bifurcation analysis of a generalized gause model with prey harvesting and a generalized holling response function of type iii abstract in this paper we study a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 the goal of our study is to give the bifurcation diagram of the model for this we need to study saddle node bifurcations hopf bifurcation of codimension 1 and 2 heteroclinic bifurcation and nilpotent saddle bifurcation of codimension 2 and 3 the nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram the hopf bifurcation is studied by means of a generalized lienard system and for b 0 we discuss the potential integrability of the system the nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4 but because it occurs with an invariant line the effective highest codimension is 3 we develop normal forms in which the invariant line is preserved for studying of the nilpotent saddle bifurcation for b 0 the reversibility of the nilpotent saddle is discussed we study the type of the heteroclinic loop and its cyclicity the phase portraits of the bifurcations diagram partially conjectured via the results obtained allow us to give a biological interpretation of the behavior of the two species

a relaxation method for constructing a beltrami flow in a bounded domain it is shown that a time dependent equation generates a beltrami flow or a force free field at time t from non beltrami initial data in an arbitrary bounded domain in three dimensions the generated beltrami flow has a zero normal component on the boundary and its helicity is equal to a prescribed value the existence and uniqueness of a solution to the equation are guaranteed globally in time and no finite time singularity occurs

adaptive context trees and text clustering in the finite alphabet context we propose four alternatives to fixed order markov models to estimate a conditional distribution they consist in working with a large class of variable length markov models represented by context trees and building an estimator of the conditional distribution with a risk of the same order as the risk of the best estimator for every model simultaneously in a conditional kullback leibler sense such estimators can be used to model complex objects like texts written in natural language and define a notion of similarity between them this idea is illustrated by experimental results of unsupervised text clustering

on the vanishing viscosity limit for the 3d navier stokes equations with a slip boundary condition where and below and denote the div and curl operators respectively n is the outward normal and is the unit tangential vector of the investigation of vanishing viscosity limit of solutions of the navier stokes equations both in the two and three spacial dimensional cases is a classical issue there are two related questions arising from here one is how to describe the inviscid limiting behavior of the navier stokes equation and the other is that does the euler equation can be approximated by the navier stokes equations in the case that the solution to the this research is supported in part by zheng ge ru foundation and hong kong rgc earmarked research grants cuhk 4028 04p cuhk 4040 02p and cuhk 4279 00p

the cyclicity of period annuli of a class of quintic hamiltonian systems abstract in this paper we prove that the cyclicity of period annuli for system d x d t y d y d t a x b x 3 c x 5 under perturbations of polynomials with degree n is not more than 54 n 13 taking into account the multiplicity where a b c r with c 0 such that the unperturbed systems have at least one center

on a selfadjoint realization of curl in exterior domains the first order vectorial operator curl occurs in many areas of applications e g in electrodynamics fluid mechanics hydrodynamics magnetohydrodynamics and elasticity theory it is the purpose of this note to discuss a particular instance of curl where a domain can be given to curl so that it becomes a selfadjoint operator the particular domain of selfadjointness under consideration is singled out by its relevance in applications to establish this operator we initially consider curl as an operator curl c read restricted to in l2 defined by curl c c l2 l2 0 1

helicity is the only integral invariant of volume preserving transformations we prove that any regular integral invariant of volume preserving transformations is equivalent to the helicity specifically given a functional i defined on exact divergence free vector fields of class c 1 on a compact 3 manifold that is associated with a well behaved integral kernel we prove that i is invariant under arbitrary volume preserving diffeomorphisms if and only if it is a function of the helicity

asymptotic massey products induced currents and borromean torus links we introduce a class of currents which allows a new and very explicit form for the massey product of a third order link as a line integral the explicit form permits the introduction of an asymptotic massey product analogous to that introduced previously for gauss s integral by v arnold the average third order asymptotic massey product is shown to be equal to berger s third order helicity for divergence free vector fields in linked tori

a new description of force free magnetic fields the moses curl eigenfunctions are used to describe linear force free magnetic fields enabling a proof that such fields are defined entirely by the value of their curl transform on the unit hemisphere in transform space this change of viewpoint suggests an orderly approach to the exploration and classification of the properties of such fields which is briefly sketched the simplest force free fields defined on zero one and two dimensional sets on the transform sphere are exhibited and the possibility of fields with fractal support sets suggested connections with other mathematical descriptions are pointed out as well as several promising directions for further exploration all results apply equally well to the description of the trkalian subset of beltrami fields in fluid dynamics

