dynamical systems defining jacobi s ϑ constants we propose a system of equations that defines weierstrass jacobi s eta and theta constant series in a differentially closed way this system is shown to have a direct relationship to a little known dynamical system obtained by jacobi the classically known differential equations by darboux halphen chazy and ramanujan are the differential consequences or reductions of these systems the proposed system is shown to admit the lagrangian hamiltonian and nambu formulations we explicitly construct a pencil of nonlinear poisson brackets and complete set of involutive conserved quantities as byproducts of the theory we exemplify conserved quantities for the ramamani dynamical system and quadratic system of halphen brioschi

the godbillon vey invariant of a transversely homogeneous foliation a real projective foliation is a foliation 3 with a system of local coordinates transverse to 5 modelled on rp1 so that the coordinate changes are real linear fractional transformations given a closed manifold m there is but a finite set of values in f 3 m r which the godbillon vey invariant of such foliations may assume a bound on the possible values in terms of the fundamental group is computed for m an oriented circle bundle over a surface this finite set is explicitly computed in 5 c godbillon and j vey introduced a characteristic class associated to a codimension one foliation which lies in the third real cohomology group of the ambient manifold the question was then raised whether one could assign some ge ometric significance to this invariant the problem of understanding the godbillon vey invariant in such a manner was underscored by a result of thurston 12 who showed that for any closed 3 manifold there are families of codimension one folia tions whose godbillon vey invariant varies continuously in a parameter and which take on arbitrarily large values it is the main goal of this paper to show that this situation changes when one restricts to foliations which are transversely homogeneous see 1 to explain the geometric picture which emerges it is worthwhile reviewing briefly gromov s notion

bi hamiltonian structure of the kermack mckendrick model for epidemics the dynamical system proposed by kermack and mckendrick 1933 to model the spread of epidemics is shown to admit bi hamiltonian structure without any restrictions on the rate constants these two inequivalent hamiltonian structures are compatible

integrable systems and reductions of the self dual yang mills equations many integrable equations are known to be reductions of the self dual yang mills equations this article discusses some of the well known reductions including the standard soliton equations the classical painleve equations and integrable generalizations of the darboux halphen system and chazy equations the chazy equation first derived in 1909 is shown to correspond to the equations studied independently by ramanujan in 1916

hamiltonian structure and darboux theorem for families of generalized lotka volterra systems this work is devoted to the establishment of a poisson structure for a format of equations known as generalized lotka volterra systems these equations which include the classical lotka volterra systems as a particular case have been deeply studied in the literature they have been shown to constitute a whole hierarchy of systems the characterization of which is made in the context of simple algebra our main result is to show that this algebraic structure is completely translatable into the poisson domain important poisson structures features such as the symplectic foliation and the darboux canonical representation rise as a result of rather simple matrix manipulations

hamiltonians and variational principles for alfven simple waves the evolution equations for the magnetic field induction b with the wave phase for alfven simple waves are expressed as variational principles and in the hamiltonian form the evolution of b with the phase which is a function of the space and time variables depends on the generalized frenet serret equations in which the wave normal n which is a function of the phase is taken to be tangent to a curve x in a 3d cartesian geometry vector space the physical variables the gas density fluid velocity gas pressure and magnetic field induction in the wave depend only on the phase three approaches are developed one approach exploits the fact that the frenet equations may be written as a 3d hamiltonian system which can be described using the nambu bracket it is shown that b as a function of the phase satisfies a modified version of the frenet equations and hence the magnetic field evolution equations can be expressed in the hamiltonian form a second approach develops an euler poincare variational formulation a third approach uses the frenet frame formulation in which the hodograph of b moves on a sphere of constant radius and uses a stereographic projection transformation due to darboux the equations for the projected field components reduce to a complex riccati equation by using a cole hopf transformation the riccati equation reduces to a linear second order differential equation for the new variable a hamiltonian formulation of the second order differential equation then allows the system to be written in the hamiltonian form alignment dynamics equations for alfven simple waves give rise to a complex riccati equation or equivalently to a quaternionic riccati equation which can be mapped onto the riccati equation obtained by stereographic projection

generalization of solutions of the jacobi pdes associated to time reparametrizations of poisson systems the determination of solutions of the jacobi partial differential equations pdes for finite dimensional poisson systems is considered in particular a novel procedure for the construction of solution families is developed such a procedure is based on the use of time reparametrizations preserving the existence of a poisson structure as a result a method which is valid for arbitrary values of the dimension and the rank of the poisson structure under consideration is obtained in this article two main families of time reparametrizations of this kind are characterized in addition these results lead to a novel application which is also developed namely the global and constructive determination of the darboux canonical form for poisson systems of arbitrary dimension and rank two thus improving the local result provided by darboux theorem for such a case

