--quickly--and illustrates them using an eclectic array of problems. A stimulating follow-up to Guesstimation, this is the must-have book for anyone preparing for a job interview in technology or finance, where more and more leading businesses test applicants using estimation questions just like these. The ability to guesstimate on your feet is an essential skill to have in today's world, whether you're trying to distinguish between a billion-dollar subsidy and a trillion-dollar stimulus, a megawatt wind turbine and a gigawatt nuclear plant, or parts-per-million and parts-per-billion contaminants. Lawrence Weinstein begins with a concise tutorial on how to solve these kinds of order of magnitude problems, and then invites readers to have a go themselves. The book features dozens of problems along with helpful hints and easy-to-understand solutions. It also includes appendixes containing useful formulas and more. Guesstimation 2.0 shows how to estimate everything from how closely you can orbit a neutron star without being pulled apart by gravity, to the fuel used to transport your food from the farm to the store, to the total length of all toilet paper used in the United States. It also enables readers to answer, once and for all, the most asked environmental question of our day: paper or plastic?

In this paper, we introduce some notions of invariancy for the Ricci tensor on real hypersurfaces in complex two-plane Grassmannians G 2 ( C m + 2 ) , namely, F -invariant and invariant Ricci tensor. Using these notions, we give non-existence theorem and characterization for the special case among the real hypersurfaces of Type ( A ) in G 2 ( C m + 2 ) , respectively. Here the distribution F is defined by F = [ ξ ] ∪ D ⊥ where D ⊥ = Span { ξ 1 , ξ 2 , ξ 3 } and [ ξ ] = Span { ξ } .

One century after the seminal work by Leonor Michaelis and Maud Menten devoted to the theoretical study of the enzymatic reactions, in this paper, we give an overview of the most recent trends concerning the mathematical modeling of several enzymatic mechanisms, focusing on its asymptotic analysis, which needs the use of advanced mathematical tools, such as center manifold theory, normal forms, and bifurcation theory. Moreover, we present some perspectives, linking the models here presented with similar models, arising from different research fields.

A. Yu. Ishlinsky’s scientific publications impress one by the extremely wide scope of his scientific interests. Most of his results lie in two fundamental fields of research, mechanics of deformable bodies and mechanics of gyroscope and navigation systems; chronologically, they appeared mainly in this order. One of his first important results that became widely known was the construction of an exact theory of the Brinell test (1942). This theory dealt with determining the stress state under a rigid punch of spherical or plane shape penetrating into a perfectly plastic medium. The problem was solved in the rigorous statement in the framework of the Haar–von Karman complete plasticity hypothesis containing two relations between the principal stresses rather than one relation as in the Saint-Venant theory. The subsequent development of this problem led (1954) to construction of a general theory of plasticity which, in particular, allowed one to overcome the difficulties related to the derivation of the 3D strain equations beyond the elasticity range. A series of Ishlinsky’s papers (1940–1946) deals with imperfect elasticity, fracture theory, and plasticity theory. In this series, he constructed a description of aftereffect and relaxation generalizing the Maxwell and Thomson concepts on the basis of a linear differential relation between the stress and the strain. Plastic fracture models were also studied in these works. In particular, one of the fracture models was constructed for the example of a body whose one-dimensional strain obeys the law dσ/dt+ rσ= bde/dt+ bne, where σ is the stress and e is the strain. This model also admits a simple and illustrative interpretation as a damper with springs, which permits readily understanding the introduced fracture hypotheses and analyzing specific examples for various loading laws. Ishlinsky returned to plasticity and fracture theories in 1965–1968. The problem on the impact of a viscoplastic rod on a rigid obstacle is of interest. In the model accepted by Ishlinsky, the motion of the rod cross-sections is governed by heat equations with complex boundary conditions. It turned out unexpectedly that on impact the viscoplastic state does not instantly extend to the entire rod; instead, it runs away from the impact point and then returns without reaching the free end, thus leaving a rigid segment of the rod. In fracture theory of this time period, one should mention Ishlinsky’s paper of 1968, where two crack development models are compared. One model is the Griffith–Irvin model, and the other, which was developed later, is due to S. A. Khristianovich and G. I. Barenblatt. Ishlinsky showed that the latter model tends to the former as the range of the adhesion forces postulated in the latter tends to zero. Ishlinsky’s papers on the stability of elastic systems are of extreme importance. In a joint 1949 paper with M. A. Lavrent’ev, he theoretically explained the spectacular experiment in which a tube subjected to an underwater explosion loses its shape in such a way that the number of corrugations is the greater the closer they are to the explosion site. It turned out that the relationship between the characteristic exponents, the equilibrium mode number, and the applied external load constructed when studying the equilibrium modes permits one to maximize the buckling rate with respect to the mode number. The greater the applied force, the higher the mode number maximizing the characteristic exponent is. This result has numerous practical applications. The buckling problem for elastic systems, which goes back to Euler, is usually solved by methods of the theory of strength of materials for simplified models of rod or plate type. Ishlinsky studied this