abelian integrals in unfoldings of codimension 3 singularities with nilpotent linear parts abstract this paper is concerned with the upper bound of the number of limit cycles in unfolding of codimension 3 planar singularities with nilpotent linear parts after making a central rescaling the problem reduces to a perturbation problem of a one parameter family of quadratic reversible systems as the parameter a 1 1 0 is rational except the case a 2 3 based on the chebyshev criterion for abelian integrals and a rationalizing transformation the problem could be solved theoretically to illustrate our approaches two particular cases corresponding to nilpotent codimension 3 saddle and elliptic case respectively are proved where the upper bound of the number of limit cycles is two

laplace s law of succession and universal encoding this correspondence shows that the known add half rule is not asymptotically optimal for predicting the n 1 st symbol after a sequence of n symbols whereas the add spl beta sub 0 rule spl beta sub 0 0 50922 spl middot spl middot spl middot is

existence of knotted vortex tubes in steady euler flows we prove the existence of knotted and linked thin vortex tubes for steady solutions to the incompressible euler equation in mathbb r 3 more precisely given a finite collection of possibly linked and knotted disjoint thin tubes in mathbb r 3 we show that they can be transformed with a c m small diffeomorphism into a set of vortex tubes of a beltrami field that tends to zero at infinity the structure of the vortex lines in the tubes is extremely rich presenting a positive measure set of invariant tori and infinitely many periodic vortex lines the problem of the existence of steady knotted thin vortex tubes can be traced back to lord kelvin

adaptive coding and prediction of sources with large and infinite alphabets the problem of predicting a sequence x1 x2 generated by a discrete source with unknown statistics is considered each letter xt 1 is predicted using the information on the word x1x2 hellip xt only this problem is of great importance for data compression because of its use to estimate probability distributions for ppm algorithms and other adaptive codes on the other hand such prediction is a classical problem which has received much attention its history can be traced back to laplace we address the problem where the sequence is generated by an independent and identically distributed i i d source with some large or even infinite alphabet and suggest a class of new methods of prediction

finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type abstract in this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside c families of planar vector fields in some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3 by blowing up the family of vector fields we obtain all the limit periodic sets we calculate two different types of dulac maps in the blown up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point the finite cyclicity theorems are derived by a generalized derivation division method introduced by roussarie

bifurcation analysis of a predator prey system involving group defence a class of odes of generalized gause type modeling predator prey interaction is considered the prey are assumed to exhibit a phenomenon called group defence that is predation is decreased or even eliminated due to the ability of the prey to defend or disguise themselves as their numbers increase using the carrying capacity of the environment as the bifurcation parameter it is shown that the model undergoes a sequence of bifurcations that includes a homoclinic bifurcation as well as a hopf bifurcation conditions that hold even in the case of no group defence that ensure a subcritical hopf bifurcation and also the spontaneous appearance of a semistable periodic orbit that splits into a pair one stable and one unstable of periodic orbits are given ecological ramifications are considered unlike the classical model sufficient enrichment of the environment combined with group defence leads to extinction of the predator deterministically for almost all initial conditions providing strong support fo

global stability for a class of predator prey systems this paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator prey systems the dulac s criterion is applied and liapunov functions are constructed to establish the global stability

achievability of asymptotic minimax regret by horizon dependent and horizon independent strategies the normalized maximum likelihood distribution achieves minimax coding log loss regret given a fixed sample size or horizon n it generally requires that n be known in advance furthermore extracting the sequential predictions from the normalized maximum likelihood distribution is computationally infeasible for most statistical models several computationally feasible alternative strategies have been devised we characterize the achievability of asymptotic minimaxity by horizon dependent and horizon independent strategies we prove that no horizon independent strategy can be asymptotically minimax in the multinomial case a weaker result is given in the general case subject to a condition on the horizon dependence of the normalized maximum likelihood motivated by these negative results we demonstrate that an easily implementable bayes mixture based on a conjugate dirichlet prior with a simple dependency on n achieves asymptotic minimaxity for all sequences simplifying earlier similar proposals our numerical experiments for the bernoulli model demonstrate improved finite sample performance by a number of novel horizon dependent and horizon independent algorithms

limit cycle bifurcation by perturbing a cuspidal loop of order 2 in a hamiltonian system abstract this paper deals with the analytical property of the first melnikov function for general hamiltonian systems possessing a cuspidal loop of order 2 and its expansion at the hamiltonian value corresponding to the loop the explicit formulas for the first coefficients of the expansion have been given we prove that at least 13 limit cycles can bifurcate from the cuspidal loop of order 2 under certain conditions then we consider the cyclicity of a cuspidal loop in some lienard and hamiltonian systems and determine the number of limit cycles that can bifurcate from the perturbed system

a priori estimates for a singular limit approximation of the constitutive laws for chiral media in the time domain abstract the equations for the evolution of electromagnetic fields in chiral media in the time domain are nonlocal in time in this work we study the validity of a singular limit local in time approximation for these nonlocal in time equations by estimating the size of the difference of the fields as predicted by both models in particular we establish an a priori estimate for this difference depending on the time horizon properties of the domain spatial properties of the initial data and the source terms and the chirality measure of the approximating model

