weak focus limit cycles and bifurcations for bounded quadratic systems abstract we study the bounded quadratic systems with either two weak foci or a weak focus of order 2 from the first case we obtain 1 1 configuration of limit cycles and in the second case we prove that there is no limit cycle surrounding the weak focus of order 2 also we unfold the bounded quadratic system with a center and the maximum number of limit cycles in this case is two

the covariance matrix of the potts model a random cluster analysis we consider the covariance matrix g mm q2 of thed dimensionalq states potts model rewriting it in the random cluster representation of fortuin and kasteleyn in any of theq ordered phases we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances thereby attributing algebraic significance to these stochastic geometric quantities we also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers for dimensiond 2 we show that this correlation length and the correlation length of the two point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two for systems with first order transitions this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self dual point o for systems with second order transitions this relation implies the equality of the correlation length exponents from above and below threshold as well as an amplitude ratio of two in the course of proving the above results we establish several properties of independent interest including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions and left continuity of the two dimensional free boundary condition percolation probability at o we also introduce dlr equations for the random cluster model and use them to establish ergodicity of the free measure in order to prove these results we introduce a new class of events which we call decoupling events and two inequalities for these events the first is similar to the fkg inequality but holds for events which are neither increasing nor decreasing the second is similar to the van den berg kesten inequality in standard percolation both inequalities hold for an arbitrary fkg measure

homoclinic loop and multiple limit cycle bifurcation surfaces this paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces z and multiplicity two limit cycle bifurcation surfaces f for planar systems depending on two or more parameters it determines the side of or f on which limit cycles occur and it shows that if x and intersect then typically they do so at a flat contact

bounded polynomial vector fields in r2 and rn abstract we characterize the bounded polynomial vector fields in r 2 additionally we provide a necessary condition but not sufficient which must be satisfied by bounded polynomial vector fields in r n

limit cycles of planar quadratic differential equations abstract since hilbert posed the problem of systematically counting and locating the limite cycles of polynomial systems on the plane in 1900 much effort has been expended in its investigation a large body of literature chiefly by chinese and soviet authors has addressed this question in the context of differential equations whose field is specified by quadratic polynomials this paper considers the class of quadratic differential equations which admit a unique equilibrium state and establish sufficient conditions for the existence and uniqueness of limit cycles the work is based upon insights and techniques developed in an earlier analysis of such systems motivated by questions from mathematical control theory

percolation in the random cluster process and q state potts model the critical probabilities pt pt q and ph ph q of the random cluster model with parameter q or 1 on the square lattice are shown to satisfy pt or square root q 1 square root q or ph ph or q 1 pt pt q 1 pt the last inequality is the q analogue of the relation pt ph or 1 of ordinary q 1 percolation though of course in this case equality holds

computation of lyapunov values for two planar polynomial differential systems the complex formulas of computing the lyapunov values are investigated for two planar polynomial systems with a pair of pure imaginary eigenvalues the relations between the two systems and their standard forms are discussed and the transformation formulas are given the corresponding complex formulas of computing the lyapunov values are obtained the results obtained here can be applied to judging the type of singular points and the number of limit cycles in planar polynomial systems

the number of limit cycles of certain polynomial differential equations two dimensional differential systems are considered where p and q are polynomials the question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of p and q an algorithm is described for determining a so called focal basis this can be implemented on a computer estimates can then be obtained for the number of small amplitude limit cycles the technique is applied to certain cubic systems a class of examples with exactly five small amplitude limit cycles is constructed quadratic systems are also considered

comparison and disjoint occurrence inequalities for random cluster models a principal technique for studying percolation ferromagnetic ising potts and random cluster models is the fkg inequality which implies certain stochastic comparison inequalities for the associated probability measures the first result of this paper is a new comparison inequality proved using an argument developed elsewhere in order to obtain strict inequalities for critical values as an application of this inequality we prove that the critical pointp c q of the random cluster model with cluster weighting factorq 1 is strictly monotone inq our second result is a bk inequality for the disjoint occurrence of increasing events in a weaker form than that available in percolation theory