In 2013, Entropy instituted the “Best Paper” award to recognize outstanding papers in the area of entropy and information studies published in Entropy [1]. We are pleased to announce the “Entropy Best Paper Award” for 2014. Nominations were selected by the Editor-in-Chief and designated Editorial Board Members from all the papers published in 2010.

The lattice dynamics for rock-paper-scissors games is related to population theories in ecology. In most cases, simulations are performed by local and global interactions. It is known in the former case that the dynamics is usually stable. We find two types of non-random distributions in the stationary state. One is a cluster formation of endangered species: when the density of a species approaches zero, its clumping degree diverges to infinity. The other is the strong aggregations of high-density species. Such spatial pattern formations play important roles in population dynamics.

The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

Recently, Alaca et al. [Alaca et al., Chaos, Solitons & Fractals 2006;29:1073–9] proved some fixed point theorems in intuitionistic fuzzy metric spaces by a strong definition of Cauchy sequence (see [George and Veeramani, Fuzzy Sets Syst 1994;64:395–9] and [Veeramani and Vasuki, Fuzzy Sets Syst 2003;135:409–13]), also the intuitionistic fuzzy metric space has extra conditions (see [Gregori et al., Chaos, Solitons & Fractals, 2006;28:902–5]). In this paper, we consider generalized intuitionistic fuzzy metric spaces i.e., L -fuzzy metric spaces and prove the fuzzy version of Banach and Edelstein contraction theorems in these spaces for modified definition of Cauchy sequence.

In July 1925 Heisenberg published a paper [Z. Phys. 33, 879-893 (1925)] which ended the period of `the Old Quantum Theory' and ushered in the new era of Quantum Mechanics. This epoch-making paper is generally regarded as being difficult to follow, perhaps partly because Heisenberg provided few clues as to how he arrived at the results which he reported. Here we give details of calculations of the type which, we suggest, Heisenberg may have performed. We take as a specific example one of the anharmonic oscillator problems considered by Heisenberg, and use our reconstruction of his approach to solve it up to second order in perturbation theory. We emphasize that the results are precisely those obtained in standard quantum mechanics, and suggest that some discussion of the approach - based on the direct computation of transition amplitudes - could usefully be included in undergraduate courses in quantum mechanics., Comment: 24 pages, no figures, Latex, submitted to Am. J. Phys

In this commentary, we try to make clearer the state of the art concerning the relation between mechanical contact interactions and the different notions of stresses. We emphasize the importance of the concept of virtual displacements. Its role has been recognized in Mechanics and in Continuum Mechanics long ago (see e.g., Vailati in Il principio dei lavori virtuali da Aristotele a Erone d’Alessandria, 113–128, 1987; Russo in The forgotten revolution, Springer, Berlin, 2003, or Cosserat and Cosserat in Sur la Theorie des Corps Deformables, Herman, Paris, 1909; Cosserat and Cosserat in Note sur la theorie de l.action euclidienne, Gauthier-Villars, Paris, 1908), and it is central as well when starting with an expression of the power expended by internal stresses and deducing the form of contact interactions as when starting with some form of the contact interactions and developing a representation theorem for these contact interactions based on the Cauchy tetrahedron construction.

In this paper we rigorously prove some statements on the symbolic dynamics for Bernoulli coupled map lattices studied by Kawasaki and Sasa. The advantage of our approach is that it is purely topological and it gives a simultaneous proof for the statements.

