integral invariants and limit sets of planar vector fields we show that if a planar system of differential equations admits an inverse integrating factor v defined in a neighborhood of a singular point with exactly one zero eigenvalue then v vanishes along any separatrix of the singular point additionally we prove that if k is a compact or limit set that contains a regular point or has an elliptic or parabolic sector if not and if v is defined on a neighborhood of k then v vanishes at at least one point of k and on all of k if v is real analytic or morse

fluctuation theorems and entropy production with odd parity variables we show that the total entropy production in stochastic processes with odd parity variables under time reversal is separated into three parts only two of which satisfy the integral fluctuation theorems in general one is the usual excess contribution that can appear only transiently and is called nonadiabatic another one is attributed solely to the breakage of detailed balance the last part that does not satisfy the fluctuation theorem comes from the steady state distribution asymmetry for odd parity variables that is activated in a nontransient manner the latter two parts combine together as the housekeeping adiabatic contribution whose positivity is not guaranteed except when the excess contribution completely vanishes our finding reveals that the equilibrium requires the steady state distribution symmetry for odd parity variables independently in addition to the usual detailed balance

universal property of the housekeeping entropy production the entropy production of a nonequilibrium system with broken detailed balance is a random variable whose mean value is nonnegative the housekeeping entropy production which is a part of total entropy production is associated with the heat dissipation in maintaining a nonequilibrium steady state we derive a langevin type stochastic differential equation for the housekeeping entropy production the equation allows us to define a housekeeping entropic time ensuremath tau remarkably it turns out that the probability distribution of the housekeeping entropy production at a fixed value of ensuremath tau is given by the gaussian distribution regardless of system details the gaussian distribution is universal for any systems whether in the steady state or in the transient state and whether they are driven by time independent or time dependent driving forces we demonstrate the universal distribution numerically for model systems

experimental study of the fluctuation theorem in a nonequilibrium steady state the fluctuation theorem ft quantifies the probability of second law violations in small systems over short time scales while this theorem has been experimentally demonstrated for systems that are perturbed from an initial equilibrium state there are a number of studies suggesting that the theorem applies asymptotically in the long time limit to systems in a nonequilibrium steady state the asymptotic application of the ft to such nonequilibrium steady states has been referred to in the literature as the steady state fluctuation theorem or ssft in this paper we demonstrate experimentally the application of the ft to nonequilibrium steady states using a colloidal particle localized in a translating optical trap furthermore we show for this colloidal system that the ft holds under nonequilibrium steady states for all time and not just in the long time limit as in the ssft

the inverse integrating factor and the poincare map this work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit we show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit when the regular orbit is a limit cycle we can determine its associated poincare return map in terms of the inverse integrating factor in particular we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it we also apply this result to study the homoclinic loop bifurcation we only consider homoclinic loops whose critical point is a hyperbolic saddle and whose poincare return map is not the identity a local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable that is the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

nonequilibrium thermodynamics of stochastic systems with odd and even variables the total entropy production of stochastic systems can be divided into three quantities the first corresponds to the excess heat while the second two comprise the housekeeping heat we denote these two components the transient and generalized housekeeping heat and we obtain an integral fluctuation theorem for the latter valid for all markovian stochastic dynamics a previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal which restricts consideration of directional variables such as velocity

a model calibration approach to using complete auxiliary information from survey data suppose that the finite population consists of n identifiable units associated with the ith unit are the study variable yi and a vector of auxiliary variables xi the values x1 x2 xn are known for the entire population i e complete but yi is known only if the ith unit is selected in the sample one of the fundamental questions is how to effectively use the complete auxiliary information at the estimation stage in this article a unified model assisted framework has been attempted using a proposed model calibration technique the proposed model calibration estimators can handle any linear or nonlinear working models and reduce to the conventional calibration estimators of deville and sarndal and or the generalized regression estimators in the linear model case the pseudoempirical maximum likelihood estimator of chen and sitter when used in this setting gives an estimator that is asymptotically equivalent to the model calibration estimator but with positive weights some existing estimators