new solution family of the jacobi equations characterization invariants and global darboux analysis a new family of skew symmetric solutions of the jacobi partial differential equations for finite dimensional poisson systems is characterized and analyzed such family has some remarkable properties firstly it is defined for arbitrary values of the dimension and the rank secondly it is described in terms of arbitrary differentiable functions namely it is not limited to a given degree of nonlinearity additionally it is possible to determine explicitly the fundamental properties of those solutions such as their casimir invariants and the algorithm for the reduction to the darboux canonical form which have been reported only for a very limited sample of finite dimensional poisson structures moreover such analysis is carried out globally in phase space thus improving the usual local scope of the darboux theorem

nonequilibrium statistical mechanics of a susceptible infected recovered epidemic model the liouville and stochastic liouville equations have been investigated with respect to the susceptible infected recovered sir epidemic model described in terms of extended nambu mechanics it has been shown that along with the equilibrium distribution functions the kubo formula can be derived for the sir model although the theory undergoes a few restrictions imposed by the nonlinear nature of the model

quantization with maximally degenerate poisson brackets the harmonic oscillator nambu s construction of multi linear brackets for super integrable systems can be thought of as degenerate poisson brackets with a maximal set of casimirs in their kernel by introducing privileged coordinates in phase space these degenerate poisson brackets are brought to the form of heisenberg s equations we propose a definition for constructing quantum operators for classical functions which enables us to turn the maximally degenerate poisson brackets into operators they pose a set of eigenvalue problems for a new state vector the requirement of the single valuedness of this eigenfunction leads to quantization the example of the harmonic oscillator is used to illustrate this general procedure for quantizing a class of maximally super integrable systems

hamiltonian structure of compartmental epidemiological models any epidemiological compartmental model with constant population is shown to be a hamiltonian dynamical system in which the total population plays the role of the hamiltonian function moreover some particular cases within this large class of models are shown to be bi hamiltonian new interacting compartmental models among different populations which are endowed with a hamiltonian structure are introduced the poisson structures underlying the hamiltonian description of all these dynamical systems are explicitly presented and their associated casimir functions are shown to provide an efficient tool in order to find exact analytical solutions for epidemiological models such as the ones describing the dynamics of the covid 19 pandemic

new global solutions of the jacobi partial differential equations abstract a new family of solutions of the jacobi partial differential equations for finite dimensional poisson systems is investigated this family is mathematically remarkable as the functional dependences of the solutions appear to be associated to the distinguished invariants of the solutions themselves this kind of poisson structure termed distinguished solutions or d solutions is defined for every nontrivial combination of values of the dimension and the rank and is also determined in terms of functions of arbitrary nonlinearity properties usually not present simultaneously in the already known solution families in addition d solutions display several properties allowing the generation of an infinity of d solutions from a given one which is an uncommon feature in the framework of the jacobi equations furthermore a special family of d solutions complying with the previous requirements is constructively characterized and analyzed examples are discussed focusing on physical implications and including an application for the global construction of the darboux canonical form

hamiltonian and quasi hamiltonian systems nambu poisson structures and symmetries the theory of hamiltonian and quasi hamiltonian systems with respect to nambu poisson structures is studied it is proved that if a dynamical system is endowed with certain properties related to the theory of symmetries then it can be considered as a quasi hamiltonian or hamiltonian system with respect to an appropriate nambu poisson structure several examples of this construction are presented these examples are related to integrability and also to superintegrability

generalized hamiltonian structures for ermakov systems we construct poisson structures for ermakov systems using the ermakov invariant as the hamiltonian two classes of poisson structures are obtained one of them is degenerate in which case we derive the casimir functions in some situations the existence of casimir functions can give rise to superintegrable ermakov systems finally we characterize the cases where linearization of the equations of motion is possible

phase transition and landscape statistics of the number partitioning problem the phase transition in the number partitioning problem npp i e the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution is investigated by examining the energy landscape of this classic optimization problem this is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure termed a barrier tree the leaves and internal nodes of which represent respectively the minima and the lowest energy saddles connecting those minima here we apply several measures of shape balance and symmetry as well as of branch lengths barrier heights to the barrier trees that result from the landscape of the npp aiming at identifying traces of the easy hard transition we find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights only the difficulty measure given by the maximum value of the ratio between the barrier height and the energy surplus of local minima succeeded in detecting traces of the phase transition in the tree in addition we show that the barrier trees associated with the npp are very similar to random trees contrasting dramatically with trees associated with the p spin glass and random energy models we also examine critically a recent conjecture on the equivalence between the npp and a truncated random energy model

