a new approach to the converse of noether s theorem the concepts of vector fields and forms along a map are used to establish a condition characterising symmetries of the hamiltonian system associated with a regular lagrangian this condition does not mention any second order differential equation field but is expressed in terms of the geometry of the second order tangent bundle this result is also generalised to the case of lagrangian functions depending on higher order derivatives

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

internal deformation of lie algebroids and symplectic realizations given a lie algebroid and a bundle over its base which is endowed with a localizable poisson structure and a flat connection we construct an extended bundle whose dual is endowed with an almost poisson structure that is a quadratic poisson structure when a certain compatibility property is satisfied this new formalism on lie algebroids describes systems with internal degrees of freedom

some new characters on the wire tap channel of type ii the noiseless wire tap channel of type ii with coset coding scheme was provided by ozarow and wyner in this correspondence the user is split into multiple parties who are coordinated in coding their data symbols by using the same encoder the adversary can tap not only partial transmitted symbols but also partial data symbols we are interested in the equivocation of the data symbols to this adversary who has more power than that of ozarow and wyner the generalized hamming weight of wei and the dimension length profile dlp of forney are extended to two code formats relative generalized hamming weight and relative dimension length profile rdlp upper and lower bounds of the new concepts are investigated they are useful to design a perfect secrecy coding scheme for the coordinated multiparty model under a general secrecy standard the coordinated model can provide a higher transmission rate than an uncoordinated time sharing model

on the dimension of subfield subcodes a new lower bound for the dimension of a subfield subcode is given this bound improves the well known lower bound of p delsarte 1975 and coincides in many cases with the true dimension of the code alternant codes and geometric goppa codes are discussed as special cases

on the differential geometry of the euler lagrange equations and the inverse problem of lagrangian dynamics the conditions for a system of second order differential equations to be derivable from a lagrangian the conditions of self adjointness in the terminology of santilli 1978 and others are related in the time independent case to the differential geometry of the tangent bundle of configuration space these conditions are simply expressed in terms of the horizontal distribution which is associated with any vector field representing a system of second order differential equations necessary and sufficient conditions for such a vector field to be derivable from a lagrangian may be stated as the existence of a two form with certain properties it is interesting that it is a deduction not an assumption that this two form is closed and thus defines a symplectic structure some other differential geometric properties of euler lagrange second order differential equations are described

almost hermitian geometries that characterize hamilton jacobi distributions the author considers horizontal lagrangian sub bundles of a cotangent bundle t n that contain the dynamical vector field of a hamiltonian system it is shown that if the hamiltonian is quadratic on the fibres of t n then all such sub bundles can be characterized as solutions to a system of algebro partial differential equations that are defined solely in terms of vertical quantities this system of equations has a universal solution the universal solution is applied to the study of the geometry of the lorentz force law it is shown that the dynamical flow of the lorentz force law on t n is a geodesic vector field for a metric connection on t n with torsion the torsion of this connection is given by an extension of cartan s first structure equation to non linear connections

maximal symmetry group of the hamilton jacobi equation relativistic particle in flat spacetime lie s extended group method has been used to obtain the maximal symmetry group of the hamilton jacobi equation for a relativistic particle moving in flat spacetime for a massive particle it is the 21 parameter inhomogeneous pseudo orthogonal group io 4 1 of signature 4 1 for a zero mass particle like a photon or a neutrino this group is an infinite parameter lie group with an infinite parameter invariant subgroup such that the factor group is isomorphic to the inhomogeneous pseudo orthogonal group io 3 1 of signature 3 1

on the many time formulation of classical particle dynamics starting from the standard one time dynamics of n nonrelativistic particles the n time equations of motion are inferred and a variational principle is formulated a suitable generalization of the classical lie konig theorem is demonstrated which allows the determination of all the associated presymplectic structures the conditions under which the action of an invariance group is canonical are studied and a corresponding noether theorem is deduced a formulation of the theory in terms of n first class constraints is recovered by means of coisotropic imbeddings the proposed approach also provides for a better understanding of the relativistic particle dynamics since it shows that the different roles of the physical positions and the canonical variables is not peculiar to special relativity but rather to any n time approach indeed a nonrelativistic no interaction theorem is deduced