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems this paper presents a survey of results on the bifurcation of limit cycles from centers and separatrix cycles of perturbed planar analytic systems and contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of such systems the basic theme throughout the paper is that the number positions and multiplicities of the limit cycles that bifurcate under perturbations are related to the number positions and multiplicities of the zeros of the melnikov function for the system the general theory is illustrated by a number of examples from the literature some of which are extended to include new results

some theorems on the existence uniqueness and nonexistence of limit cycles for quadratic systems abstract given a quadratic system qs with a focus or a center at the origin we write it in the form y p 2 x y x dy q 2 x y where p 2 and q 2 are homogeneous polynomials of degree 2 if we define f x y x dy p 2 x y yq 2 x y and g x y xq 2 x y yp 2 x y we give results of existence nonexistence and uniqueness of limit cycles if f x y g x y does not change of sign then by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of qs i e the qs with a unique finite singularity and the bounded qs with either one or two finite singularities

strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

uniqueness of limit cycles for polynomial first order differential equations abstract we study the uniqueness of limit cycles periodic solutions that are isolated in the set of periodic solutions in the scalar ode x k 1 m a k sin i k t cos j k t x n k in terms of i k j k n k our main result characterizes under some additional hypotheses the exponents i k j k n k such that for any choice of a 1 a m r the equation has at most one limit cycle the obtained results have direct application to rigid planar vector fields thus planar systems of the form x y x r x y y x y r x y where r x y k 1 m a k x i k y j k concretely when the set i k j k k 1 m has at least three elements or exactly one and another technical condition is satisfied we characterize the exponents i k j k such that the origin of the rigid system is a center for any choice of a 1 a m r and also when there are no limit cycles surrounding the origin for any choice of a 1 a m r

multiple bifurcation in a predator prey system with nonmonotonic predator response a model of a predator prey system showing group defence on the part of the prey is formulated and reduced to a three parameter family of quartic polynomial systems of equations mathematically this system contains the volterra lotka system and yields numerous kinds of bifurcation phenomena including a codimension two singularity of cusp type in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation biologically the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant so that the system exhibits an enrichment paradox and for certain choices of parameters coexistence of stable oscillation and a stable equilibrium

the phase transitions of the planar random cluster and potts models with q ge 1 are sharp we prove that random cluster models with q ge 1 on a variety of planar lattices have a sharp phase transition that is that there exists some parameter p c below which the model exhibits exponential decay and above which there exists a s an infinite cluster the result may be extended to the potts model via the edwards sokal coupling our method is based on sharp threshold techniques and certain symmetries of the lattice in particular it makes no use of self duality part of the argument is not restricted to planar models and may be of some interest for the understanding of random cluster and potts models in higher dimensions due to its nature this strategy could be useful in studying other planar models satisfying the fkg lattice condition and some additional differential inequalities

strict inequality for critical values of potts models and random cluster processes we prove that the critical value c of a ferromagnetic potts model is astrictly decreasing function of the strengths of interaction of the process this is achieved in the more general context of the random cluster representation of fortuin and kasteleyn by deriving and utilizing a formula which generalizes the technique known in percolation theory as russo s formula as a byproduct of the method we present a general argument for showing that at any given point on the critical surface of a multiparameter process the values of a certain critical exponent do not depend on the direction of approach of that point our results apply to all random cluster processes satisfying the fkg inequality

exponential decay for subcritical contact and percolation processes we study the contact process together with a version of the percolation process with one continuously varying coordinate it is proved here that the radius of the infected cluster has an exponentially decaying tail throughout the subcritical phase the same is true of the lebesgue measure in space time of this cluster certain critical exponent inequalities are derived and the critical point of the percolation process in two dimensions is determined exactly

random cluster representation of the blume capel model the so called diluted random cluster model may be viewed as a random cluster representation of the blume capel model it has three parameters a vertex parameter a an edge parameter p and a cluster weighting factor q stochastic comparisons of measures are developed for the vertex marginal when q 1 2 and the edge marginal when q 1 taken in conjunction with arguments used earlier for the random cluster model these permit a rigorous study of part of the phase diagram of the blume capel model

on sobolev type integral inequalities on etablit de nouvelles inegalites integrales du type de sobolev comportant des fonctions de plusieurs variables independantes on obtient de nouvelles estimations pour ce type d inegalite