perturbed euler top and bifurcation of limit cycles on invariant casimir surfaces abstract analytical perturbations of the euler top are considered the perturbations are based on the poisson structure for such a dynamical system in such a way that the casimir invariants of the system remain invariant for the perturbed flow by means of the poincare pontryagin theory the existence of limit cycles on the invariant casimir surfaces for the perturbed system is investigated up to first order of perturbation providing sharp bounds for their number examples are given

evolutionary algorithms with preference polyhedron for interval multi objective optimization problems multi objective optimization problems mops with interval parameters are ubiquitous in real world applications existing evolutionary optimization methods however aim to obtain a set of well converged and evenly distributed pareto optimal solutions in this paper we presented a novel evolutionary algorithm ea that interacts with a decision maker dm during the optimization process to obtain the most preferred solution the theory of preference polyhedron for the optimization problem with interval parameters was developed first an interactive evolutionary algorithm iea for mops with interval parameters was then proposed based on the theory of preference polyhedron the algorithm periodically provides a set of non dominated solutions to the dm a preference polyhedron is constructed in the objective space by taking the worst solution chosen by the dm as the vertex the solutions with the same rank are sorted based on the polyhedron constructed finally our method was empirically evaluated on several mops with interval parameters where the value functions were used to simulate the dm s responses the numerical results indicated that our method is simpler and more efficient than the a posteriori method

poisson structure of dynamical systems with three degrees of freedom it is shown that the poisson structure of dynamical systems with three degrees of freedom can be defined in terms of an integrable one form in three dimensions advantage is taken of this fact and the theory of foliations is used in discussing the geometrical structure underlying complete and partial integrability techniques for finding poisson structures are presented and applied to various examples such as the halphen system which has been studied as the two monopole problem by atiyah and hitchin it is shown that the halphen system can be formulated in terms of a flat sl 2 r valued connection and belongs to a nontrivial godbillon vey class on the other hand for the euler top and a special case of three species lotka volterra equations which are contained in the halphen system as limiting cases this structure degenerates into the form of globally integrable bi hamiltonian structures the globally integrable bi hamiltonian case is a linear and the sl 2 r structure is a quadratic unfolding of an integrable one form in 3 1 dimensions it is shown that the existence of a vector field compatible with the flow is a powerful tool in the investigation of poisson structure and some new techniques for incorporating arbitrary constants into the poisson one form are presented herein this leads to some extensions analogous to q extensions of poisson structure the kermack mckendrick model and some of its generalizations describing the spread of epidemics as well as the integrable cases of the lorenz lotka volterra may leonard and maxwell bloch systems admit globally integrable bi hamiltonian structure

nambu dirac manifolds we present a generalization of the dirac structure in the direction of the nambu poisson structure it is shown that with every integrable nambu dirac structure there is associated a leibniz algebroid which yields a singular foliation endowed with a closed form the examples of nambu dirac manifolds are dirac manifolds nambu poisson manifolds and manifolds with closed forms a different leibniz algebroid structure associated with a nambu poisson structure is adopted for this generalization

composite integrators for bi hamiltonian systems abstract symmetric composition methods are applied to dynamical systems in bi hamiltonian form to lotka volterra equations to the completely integrable lorenz equation and to the periodic toda lattice the numerical results obtained show that the hamiltonians are preserved with high accuracy and the periodicity of the solutions are retained

first integrals of a generalized darboux halphen system a third order system of nonlinear ordinary differential equations depending on three arbitrary parameters is analyzed the system arises in the study of su 2 invariant hypercomplex manifolds and is a dimensional reduction of the self dual yang mills equation the general solution first integrals and the nambu poisson structure of the system are explicitly derived it is shown that the first integrals are multi valued on the phase space even though the general solution of the system is single valued for special choices of parameters

robust static output feedback controller synthesis using kharitonov s theorem and evolutionary algorithms this paper presents a simple but effective tuning strategy for robust static output feedback sof controllers with minimal quadratic cost in the context of multiple parametric uncertainties finding this type of controller is known to be computationally intractable using conventional techniques this is mainly due to the non convexity of the resulting control problem which has a fixed structure to solve this kind of control problem easily and directly without using any complicated mathematical manipulations we utilize kharitonov s theorem and an evolutionary algorithm ea for the resolution of the underlying constrained optimization problem using kharitonov s theorem a family of bounded robustly stable static output feedback controllers can be defined and ea is used to select the controller that ensures a minimal quadratic cost within this family the resulting tuning strategy is applicable to both stable and unstable systems without any limitations on the order of the process to be controlled a numerical study was conducted to demonstrate the validity of the proposed tuning procedure