isometries for rank distance and permutation group of gabidulin codes the rank distance was introduced by e m gabidulin see probl pered inform vol 21 p 1 12 1985 he determined an upper bound for the minimum rank distance of a code moreover he constructed a class of codes which meet this bound the so called gabidulin codes we first characterize the linear and semilinear isometries for the rank distance then we determine the isometry group and the permutation group of gabidulin codes of any length we give a characterization of equivalent gabidulin codes finally we prove that the number of equivalence classes of gabidulin codes is exactly the number of equivalence classes of vector spaces of dimension n contained in gf p sup m under some particular relations

n dimensional equations with the maximum number of symmetry generators the authors obtain the general class of ordinary differential equations that can be reduced by a point transformation to the free particle equation and given the general form of the lie symmetry generators for these equations they apply these results to obtain the symmetry generators for the n dimensional harmonic oscillator and for a particle in a constant magnetic field

maximum rank array codes and their application to crisscross error correction a mu n n k array code c over a field f is a k dimensional linear space of n n matrices over f such that every nonzero matrix in c has rank or mu it is first shown that the dimension of such array codes must satisfy the singleton like bound k

symmetries of nonlinear differential equations and linearisation a non linear ordinary differential equation is linearisable if it possesses sl 3 r symmetry the conditions under which the abelian two dimensional subalgebras of sl 3 r are sufficient for linearisation are established

the helmholtz conditions revisited a new approach to the inverse problem of lagrangian dynamics deals with the general problem of finding a multiplier matrix that can give to a prescribed system of second order ordinary equations the structure of euler lagrange equations the approach is based on a generalisation of previous studies on linear systems the main result concerns a set of necessary and sufficient conditions for the existence of a multiplier which contains an infinite set of algebraic equations the coefficients of which can be used to derive necessary conditions involving only the given right hand sides of the differential equations an outline is given of interesting points for future studies and an example is presented for which all multipliers are explicitly constructed

extended tangent bundle formalism for time dependent lagrangian systems the lagrangian formalism for time dependent systems is developed using vector fields defined in the extended tangent bundle t q r the definition of the time independent extended lagrangian function l associated to a time dependent lagrangian l is given and then the techniques of symplectic mechanics are used to prove some properties of the equations determining the euler lagrange vector field the relations between the time independent and the time dependent helmholtz conditions are analyzed from a geometric perspective and finally the noether theorem for the extended lagrangian l is considered

lagrangian systems with constraints a geometric approach to the method of lagrange multipliers a geometric approach to the method of lagrange multipliers is presented using the framework of the tangent bundle geometry the nonholonomic systems with constraint functions linear in the velocities are studied in the first place and then and using this study of the nonholonomic mechanical systems as a previous result the holonomic systems are considered the lagrangian inverse problem is also analysed and finally the theory is illustrated with several examples

on list decodability of random rank metric codes and subspace codes codes in rank metric have a wide range of applications to construct such codes with better list decoding performance explicitly it is of interest to investigate the listdecodability of random rank metric codes it is shown that if n m b is a constant then for every rank metric code in fm n q with rate r and list decoding radius must obey the gilbert varshamov bound that is r 1 1 b otherwise the list size can be exponential and hence no polynomial time list decoding is possible on the other hand for arbitrary 0 0 with e and being independent of each other with high probability a random rank metric code with rate r 1 1 b can be efficiently list decoded up to a fraction of rank errors with constant list size o 1 e we establish similar results for constant dimension subspace codes moreover we show that with high probability the list decoding radius of random fq linear rank metric codes also achieve the gilbert varshamov bound with constant list size o exp 1 e

second order equation fields and the inverse problem of lagrangian dynamics the transformation properties of determined autonomous systems of second order ordinary differential equations identified as vector fields on the tangent bundle of the space of dependent variables are derived and studied the inverse problem of lagrangian dynamics is studied from this transformation viewpoint as well as the problem of alternative lagrangians in particular regular lagrangians which are analytic as functions of the first derivatives are considered finally the inverse problem for second order systems corresponding to the geodesic flow of a symmetric linear connection is investigated