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

quadratic models for generic local 3 parameter bifurcations on the plane the first chapter deals with singularities occurring in quadratic planar vector fields we make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated or hamiltonian or integrable or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers in the second chapter we provide quadratic models for all the known versal k parameter unfoldings with k 1 2 3 except for the nilpotent focus which cannot occur as a quadratic system we finally show that a certain type of elliptic points of codimension 4 does not have a quadratic versal unfolding introduction how complicated can local bifurcations be in quadratic systems or in other terms which phenomena can be described by germs of families of quadratic vector fields on the plane we do not give a complete answer to this question but we essentially treat two exemplary aspects the first one concerns the singularities themselves emphasizing their codimension as a quadratic system and as a general system the second one concerns the relation between the quadratic unfolding of the singularity and its versal unfolding as a general system the complete treatment of the first problem is described in chapter 1 we make a distinction between singularities having a finite codimension or having infinite codimension as a general system the singularities of the first kind are listed completely while those of the second kind are only checked to be nonisolated or isolated but hamiltonian or integrable or having an axis of symmetry after a linear coordinate change or approachable by centers these cases are of course not mutually distinct for the singularities of the first kind the quadratic codimension reveals to be the same as the general one it can reach at most 5 and this only happens if the vector field is homogeneous different kinds of singularities of codimension 4 however occur received by the editors december 1 1989 1980 mathematics subject classification 1985 revision primary 58fxx 34cxx

the stability of second order quadratic differential equations this paper investigates the stability properties of second order systems dot x f x where f x contains either quadratic terms system 1 or linear and quadratic terms system 2 in x the principal contributions are summarized in two theorems which give necessary and sufficient conditions for stability and asymptotic stability in the large of systems 1 and 2 respectively

multiple limit cycle bifurcation surfaces and global families of multiple limit cycles abstract sufficient conditions are given for the local exitence of multiplicity m limit cycle bifurcation surfaces cm of planar analytic systems depending on n parameters with n m 2 in the generic case the surfaces c2 c3 and c4 are the familiar saddle node cusp and swallow tail bifurcation surfaces respectively the author s termination principle for global one parameter families of simple limit cycles of relatively prime planar analytic systems is generalized to a termination principle for global one parameter families of multiple limit cycles which implies that the boundary of a global limit cycle bifurcation surface typically consists of hopf bifurcation surfaces and or homoclinic or heteroclinic loop bifurcation surfaces

radial equivalence and study of self similarity for two very fast diffusion equations abstract in this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p laplacian equation begun in a previous work we treat the cases m 0 and p 1 we perform an exhaustive study of self similar solutions for both equations based on a phase plane analysis and the correspondences we discover we also obtain special correspondence relations and self maps for the limit case m 1 p 0 which is particularly important in applications in image processing we also find self similar solutions for the very fast p laplacian equation that have finite mass and in particular some of them that conserve mass while this phenomenon is not true for the very fast diffusion equation

a class of vectorfields on s2 that are topologically equivalent to polynomial vectorfields abstract let x be a c1 vectorfield on s2 x y z r x2 y2 z2 1 such that no open subset of s2 is the union of closed orbits of x if x has only a finite number of singular orbits and satisfies one additional condition then it is shown that x is topologically equivalent to a polynomial vectorfield a corollary is that a foliation f of the plane is topologically equivalent to a foliation by orbits of a polynomial vectorfield if and only if f has only a finite number of inseparable leaves

a note on sobolev type inequalities in two independent variables abstract the aim of the present note is to establish some new integral inequalities of the sobolev type involving functions of two independent variables and their first order partial derivatives the analysis used in the proofs is quite elementary and based on some simple observations and applications of the fundamental inequalities

separatrix and limit cycles of quadratic systems and dulac s theorem separatrix cycles for a planar quadratic vector field are studied the results obtained are used to show that in any bounded region of the plane a quadratic vector field has at most a finite number of limit cycles

local bifurcations and a survey of bounded quadratic systems abstract this paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems it is shown that the only finite codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle node and the hopf takens bifurcations of codimensions 1 and 2 and the bogdanov takens bifurcations of codimensions 2 and 3 furthermore it is shown that whenever a bounded quadratic system has one of these critical points then a full generic unfolding of the critical point exists in the class of bounded quadratic systems finally we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system

phase portraits for quadratic systems with a higher order singularity with two zero eigenvalues abstract this paper presents a classification of all possible phase portraits of quadratic systems of differential equations with a singular point with two zero eigenvalues the singular points include the fourth order saddle node the third order point having an elliptic and a hyperbolic sector the third order saddle point and the second order cusp point

decay of correlations in random cluster models we prove exponential decay for the tail of the radius r of the cluster at the origin for subcritical random cluster models under an assumption slightly weaker than that here d is the number of dimensions specifically if throughout the subcritical phase then for some 0 this implies the exponential decay of the two point correlation function of subcritical potts models subject to a hypothesis of at least polynomial decay of this function similar results are known already for percolation and ising models and for potts models when the number q of available states is sufficiently large indeed the hypothesis of polynomial decay has been proved rigorously for these cases in two dimensions the hypothesis that is weaker than requiring that the susceptibility be finite i e that the two point function be summable the principal new technique is a form of russo s formula for random cluster models reported by bezuidenhout grimmett and kesten for the current application this leads to an analysis of a first passage problem for random cluster models and a proof that the associated time constant is strictly positive if and only if the tail of r decays exponentially