global analysis in a predator prey system with nonmonotonic functional response a predator prey system with nonmonotonic functional response is considered global qualitative and bifurcation analyses are combined to determine the global dynamics of the model the bifurcation a

eigenfunctions of the curl operator in spherical coordinates the eigenfunctions of the curl operator are obtained by separation of variables in spherical coordinates making use of the spin weighted spherical harmonics it is shown that the eigenfunctions of the curl operator with vanishing divergence can be written in terms of a single scalar potential that satisfies the helmholtz equation it is also shown that these eigenfunctions give a complete basis for the divergenceless vector fields

entropy computations via analytic depoissonization we investigate the basic question of information theory namely evaluation of shannon entropy and a more general renyi 1961 entropy for some discrete distributions e g binomial negative binomial etc we aim at establishing analytic methods i e those in which complex analysis plays a pivotal role for such computations which often yield estimates of unparalleled precision the main analytic tool used here is that of analytic poissonization and depoissonization we illustrate our approach on the entropy evaluation of the binomial distribution that is we prove that for binomial n p distribution shannon s h sub n becomes h sub n spl ap 1 2 ln n 1 2 ln spl radic 2 spl pi p 1 p spl sigma sub k spl ges 1 a sub k n sup k where a sub k are explicitly computable constants moreover we argue that analytic methods e g complex asymptotics such as rice s method and singularity analysis mellin transforms poissonization and depoissonization can offer new tools for information theory especially for studying second order asymptotics e g redundancy in fact there has been a resurgence of interest and a few successful applications of analytic methods to a variety of problems of information theory therefore we propose to name such investigations as analytic information theory

eigenfunctions of the curl in cylindrical geometry eigenfunctions of the equation b b are found for finite cylindrical geometry with normal boundary condition b n 0 and nonaxisymmetric modes eim m 0 the vector field b can be represented by a scalar generating function of the chandrasekhar kendall type with radial bessel functions for the nondegenerate cases a general set of solutions can also be generated by transformation of variables a series solution in terms of radial bessel functions is found which has excellent convergence properties an 1 n4 and a robust method of locating eigenvalues is described

nilpotent singularities and dynamics in an sir type of compartmental model with hospital resources abstract an sir type of compartmental model with a standard incidence rate and a nonlinear recovery rate was formulated to study the impact of available resources of public health system especially the number of hospital beds cusp focus and elliptic type of nilpotent singularities of codimension 3 are discovered and analyzed in this three dimensional model complex dynamics of disease transmission including multi steady states and multi periodicity are revealed by bifurcation analysis large amplitude oscillations found in our model provide a more reasonable explanation for disease recurrence with clinical data our studies have practical implications for the prevention and control of infectious diseases

bifurcation analysis in a host generalist parasitoid model with holling ii functional response abstract in this paper we study a host generalist parasitoid model with holling ii functional response where the generalist parasitoids are introduced to control the invasion of the hosts it is shown that the model can undergo a sequence of bifurcations including cusp focus and elliptic types degenerate bogdanov takens bifurcations of codimension three and a degenerate hopf bifurcation of codimension at most two as the parameters vary and the model exhibits rich dynamics such as the existence of multiple coexistent steady states multiple coexistent periodic orbits homoclinic orbits etc moreover there exists a critical value for the carrying capacity of generalist parasitoids such that i when the carrying capacity of the generalist parasitoids is smaller than the critical value the invading hosts can always persist despite of the predation by the generalist parasitoids i e the generalist parasitoids cannot control the invasion of hosts ii when the carrying capacity of the generalist parasitoids is larger than the critical value the invading hosts either tend to extinction or persist in the form of multiple coexistent steady states or multiple coexistent periodic orbits depending on the initial populations i e whether the invasion can be stopped and reversed by the generalist parasitoids depends on the initial populations iii in both cases the generalist parasitoids always persist numerical simulations are presented to illustrate the theoretical results

the loop quantities and bifurcations of homoclinic loops abstract the stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles for a homoclinic loop of a given planar vector field a sequence of quantities the homoclinic loop quantities were defined to study the stability and bifurcations of the loop among the sequence of the loop quantities the first nonzero one determines the stability of the homoclinic loop there are formulas for the first three and the fifth loop quantities in this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops as applications we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability switching

on the representation of inhomogeneous linear force free fields it is shown that there is a false assumption hidden in the description of a relaxed state with inhomogeneous boundary conditions as the vector sum of a potential field satisfying the boundary conditions and a sum of eigenfunctions of the associated eigenvalue problem expanded by certain coefficients in particular although the jensen and chu formula 1984 can provide the correct expansion coefficients it contains an implicit paradox in its derivation according to a general vector theorem the same paradox led chu et al 1999 to be concerned about a contradiction obtained by taking the curl of their magnetic field expansion which if permitted becomes inconsistent with a current normal to the surface the assumption that the curl can be commuted across an infinite sum of terms is the mechanism leading to these apparently paradoxical conclusions two mechanisms for resolving this apparent paradox are possible one of which will be described in some detail below and the other discussed further in a forthcoming more theoretical paper laurence et al 2000 the decomposition of the magnetic field above is valid with convergence in the mean squared sense but a decomposition of the current needs to be reinterpreted in terms of negative sobolev spaces to avoid this and remain in a more easily managable and familiar setting we derive the expansion coefficients in a way that involves the commuting of the inverse curl as opposed to the curl and the series the resulting series converges in a mean square sense when this is done the calculation can conform to the general vector theorem and a new gauge invariant expression for the coefficients is obtained however the consequence of the non commutability is nullified in the jensen and chu formula in both simply and multiply connected domains by the important extra requirement of a boundary condition on the vector potential eigenfunctions this excludes magnetic field eigenfunctions that carry flux but there remains a complete set for the expansion and all flux is carried by the potential field the two formulas are then identical on a different issue it is shown that if the general expansion is taken over a half space by combining positive and negative eigenvalue terms then the coefficients are anisotropic that is they are tensors except when evaluated at the first eigenvalue a specific example is presented to illustrate the situation and to validate the new method of deriving the coefficients

canard cycles for predator prey systems with holling types of functional response abstract by using the singular perturbation theory developed by dumortier and roussarie and recent work of de maesschalck and dumortier we study the canard phenomenon for predator prey systems with response functions of holling types we first develop a formula for computing the slow divergence integrals by using the formula we prove that for the systems with the response function of holling types iii and iv the cyclicity of any limit periodic set is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations we also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two

universal prediction this paper consists of an overview on universal prediction from an information theoretic perspective special attention is given to the notion of probability assignment under the self information loss function which is directly related to the theory of universal data compression both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings

pac bayesian analysis of co clustering and beyond we derive pac bayesian generalization bounds for supervised and unsupervised learning models based on clustering such as co clustering matrix tri factorization graphical models graph clustering and pairwise clustering we begin with the analysis of co clustering which is a widely used approach to the analysis of data matrices we distinguish among two tasks in matrix data analysis discriminative prediction of the missing entries in data matrices and estimation of the joint probability distribution of row and column variables in co occurrence matrices we derive pac bayesian generalization bounds for the expected out of sample performance of co clustering based solutions for these two tasks the analysis yields regularization terms that were absent in the previous formulations of co clustering the bounds suggest that the expected performance of co clustering is governed by a trade off between its empirical performance and the mutual information preserved by the cluster variables on row and column ids we derive an iterative projection algorithm for finding a local optimum of this trade off for discriminative prediction tasks this algorithm achieved state of the art performance in the movielens collaborative filtering task our co clustering model can also be seen as matrix tri factorization and the results provide generalization bounds regularization terms and new algorithms for this form of matrix factorization the analysis of co clustering is extended to tree shaped graphical models which can be used to analyze high dimensional tensors according to the bounds the generalization abilities of tree shaped graphical models depend on a trade off between their empirical data fit and the mutual information that is propagated up the tree levels we also formulate weighted graph clustering as a prediction problem given a subset of edge weights we analyze the ability of graph clustering to predict the remaining edge weights the analysis of co clustering easily extends to this problem and suggests that graph clustering should optimize the trade off between empirical data fit and the mutual information that clusters preserve on graph nodes

universal coding information prediction and estimation a connection between universal codes and the problems of prediction and statistical estimation is established a known lower bound for the mean length of universal codes is sharpened and generalized and optimum universal codes constructed the bound is defined to give the information in strings relative to the considered class of processes the earlier derived minimum description length criterion for estimation of parameters including their number is given a fundamental information theoretic justification by showing that its estimators achieve the information in the strings it is also shown that one cannot do prediction in gaussian autoregressive moving average arma processes below a bound which is determined by the information in the data

eigenfunctions of the curl in annular cylindrical and rectangular geometry a method for determining the eigenfunctions of the equation b b for finite cylindrical geometry with normal boundary condition b n 0 and nonaxisymmetric modes eim m 0 was given by morse j math phys 46 113511 2005 this method uses series solution of a chandrasekhar kendall astrophys j 126 157 1957 function here this method is generalized to include annular coaxial cylindrical geometry of finite length the method is also applied to the case of rectangular duct geometry with periodic z dependence for slender aspect ratios in the annular cylinder the eigenfunctions and eigenfields correspond to the rectangular duct the eigenfunctions for finite length rectangular boxes can also be obtained from the rectangular duct case the robust convergence properties an 1 n4 and simple eigenvalue equations from morse are retained in these other geometries