new four dimensional solutions of the jacobi equations for poisson structures a four dimensional family of skew symmetric solutions of the jacobi equations for poisson structures is characterized as a consequence previously known types of poisson structures found in a diversity of physical situations appear to be obtainable as particular cases of new family of solutions additionally it is possible to apply constructive methods for the explicit determination of fundamental properties of those solutions such as their casimir invariants symplectic structure and the algorithm for the reduction to the darboux canonical form which have been reported only for a limited sample of known finite dimensional poisson structures moreover the results developed are valid globally in phase space thus ameliorating the usual scope of darboux theorem which is of local nature

optimizing transformations of structural balance in signed networks with potential relationships abstract a signed network includes positive edges negative edges and 0 edges the last of which denote potential relationships however 0 edges are commonly ignored in transformations of unbalanced networks in this paper we take 0 edges into account and solve the optimization problem in a more comprehensive way we transform the unbalanced network by not only changing signs of edges but also changing edges into potential the experimental results show that our method can solve this problem efficiently and that our solutions are cost saving

line integral methods which preserve all invariants of conservative problems recently the class of hamiltonian boundary value methods hbvms has been introduced with the aim of preserving the energy associated with polynomial hamiltonian systems and more in general with all suitably regular hamiltonian systems however many interesting problems admit other invariants besides the hamiltonian function it would be therefore useful to have methods able to preserve any number of independent invariants this goal is achieved by generalizing the line integral approach which hbvms rely on thus obtaining a number of generalizations which we collectively name line integral methods in fact it turns out that this approach is quite general so that it can be applied to any numerical method whose discrete solution can be suitably associated with a polynomial such as a collocation method as well as to any conservative problem in particular a completely conservative variant of both hbvms and gauss collocation methods is presented numerical experiments confirm the effectiveness of the proposed methods

quasi hamiltonian structure and hojman construction abstract given a smooth vector field and assuming the knowledge of an infinitesimal symmetry x hojman s hojman the construction of a poisson structure out of a symmetry and a conservation law of a dynamical system j phys a math gen 29 1996 667 674 proposed a method for finding both a poisson tensor and a function h such that is the corresponding hamiltonian system in this paper we approach the problem from geometrical point of view the geometrization leads to the clarification of several concepts and methods used in hojman s paper in particular the relationship between the nonstandard hamiltonian structure proposed by hojman and the degenerate quasi hamiltonian structures introduced by crampin and sarlet m crampin w sarlet bi quasi hamiltonian systems j math phys 43 2002 2505 2517 is unveiled in this paper we also provide some applications of our construction

bi hamiltonian dynamical systems related to low dimensional lie algebras abstract a method of generation of bi hamiltonian systems based on the use of low dimensional lie algebras is presented new nontrivial 3 4 and 5 dimensional bi hamiltonian systems are found

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

what statistical mechanics has to say to computer scientists abstract we discuss the relationship between the large n limit of statistical mechanics and the large n limit of computer complexity theory the cost function of many computer optimization problems may be treated as a hamiltonian whose ground state is the solution of the problem and statistical mechanics gives us many insights into the structure of possible algorithms or solutions

clustering to minimize the maximum intercluster distance the problem of clustering a set of points so as to minimize the maximum intercluster distance is studied an o kn approximation algorithm where n is the number of points and k is the number of clusters that guarantees solutions with an objective function value within two times the optimal solution value is presented this approximation algorithm succeeds as long as the set of points satisfies the triangular inequality we also show that our approximation algorithm is best possible with respect to the approximation bound if pz np

hamiltonian equations in r3 the hamiltonian formulation of n 3 systems is considered in general the most general solution of the jacobi equation in r3 is proposed the form of the solution is shown to be valid also in the neighborhood of some irregular points compatible poisson structures and corresponding bi hamiltonian systems are also discussed hamiltonian structures the classification of irregular points and the corresponding reduced first order differential equations of several examples are given

quantum algebras in classical mechanics it is shown that the equations of motion do not determine their hamiltonian descriptions uniquely in particular different poisson bracket relations are admissible for a given physical problem the free symmetric top and a charged particle moving on a homogeneous and constant magnetic field for instance may be described by a classical hamiltonian theory with an su 2 q poisson bracket structure

bi hamiltonian structure in 2 d field theory we exhibit the bi hamiltonian structure of the equations of associativity witten dijkgraaf verlinde verlinde dubrovin equations in 2 d topological field theory which reduce to a single equation of monge ampere type f ttt f xxt 2 f xxx f xtt in the case of three primary fields the first hamiltonian structure of this equation is based on its representation as a 3 component system of hydrodynamic type and the second hamiltonian structure follows from its formulation in terms of a variational principle with a degenerate lagrangian

quadratic poisson algebras and their infinite dimensional extensions a certain generalization of the classical modified yang baxter equation is presented also certain poisson algebras arising in the calculus of variations are investigated

lie symmetries for lie systems a lie system is a nonautonomous system of first order differential equations admitting a superposition rule i e a map expressing its general solution in terms of a generic family of particular solutions and some constants using that a lie system can be considered as a curve in a finite dimensional lie algebra of vector fields a so called vessiot guldberg lie algebra we associate every lie system with a lie algebra of lie point symmetries induced by the vessiot guldberg lie algebra this enables us to derive lie symmetries of relevant physical systems described by first and higher order systems of differential equations by means of lie systems in an easier way than by standard methods a generalization of our results to partial differential equations is introduced among other applications lie symmetries for several new and known generalizations of the real riccati equation are studied

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

new solutions of the jacobi equations for three dimensional poisson structures a systematic investigation of the skew symmetric solutions of the three dimensional jacobi equations is presented as a result three disjoint and complementary new families of solutions are characterized such families are very general thus unifying many different and well known poisson structures seemingly unrelated which now appear embraced as particular cases of a more general solution this unification is not only conceptual but allows the development of algorithms for the explicit determination of important properties such as the symplectic structure the casimir invariants and the darboux canonical form which are known only for a limited sample of poisson structures these common procedures are thus simultaneously valid for all the particular cases which can now be analyzed in a unified and more economic framework instead of using a case by case approach in addition the methods developed are valid globally in phase space thus ameliorating the usual scope of darboux reduction which is only of

bi hamiltonian structure in frenet serret frame abstract we reduce the problem of constructing bi hamiltonian structure in three dimensions to the solutions of a riccati equation in moving coordinates of frenet serret frame all explicitly constructed examples in the literature are exhausted by constant solutions we conclude that in three dimensions vector fields which are not eigenvectors of the curl operator are locally bi hamiltonian

jacobi structures in r3 the most general jacobi brackets in r3 are constructed after solving the equations imposed by the jacobi identity two classes of jacobi brackets were identified according to the rank of the jacobi structures the associated hamiltonian vector fields are also constructed

evolutionary algorithms for optimization problems with uncertainties and hybrid indices many optimization problems in real world applications contain both explicit quantitative and implicit qualitative indices that usually contain uncertain information how to effectively incorporate uncertain information in evolutionary algorithms is one of the most important topics in information science in this paper we study optimization problems with both interval parameters in explicit indices and interval uncertainties in implicit indices to incorporate uncertainty in evolutionary algorithms we construct a mathematical uncertain model of the optimization problem considering the uncertainties of interval objectives and then we transform the model into a precise one by employing the method of interval analysis finally we develop an effective and novel evolutionary optimization algorithm to solve the converted problem by combining traditional genetic algorithms and interactive genetic algorithms the proposed algorithm consists of clustering of a large population according to the distribution of the individuals and estimation of the implicit indices of an individual based on the similarity among individuals in our experiments we apply the proposed algorithm to an interior layout problem a typical optimization problem with both interval parameters in the explicit index and interval uncertainty in the implicit index our experimental results confirm the feasibility and efficiency of the proposed algorithm

poisson lie groups bi hamiltonian systems and integrable deformations given a lie poisson completely integrable bi hamiltonian system on we present a method which allows us to construct under certain conditions a completely integrable bi hamiltonian deformation of the initial lie poisson system on a non abelian poisson lie group of dimension n where is the deformation parameter moreover we show that from the two multiplicative poisson lie hamiltonian structures on that underly the dynamics of the deformed system and by making use of the group law on one may obtain two completely integrable hamiltonian systems on by construction both systems admit reduction via the multiplication in to the deformed bi hamiltonian system in the previous approach is applied to two relevant lie poisson completely integrable bi hamiltonian systems the lorenz and euler top systems

probabilistic analysis of the number partitioning problem given a sequence of n positive real numbers the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of over the two sets is minimized in the case in which the s are statistically independent random variables uniformly distributed in the unit interval this np complete problem is equivalent to the problem of finding the ground state of an infinite range random antiferromagnetic ising model we employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large n limit furthermore we calculate analytically the fraction of metastable states i e states that are stable against all single spin flips and found that it vanishes like