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

widely separated frequencies in coupled oscillators with energy preserving quadratic nonlinearity in this paper we present an analysis of a system of coupled oscillators suggested by atmospheric dynamics we make two assumptions for our system the first assumption is that the frequencies of the characteristic oscillations are widely separated and the second is that the nonlinear part of the vector field preserves the distance to the origin using the first assumption we prove that the reduced normal form of our system has an invariant manifold which exists for all values of the parameters this invariant manifold cannot be perturbed away by including higher order terms in the normal form using the second assumption we view the normal form as an energy preserving three dimensional system which is linearly perturbed restricting ourselves to a small perturbation the flow of the energy preserving system is used to study the flow in general we present a complete study of the flow of the energy preserving system and its bifurcations using these results we return to the dissipative system and provide the condition for having a hopf bifurcation of one of the two equilibria of the perturbed system we also numerically follow the periodic solution created via the hopf bifurcation and find a sequence of period doubling and fold bifurcations also a torus or neimark sacker bifurcation

the asymmetric random cluster model and comparison of ising and potts models we introduce the asymmetric random cluster or arc model which is a graphical representation of the potts lattice gas and establish its basic properties the arc model allows a rich variety of comparisons in the fkg sense between models with different parameter values we give for example values h for which the 0 s configuration in the potts lattice gas is dominated by the configuration of the h ising model the potts model with possibly an external field applied to one of the spins is a special case of the potts lattice gas which allows our comparisons to yield rigorous bounds on the critical temperatures of potts models for example we obtain 0 571 1 exp c 0 600 for the 9 state potts model on the hexagonal lattice another comparison bounds the movement of the critical line when a small potts interaction is added to a lattice gas which otherwise has only interparticle attraction arc models can also be compared to related models such as the partial fk model obtained by deleting a fraction of the nonsingleton clusters from a realization of the fortuin kasteleyn random cluster model this comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the potts model

the wetting transition in a random surface model we continue our analysis of the phase diagram of a discrete random surface with no downward fingers lying above a flat two dimensional substrate the surface is closely related to the 2d ising model and its free energy is exactly solvable in much but not all of the phase diagram there is a transition at temperaturet w from a high t infinite height or wet phase to a low t finite height or partially wet phase previously it was shown that when a parameterb related to the contact interaction is positive t w is independent ofb and there is a logarithmic specific heat divergence ast w is approached fromeither side here we show that forb 0 t w does depend onb and there isno thermodynamic singularity from the wet phase the partially wet phases forb 0 andb 0 differ in the absence or presence of a monolayer covering the entire substrate this results in a first order transition across the lineb 0 t t w

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

limit cycles of a class of polynomial vector fields in the plane on montre que pour 0 suffisamment petit le systeme quadratique dx dt x y x x y dy dt x y xy a une source spirale hyperbolique unique a l origine et un cycle limite unique stable entourant l origine

the cyclicity of triangles and segments in quadratic systems abstract the paper deals with quadratic systems having separatrix contour in the form of a triangle or a conic segment the limit cycles appearing near this configuration after quadratic perturbation is studied the maximal number of limit cycles is 3 for the bounded triangle and 2 for the unbounded triangle and for the bounded segment

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

limit cycles of polynomial systems with homogeneous non linearities abstract we study the class of vector fields of the form x p1 x y pn x y y q1 x y qn x y where pk and qk are real homogeneous polynomials of degree k n 2 and the origin is a non degenerate critical point of index 1 for this class of vector fields we give four theorems about their limit cycles

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

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

phases coexistence and surface tensions for the potts model theq states potts model exhibits a first order phase transition at some inverse temperature t between ordered and disordered phases forq large as proved in 1 in space dimension 2 we use theduality transformation as aninternal symmetry of the partition function at t to derive an estimate on the probability of a contour this enables us to prove the preceding result and the following new results i the discontinuity of the mass gap at t ii the existence of astrictly positive surface tension between two ordered phases up to t iii the existence of a non zero surface tension between an ordered and the disordered phase at t

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems