recursion operator for the stationary nizhnik veselov novikov equation we present a new general construction of a recursion operator from the zero curvature representation using it we find a recursion operator for the stationary nizhnik veselov novikov equation and present a few low order symmetries generated with the help of this operator

darboux transformation with dihedral reduction group we construct the darboux transformation with dihedral reduction group for the 2 dimensional generalisation of the periodic volterra lattice the resulting backlund transformation can be viewed as a nonevolutionary integrable differential difference equation we also find its generalised symmetry and the lax representation for this symmetry using formal diagonalisation of the darboux matrix we obtain local conservation laws of the system

rational recursion operators for integrable differential difference equations in this paper we introduce the concept of prehamiltonian pairs of difference operators demonstrate their connections with nijenhuis operators and give a criteria for the existence of weakly nonlocal inverse recursion operators for differential difference equations we begin with a rigorous setup of the problem in terms of the skew field of rational pseudo difference operators over a difference field with a zero characteristic subfield of constants and the principal ideal ring of matrix rational pseudo difference operators in particular we give a criteria for a rational operator to be weakly nonlocal a difference operator is called prehamiltonian if its image is a lie subalgebra with respect to the lie bracket on the difference field two prehamiltonian operators form a prehamiltonian pair if any linear combination of them is prehamiltonian then we show that a prehamiltonian pair naturally leads to a nijenhuis operator and a nijenhuis operator can be represented in terms of a prehamiltonian pair this provides a systematic method to check whether a rational operator is nijenhuis as an application we construct a prehamiltonian pair and thus a nijenhuis recursion operator for the differential difference equation recently discovered by adler and postnikov the nijenhuis operator obtained is not weakly nonlocal we prove that it generates an infinite hierarchy of local commuting symmetries we also illustrate our theory on the well known examples including the toda the ablowitz ladik and the kaup newell differential difference equations

on the density of lyndon roots in factors this work takes another look at the number of runs that a string may contain and provides an alternative proof for the bound we also propose another stronger conjecture that states the following for a fixed order on the alphabet within every factor of a word there are at most as many occurrences of lyndon roots corresponding to runs in the word as the length of the factor only first occurrences of roots in each run are considered

classification of third order integrable evolution equations abstract the classification of scalar quasilinear integrable evolution equations of third order is completed under the symmetry approach a nonstandard way to represent an evolution equation in the form of a system is used in order to simplify the calculations the integrable nonlinearizable quasilinear equations are found to be related to the classical kdv and kn equations through differential substitutions we formulate the conjecture that all the third order integrable evolution equations are related to these two equations or to the linear one through miura type differential substitutions

boundary conditions for integrable equations the problem of constructing boundary conditions for nonlinear equations compatible with higher symmetries is considered in particular this problem is discussed for the sine gordon jiber shabat liouville and kdv equations new results are obtained for the last two ones the boundary condition for the kdv contains two arbitrary constants the substitution maps it onto the boundary condition with linear dependence on t for the potentiated kdv

integrable equations on time scales integrable systems are usually given in terms of functions of continuous variables on r in terms of functions of discrete variables on z and recently in terms of functions of q variables on kq we formulate the gel fand dikii gd formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations in particular they yield integrable equations over integers difference equations and over q numbers q difference equations we formulate the gd formalism also in terms of shift operators for all regular discrete time scales we give a method allowing to construct the recursion operators for integrable systems on time scales finally we give a trace formula on time scales and then construct infinitely many conserved quantities casimirs of the integrable systems on time scales

lenard scheme for two dimensional periodic volterra chain we prove that for compatible weakly nonlocal hamiltonian and symplectic operators hierarchies of infinitely many commuting local symmetries and conservation laws can be generated under some easily verified conditions no matter whether the generating nijenhuis operators are weakly nonlocal or not we construct a recursion operator of the two dimensional periodic volterra chain from its lax representation and prove that it is a nijenhuis operator furthermore we show that this system is a generalized bi hamiltonian system rather surprisingly the product of its weakly nonlocal hamiltonian and symplectic operators gives rise to the square of the recursion operator

