consistent estimation of a general nonparametric regression function in time series abstract we propose an estimator of the conditional distribution of x t x t 1 x t 2 and the corresponding regression function e x t x t 1 x t 2 where the conditioning set is of infinite order we establish consistency of our estimator under stationarity and ergodicity conditions plus a mild smoothness condition

on testing the validity of sequential probability forecasts events are observed sequentially and at each stage a probability for the next event is assessed we consider the relationship between the sequence of probability forecasts and the sequence of outcomes we argue that the forecasts may be considered empirically valid when both these sequences are consistent with a common joint distribution for the events to aid in assessing validity we introduce various test statistics that measure in a natural way the empirical performance of the probability forecasts in the light of the outcomes obtained it is shown that under the null hypothesis of forecast validity such statistics will after suitable normalization have a standard normal or chi squared distribution virtually irrespective of the common joint distribution supposed to underlie both sequences

determination of limit cycles for two dimensional dynamical systems we consider in this paper two dimensional dynamical systems of the form x ifmmode dot else fi p x y y ifmmode dot else fi q x y where p and q are analytic functions we introduce a method for finding the limit cycles of the system this method consists of searching for a power series solution of the equation p ensuremath partial v ensuremath partial x q ensuremath partial v ensuremath partial y ensuremath partial p ensuremath partial x ensuremath partial q ensuremath partial y v the limit cycles are determined from the condition v x y 0

a class of polynomial planar vector fields with polynomial first integral abstract we give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity in the affirmative case our algorithm computes a minimal first integral in addition we solve the poincare problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field

integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities abstract we consider the class of polynomial differential equations x x y p n x y y x y q n x y where p n and q n are homogeneous polynomials of degree n these systems have a focus at the origin if 0 and have either a center or a focus if 0 inside this class we identify a new subclass of darbouxian integrable systems having either a focus or a center at the origin additionally under generic conditions such darbouxian integrable systems can have at most one limit cycle and when it exists is algebraic for the case n 2 and 3 we present new classes of darbouxian integrable systems having a focus

monodromy and stability of a class of degenerate planar critical points abstract necessary and also sufficient monodromy conditions for a large class of degenerate singular points of planar differential systems are given for these systems we also find a computable expression of the principal term of the asymptotic expansion of the return map which gives the stability of the point

invariant algebraic curves and conditions for a centre conditions for the existence of a centre in two dimensional systems are considered along the lines of darboux we show how these methods can be used in the search for maximal numbers of bifurcating limit cycles we also extend the method to include more degenerate cases such as are encountered in less generic systems these lead to new classes of integrals in particular the kukles system is considered and new centre conditions for this system are obtained

integrability degenerate centers and limit cycles for a class of polynomial differential systems we consider the class of polynomial differential equations email protected p n x y p n 1 x y p n 2 x y email protected q n x y q n 1 x y q n 2 x y for n 1 and where p i and q i are homogeneous polynomials of degree i these systems have a linearly zero singular point at the origin if n 2 inside this class we identify a new subclass of darboux integrable systems and some of them having a degenerate center i e a center with linear part identically zero moreover under additional conditions such darboux integrable systems can have at most one limit cycle we provide the explicit expression of this limit cycle

some new results on darboux integrable differential systems abstract we deal with complex planar differential systems having a darboux first integral h we present a definition of remarkable values and remarkable curves associated to h and characterize the existence of a polynomial inverse integrating factor for these systems furthermore we study the relation between the characteristic polynomial f and the inverse integrating factors of the system and show the importance of the numerator of the exponential factor of h in the construction of f

a logic of probability with application to the foundations of statistics a simple example shows that the classical theory of probability implies more than one can deduce via kolmogorov s calculus of probability developing dawid s ideas i propose a new calculus of probability which is free from this drawback this calculus naturally leads to a new interpretation of probability i argue that attempts to create a general empirical theory of probability should be abandoned and we should content ourselves with the logic of probability establishing relations between probabilistic theories and observations my approach to the logic of probability is based on a variant of ville s principle of the excluded gambling strategy

a fast algorithm for optimally increasing the edge connectivity let g v e be an undirected unweighted graph with n nodes m edges and edge connectivity lambda given an input parameter delta the edge augmentation problem is to find the smallest set of edges to add to g so that its edge connectivity is increased by delta in this paper we present a solution to this problem which runs in o delta 2 nm delta 3 n 2 n f g where f g is the time to perform one maximum flow on g in fact our solution gives the optimal augmentation for every delta 1 le delta le delta in the same time bound by introducing minor modifications to the solution we can solve the problem without knowing delta in advance and we can also solve the node weighted version and the degree constrained version of the problem if delta 1 then our solution is particularly simple it runs in o nm time and it is a natural generalization of the algorithm in k eswaran and r e tarjan siam j comput 5 1976 pp 653 665 for the case where lambda delta 2 we also solve the converse problem in the same time bound given an input number k increase the connectivity of g as much as possible by adding at most k edges our solution makes extensive use of the structure of particular sets of cuts