darboux integrability and the inverse integrating factor abstract we mainly study polynomial differential systems of the form dx dt p x y dy dt q x y where p and q are complex polynomials in the dependent complex variables x and y and the independent variable t is either real or complex we assume that the polynomials p and q are relatively prime and that the differential system has a darboux first integral of the form h f 1 1 f p p exp h 1 g 1 n 1 1 exp h q g q n q q where the polynomials f i and g j are irreducible the polynomials g j and h j are coprime and the i and j are complex numbers when i 1 p and j 1 q prelle and singer proved that these systems have a rational integrating factor we improve this result as follows assume that h is a rational function which is not polynomial following to poincare we define the critical remarkable values of h then we prove that the system has a polynomial inverse integrating factor if and only if h has at most two critical remarkable values under some assumptions over the darboux first integral h we show first that the system has a polynomial inverse integrating factor and secondly that if the degree of the system is m the homogeneous part of highest degree of h is a multi valued function and the functions exp h j g j are exponential factors for j 1 q then the system has a polynomial inverse integrating factor of degree m 1 we also present versions of these results for real polynomial differential systems finally we apply these results to real polynomial differential systems having a darboux first integral and limit cycles or foci

empirical likelihood estimation for finite populations and the effective usage of auxiliary information summary in finite population inference problems auxiliary population information is often available we show in this paper that the empirical likelihood method can be naturally applied to such problems to make effective use of the auxiliary information we prove that the resulting estimates have smaller asymptotic variances than the usual estimates which do not use auxiliary information a bahadur type representation for empirical likelihood sample quantiles is given simulation results show that the empirical likelihood estimates perform well among a number of competitors and are model robust

efficient estimation of quadratic finite population functions in the presence of auxiliary information by viewing quadratic and other second order finite population functions as totals or means over a derived synthetic finite population we show that the recently proposed model calibration and pseudoempirical likelihood methods for effective use of auxiliary information from survey data can be readily extended to obtain efficient estimators of quadratic and other second order finite population functions in particular estimation of a finite population variance covariance or variance of a linear estimator can be greatly improved when auxiliary information is available the proposed methods are model assisted in that the resulting estimators are asymptotically design unbiased irrespective of the correctness of a working model but very efficient if the working model is nearly correct they have a number of attractive features which include applicability to a general sampling design incorporation of information on possibly multivariate auxiliary variables and the ability to entertain linear or nonlinear

analytic integrability of cubic linear planar polynomial differential systems for the cubic linear polynomial planar differential systems with a finite singular point we classify the ones which have a local analytic first integral around the origin and the ones that have a global analytic first integral

finite time generalization of the thermodynamic uncertainty relation for fluctuating currents in nonequilibrium steady states the recently discovered thermodynamic uncertainty relation expresses a fundamental relation between their variance and the overall entropic cost associated with the driving we show that this relation holds not only for the long time limit of fluctuations as described by large deviation theory but also for fluctuations on arbitrary finite time scales this generalization facilitates applying the thermodynamic uncertainty relation to single molecule experiments for which infinite time scales are not accessible importantly often this finite time variant of the relation allows inferring a bound on the entropy production that is even stronger than the one obtained from the long time limit we illustrate the relation for the fluctuating work that is performed by a stochastically switching laser tweezer on a trapped colloidal particle

numerical integration of stochastic differential equations several numerical methods for treating stochastic differential equations are considered both the convergence in the mean square limit and the convergence of the moments is discussed and the generation of appropriate random numbers is treated the necessity of simulations at various time steps with an extrapolation to time step zero is emphasized and demonstrated by a simple example

a method for characterizing nilpotent centers to characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest first for the problem of distinguishing between a focus and a center and after for studying the bifurcation of limit cycles from it or from its period annulus we give an effective algorithm in the search of necessary conditions for detecting nilpotent centers based in recent developments moreover we survey the last results on this problem and illustrate our approach by means of examples

empirical likelihood confidence intervals for complex sampling designs we define an empirical likelihood approach which gives consistent design based confidence intervals which can be calculated without the need of variance estimates design effects resampling joint inclusion probabilities and linearization even when the point estimator is not linear it can be used to construct confidence intervals for a large class of sampling designs and estimators which are solutions of estimating equations it can be used for means regressions coefficients quantiles totals or counts even when the population size is unknown it can be used with large sampling fractions and naturally includes calibration constraints it can be viewed as an extension of the empirical likelihood approach to complex survey data this approach is computationally simpler than the pseudoempirical likelihood and the bootstrap approaches the simulation study shows that the confidence interval proposed may give better coverages than the confidence intervals based on linearization bootstrap and pseudoempirical likelihood our simulation study shows that under complex sampling designs standard confidence intervals based on normality may have poor coverages because point estimators may not follow a normal sampling distribution and their variance estimators may be biased