hydrodynamic type integrable equations on a segment and a half line the concept of integrable boundary conditions is applied to hydrodynamic type systems examples of such boundary conditions for dispersionless toda systems are obtained the close relation of integrable boundary conditions with integrable reductions in multifield systems is observed the problem of consistency of boundary conditions with the hamiltonian formulation is discussed examples of hamiltonian integrable hydrodynamic type systems on a segment and a semiline are presented

the r matrix approach to integrable systems on time scales a general unifying framework for integrable soliton like systems on time scales is introduced the r matrix formalism is applied to the algebra of differential operators in terms of which one can construct an infinite hierarchy of commuting vector fields the theory is illustrated by two infinite field integrable hierarchies on time scales which are differential counterparts of kp and mkp the difference counterparts of akns and kaup broer soliton systems are constructed as related finite field restrictions

factorization of operators ii we extend the methods of a previous paper j math phys 21 2508 1980 factorizing the general scalar third order differential operator and obtain a miura transformation for the boussinesq equation we give a general factorized eigenvalue problem we also give a hamiltonian structure associated with the factorized eigenvalue problem we derive several isospectral flows some of klein gordon type

multilayer dielectric filter design using a multiobjective evolutionary algorithm design of multilayer dielectric filters involve the identification of suitable dielectric material and appropriate thicknesses of the layers that best satisfies the desired frequency response for the application such problems like any other practical design optimization problem require simultaneous consideration of multiple objectives and constraints in this paper we introduce a multiobjective evolutionary algorithm that is capable of handling unconstrained and constrained single and multiobjective problems without any restriction on the number and nature of variables constraints and objectives the algorithm handles constraints and objectives separately using two fitness measures derived out of nondominance unlike most of its counterparts which use a single fitness measure unlike most evolutionary algorithms where only the good parents participate in mating our algorithm ensures that all solutions participate in mating which is useful for exploring highly nonlinear search spaces the diversity of the solutions is controlled by the partner selection scheme that prefers elites with distant neighbors as mating partners the results of two multiobjective test problems three multilayer dielectric filter designs low pass bandpass and stopband and one variable layer low pass filter design are presented in this paper to highlight the benefits offered by our algorithm in terms of modeling flexibility computational efficiency and its ability to arrive at competitive nondominated designs a comparison of our results with those obtained using a single objective aggregated formulation for the stopband filter design is also presented we have also compared the performance of our algorithm with nondominated sorting genetic algorithm nsga ii for the low pass filter design where our results are better

the number of runs in a string a run in a string is a nonextendable with the same minimal period periodic segment in a string the set of runs corresponds to the structure of internal periodicities in a string periodicities in strings were extensively studied and are important both in theory and practice combinatorics of words pattern matching computational biology let n be the maximal number of runs in a string of length n it has been shown that n o n the proof was very complicated and the constant coefficient in o n has not been given explicitly we demystify the proof of the linear upper bound for n and propose a new approach to the analysis of runs based on the properties of subperiods the periods of periodic parts of the runs we show that n n and there are at most o 67n runs with periods larger than 87 this supports the conjecture that the number of all runs is smaller than n we also give a completely new proof of the linear bound and discover several new interesting periodicity lemmas

two way string matching a new string matching algorithm is presented which can be viewed as an intermediate between the classical algorithms of knuth morris and pratt on the one hand and boyer and moore on the other hand the algorithm is linear in time and uses constant space as the algorithm of galil and seiferas it presents the advantage of being remarkably simple which consequently makes its analysis possible the algorithm relies on a previously known result in combinatorics on words called the critical factorization theorem which relates the global period of a word to its local repetitions of blocks

hamiltonian theory over noncommutative rings and integrability in multidimensions a generalization of the adler gel fand dikii scheme is used to generate bi hamiltonian structures in two spatial dimensions in order to implement this scheme a hamiltonian theory is built over a noncommutative ring namely the ring of formal pseudodifferential operators bi hamiltonian structures generated in this way can be used for the kadomtsev petviashvili equation as well as other integrable equations in 2 1