network generalized hamming weight in this work we extend the notion of generalized hamming weight for classical linear block codes to linear network codes by introducing the network generalized hamming weight nghw for a given network with respect to a fixed linear network code the basic properties of nghw are studied we further show that nghw can be used as a tool to characterize the security performance of a linear network code on a wiretap network we also introduce the notion of network maximum distance separable code nmds code by extending the notion of maximum distance separable code in classical algebraic coding theory we prove that nmds codes play an important role in minimizing the information that an eavesdropper can obtain from the network

variational and integrable connections the two dimensional symmetric connections whose geodesic equations are derivable from a lagrangian function are divided into five classes this classification is compared with that of douglas for more general classes of systems of ordinary differential equations three of the five classes of connections are further investigated and in most cases specific lagrangians are exhibited in particular those connections that are engendered by lagrangians homogeneous in velocities are characterized in terms of the ricci tensor of each connection finally several examples of variational connections that possess integrals of motion are given thereby extending the known class of completely integrable systems

on metrics for error correction in network coding the problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model for coherent network coding where knowledge of the network topology and network code is assumed at the source and destination nodes the error correction capability of an outer code is succinctly described by the rank metric as a consequence it is shown that universal network error correcting codes achieving the singleton bound can be easily constructed and efficiently decoded for noncoherent network coding where knowledge of the network topology and network code is not assumed the error correction capability of a subspace code is given exactly by a new metric called the injection metric which is closely related to but different than the subspace metric of koumltter and kschischang in particular in the case of a non constant dimension code the decoder associated with the injection metric is shown to correct more errors then a minimum subspace distance decoder all of these results are based on a general approach to adversarial error correction which could be useful for other adversarial channels beyond network coding

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

equivalent lagrangians multidimensional case we generalize a theorem known for one dimensional nonsingular equivalent lagrangians l and to the multidimensional case in particular we prove that the matrix which relates the left hand sides of the euler lagrange equations obtained from l and is such that the trace of all its integer powers are constants of the motion we construct several multidimensional examples in which the elements of are functions of position velocity and time and prove that in some cases equivalence prevails even if det 0

relative generalized matrix weights of matrix codes for universal security on wire tap networks universal security over a network with linear network coding has been intensively studied however previous linear codes and code pairs used for this purpose were linear over a larger field than that used on the network which restricts the possible packet lengths of optimal universal secure codes does not allow to apply known list decodable rank metric codes and requires performing operations over a large field in this paper we introduce new parameters relative generalized matrix weights and relative dimension rank support profile for code pairs that are linear over the field used in the network and show that they measure the universal security performance of these code pairs for one code and non square matrices generalized matrix weights coincide with the existing delsarte generalized weights hence we prove the connection between these latter weights and secure network coding which was left open as main applications the proposed new parameters enable us to 1 obtain optimal universal secure linear codes on noiseless networks for all possible packet lengths in particular for packet lengths not considered before 2 obtain the first universal secure list decodable rank metric code pairs with polynomial sized lists based on a recent construction by guruswami et al and 3 obtain new characterizations of security equivalences of linear codes finally we show that our parameters extend relative generalized hamming weights and relative dimension length profile respectively and relative generalized rank weights and relative dimension intersection profile respectively

bi invariant and noninvariant metrics on lie groups a restricted version of the inverse problem of lagrangian dynamics for the canonical linear connection on a lie group is studied specifically for solvable lie algebras of dimension up to and including six all algebras for which there is a compatible pseudo riemannian metric on the corresponding linear lie group are found of the 19 such metrics four are bi invariant the lie algebras are taken from tables compiled originally by mubarakzyanov izv vyssh uchebn zaved mat 4 104 116 1963 and morozov izv vyssh uchebn zaved mat 4 161 171 1958

the inverse problem of the calculus of variations for discrete systems fil barbero linan maria instituto de ciencias matematicas espana universidad politecnica de madrid espana

on the similarities between generalized rank and hamming weights and their applications to network coding rank weights and generalized rank weights have been proved to characterize error and erasure correction and information leakage in linear network coding in the same way as hamming weights and generalized hamming weights describe classical error and erasure correction and information leakage in wire tap channels of type ii and code based secret sharing although many similarities between both the cases have been established and proved in the literature many other known results in the hamming case such as bounds or characterizations of weight preserving maps have not been translated to the rank case yet or in some cases have been proved after developing a different machinery the aim of this paper is to further relate both weights and generalized weights show that the results and proofs in both cases are usually essentially the same and see the significance of these similarities in network coding some of the new results in the rank case also have new consequences in the hamming case

integrals and symmetries the bernoulli laplace lenz vector the notions of independent constants of motion are clarified emphasising the role of precise definition of the background space using a well defined algorithm that relates symmetries of equations of motion of a classical system newtonian lagrangian or hamiltonian to a conserved object in one to one fashion the symmetries of the coulomb and linear force problems are determined

extended covariance for the lagrange equations of motion a geometric analysis the infinitesimal transformations which leave the lagrangian structure of the equations of motion unchanged are intrinsically characterized a new condition is given with which to obtain alternative lagrangians

deformation of lie group representations and linear filters part i deformation theory is a branch of mathematics which studies the geometry of dependence on parameters of geometric and physical systems material arising from lie group theory and mathematical physics e g in the study of asymptotic behavior of angular momentum is applied to study the asymptotic behavior of certain linear filters depending on parameters a mathematical machine which unifies many of these problems will be developed in this series

alternative lagrangians for spherically symmetric potentials a new geometrical account of the existence of alternative s equivalent lagrangians is given for a particle moving in three dimensional euclidean space under the influence of a central force the fact that the lagrangian for this familiar problem is not unique was pointed out by henneaux and shepley j math phys 23 2101 1982 the present analysis makes the reasons for the phenomenon more transparent than theirs

symmetries first integrals and the inverse problem of lagrangian mechanics deals with the following question given a symmetry vector field y of a system of second order ordinary differential equations and an associated constant of the motion f is it possible to find a lagrangian l for the system such that y becomes a noether symmetry with respect to l and f its implied noether constant for one degree of freedom systems the answer to this question is affirmative in addition attention is paid to the construction of a suitable constant of the motion f for given symmetry y and vice versa several examples are discussed

relative generalized rank weight of linear codes and its applications to network coding by extending the notion of minimum rank distance this paper introduces two new relative code parameters of a linear code mathcal c 1 of length n over a field extension mathbb f q m and its subcode mathcal c 2 subsetneqq mathcal c 1 one is called the relative dimension intersection profile rdip and the other is called the relative generalized rank weight rgrw we clarify their basic properties and the relation between the rgrw and the minimum rank distance as applications of the rdip and the rgrw the security performance and the error correction capability of secure network coding guaranteed independently of the underlying network code are analyzed and clarified we propose a construction of secure network coding scheme and analyze its security performance and error correction capability as an example of applications of the rdip and the rgrw silva and kschischang showed the existence of a secure network coding in which no part of the secret message is revealed to the adversary even if any dim mathcal c 1 1 links are wiretapped which is guaranteed over any underlying network code however the explicit construction of such a scheme remained an open problem our new construction is just one instance of secure network coding that solves this open problem

maximal subalgebra associated with a first integral of a system possessing sl 3 r algebra conventionally the symmetries of a dynamical system are used to determine the first integrals associated with the system in this paper the reverse procedure is adopted for one dimensional systems possessing sl 3 r symmetry the equivalence of such systems enables the free particle equation to be used as a vehicle it is observed that for three particular first integrals each gives rise to a triplet of generators having isomorphic algebras it is then shown how the knowledge of a first integral and its associated triplet enables one to obtain the remaining integrals and triplets

spatiotemporal wavelets a group theoretic construction for motion estimation and tracking this paper presents a new group theoretic perspective for signal and image analysis it addresses the problem of motion analysis and trajectory determination the construction exploits the properties of motion parameters to be structured in lie algebras and lie groups the motion models are provided by the group structure which carries an entire theoretical build up this theoretical construction is based on a natural association of lie group representations minimum mean squared error estimations and variational principles of optimality these concepts naturally provide a corresponding association of tools based on continuous wavelet transforms kalman filters and lagrangians these tools result in highly parallelizable algorithms based on ffts gradients and dynamic programming the core of the construction is made of spatiotemporal continuous wavelets that are tuned to the motion parameters to perform motion estimation the motion parameters consist of scale orientation location velocity acceler