The German engineer H.A. Janssen gave one of the first accounts of the often peculiar behavior of granular material in a paper published in German in 1895. From simple experiments with corn he inferred the saturation of pressure with height in a granular system. Subsequently, Janssen derived the equivalent of the barometric formula for granular material from the main assumption that the walls carry part of the weight. The following is a translation of this article. The wording is chosen as close as possible to the original. While drawings are copied from the original, figures displaying data are redone for better readability. The translation is complemented by some bibliographical notes and an assessment of earlier work, wherein Hagen predicted the saturation of pressure with depth in 1852, and Huber-Burnand demonstrated that saturation qualitatively as early as in 1829. We conclude with a brief discussion of more recent developments resting on Janssen's work., Comment: 7 pages, 15 figures, accepted for publication in Granular Matter, original article (German) can be found under http://www.phy.duke.edu/~msperl/Janssen/

Many solid materials exhibit stress–induced phase transformations. Such phenomena can be modelled with the aid of the nonlinear elasticity theory with appropriate choices of the strain–energy function. It is known that if a two–phase deformation (with gradient F ) in a finite elastic body is a local energy minimizer, then given any point p of the surface of discontinuity, the piecewise–homogeneous deformation corresponding to the two values F ± ( p ) of F ( p ) is a global energy minimizer. Thus, instability of the latter state would imply instability of the former state. In this paper we investigate the stability properties of such piecewise–homogeneous deformations. More precisely, we are concerned with two joined half–spaces that correspond to two different phases of the same material. We first show how such a two–phase deformation can be constructed. Then the stability of the piecewise–homogeneous deformation is investigated with the aid of two test criteria. One is a kinetic stability criterion based on a quasi–static approach and on the growth/decay behaviour of the interface in the undeformed configuration when it is perturbed; the other, referred to as the energy criterion, is used to determine whether the deformation is a minimizer of the total energy with respect to perturbations of the interface in both the current and undeformed configurations. We clarify the differences between the two criteria, and provide a compact formula which can be used to establish the stability/instability of any two–phase piecewise–homogeneous deformations.

In recent years, after a prolonged period of intense activity, the study of numerical methods for solving stiff initial-value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity. There now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems. In this paper, we sketch some of the main theory which underpins stiff integration methods and we use this to describe, and put into context, some of the best codes currently available. By referencing only codes which have been thoroughly tested and are widely available, our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type. An additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated.

It has been conjectured that for ≥0 the entire spectrum of the non-Hermitian -symmetric Hamiltonian HN = p2 + x2(ix), where N = 2 + , is real. Strong evidence for this conjecture for the special case N = 3 was provided in a recent paper by Mezincescu (Mezincescu G A 2000 J. Phys. A: Math. Gen. 33 4911) in which the spectral zeta function Z3(1) for the Hamiltonian H3 = p2 + ix3 was calculated exactly. Here, the calculation of Mezincescu is generalized from the special case N = 3 to the region of all N≥2 (≥0) and the exact spectral zeta function ZN(1) for HN is obtained. Using ZN(1) it is shown that to extremely high precision (about three parts in 1018) the spectrum of HN for other values of N such as N = 4 is entirely real.

I give a simple analysis of the game that I previously published in Scientific American which shows the paradoxical behavior whereby two losing games randomly combine to form a winning game. The game, modeled on a random walk, requires only two states and is described by a first-order Markov process.

In this paper, it is shown that von Neumann continuous geometry may be regarded as the first attempt towards formulating a general quantum spacetime geometry akin to that of Cantorian spacetime E(∞) and noncommutative geometry.

The equations governing the motion of the paper sheet in a press nip and the anisotropic percolation of the water in the sheet, are derived in invariant form. In order to avoid restrictive assumptions on the configuration of the paper sheet, the system of coordinates is generated by the flow of fibrous material through the nip region. With the corresponding constitutive equations, the formulation consists of a system of partial differential equations for the metric tensor of the coordinate system, and for the water velocity. For practical use, the solution is then mapped back into a cartesian frame of reference. Quantities of industrial interest, such as the residual water content and stresses, as well as the press-induced anisotropy, can be calculated in principle. A Galerkin finite-element approximation is implemented using rectangular linear elements. Two case studies are presented, for large and small permeabilities, and the corresponding differences in water pressure and velocity are in general qualitative agreement with the observations. Finally, the predictive value of the model is demonstrated by the dependence of the solution on the imposed shear stress and its gradient across the sheet.