darboux theory of integrability in cn taking into account the multiplicity abstract darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces called now the darboux theory of integrability in 1979 jouanolou successfully improved the darboux theory of integrability characterizing the existence of rational first integrals for this he used sophisticated tools of algebraic geometry the aim of this paper is to improve the classical result of darboux and the new one of jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces additionally our proof of the improved result of jouanolou is much simpler and elementary than the original one some examples show that the improved darboux theory of integrability with multiplicity is much useful than the classical one

a topological characterization of the ω limit sets for analytic flows on the plane the sphere and the projective plane in this paper the limit sets for analytic and polynomial differential equations on the plane are characterized up to homeomorphisms the analogous problem is solved in full detail for analytic flows on the sphere and the projective plane we also explain how to carry on the same program for analytic flows defined on open subsets of these surfaces

the well calibrated bayesian abstract suppose that a forecaster sequentially assigns probabilities to events he is well calibrated if for example of those events to which he assigns a probability 30 percent the long run proportion that actually occurs turns out to be 30 percent we prove a theorem to the effect that a coherent bayesian expects to be well calibrated and consider its destructive implications for the theory of coherence

cyclicity of a simple focus via the vanishing multiplicity of inverse integrating factors abstract first we provide new properties about the vanishing multiplicity of the inverse integrating factor of a planar analytic differential system at a focus after we use this vanishing multiplicity for studying the cyclicity of foci with pure imaginary eigenvalues and with homogeneous nonlinearities of arbitrary degree having either its radial or angular speed independent of the angle variable in polar coordinates after we study the cyclicity of a class of nilpotent foci in their analytic normal form

configurations of limit cycles in lienard equations abstract we show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial lienard equation the related vector field x is morse smale moreover it has the minimum number of singularities required for realizing the configuration in a lienard equation we provide an explicit upper bound on the degree of x which is lower than the results obtained before obtained in the context of general polynomial vector fields

algebraic limit cycles in polynomial systems of differential equations using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4 5 and 6 we also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3 moreover we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3 as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4

testing eulerianity and connectivity in directed sparse graphs property testing problems are relaxations of decision problems a property testing algorithm referred to as a testing algorithm or tester has to decide if a given object has a prespecified property or is far from the property for a given distance parameter and for a prespecified distance measure the tester is given query access to the input and is required to run in sublinear time in this paper we focus on testing properties of directed graphs digraphs in particular we present the following results where n is the number of vertices in the graph d is the maximum degree and davg is the average degree we present a testing algorithm for the property of eulerianity in bounded degree digraphs which runs in time11the notation o g k stands for o g k polylog g k o 1 for unbounded degree digraphs we show a lower bound of n and give a testing algorithm that runs in time o n 3 2 we study the property of k vertex connectivity and give testing algorithms for both bounded degree and unbounded degree digraphs that run in time o ck d kd and o ck davg k 1 respectively where c 1 is a constant in addition we give a simpler analysis of the testing algorithm for k vertex connectivity in bounded degree undirected graphs that was shown by yoshida and ito y yoshida h ito property testing on k vertex connectivity of graphs in icalp 08 proceedings of the 35th international colloquium on automata languages and programming part i springer verlag berlin heidelberg 2008 pp 539 550 and extend the result to unbounded degree undirected graphs we consider the property of k edge connectivity in digraphs and simplify the analysis of the algorithm of yoshida and ito y yoshida h ito testing k edge connectivity of digraphs journal of system science and complexity 23 1 2010 91 101 for this property in addition we give a simpler analysis for the correctness of the testing algorithm for k edge connectivity in undirected graphs that was introduced by goldreich and ron o goldreich d ron property testing in bounded degree graphs algorithmica 2002 302 343