on the existence of local and global lagrangians for ordinary differential equations necessary and sufficient conditions for the existence of local and global lagrangians for ordinary differential equations of arbitrary order are described in terms of the geometry of higher order tangent bundles the results are applied to the study of gauge invariant differential equations and the second order differential equation defined by the 2 1 dimensional yang mills lagrangian with the chern simons term is discussed

time dependent lagrangian systems a geometric approach using semibasic forms a new presentation of the geometric formalism in r tq of the time dependent lagrangian systems is given the formalism is used to prove some properties of the equations determining the euler lagrange vector field xl for relating the theory with the time independent case and for studying the time dependent lagrangian inverse problem

quantum dynamics of parasupersymmetric and shape invariant coupled systems a class of bound state problems which represents the coupling of a three level atom with a two dimensional system involving two shape invariant potentials was introduced in a previous paper in this paper we considered second order parasupersymmetric quantum mechanical models and two possible kinds for the coupling hamiltonian linear and nonlinear in the potential ladder operators in the present paper using an algebraic formulation for shape invariant potential systems we study the quantum dynamics of these coupled systems and obtain the temporal behaviour of some dynamical variables related with the atom and the coupling potentials an application is given for a couple of shape invariant potentials poschl teller self similar potentials

a separability theorem for dynamical systems admitting alternative lagrangian descriptions by generalising earlier results the authors prove that whenever a lagrangian dynamical system can be associated with a 1 1 type tensor field which is left invariant by the dynamics satisfies the nijenhuis condition and is compatible with the tangent bundle structure the system necessarily decomposes into a collection of lower dimensional mutually non interacting lagrangian subsystems

a geometric approach to a generalized virial theorem the virial theorem introduced by clausius in the field of statistical mechanics and later applied in both classical mechanics and quantum mechanics is studied by making use of symplectic formalism as an approach in the case of both the hamiltonian and lagrangian systems the possibility of establishing virial like theorems from one parameter groups of non strictly canonical transformations is analysed the case of systems with a position dependent mass is also discussed using the modern symplectic approach to quantum mechanics we arrive at the quantum virial theorem in full analogy with the classical case

a linear connection for higher order ordinary differential equations a linear connection is defined on the space of n jets of sections of where b is one dimensional this is a first step toward classifying n 1 th order time dependent ordinary differential equations the module of vector fields on splits into coordinate invariant components n 1 of them isomorphic and one component generated by the differential equation field the connection is shown to uniquely determine the differential equation field to which it is associated

tangent bundle geometry lagrangian dynamics various aspects of the differential geometry of the tangent bundle of a differentiable manifold are examined and the results applied to time independent lagrangian dynamics it is shown that a certain type 1 1 tensor field which is part of the intrinsic geometry of a tangent bundle being a tensorial equivalent of the projection map of tangent vectors plays a role in lagrangian theory scarcely less important than that of the canonical one form on a cotangent bundle in hamiltonian theory recent results in lagrangian theory are interpreted from this new viewpoint

sl 3 r as the group of symmetry transformations for all one dimensional linear systems the converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x f2 t x f1 t x f0 t the eight parameter realizations of the symmetry group are obtained in the form f 1p2 f where f stands for transformations of t x that depend exclusively on the fundamental solutions of the equation and where p2 is an arbitrary projective transformation in the plane thus it is shown that the full point symmetry group corresponds to sl 3 r indeed without recourse to the lie algebra also a technique is obtained for calculating the finite point symmetry realization of sl 3 r for any given one dimensional linear system some miscellaneous examples are given

quantum revivals in the jaynes cummings model the author obtains an exact integral representation of the sum describing the evolution of the inversion of the two level atom in the jaynes cummings model when the field is initially prepared in a coherent state or a state generated by classical sources at non zero temperature the author uses the saddle point method to estimate the integrals