The Hirsch index or h-index is widely used to quantify the impact of an individual's scientific research output, determining the highest number h of a scientist's papers that received at least h citations. Several variants of the index have been proposed in order to give more or less preference to highly cited papers. I analyse the citation records of 26 physicists discussing various suggestions, in particular A, e, f, g, h(2), h_w, h_T, \hbar, m, {\pi}, R, s, t, w, and maxprod. The total number of all and of all cited publications as well as the highest and the average number of citations are also compared. Advantages and disadvantages of these indices and indicators are discussed. Correlation coefficients are determined quantifying which indices and indicators yield similar and which yield more deviating rankings of the 26 datasets. For 6 datasets the determination of the indices and indicators is visualized., Comment: 19 pages, including 6 tables and 3 figures accepted for publication in Annalen der Physik Berlin

A paper by Kiya et al. (1992, Fluid Dyn. Res. 10 , 117–131) on the numerical simulation of pseudo-elliptical vortex ring using the viscous version of the 3-D vortex particle method (Winckelmans 1989 (Ph.D. Thesis); Winckelmans and Leonard 1989 ( Proc. Workshop SIAM , 1988), 1992a (to appear in J. Comput. Phys. )) was recently published in this journal. The blame for the poor quality of the numerical results was put on the “Winckelmans-Leonard viscous scheme” (actually, the credit for the viscous method by exchange of particle strengths should go to Degond and Mas-Gallic (1989, Math. Comput. 53, 485–526)). It is the author's hope that this short note will convince the reader that the method is perfectly sound as long as it is used with care.

We point out that the analysis of Piotrowski and Sladkowski of the paradox described by Astumian has a flaw since they consider a different game.

A list of the 100 most popular scientific papers that were discussed on social media in 2013 – those that had the most "likes" on Facebook, Tweets or coverage in blogs – is dominated by papers in biology and the social sciences, with physics and mathematics papers only taking up a couple of places on the list.

The rock—paper—scissors (RPS) game is a nice model to study the biodiversity in an ecosystem. However, in the previous studies only the nearest-neighbor interaction among the species was considered. In this paper, taking the long-range migration into account, the effects of the interplay between nearest-neighbor-interaction and long-range-interaction given by Levy flight with distance distribution lh (−3 ≤ h < −1) in the spatial RPS game are investigated. Taking the probability, exchange rate, and power-law exponent of Levy flight as parameters, the coexistence conditions of three species are given. The critical curves for stable coexistence of three species in the parameter space are presented. It is also found that Levy flight has interesting effects on the final spatiotemporal pattern of the system. The results reveal that the long-range-interaction given by Levy flight exhibits pronounced effects on biodiversity of the ecosystem.

This special issue of Fluid Dynamics Research includes nine papers which are based on nine of the presentations at the Second UK–Japan bilateral Workshop and First ERCOFTAC Workshop on 'Turbulent flows generated/designed in multiscale/fractal ways: fundamentals and applications' held from 26 to 27 March 2012 at Imperial College London, UK. The research area of fractal-generated turbulent flows started with a chapter published in 2001 in one of the conference proceedings which came out of the 1999 Isaac Newton Institute 6 month Programme on Turbulence in Cambridge (UK). However, the first results which formed the basis of much of the work reported in this special issue started appearing from 2007 onwards and progress since then could perhaps be described as not insignificant. Research in this area has resulted in the following six notable advances: (a) the definition of two new length-scales characterizing grid-generated turbulence; (b) enhanced and energy-efficient stirring and scalar transfer by fractal grid and fractal openings/flanges with applications, in particular, to improved turbulence generation for combustion; (c) the non-equilibrium turbulent dissipation law; (d) non-equilibrium axisymmetric wake laws; (e) insights into the dependence of drag forces and vortex shedding on the fractal geometry of fractal objects and simulation methods for the calculation of drag of fractal trees; and (f) the invention and successful proof of concept of fractal spoilers and fractal fences. The present special issue contains papers directly related to these advances and can be seen as a reflection of the current research in the field of fractal-generated turbulent flows and their differences and commonalities with other turbulent flows. The financial support from the Japan Society for the Promotion of Science has been decisive for the organization and success of this workshop. We are also grateful to ERCOFTAC who put in place the EU-wide Special Interest Group on multiscale-generated turbulence and for supporting the workshop both financially and by way of advertisement. Last but by no means least, we express our deep gratitude to the editors of FDR who handled this issue: M Funakoshi, A D Gilbert, L B Mydlarski and K Suga.

The spatial rock-scissors-paper game (or cyclic Lotka-Volterra system) is extended to study how the spatiotemporal patterns are affected by the constructed backgrounds providing uniform number of neighbors (degree) at each site. On the square lattice this system exhibits a self-organizing pattern with equal concentration of the competing strategies (species). If the quenched background is constructed by substituting random links for the nearest neighbor bonds of a square lattice then a limit cycle occurs when the portion of random links exceeds a threshold value. This transition can also be observed if the standard link is replaced temporarily by a random one with a probability $P$ at each step of iteration. Above a second threshold value of $P$ the amplitude of global oscillation increases with time and finally the system reaches one of the homogeneous (absorbing) states. In this case the results of Monte Carlo simulations are compared with the predictions of the dynamical cluster technique evaluating all the configuration probabilities on one-, two-, four-, and six-site clusters., Comment: 6 pages, 6 figures

This article reviews (I) Philosophy of Mathematics and Natural Science Hermann Weyl. 336 pp. , Princeton, NJ, 2009. Price: $35.00(paper). ISBN 978-0-691-14120-6.

Gaemer's technique for recovering a function ⨍(x) from its power moments μ n = ∝ 1 0 ⨍(x)x n dx specified at n = 0, 1, 2,…N, is rederived using an expansion in Chebyshev polynomials of the second kind. Advantages of the present derivation over the original one are pointed out.

A simple paper-folding construction for a parabola is proposed. Besides providing a concrete, hands-on activity, the construction gives intuitive meaning to the role of the directrix and leads to a simple noncalculus proof of the reflection property of the parabola. The paper-folding technique can be generalized to ellipses and hyperbolas, and leads to their reflection properties as well.

This special issue of Journal of Physics A: Mathematical and Theoretical is dedicated to Darryl Holm on the occasion of his 60th birthday. The papers were largely drawn from the lectures at a meeting held in his honour at the Bernoulli Center at the EPFL in Lausanne, Switzerland, 22–28 July, 2007. The papers cover, as much as it is possible, the tremendous range and depth of work done by Darryl over the years. These topics include the Camassa–Holm equation and waves, geometric mechanics, symmetry and integrable systems, fluid and plasma dynamics, including stability problems, regularized incompressible fluid models (such as the α- or LANS models), including turbulence and multiscale methods, ocean dynamics, the EPDiff equation (a higher dimensional version of the CH equation) and computational anatomy. Perhaps a little less well known is Darryl's influential work in adiabatic fluids in general relativity, liquid crystals and other topics. But underlying all of Darryl's work is the geometric view of mechanics and physics, which was already kindled in his 1976 PhD thesis at Michigan. One of Darryl's and our favorite accomplishments is the development of the Camassa–Holm equation, a remarkable integrable PDE with, amazingly, spatially non-smooth, peaked, solitions. This is understandable—the equation caught on like wildfire because of its fascinating properties and the ideas in that paper proved to be a gold mine of research. It is his most cited paper with nearly a thousand citations, according to Google Scholar. We are pleased that our joint 1985 paper on fluid and plasma stability comes in second and the paper that was a precursor to the alpha model series of papers, namely our Euler–Poincare paper, comes in third. Of course having an equation named after you is a very nice bonus. Congratulations Darryl on a brilliant career that shows no signs of slowing down!

We disagree with Bucy’s comments (Bucy 1994) on our (Udwadia &; Kalaba 1992) paper. The reasons are as follows. 1. Just as the Gibbs-Appell equations (which use quasi-coordinates) are not the same as the Lagrange equations (which use Lagrange multipliers), the equations of motion obtained by us are not the same as the Gibbs-Appell equations. Some of the steps required to obtain the Gibbs-Appell equations are (Pars 1979; Whittaker 1937): (1) choice of quasi-coordinates; (2) elimination of certain quasicoordinates in terms of other preferred quasi-coordinates; (3) setting up of the Gibbs function; and (4) differentiation of the Gibbs function with respect to the preferred quasi-coordinates. None of these steps is used in obtaining our equations of motion. 2. Yet, the equations of motion obtained by us are equivalent to the Lagrange equations with multipliers (Kalaba et al. 1995) and to the Gibbs-Appell equations (Udwadia & Kalaba 1996). By equivalent we mean each set of equations implies the other. In fact all these different sets of equations are simply different, yet equivalent, ways of stating D’Alembert’s principle for bilinear constraints. Thus what applies to one set of equations applies to the others. Any deficiency in the equations derived in our paper (say regarding rank changes of the matrix A) is therefore present in the Gibbs-Appell equations and Lagrange’s equations as well, because of their equivalence.