the center problem a view from the normal form theory abstract in this work we analyze some aspects of the center problem from the perspective of the normal form theory we provide alternative proofs of some well known results in the case of non vanishing linear part nondegenerate and nilpotent centers moreover some new results are also derived in particular we show that the unique characterization for centers in this situation is the orbital reversibility also we include some examples illustrating that this characterization is no longer valid for the case of null linear part

integral fluctuation theorem for the housekeeping heat the housekeeping heat qhk is the dissipated heat necessary to maintain the violation of detailed balance in nonequilibrium steady states by analysing the evolution of its probability distribution we prove an integral fluctuation theorem exp qhk 1 valid for arbitrary driven transitions between steady states we discuss gaussian limiting cases and the difference between the new theorem and both the hatano sasa and the jarzynski relation

quasi homogeneous linearization of degenerate vector fields abstract we characterize the analytic planar vector fields orbitally equivalent to its quasi homogeneous leader term by means the existence of a class of inverse integrating factors such a class of inverse integrating factors is determined by providing a normal form of two dimensional scalar functions this fact allows us to give some relevant criteria on t linearization analytical integrability and characterization of the centers for several families of planar vector fields we study the systems whose leader term is quadratic or cubic and we analyze also a class of nilpotent systems

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

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

time reversal and entropy there is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time reversal symmetry and the entropy production in such processes we explain on an elementary mathematical level the relations between entropy production phase space contraction and time reversal starting from a deterministic dynamics both closed and open systems in the transient and in the steady regime are considered the main result identifies under general conditions the statistical mechanical entropy production as the source term of time reversal breaking in the path space measure for the evolution of reduced variables this provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful in equalities we also discuss the markov approximation important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior

invariant curves and analytic integrability of a planar vector field abstract we give an expression of the irreducible invariant curves at the singular point for analytically integrable systems we provide an expression of its primitive first integral this fact allows us to obtain necessary conditions of analytic integrability at degenerate singular points

period function and normalizers of vector fields in rn with n 1 first integrals we extend the theory that relates period function normalizers and first integrals which has been widely developed in r2 to any dimension in particular we show how to find n 1 independent normalizers of vector fields in rn with n 1 independent first integrals a formula for the derivatives of the period function is obtained and a necessary and sufficient condition for isochronicity is also proved these results generalize previous works of freire gasull and guillamon sabatini and villarini

the reversibility and the center problem abstract in this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem it is well known that an analytic system having either a non degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible respectively in this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a c inverse integrating factor into a reversible linear system after rescaling the time moreover if the degenerate center has an analytic or a c reversing symmetry then the transformed system by the map also has a reversing symmetry from the knowledge of a first integral near the center we give a procedure to detect reversing symmetries

existence of inverse integrating factors and lie symmetries for degenerate planar centers abstract we prove that any possibly degenerate center of an analytic planar system admits a c inverse integrating factor and a c lie symmetry in a neighborhood of the singularity we also show that there exists an analytic degenerate center for which any c inverse integrating factor is flat at the singular point and that a sufficient condition for the existence of an analytic inverse integrating factor is the existence of an analytic first integral

proof of the finite time thermodynamic uncertainty relation for steady state currents this paper presents the finite time version of the thermodynamic uncertainty relation for nonequilibrium steady states which sets a bound on the magnitude of current fluctuations the authors use a methodology that is similar to the proof for the long time limit but with large deviation theory applied to an ensemble of many copies

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

generic properties of stochastic entropy production we derive an ito stochastic differential equation for entropy production in nonequilibrium langevin processes introducing a random time transformation entropy production obeys a one dimensional drift diffusion equation independent of the underlying physical model this transformation allows us to identify generic properties of entropy production it also leads to an exact uncertainty equality relating the fano factor of entropy production and the fano factor of the random time which we also generalize to non steady state conditions

speed limit for classical stochastic processes we consider the speed limit for classical stochastic markov processes with and without the local detailed balance condition we find that for both cases a trade off inequality exists between the speed of the state transformation and the entropy production the dynamical activity is related to a time scale and plays a crucial role in the inequality for the dynamics without the local detailed balance condition we use the hatano sasa entropy production instead of the standard entropy production our inequalities consist of the quantities that are commonly used in stochastic thermodynamics and explicitly show underlying physical mechanisms

empirical likelihood inference under stratified random sampling using auxiliary population information summary the empirical likelihood method under stratified random sampling is used for making inferences on finite population parameters our results show that it can lead to efficient estimators by making effective use of auxiliary population information in the form of overall population totals or means ascertained from external sources empirical likelihood ratio confidence intervals for the population mean are also studied and adjusted intervals with asymptotically correct coverage rates are obtained