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

geometric lagrangian approach to first order systems and applications the geometric theory of presymplectic systems is developed to study both the lagrangian and hamiltonian formulations of a system described by a lagrangian linear in the velocities the results are used to study some related problems for second order differential equations and regular lagrangians

a direct approach to the construction of standard and non standard lagrangians for dissipative like dynamical systems with variable coefficients we present a direct approach to the construction of lagrangians for a large class of one dimensional dynamical systems with a simple dependence monomial or polynomial on the velocity we rederive and generalize some recent results and find lagrangian formulations which seem to be new some of the considered systems e g motions with the friction proportional to the velocity and to the square of the velocity admit infinite families of different explicit lagrangian formulations

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

the constraint algorithm for time dependent lagrangians the aim of this paper is to develop a constraint algorithm for time dependent lagrangian systems which permit us to solve the motion equations this algorithm extends the gotay and nester algorithm for autonomous lagrangians which is in fact a particular case to do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used

alternative lagrangians for central potentials two methods for the construction of alternative lagrangians to the standard lagrangian l t v where v is a central potential are presented the direct integration of the hessian matrix and the use of the noether symmetries of l the lagrangians obtained are of the so called nonlinear homogenous nlh type and the study is centered in those lagrangians not preserving the spherical symmetry of l finally the associated hamiltonian formalism is presented

lie symmetries for the charge monopole problem the authors obtain the lie symmetries for the equation of motion of an electric charge interacting with a magnetic monopole fixed at the origin they also discuss the construction of first integrals for this problem

origin of the lagrangian constraints and their relation with the hamiltonian formulation the theory of presymplectic systems is used for the study of mechanical systems described by singular lagrangians in order to clarify the geometric meaning of the euler lagrange equations for such systems the two different types of primary constraint functions arising in the lagrangian formulation are analyzed by means of the relation between the image under the vertical endomorphism of the kernel of the presymplectic form l and its vertical part the connection with the hamiltonian dirac theory is also studied and the theory is illustrated with several examples

singular lagrangians affine in velocities the properties of lagrangians affine in velocities are analysed in a geometric way these systems are necessarily singular and exhibit in general gauge invariance the analysis of constraint functions and gauge symmetry leads us to a complete classification of such lagrangians

a resummation formula for collapse and revival in the jaynes cummings model we present a new approach to the study of the function of atomic inversion in the model of interaction of a single two level atom with a single mode of the quantized electromagnetic field in the coherent state in an ideal resonator the approach suggested is based on an application of certain number theoretic results to the approximation of the trigonometric sums of a special form in particular of the functional equation of the jacobi theta function new asymptotic formulae for the atomic inversion are found the asymptotics that we obtain make it possible to predetermine the details of the behavior of the inversion on various time intervals depending on the parameters of the system

on the lie symmetries of the classical kepler problem the standard version of noether s theorem when applied to the classical kepler problem leads to the constants of energy and angular momentum but does not give the hidden symmetry known as the runge lenz vector lie s theory of differential equations is used to obtain all three constants of motion the transformations of solutions under the point transformations to which these constants correspond are studied the results are generalised to n dimensions

quantum dynamics of a two level system coupled to a shape invariant potential the quantum dynamics of a two level system coupled to a shape invariant potential is investigated shape invariant potentials share an integrability condition called shape invariance which identifies an underlying algebraic structure in general infinite dimensional that transforms the potential parameters such as strength range and diffuseness we determine the time evolution operator the density operator of the system and obtain the temporal behaviour of various dynamical variables by considering either pure or mixed initial states of the system constructed with the generalized coherent state of the shape invariant coupling potential we consider specific examples of shape invariant coupling potentials harmonic oscillator poschl teller and self similar potentials the results obtained for all dynamical variables exhibit rapid oscillations which periodically collapse and regenerate in different ways depending on the coupling potential nature

a rank metric approach to error control in random network coding the problem of error control in random linear network coding is addressed from a matrix perspective that is closely related to the subspace perspective of rotter and kschischang a large class of constant dimension subspace codes is investigated it is shown that codes in this class can be easily constructed from rank metric codes while preserving their distance properties moreover it is shown that minimum distance decoding of such subspace codes can be reformulated as a generalized decoding problem for rank metric codes where partial information about the error is available this partial information may be in the form of erasures knowledge of an error location but not its value and deviations knowledge of an error value but not its location taking erasures and deviations into account when they occur strictly increases the error correction capability of a code if mu erasures and delta deviations occur then errors of rank t can always be corrected provided that 2t les d 1 mu delta where d is the minimum rank distance of the code for gabidulin codes an important family of maximum rank distance codes an efficient decoding algorithm is proposed that can properly exploit erasures and deviations in a network coding application where n packets of length m over f q are transmitted the complexity of the decoding algorithm is given by o dm operations in an extension field f qn

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

some algebraic geometric and system theoretic properties of the special functions of mathematical physics it is known that many of the special functions of mathematical physics appear as matrix elements of lie group representations this paper is concerned with a beginning attack on the converse problem i e finding conditions that a given function be a matrix element the methods used are based on a combination of ideas from system theory functional analysis lie theory differential algebra and linear ordinary differential equation theory a key idea is to attach a symbol as an element of a commutative algebra in favorable cases this symbol defines a riemann surface and a meromorphic differential form on that surface the topological and analytical invariants attached to this form play a key role in system theory the lie algebras of the groups appear as linear differential operators on this riemann surface finally it is shown how the picard vessiot infeld hull theory of factorization of linear differential operators leads to realization of many special functions as matrix representations of group

application of the hamilton jacobi method to linear nonconservative vibration theory in this paper it is shown how linear strictly nonconservative oscillatory systems with one and many degrees of freedom may be analyzed by application of the theory of hamilton and jacobi

extended legendre transformation approach to the time dependent hamiltonian formalism the hamiltonian formalism for time dependent systems is developed using vector fields defined in the extended cotangent bundle t q r the extended hamiltonian h is obtained using the extended legendre transformation associated to an extended lagrangian l previously defined in t q r and then the techniques of symplectic mechanics are employed in t q r to study the properties of the extended hamiltonian system finally this approach is related with the multisymplectic formalism and other extended approaches

the helmholtz conditions in terms of constants of motion in classical mechanics the helmholtz conditions are the necessary and sufficient conditions for a set of second order differential equations to be equivalent to a variational principle in this work an alternative approach to the inverse problem in classical mechanics is described it is proven that the helmholtz conditions can be transformed into a set of conditions for a nonsingular antisymmetric matrix whose entries are constants of motion of the problem in question

cartan s method of equivalence and second order equation fields the theory underlying cartan s method of equivalence is outlined and applied to second order equation fields in particular the technique is used to obtain the canonical horizontal distribution associated to a second order equation field a construction which has been presented before but in a rather ad hoc manner

quasi coordinates from the point of view of lie algebroid structures a geometrical description of the lagrangian dynamics in quasi coordinates on the tangent bundle using the lie algebroid framework is given linear nonholonomic systems on lie algebroids are solved in local coordinates adapted to the constraints through lagrangian multipliers and gibbs appell generalized methods

on the inverse problem of the calculus of variations in field theory the inverse problem of the calculus of variations is investigated in the case of field theory uniqueness of the action principle is demonstrated for the vector laplace equation in a non decomposable riemannian space as well as for the harmonic map equation

the symmetry in the structure of dynamical and adjoint symmetries of second order differential equations with each second order differential equation z in the evolution space j1 mn 1 we associate using a new differential operator az four families of vector fields and 1 forms on j1 mn 1 providing a natural set up for the study of symmetries first integrals and the inverse problem for z we analyse the relations between the four families pointing out the symmetric structure of this set up when a lagrangian for z exists the bijection between dynamical and dual symmetries is included in the whole context suggesting the corresponding bijection between dual adjoint and adjoint symmetries as an application we show how some results of the inverse problem can be framed naturally in this geometrical context

generalizations of noether s theorem in classical mechanics in this paper a review is presented of various approaches to the generalization of the version of noether s theorem which is presented in most textbooks on classical mechanics its motivation is the controversy still persisting around the possible scope of a noether type theorem allowing for velocity dependent transformations our analysis is centered around the one factor common to all known treatments namely the structure of the related first integral we first discuss the most general framework in which a function of the above mentioned structure constitutes a first integral of a given lagrangian system and show that one cannot really talk about an interrelationship between symmetries and first integrals there we then compare different proposed generalizations of noether s theorem by describing the nature of the restrictions which characterize them when they are situated within the broadest framework we prove a seemingly new equivalence result between the two main approaches that of invaria