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

note on a paper of j llibre and g rodriguez concerning algebraic limit cycles abstract in a recent paper of llibre and rodriguez j differential equations 198 2004 374 380 it is proved that every configuration of cycles in the plane is realizable up to homeomorphism by a polynomial vector field of degree at most 2 n r 1 where n is the number of cycles and r the number of primary cycles a cycle c is primary if there are no other cycles contained in the bounded region limited by c in this letter we prove the same theorem by using an easier construction but with a greater polynomial bound the vector field we construct has degree at most 4 n 1 by using the same technique we also construct r 3 polynomial vector fields realizing up to homeomorphism any configuration of limit cycles which can be linked and knotted in r 3 this answers a question of r sverdlove

on prequential model assessment in life history analysis summary we extend the prequential approach introduced by a p dawid to continuous time marked point processes a prequential forecasting system is expressed by means of the stochastic intensities of the point process the exact independence and 1 exponentiality of the derived compensators are used to assess the predictive performance of the model and the inferential procedure

calibration based empirical probability probability forecasts for a sequence of uncertain events may be compared with the outcomes of those events by means of a natural criterion of empirical validity calibration it is shown that any two sequences of forecasts which both meet this criterion must be in asymptotic agreement these agreed values can then be considered as correct objective probability forecasts for the particular sequence of outcome results obtained however the objective forecasts vary with the extent of the information taken into account when they are formulated we thus obtain a general theory of empirical probability relative to an information base this theory does not require that such probabilities be interpreted in terms of repeated trials of the same event some implications of this theory are discussed

the poincare problem algebraic integrability and dicritical divisors abstract we solve the poincare problem for plane foliations with only one dicritical divisor moreover in this case we give a simple algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case we also provide an algorithm to compute a rational first integral of prefixed genus g 1 of any type of plane foliation f when the number of dicritical divisors dic f is larger than 2 this algorithm depends on suitable families of invariant curves when dic f 2 it proves that the degree of the rational first integral can be bounded only in terms of g the degree of f and the local analytic type of the dicritical singularities of f

a characterization of centres via first integrals on considere l existence d une integrale premiere lisse et l existence d une mesure invariante equivalente a la mesure de lebesgue pour des systemes differentiels dans le plan

using melnikov functions of any order for studying limit cycles abstract in this work we study a class of planar hamiltonian systems perturbed with general quadratic polynomials in order to identify the number of limit cycles employing melnikov functions of any order we show that the class of systems can have at most three limit cycles and the limit is reached

on the integrability of two dimensional flows this paper deals with the notion of integrability of flows or vector fields on two dimensional manifolds we consider the following two key points about first integrals 1 they must be defined on the whole domain of definition of the flow or vector field or defined on the complement of some special orbits of the system 2 how are they computed we prove that every local flow on a two dimensional manifold m always has a continuous first integral on each component of m where is the set of all separatrices of we consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two dimensional flow or vector field finally we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus of a non resonant hyperbolic node and of a siegel hyperbolic saddle

planar polynomial vector fields having first integrals and algebraic limit cycles with the help of abel differential equations we obtain a new class of darboux integrable planar polynomial differential systems which have degenerate infinity moreover such integrable systems may have algebraic limit cycles also we present the explicit expressions of these algebraic limit cycles for quintic systems

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

sufficient conditions for the existence of at least n or exactly n limit cycles for the lienard differential systems in this paper we study the limit cycles of the lienard differential system of the form x f x x g x 0 or its equivalent system x y f x y g x we provide sufficient conditions in order that the system exhibits at least n or exactly n limit cycles

darboux theory of integrability in the sparse case darboux s theorem and jouanolou s theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field these results are given in terms of the degree of the polynomial vector field here we show that we can get the same kind of results if we consider the size of a newton polytope associated to the vector field furthermore we show that in this context the bound is optimal

a class of reversible cubic systems with an isochronous center abstract we study cubic polynomial differential systems having an isochronous center and an inverse integrating factor formed by two different parallel invariant straight lines such systems are time reversible we find nine subclasses of such cubic systems see theorem 8 we also prove that time reversible polynomial differential systems with a nondegenerate center have half of the isochronous constants equal to zero see theorem 3 we present two open problems

morphisms and inverse problems for darboux integrating factors polynomial vector fields which admit a prescribed darboux integrat ing factor are quite well understood when the geometry of the underlying curve is nondegenerate in the general setting morphisms of the affine plane may remove degeneracies of the curve and thus allow more structural insight in the present paper we establish some properties of integrating factors subjected to morphisms and we discuss in detail one particular class of morphisms related to finite reflection groups the results indicate that degeneracies for the underlying curve generally impose restrictions on the nontrivial vector fields which admit a given integrating factor

a characterization of theω limit sets of planar continuous dynamical systems abstract if the family of curves g i i and the set of points s are given we find necessary and sufficient conditions for the existence of a c k map f from the plane into itself 0 k if k 0 we also assume that f is locally lipschitz such that the curves from g and the points from s are respectively nondegenerate trajectories and singular points of the system x f x and additionally s i i is the limit set of some trajectory of the system in such cases we also provide a detailed geometric description of s i i after contraction of the connected components of s

directed vertex connectivity augmentation the problem of finding a smallest set of new edges to be added to a given directed graph to make it k vertex connected was shown to be polynomially solvable recently in 6 for any target connectivity k 1 however the algorithm given there relied on the ellipsoid method here we refine some results of 6 which makes it possible to construct a combinatorial algorithm which is polynomial for any fixed k short proofs for extensions of some earlier results related to this problem will also be given

realization problems for limit cycles of planar polynomial vector fields abstract we show that for any finite configuration of closed curves r 2 one can construct an explicit planar polynomial vector field that realizes up to homeomorphism as the set of its limit cycles with prescribed periods multiplicities and stabilities the only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities the constructed vector fields are darboux integrable and admit a polynomial inverse integrating factor

the problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems abstract in this work we study the centers of planar analytic vector fields which are limit of linear type centers it is proved that all the nilpotent centers are limit of linear type centers and consequently the poincare liapunov method to find linear type centers can be also used to find the nilpotent centers moreover we show that the degenerate centers which are limit of linear type centers are also detectable with the poincare liapunov method

configurations of limit cycles and planar polynomial vector fields abstract we show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field moreover the realization can be made by algebraic limit cycles and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles

inverse problems for dynamical systems in this paper we consider the problem of reconstructing a structurally stable vector field from its global phase portrait or from certain local pieces of the phase portrait we show that for arbitrary finite sets of points and of simple closed curves we can construct a structurally stable system having the elements of those sets as its critical points and limit cycles and a prescribed topological type in a neighborhood of each of them given a global structurally stable phase portrait we can construct a structurally stable polynomial system which is topologically equivalent to it some of our results were originally announced in 28 1 the general theory of dynamical systems orginated in the study of solution curves for systems of ordinary differential equations in a phase space this qualitative theory had its first major development in the work of poincare on the topology of integral curves a typical problem is to give a geometric description of the solutions of a particular system or class of systems with emphasis on the location of special features such as critical points and periodic orbits an inverse problem then is the construction of a system of equations given geometric properties of the solutions which may be in the form of a picture of the flow a schematic diagram of phase space or a list of critical orbits defined analytically a method for solving such problems is potentially useful in building models in the various fields in which the theory of dynamical systems is now being applied it could provide a basic model to which statistical identification techniques 141 could then be applied there is also an intrinsic interest in methods for constructing polynomial systems of prescribed type which is related to the lack of any progress toward a solution of hilbert s sixteenth problem the recent appearance of a counterexample 27 to the petrovsky landis conjecture for quadratic systems leaves the problem completely open one of the earliest results of this kind was obtained by forster in 1938

on the 16th hilbert problem for algebraic limit cycles abstract for a polynomial planar vector field of degree n 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 n 1 n 2 2 when n is even and n 1 n 2 2 when n is odd furthermore these upper bounds are reached

the 16th hilbert problem on algebraic limit cycles abstract for real planar polynomial differential systems there appeared a simple version of the 16th hilbert problem on algebraic limit cycles is there an upper bound on the number of algebraic limit cycles of all polynomial vector fields of degree m in j llibre r ramirez n sadovskaia on the 16th hilbert problem for algebraic limit cycles j differential equations 248 2010 1401 1409 llibre ramirez and sadovskaia solved the problem providing an exact upper bound in the case of invariant algebraic curves generic for the vector fields and they posed the following conjecture is 1 m 1 m 2 2 the maximal number of algebraic limit cycles that a polynomial vector field of degree m can have in this paper we will prove this conjecture for planar polynomial vector fields having only nodal invariant algebraic curves this result includes the llibre et al s as a special one for the polynomial vector fields having only non dicritical invariant algebraic curves we answer the simple version of the 16th hilbert problem

on the remarkable values of the rational first integrals of polynomial vector fields abstract the remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by poincare he was mainly interested in their algebraic aspects here we are interested in their dynamic aspects i e how they contribute to the phase portrait of the system to its separatrices to its singular points etc the relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor

darboux theory of integrability for a class of nonautonomous vector fields the goal of this paper is to extend the classical darboux theory of integrability from autonomous polynomial vector fields to a class of nonautonomous vector fields we also provide sufficient conditions for applying this theory of integrability and we illustrate this theory in several examples