the cyclicity of period annuli for a class of cubic hamiltonian systems with nilpotent singular points abstract this paper deals with the limit cycles of a class of cubic hamiltonian systems under polynomial perturbations we suppose that the corresponding hamiltonian system which has at least one center has finite singular points and is symmetrical with respect to both x axis and y axis and also the origin is a nilpotent singular point hence the hamiltonian h x y can be written as h x y x 2 a x 4 b x 2 y 2 c y 4 a b c r 3 c 0 for the above h x y the corresponding vector field is x 2 y b x 2 2 c y 2 y 2 x 1 2 a x 2 b y 2 we first obtain that the above vector fields with at least one center can be divided into 8 classes by its topological phase portraits for the following perturbed system x 2 y b x 2 2 c y 2 e f x y y 2 x 1 2 a x 2 b y 2 e g x y where 0 e 1 f x y and g x y are polynomials in x y with degree n we give an estimation of the number of isolated zeros of the corresponding abelian integral the number of limit cycles follows from poincare pontryagin theorem

perturbation from an asymmetric cubic hamiltonian this paper concerns with the number of limit cycles from an asymmetric hamiltonian of degree three under cubic perturbation eleven limit cycles are found and three different distributions are given by using the methods of bifurcation theory and qualitative analysis two of which are new

pp graphics with a nilpotent elliptic singularity in quadratic systems and hilbert s 16th problem abstract this paper is part of the program launched in j differential equations 110 1 1994 86 to prove the finiteness part of hilbert s 16th problem for quadratic system which consists in proving that 121 graphics have finite cyclicity among quadratic systems we show that any pp graphic through a multiplicity 3 nilpotent singularity of elliptic type which does not surround a center has finite cyclicity such graphics may have additional saddles and or saddle nodes altogether we show the finite cyclicity of 15 graphics of j differential equations 110 1 1994 86 in particular we prove the finite cyclicity of a pp graphic with an elliptic nilpotent singular point together with a hyperbolic saddle with hyperbolicity 1 which appears in generic 3 parameter families of vector fields and hence belongs to the zoo of kotova and stanzo concerning the hilbert 16th problem american mathematical society translation series 2 vol 165 american mathematical society providence ri 1995 pp 155 201

local representation and construction of beltrami fields ii solenoidal beltrami fields and ideal mhd equilibria abstract object of the present paper is the local theory of solution for steady ideal euler flows and ideal mhd equilibria the present analysis relies on the lie darboux theorem of differential geometry and the local theory of representation and construction of beltrami fields developed in sato and yamada 2019 a theorem for the construction of harmonic orthogonal coordinates is proved using such coordinates families of solenoidal beltrami fields with different topologies are obtained in analytic form existence of global solenoidal beltrami fields satisfying prescribed boundary conditions while preserving the local representation is considered it is shown that only singular solutions are admissible an explicit example is given in a spherical domain and a theorem on existence of singular solutions is proven local conditions for existence of solutions and local representation theorems are derived for generalized beltrami fields ideal mhd equilibria and general steady ideal euler flows the theory is applied to construct analytic examples

helical bifurcation and tearing mode in a plasma a description based on casimir foliation the relation between the helical bifurcation of a taylor relaxed state a beltrami equilibrium and a tearing mode is analyzed in a hamiltonian framework invoking an eulerian representation of the hamiltonian the symplectic operator defining a poisson bracket becomes non canonical i e the symplectic operator has a nontrivial cokernel dual to its nullspace foliating the phase space into level sets of casimir invariants a taylor relaxed state is an equilibrium point on a casimir helicity leaf changing the helicity equilibrium points may bifurcate to produce helical relaxed states a necessary and sufficient condition for bifurcation is derived tearing yields a helical perturbation on an unstable equilibrium producing a helical structure approximately similar to a helical relaxed state a slight discrepancy found between the helically bifurcated relaxed state and the linear tearing mode viewed as a perturbed singular equilibrium state is attributed to a casimir element named helical flux pertinent to a resonance singularity of the non canonical symplectic operator while the helical bifurcation can occur at discrete eigenvalues of the beltrami parameter the tearing mode being a singular eigenfunction exists for an arbitrary beltrami parameter bifurcated beltrami equilibria appearing on the same helicity leaf are isolated by the helical flux casimir foliation the obstacle preventing the tearing mode to develop in the ideal limit turns out to be the shielding current sheet on the resonant surface preventing the release of the potential energy when this current is dissipated by resistivity reconnection is allowed and tearing instability occurs the criterion for linear tearing instability of beltrami equilibria is shown to be directly related to the spectrum of the curl operator

eigenfunction expansions associated with the curl derivatives in cylindrical geometries completeness of chandrasekhar kendall eigenfunctions eigenfunctions of the curl derivatives are used in many different fields of theoretical physics for example in plasma physics an eigenfunction of curl represents a certain eigenstate of a plasma that has the force free property the present theory proves that the spectral resolution of the curl operator gives a complete eigenfunction expansion of a solenoidal vector field with homogeneous boundary and flux conditions useful decomposition theories and a previous abstract result math z 204 235 1990 are concisely summarized explicit eigenfunctions have been studied in cylindrical geometries chandrasekhar kendall functions astrophys j 126 457 1957 which are eigenfunctions of the curl derivatives in a cylindrical domain give an orthogonal complete basis of the hilbert space that is the orthogonal complement of the space of irrotational vector fields

minimax quantum tomography estimators and relative entropy bounds a minimax estimator has the minimum possible error risk in the worst case we construct the first minimax estimators for quantum state tomography with relative entropy risk the minimax risk of nonadaptive tomography scales as o 1 sqrt n in contrast to that of classical probability estimation which is o 1 n where n is the number of copies of the quantum state used we trace this deficiency to sampling mismatch future observations that determine risk may come from a different sample space than the past data that determine the estimate this makes minimax estimators very biased and we propose a computationally tractable alternative with similar behavior in the worst case but superior accuracy on most states

hopf takens bifurcations and centres the paper deals with hopf takens bifurcations both in generic families and in families containing centres attention goes to the relation between normal forms lyapunov coefficients and the bautin ideal in case the bautin ideal has only one generator we pay attention to the first non zero melnikov function

a special class in vanishing helicity ideal flows the derivation of a lagrangian invariant so called spirality is reviewed through the lagrangian coordinates the value of the spirality is fixed up to a gauge transformation the helicity conservation follows directly from this invariant among all ideal flows with zero helicity in a domain of flow frozen into the fluid motion a special class is introduced this special topological class has the possibility to transform its spirality to be identically zero everywhere in the domain for those simply connected domains of motion a necessary and a sufficient condition is presented for these zero spirality flows

bifurcation of limit cycles for a quartic near hamiltonian system by perturbing a nilpotent center hopf bifurcation which produces oscillations is a very important phenomena in the theory and application of dynamical systems almost all works available about hopf bifurcations are related to a non degenerate focus or center for the case of a degenerate focus or center the study of the bifurcations becomes challenge in this paper we consider the bifurcation of limit cycles for a quartic near hamiltonian system by perturbing a nilpotent center we take coefficients as parameters then we can get six limit cycles

numerical computation of the helical chandrasekhar kendall modes a new formulation is presented for numerically computing the helical chandrasekhar kendall modes in an axisymmetric torus it explicitly imposes b 0 and yields a standard matrix eigenvalue problem which can then be solved by standard matrix eigenvalue techniques numerical implementation and computational results are shown for an axisymmetric torus typical of reversed field pinch and spherical tokamak

a method of constructing cycles without contact around a weak focus has a weak focus then limit cycles can be made to appear in its neighborhood by varying slightly the right sides 11 frommer 2 bautin 3 chin et al 4 and tung 5 are examples of excellent expositions of poincare s idea it is necessary to go from numerical examples to general theory in the first part of this paper we present a method of constructing cycles without contact around a weak focus which makes clear poincart s idea used in the above work 2 51 in the second part another method to analyze global phase portraits is used quadratic systems with at least four limit cycles are produced by means of both the above methods this contradicts the results of petrovskii and landis 6

on the ideal compressible magnetohydrodynamic equations abstract in this paper we consider the short time strong solutions to the ideal compressible magnetohydrodynamic equations with initial vacuum where the velocity field satisfies the navier slip condition inspired by kato and lax s idea we use the contraction mapping argument to prove the local existence moreover under the navier slip condition we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of l norm of the deformation tensor d u

bifurcations of a ratio dependent predator prey system with constant rate harvesting the ratio dependent predator prey model exhibits rich interesting dynamics due to the singularity of the origin the objective of this paper is to study the dynamical properties of the ratio depend

bifurcations in a predator prey system of leslie type with generalized holling type iii functional response abstract we consider a predator prey system of leslie type with generalized holling type iii functional response p x m x 2 a x 2 b x 1 by allowing b to be negative b 2 a p x is monotonic for b 0 and nonmonotonic for b 0 when x 0 the model has two non hyperbolic positive equilibria one is a multiple focus of multiplicity one and the other is a cusp of codimension 2 for some values of parameters and a degenerate bogdanov takens singularity focus or center case of codimension 3 for other values of parameters when there exist a multiple focus of multiplicity one and a cusp of codimension 2 we show that the model exhibits subcritical hopf bifurcation and bogdanov takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria respectively different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have i a stable limit cycle enclosing two non hyperbolic positive equilibria ii a stable limit cycle enclosing an unstable homoclinic loop iii two limit cycles enclosing a hyperbolic positive equilibrium iv one stable limit cycle enclosing three hyperbolic positive equilibria or v the coexistence of three stable states two stable equilibria and a stable limit cycle when the model has a bogdanov takens singularity of codimension 3 we prove that the model exhibits degenerate focus type bogdanov takens bifurcation of codimension 3 these results not only demonstrate that the dynamics of this model when b 2 a are much more complex and far richer than the case when b 0 but also provide new bifurcation phenomena for predator prey systems

bifurcations of invariant tori in predator prey models with seasonal prey harvesting in this paper we study bifurcations in predator prey systems with seasonal prey harvesting first when the seasonal harvesting reduces to constant yield it is shown that various kinds of bifurcat

on the existence of force free magnetic fields with small nonconstant α in exterior domains the existence of force free magnetic fields in the exterior domain of some compact simply connected surface s is proved via an iteration scheme the iteration starts with an arbitrary exterior vacuum field which contains flux tubes originating and ending on s at one cross section of such a flux tube with s an arbitrary function is prescribed for small values of in the holder norm 1 0 1 the iteration is shown to converge to a force free field with the prescribed values of in a flux tube which is close to the vacuum flux tube and 0 outside the force free field is close in the holder norm 1 to the starting vacuum field in particular it has the same field line topology the same boundary values on s and satisfies the same decay conditions in spatial infinity it is in general three dimensional and requires no continuous symmetries

moduli spaces of vortex knots for an exact fluid flow the moduli spaces s d of non isotopic vortex knots are introduced for the ideal fluid flows in invariant domains d the analogous moduli spaces of the magnetic fields b knots are defined we derive and investigate new exact fluid flows and analogous plasma equilibria satisfying the beltrami equation which have nested invariant balls b k 3 with radii rk k 1 k the first flow is z axisymmetric the other ones do not possess any rotational symmetries the axisymmetric flow has an invariant plane z 0 due to an involutive symmetry of the flow its vortex knots in the invariant half spaces z 0 and z 0 are equivalent it is demonstrated that the moduli space x1d4ae 3 for the derived fluid flow in 3 is naturally isomorphic to the set of all rational numbers p q in the interval j 1 0 25 q m 1 0 5847 where q is the safety factor for the fluid flow in the first invariant ball b 1 3 it is shown that all values of the safety factor q belong to a small interval of length l 0

blow up criterion for 3d viscous liquid gas two phase flow model abstract this work deals with the blow up criterion for the strong solution to the three dimensional viscous liquid gas two phase flow model in terms of the l 1 0 t l norm of the gradient of the velocity with two types of boundary conditions the dirichlet boundary condition and the navier slip boundary condition there is no extra restriction on viscosity coefficients the result also applies to the whole space case

cyclicity of several planar graphics and ensembles through three singular points without generic conditions abstract this paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle node p 0 and two hyperbolic saddles p 1 and p 2 for the non generic cases of r 1 0 1 r 2 0 1 and r 1 0 1 r 2 0 1 where r 1 0 and r 2 0 are the hyperbolicity ratio of the saddles p 1 and p 2 respectively for the case of r 1 0 1 r 2 0 1 we suppose that the connection from p 0 to p 2 and the connection from p 0 to p 1 keep unbroken we prove that these graphics and ensembles are of finite cyclicity respectively moreover the cyclicity is linearly dependent on the order of the neutral saddle p 1 if p 2 is contractive and r 2 0 q we also show that the nearer r 2 0 is close to 1 the more the limit cycles are bifurcated for the case of r 1 0 1 r 2 0 1 we obtain that these graphics and ensembles are of finite cyclicity respectively if p 1 is of finite order and the hp connection from p 0 to p 2 keeps unbroken

organizing center for the bifurcation analysis of a generalized gause model with prey harvesting and holling response function of type iii abstract the present note is an addendum to the paper of etoua rousseau 2010 1 which presented a study of a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 complete bifurcation diagrams were proposed but some parts were conjectural an organizing center for the bifurcation diagram was given by a nilpotent point of saddle type lying on an invariant line and of codimension greater than or equal to 3 this point was of codimension 3 when b 0 and was conjectured to be of infinite codimension when b 0 this conjecture was in line with a second conjecture that the hopf bifurcation of order 2 degenerates to a hopf bifurcation of infinite codimension when b 0 in this note we prove these two conjectures

on woltjer s variational principle for force free fields the existence of minimizers for woltjer s variational principle is established and that the minimizers are force free fields this method has the nature of a constructive implicit function theorem and handles successfully the nonconvex constraint of constant total helicity domains of arbitrary connectivity are allowed as well as nonhomogeneous boundary conditions and periods

singularity analysis and asymptotics of bernoulli sums the asymptotic analysis of a class of binomial sums that arise in information theory can be performed in a simple way by means of singularity analysis of generating functions the method developed extends the range of applicability of singularity analysis techniques to combinatorial sums involving transcendental elements like logarithms or fractional powers

bifurcation analysis of a predator prey system with nonmonotonic functional response we consider a predator prey system with nonmonotonic functional response p x frac mx ax 2 bx 1 by allowing b to be negative b 2 sqrt a p x is concave up for small values of x 0 a

saddle quantities and applications abstract in this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation ghb and the practical computational aspects for this purpose we first compare the dulac normal form with the joyal normal form these forms were both used to prove the ghb theorem but the second one is far more practical from the algorithmic point of view we then show that the information carried by these normal forms can be computed in a much simpler way using what we shall call dual lyapunov constants the coefficients of a normal form or the dual lyapunov quantities are particular cases of what we shall call saddle quantities we calculate the saddle quantities for quadratic systems and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation we also study the homoclinic loop bifurcation of order 5 appearing in a 6 parameter family close to a hamiltonian system to our knowledge this is the first time that one can find a complete description of a ghb of such high order finally we calculate the saddle quantities for a symmetric cubic vector field and we deduce a bound for the number of limit cycles that appear in a ghb

global hopf bifurcation of a general predator prey system with diffusion and stage structures abstract in this paper the dynamics of a general predator prey system with diffusion and stage structures are investigated under some assumptions the stability of positive steady state and the existence of local hopf bifurcation are obtained by analyzing the distribution of eigenvalues moreover the global existence of positive periodic solution is established by using the global hopf bifurcation result of wu finally an example and some numerical simulations are carried out for illustrating the theoretical results

local representation and construction of beltrami fields abstract a beltrami field is an eigenvector of the curl operator beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence by application of the lie darboux theorem of differential geometry we prove a local representation theorem for beltrami fields we find that locally a beltrami field has a standard form amenable to an arnold beltrami childress flow with two of the parameters set to zero furthermore a beltrami flow admits two local invariants a coordinate representing the physical plane of the flow and an angular momentum like quantity in the direction across the plane as a consequence of the theorem we derive a method to construct beltrami fields with given proportionality factor this method based on the solution of the eikonal equation guarantees the existence of beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal we construct several solenoidal and non solenoidal beltrami fields with both homogeneous and inhomogeneous proportionality factors

coexistence region and global dynamics of a harvested predator prey system the study of populational dynamics with harvesting is related to the optimal man agement of renewable resources in this paper we consider a predator prey model in which two ecologically interacting species are harvested independently with constant rates the main purpose of the present work is to offer a complete mathematical analysis for the model and to describe some of the significant qualitative results that may be expected to arise from the interplay of biological forces our study shows that the maximum safe harvest may be far less than what would be assumed from a local analysis for the equilibria it is also proved that the harvested predator prey system may exhibit very complicated dynamics such as a spontaneous appearance of a homoclinic orbit and multiple limit cycles

vortex knots for the spheromak fluid flow and their moduli spaces abstract new exact solutions to the euler hydrodynamics equations are constructed a method for the study of vortex knots is developed for a special class of ideal fluid flows the axisymmetric ones satisfying the beltrami equation curl v x v x the method is based on a construction of the moduli spaces of vortex knots s r 3 applying the method to the spheromak fluid flow we demonstrate that only those torus knots k p q are realized as vortex knots for which p q belongs to the interval i 1 0 5 m 1 0 8252 we prove that each torus knot k p q with 1 2 p q 1 2 is realized on countably many invariant tori t 2 r 3 while torus knots with 1 2 p q m 1 are realized only on finitely many tori the moduli spaces of vortex knots s m b a 3 m 1 2 are constructed for the spheromak fluid flows inside a ball b a 3 of radius a

genericity conditions for finite cyclicity of elementary graphics abstract in this paper we explore the khovanskii method for proving the finite cyclicity of elementary graphics and how it can be applied in practice the genericity conditions needed in that case form a proper subset of the usual methods moreover some of the conditions are non intrinsic and can be artificially created by action of the automorphism groups preserving the normal forms near the singularities and their action on regular transitions hence we introduce an extension of the method which treats the usual functional pfaffian systems together with the admissible changes of coordinates in the functional equations

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

degenerate and non trivial hyperbolic polycycles with two vertices abstract we study some hyperbolic polycycles with two vertices which are degenerate and non trivial and such that the maximal cyclicity in the sufficiently differentiable unfolding depends only on the vector field germ along the polycycle in these cases the maximal cyclicity and versal unfolding of the polycycle are described