on the second relative greedy weight of 4 dimensional codes based on applications to the wire tap channel of type ii with the coset coding scheme we will define optimal codes which are described as the ones achieving upper bounds on e 2 m 2 where e 2 and m 2 are the second relative greedy weight and the second relative generalized hamming weight respectively by using the finite geometry method we will first classify all the 4 dimensional codes with respect to their k 1 dimensional subcodes for k 1 1 and k 1 2 and determine the upper bounds on e 2 m 2 for each class of codes and then obtain optimal codes by explicitly constructing the value assignments which achieve upper bounds on e 2 m 2

a geometrical version of the helmholtz conditions in time dependent lagrangian dynamics appropriate geometrical machinery for the study of time dependent lagrangian dynamics is developed it is applied to the inverse problem of the calculus of variations and a set of necessary and sufficient conditions for the existence of a lagrangian are given in terms of the existence of a 2 form with suitable properties which are exactly equivalent to the helmholtz conditions

higher order noether symmetries and constants of the motion the authors identify a class of symmetries for lagrangian systems called higher order noether symmetries which yield a corresponding first integral without further integrations first a version of noether s theorem is recalled in which noether symmetries are considered to be symmetries of the two form d theta theta being the cartan form nth order noether symmetries are then defined by the requirement lynd theta 0 the picture is further generalised by exploring various ways in which the computation of repeated lie derivatives of d theta can lead to the identification of first integrals a number of illustrative examples are discussed

universal secure network coding via rank metric codes the problem of securing a network coding communication system against an eavesdropper is considered the network implements linear network coding to deliver n packets from source to each receiver and the adversary can eavesdrop on arbitrarily chosen links the objective is to provide reliable communication to all receivers while guaranteeing that the source information remains information theoretically secure from the adversary a coding scheme is proposed that can achieve the maximum possible rate of n packets the scheme which is based on rank metric codes has the distinctive property of being universal it can be applied on top of any communication network without requiring knowledge of or any modifications on the underlying linear network code the only requirement of the scheme is that the packet length be at least n which is shown to be strictly necessary for universal communication at the maximum rate a further scenario is considered where the adversary is allowed not only to eavesdrop but also to inject up to t erroneous packets into the network and the network may suffer from a rank deficiency of at most in this case the proposed scheme can be extended to achieve the rate of n 2t packets this rate is shown to be optimal under the assumption of zero error communication

a complete set of bianchi identities for tensor fields along the tangent bundle projection the various derivations defined along the tangent bundle projection tau in a series of papers by martinez carinena and sarlet 1992 are expressed as components of a single linear connection del on e the tangent bundle of the evolution space e r t m this connection is equivalent to a system of second order ordinary differential equations sode on m using the linear connection we calculate the torsion and curvature of e del the components of which are expressed in terms of the tensors along tau defined by martinez et al from these the full set of bianchi identities are calculated we also show that the generalized jacobi equation defined by several authors is precisely the horizontal component of the conventional jacobi equation along geodesics of e del finally we use this to show that if a jacobi field of the lift of a sode solution is a certain lift then it can be extended to a symmetry of the sode

the inverse problem for six dimensional codimension two nilradical lie algebras ado s theorem asserts that every real lie algebra g of dimension n has a faithful representation as a subalgebra of gl p r for some p the theorem offers no practical information about the size of p in relation to n and in principle p may be very large compared to n this article is concerned with finding representations for a certain class of six dimensional lie algebras specifically real indecomposable algebras that have a codimension two nilradical these algebras were classified by turkowski and comprise of 40 cases some of which contain up to four parameters linear representations are found for each algebra in these classes more precisely a matrix lie group is given whose lie algebra corresponds to each algebra in turkowski s list and can be found by differentiating and evaluating at the identity element of the group in addition a basis for the right invariant vector fields that are dual to the maurer cartan forms are given thereby providing an effective realization of lie s third theorem t