preliminary classification of qt f q qx qxx qxxx the preliminary classification of qt f q qx qxx qxxx is given the results are compared with fokas s symmetry and mikhailov shabat sokolov formal symmetry approaches

on matrix painleve hierarchies abstract we define a matrix first painleve hierarchy and a matrix second painleve p ii hierarchy for our matrix p ii hierarchy we also give auto backlund transformations and consider the iteration of solutions this is the first paper to define matrix painleve hierarchies and to give auto backlund transformations for a matrix painleve hierarchy we also consider amongst other results the derivation of sequences of special integrals and autonomous limits until now it has been unknown how to connect the known matrix p ii equation to the obvious candidates for related completely integrable matrix partial differential equations our matrix p ii hierarchy is placed firmly within the context of a matrix modified korteweg de vries mkdv hierarchy in deriving our matrix p ii hierarchy we make use of the hamiltonian structure of this matrix mkdv hierarchy we thus see once again the importance for painleve hierarchies of the integrability structures of related completely integrable equations

recursion operators of some equations of hydrodynamic type we give a general method for constructing recursion operators for some equations of hydrodynamic type admitting a nonstandard lax representation we give several examples for n 2 and n 3 containing the equations of shallow water waves and its generalizations with their first two general symmetries and their recursion operators we also discuss a reduction of n 1 systems to n systems of some new equations of hydrodynamic type

integrable systems and their recursion operators in this paper we discuss the structure of recursion operators we show that recursion operators of evolution equations have a nonlocal part that is determined by symmetries and cosymmetries this enables us to compute recursion operators more systematically under certain conditions which hold for all examples known to us nijenhuis operators are well defined i e they give rise to hierarchies of infinitely many commuting symmetries of the operator moreover the nonlocal part of a nijenhuis operator contains the candidates of roots and coroots

pareto optimal microwave filter design using multiobjective differential evolution microwave filters play an important role in modern wireless communications a novel method for the design of multilayer dielectric and open loop ring resonator olrr filters under constraints is presented the proposed design method is based on generalized differential evolution gde3 which is a multiobjective extension of differential evolution de gde3 algorithm can be applied for global optimization to any engineering problem with an arbitrary number of objective and constraint functions gde3 is compared against other evolutionary multiobjective algorithms like nondominated sorting genetic algorithm ii nsga ii multiobjective particle swarm optimization mopso and multiobjective particle swarm optimization with fitness sharing mopso fs for a number of microwave filter design cases in the multilayer dielectric filter design case a predefined database of low loss dielectric materials is used the results indicate the advantages of this approach and the applicability of this design method

integrable coupled kdv systems we give the conditions for a system of n coupled korteweg de vries kdv type of equations to be integrable we find the recursion operators of each subclass and give all examples for n 2

recursion operators and conservation laws for discrete lax equations in this paper a method for constructing recursion operator of discrete lax equations is proposed to illustrate the method recursion operators for discrete boussinesq equation and discrete kdv equation are obtained then the recursion operators of continuous boussinesq equation and kdv equation are successfully recovered by the discrete ones infinitely many conservation laws of discrete boussinesq equation are constructed by the discrete residue formula and the first three conserved quantities are given explicitly and verified numerically existence of recursion operators and infinite number of conservation laws proves the integrability of the discrete lax equations

integrable boundary conditions for the toda lattice the problem of construction of the boundary conditions for the toda lattice compatible with its higher symmetries is considered it is demonstrated that this problem is reduced to finding the differential constraints consistent with the zs akns hierarchy a method of their construction is offered based on the backlund transformations it is shown that the generalized toda lattices corresponding to the non exceptional lie algebras of finite growth can be obtained by imposing one of the four simplest integrable boundary conditions on both ends of the lattice this fact allows in particular the solution of the reduction problem of the series a toda lattices into the series d lattices deformations of the found boundary conditions are presented which lead to the painleve type equations

on construction of recursion operators from lax representation in this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting lax representation several new examples are given in particular we find the recursion operators for some kdv type systems of integrable equations

noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods here noncommutative hierarchies of nonlinear equations are studied they represent a generalization to the operator level of corresponding hierarchies of scalar equations which can be obtained from the operator ones via a suitable projection a key tool is the application of backlund transformations to relate different operator valued hierarchies indeed in the case when hierarchies in 1 1 dimensions are considered a backlund chart depicts links relating in particular the korteweg de vries kdv to the modified kdv mkdv hierarchy notably analogous links connect the hierarchies of operator equations the main result is the construction of an operator soliton solution depending on an infinite dimensional parameter to start with the potential kdv hierarchy is considered then backlund transformations are exploited to derive solution formulas in the case of kdv and mkdv hierarchies it is remarked that hierarchies of matrix equations of any dimension are also incorporated in the present framework

on a method for constructing the lax pairs for nonlinear integrable equations we suggest a direct algorithm for searching the lax pairs for nonlinear integrable equations it is effective for both continuous and discrete models the first operator of the lax pair corresponding to a given nonlinear equation is found immediately coinciding with the linearization of the considered nonlinear equation the second one is obtained as an invariant manifold to the linearized equation a surprisingly simple relation between the second operator of the lax pair and the recursion operator is discussed the recursion operator can immediately be found from the lax pair examples considered in the article are convincing evidence that the found lax pairs differ from the classical ones the examples also show that the suggested objects are true lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries in the case of the hyperbolic type partial differential equation our algorithm is slightly modified in order to construct the lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite laplace sequence the efficiency of the method is illustrated by application to some equations given in the svinolupov sokolov classification list for which the lax pairs and the recursion operators have not been found earlier

boundary value problems for integrable equations compatible with the symmetry algebra boundary value problems for integrable nonlinear partial differential equations are considered from the symmetry point of view families of boundary conditions compatible with the harry dym kdv and mkdv equations and the volterra chain are discussed we also discuss the uniqueness of some of these boundary conditions

a class of nonautonomous coupled kdv systems a class of nonautonomous coupled korteweg de vries kdv systems in 1 1 dimensions are considered for integrability classification integrability of the systems is associated with the existence of a certain recursion operator some new integrable nonautonomous two component kdv systems are found

toward the classification of third order integrable evolution equations a non standard way of representing an evolution equation in the form of a system is proposed this representation allows us to investigate all the different classes of third order integrable evolution equations simultaneously using this approach a preliminary classification of these equations is made

some results on the number of periodic factors in words abstract the exponent of a word is the relation between its length and its minimal period a repetition subrepetition in a word is a factor of exponent not less than 2 less than 2 a gapped repeat in a word is a pair of identical factors with a positive gap between these factors in the paper a review of some recent results concerning the number of repetitions subrepetitions and gapped repeats in words of fixed length is presented

non autonomous svinolupov jordan kdv systems non autonomous svinolupov jordan kdv systems are considered the integrability criteria of such systems are associated with the existence of recursion operators a new non autonomous kdv system and its recursion operator is obtained for all n the examples for n 2 and 3 are studied in detail some possible transformations which map some systems to autonomous ones are also discussed pacs numbers 0230i 0220 0230j

matrix korteweg de vries and modified korteweg de vries hierarchies noncommutative soliton solutions the present work continues work on kdv type hierarchies presented by s carillo and c schiebold noncommutative korteweg de vries and modified korteweg de vries hierarchies via recursion methods j math phys 50 073510 2009 10 1063 1 3155080 general solution formulas for the kdv and mkdv hierarchies are derived by means of banach space techniques both in the scalar and matrix case a detailed analysis is given of solitons breathers their countable superpositions as well as of multisoliton solutions for the matrix hierarchies

initial value problems for evolutionary partial differential equations and higher order conditional symmetries we suggest a new approach to the problem of dimensional reduction of initial boundary value problems for evolution equations in one spatial variable the approach is based on higher order generalized conditional symmetries of the equations involved it is shown that reducibility of an initial value problem for an evolution equation to a cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question we also give some examples of the solution of initial value problems for second and third order nonlinear differential equations by reduction by their conditional symmetries we give a systematic classification of general second order partial differential equations admitting second order conditional symmetries based on lie s classification of invariant second order ordinary differential equations this yields five classes of principally new initial value problems for nonlinear evolution equations which admit no lie symmetries and are reducible via second order conditional symmetries

a characterization of the squares in a fibonacci string a finite fibonacci stringfn is defined as follows f0 b f1 a for every integer n 2 fn fn 1fn 2 for n 1 the length of fn is denoted by the infinite fibonacci stringf is the string which contains every fn n 1 as a prefix apart from their general theoretical importance fibonacci strings are often cited as worst case examples for algorithms which compute all the repetitions or all the abelian squares in a given string in this paper we provide a characterization of all the squares in f hence in every prefix fn this characterization naturally gives rise to a algorithm which specifies all the squares of fn in an appropriate encoding this encoding is made possible by the fact that the squares of fn occur consecutively in runs the number of which is by contrast the known general algorithms for the computation of the repetitions in an arbitrary string require time and produce outputs when applied to a fibonacci string fn

integrable boundary value problems for elliptic type toda lattice in a disk the concept of integrable boundary value problems for soliton equations on r and r is extended to regions enclosed by smooth curves classes of integrable boundary conditions in a disk for the toda lattice and its reductions are found

a novel noncommutative kdv type equation its recursion operator and solitons a noncommutative kdv type equation is introduced extending the backlund chart in carillo et al symmetry integrability geom methods appl 12 087 2016 this equation called meta mkdv here i

variable coefficient third order korteweg de vries type of equations it is shown that the integrable subclasses of the equations q t f x t q 3 h x t q q 1 are the same as the integrable subclasses of the equations q t q 3 f q q 1

internal external and generalized symmetries backlund s theorem which characterizes contact transformations is generalized to give an analogous characterization of internal symmetries of systems of differential equations for a wide class of systems of differential equations we prove that every internal symmetry comes from a first order generalized symmetry and conversely every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry we analyze the contact conditions in detail deducing powerful necessary conditions for a system of differential equations that admit genuine internal symmetries i e ones which do not come from classical external symmetries applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to hilbert and cartan are the noncompact real form of the exceptional simple lie group g2

can we always distinguish between positive and negative hierarchies it is a common belief that nonlinearizable pdes in 1 1 dimensions cannot possess two mutually inverse positive order recursion operators and that the negative hierarchies for such pdes unlike the positive ones contain at most a finite number of local symmetries we show that the equation a generalization of the hunter saxton equation considered by manna and neveu provides a counterexample for both of these assertions namely we find two positive order integro differential recursion operators for this equation and show that the corresponding positive and negative hierarchies consist solely of local symmetries the recursion operators in question turn out to be mutually inverse on symmetries of the equation under study

painleve property auto backlund transformation lax pairs and reduction to the standard form for the korteweg de vries equation with nonuniformities it is demonstrated that the kdv equation with nonuniformities ut a t u b x t u x c t uux d t uxxx e x t 0 has the painleve property if the compatibility condition among the coefficients of it holds bt a lc b bbx dbxxx 2ah hl d c2 dh dt ce x 2a2 al d3 c4 da dt l d c l d c2 d dt l d c where l d dt lg and h t is an arbitrary function of t the auto backlund transformation and lax pairs for this equation are obtained by truncating the laurent expansion furthermore assuming the compatibility condition then the kdv equation with nonuniformities is transformable via suitable variable transformations to the standard kdv

on point transformations of evolution equations two evolution equations are considered of the form ut hi x t u ux uxx i 1 2 which are related by the general point transformation x p x t u t q x t u u r x t u it is shown that the time transformation must necessarily be of the form t q t and that if hi are polynomials in the spatial derivatives of u then x p x t

equivalence groupoid of a class of variable coefficient korteweg de vries equations we classify the admissible transformations in a class of variable coefficient korteweg de vries equations as a result a full description of the structure of the equivalence groupoid of the class is given the class under study is partitioned into six disjoint normalized subclasses the widest possible equivalence group for each subclass is found which appears to be generalized extended in five cases ways for improvement of transformational properties of the subclasses are proposed using gaugings of arbitrary elements and mapping between classes the group classification of one of the subclasses is carried out as an illustrative example

nonlocal symmetries of the hirota satsuma coupled korteweg de vries system and their applications exact interaction solutions and integrable hierarchy the nonlocal symmetry is derived from the known darboux transformation dt of the hirota satsuma coupled korteweg de vries hs ckdv system and infinitely many nonlocal symmetries are given by introducing the internal parameters by extending the hs ckdv system to an auxiliary system with five dependent variables the prolongation is found to localize the so called seed nonlocal symmetry related to the dt by applying the general lie point symmetry method to this enlarged system we obtain two main results a new type of finite symmetry transformation is derived which is different from the initial dt and can generate new solutions from old ones some novel exact interaction solutions among solitons and other complicated waves including periodic cnoidal waves and painleve waves are computed through similarity reductions in addition two kinds of new integrable models are proposed from the obtained nonlocal symmetry the negative hs ckdv hierarchy by introducing the internal parameters the integrable models both in lower and higher dimensions by restricting the symmetry constraints

the runs conjecture the runs conjecture proposed by kolpakov and kucherov 1999 7 states that the number of occurrences of maximal repetitions runs in a string of length n runs n is at most n we almost solve the conjecture by proving that runs n 1 029n this bound is obtained using a combination of theory and computer verification

integrable nonautonomous nonlinear schrodinger equations are equivalent to the standard autonomous equation a class of nonautonomous nonlinear schrodinger equations claiming to be novel integrable systems with rich properties continues to appear in the literature all such equations are shown to be not new but equivalent to the standard autonomous equation which trivially explains their integrability features

the coupled modified korteweg de vries equations similarity reduction lie backlund symmetries and integrability the lie point symmetries of n coupled modified korteweg de vries equations n cmkdv is derived and shown that the similarity reduction associated with symmetries passes the painleve property indicating its integrability the nonclassical symmetry analysis of bluman and cole and the direct method of clarkson and kruskal to n cmkdv equations show that there exists no new similarity reductions a sequence of lie backlund symmetries for n cmkdv equations is derived explicitly establishing its complete integrability the question of constructing the recursion operator of n cmkdv a characteristic of integrable systems is also discussed

how many runs can a string contain given a string x x 1 n a repetition of period p in x is a substring u r x i 1 i rp p u r 2 where neither u x i 1 i p nor x i 1 i r 1 p 1 is a repetition the maximum number of repetitions in any string x is well known to be q nlogn a run or maximal periodicity of period p in x is a substring u rt x i 1 i rp t of x where u r is a repetition t a proper prefix of u and no repetition of period p begins at position i of x or ends at position i rp t 1 in 2000 kolpakov and kucherov showed that the maximum number r n of runs in any string x 1 n is o n but their proof was nonconstructive and provided no specific constant of proportionality at the same time they presented experimental data to prompt the conjecture r n

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

on the recursion operators for the gerdjikov mikhailov and valchev system we consider the recursion operator approach to the soliton equations related to an auxiliary linear system introduced recently by gerdjikov mikhailov and valchev gmv system we discuss the recursion operators obtained for the gmv system and show that they are particular cases of the generating operators related to a zakharov shabat type linear system on the algebra sl 3 c in pole gauge

group classification and exact solutions of variable coefficient generalized burgers equations with linear damping admissible point transformations between burgers equations with linear damping and time dependent coefficients are described and used in order to exhaustively classify lie symmetries of these equations optimal systems of one and two dimensional subalgebras of the lie invariance algebras obtained are constructed the corresponding lie reductions to odes and to algebraic equations are carried out exact solutions to particular equations are found some generalized burgers equations are linearized to the heat equation by composing equivalence transformations with the hopf cole transformation

constructing conditionally integrable evolution systems in 1 1 dimensions a generalization of invariant modules approach given a generalized lie backlund vector field satisfying certain nondegeneracy assumptions we explicitly describe all 1 1 dimensional evolution systems that admit this vector field as a generalized conditional symmetry the connection with the theory of symmetries of systems of odes and with the theory of invariant modules is discussed

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

on form preserving point transformations of partial differential equations new identities are presented relating arbitrary order partial derivatives of and for the general point transformation these identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 1 partial differential equations these results facilitate the search for point symmetries both discrete and continuous lie and assist the search for point transformations which reduce equations to canonical but similar form a simple test for the existence of hodograph type transformations between equations of similar form is given

optimum design of yagi uda antennas using computational intelligence optimization of yagi uda antennas is a challenging design problem since the antenna characteristics such as gain input impedance maximum sidelobe level etc are known to be extremely sensitive to the design variables viz element lengths and their spacings this corresponds to a highly nonlinear and multimodal function space with functional and slope discontinuities that limit the use of conventional gradient based optimization approaches although stochastic zeroth order methods like genetic algorithm and evolutionary algorithm are attractive choices for such classes of problems their successful application requires scaling and aggregating parameters to handle constraints and objectives that may not be easy to provide in this paper we introduce a stochastic zeroth order optimization algorithm that handles constraints and objectives separately via pareto ranking that eliminates the problem of scaling and aggregation the algorithm is based on principles of learning and is embedded with three key learning strategies that control whom to learn from i e leader identification and leader selection and what to learn i e information acquisition in order to better guide the search the leader identification mechanism partitions the individuals into a set of leaders and a set of followers the followers interact with the leader and move toward the better performing leaders in search for better solutions as the algorithm does not require parameters for scaling or aggregation it provides the designer the true flexibility that is necessary to handle various forms of the design problem effectively and at a computational cost that is comparable to existing stochastic optimization methods results of three single objective antenna design examples a four element a 15 element and a fixed boom length 22 element design are presented and compared with published results to illustrate the behavior of the proposed algorithm and highlight its benefits in solving a wide variety of antenna design problems a new set of results are presented for a multiobjective formulation of the design problem

symmetry classification of kdv type nonlinear evolution equations group classification of a class of third order nonlinear evolution equations generalizing kdv and mkdv equations is performed it is shown that there are two equations admitting simple lie algebras of dimension three next we prove that there exist only four equations invariant with respect to lie algebras having nontrivial levi factors of dimension four and six our analysis shows that there are no equations invariant under algebras which are semi direct sums of levi factor and radical making use of these results we prove that there are three nine thirty eight fifty two inequivalent kdv type nonlinear evolution equations admitting one two three and four dimensional solvable lie algebras respectively finally we perform a complete group classification of the most general linear third order evolution equation

evolutionary programming in electromagnetic optimization a review in this paper we review recent advances in evolutionary programming ep and its implementation in various antenna microwave frequency selective surfaces and rf circuit optimization problems ep unlike the other two paradigms of evolutionary computational techniques namely genetic algorithms ga and evolution strategies es uses a mutation only variation operator where adaptive and or self adaptive techniques exist or can easily be designed for adapting the parameters of mutation operator during the evolution process we present the so called meta ep and design of its mutation operator using gaussian cauchy and poisson mutations as well as a hybrid of these mutations for continuous discrete and mixed parameter electromagnetic optimization problems in addition an efficient hybrid use of ep gradient search methods and cluster analysis as well as a novel hybrid ep ga algorithm are discussed examples presented include optimizations of corrugated horn antennas multilayered stacked microstrip antennas yagi uda arrays partially reflective surfaces dielectric filters meander line polarizer and rf duplexers