on the bilinear structure associated to bezoutians abstract this paper is partly a survey of known results on quadratic forms that are hard to find in the literature our main focus is a twisted form of a construction due to bezout this skew bezoutian is a symplectic resp quadratic space associated to a pair of reciprocal or skew reciprocal coprime polynomials of same degree the isometry group of this space turns out to contain a certain associated hypergeometric group using the skew bezoutian we construct explicit isometries of bilinear spaces with given invariants such as the characteristic polynomial or jordan form and in the quadratic case the spinor norm

generators for the hyperelliptic torelli group and the kernel of the burau representation at t 1 we prove that the hyperelliptic torelli group is generated by dehn twists about separating curves that are preserved by the hyperelliptic involution this verifies a conjecture of hain the hyperelliptic torelli group can be identified with the kernel of the burau representation evaluated at t 1 and also the fundamental group of the branch locus of the period mapping and so we obtain analogous generating sets for those one application is that each component in torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added

interconnection and damping assignment passivity based control of port controlled hamiltonian systems passivity based control pbc is a well established technique that has shown to be very powerful to design robust controllers for physical systems described by euler lagrange el equations of motion for regulation problems of mechanical systems which can be stabilized shaping only the potential energy pbc preserves the el structure and furthermore assigns a closed loop energy function equal to the difference between the energy of the system and the energy supplied by the controller thus we say that stabilization is achieved via energy balancing unfortunately these nice properties of el pbc are lost when used in other applications which require shaping of the total energy for instance in electrical or electromechanical systems or even some underactuated mechanical devices our main objective in this paper is to develop a new pbc theory which extends to a broader class of systems the aforementioned energy balancing stabilization mechanism and the structure invariance towards this end we depart from the el description of the systems and consider instead port controlled hamiltonian models which result from the network modelling of energy conserving lumped parameter physical systems with independent storage elements and strictly contain the class of el models

arithmeticity of certain symplectic hypergeometric groups we give a sufficient condition on a pair of primitive polynomials that the associated hypergeometric group monodromy group of the corresponding hypergeometric differential equation is an arithmetic subgroup of the integral symplectic group

linear differential equations and multiple zeta values iii zeta 3 we consider the hypergeometric equation 1 t t t g x3g 0 whose unique analytic solution 1 t x 1 o t near t 0 for t 1 becomes a generating function for multiple zeta values 1 1 x f3 x 1 3 x3 3 3 x6 we apply the so called wkb method to study solutions of the hypergeometric equation for large x and we calculate corresponding stokes matrices we prove that the function f3 x near x is also expressed via wkb type functions which subject to some stokes phenomenon this implies that f3 x satisfies a sixth order linear differential equation with irregular singularity at infinity

orthogonal polynomials satisfying fourth order differential equations since they are rather important and quite accessible we repeat the general theoretical facts concerning weights moments and polynomials pertaining to fourth order differential equations we then briefly discuss the squares of the differential equations of the second order giving a number of easily derived examples of fourth order problems this is followed by three new orthogonal polynomial sets satisfying fourth order differential equations but which do not satisfy second order differential equations

on the synthesis of boundary control laws for distributed port hamiltonian systems this paper is concerned with the energy shaping of 1 d linear boundary controlled port hamiltonian systems the energy casimir method is first proposed to deal with power preserving systems it is shown how to use finite dimensional dynamic boundary controllers and closed loop structural invariants to partially shape the closed loop energy function and how such controller finally reduces to a state feedback when dissipative port hamiltonian systems are considered the casimir functions do not exist anymore dissipation obstacle and the immersion via a dynamic controller reduction through invariants method cannot be applied the main contribution of this paper is to show how to use the same ideas and state functions to shape the closed loop energy function of dissipative systems through direct state feedback i e without relying on a dynamic controller and a reduction step in both cases the existence of solution and the asymptotic stability by additional damping injection of the closed loop system are proven the general theory and achievable closed loop performances are illustrated with the help of a concluding example the boundary stabilization of a longitudinal beam vibrations

dirac structures and boundary control systems associated with skew symmetric differential operators associated with a skew symmetric linear operator on the spatial domain a b we define a dirac structure which includes the port variables on the boundary of this spatial domain this dirac structure is a subspace of a hilbert space naturally associated with this dirac structure is an infinite dimensional system we parameterize the boundary port variables for which the c 0 semigroup associated with this system is contractive or unitary furthermore this parameterization is used to split the boundary port variables into inputs and outputs similarly we define a linear port controlled hamiltonian system associated with the previously defined dirac structure and a symmetric positive operator defining the energy of the system we illustrate this theory on the example of the timoshenko beam

the level four braid group by evaluating the burau representation at t 1 we obtain a symplectic representation of the braid group we study the resulting congruence subgroups of the braid group namely the preimages of the principal congruence subgroups of the symplectic group our main result is that the level 4 congruence subgroup is equal to the group generated by squares of dehn twists we also show that the image of the brunnian subgroup of the braid group under the symplectic representation is the level four congruence subgroup

algebraicity of the appell lauricella and horn hypergeometric functions we extend schwarz list of irreducible algebraic gauss functions to the four classes of appell lauricella functions in several variables and the 14 complete horn functions in two variables this gives an example of a family of functions such that for any number of variables there are infinitely many algebraic functions namely the lauricella fc functions

monodromy groups of hurwitz type problems abstract we solve the hurwitz monodromy problem for degree 4 covers that is the hurwitz space h 4 g of all simply branched covers of p 1 of degree 4 and genus g is an unramified cover of the space p 2 g 6 of 2 g 6 tuples of distinct points in p 1 we determine the monodromy of 1 p 2 g 6 on the points of the fiber this turns out to be the same problem as the action of 1 p 2 g 6 on a certain local system of z 2 vector spaces we generalize our result by treating the analogous local system with z n coefficients 3 n in place of z 2 this in turn allows us to answer a question of ellenberg concerning families of galois covers of p 1 with deck group z n 2 s 3

connected linear groups as differential galois groups theorem 1 1 let c be an algebraically closed field of characteristic zero g a connected linear algebraic group defined o er c and k a differential field containing c as its field of constants and of finite nonzero transcendence degree o er c then g can be realized as the galois group of a picard vessiot extension of k w x previous work by kovacic 17 18 reduced this problem to the case of powers of a simple connected linear algebraic group our contribution is to show that one is able to realize any connected semisimple group as a galois group and when k s c x x9 s 1 to control the number and types of singularities when one constructs a system y 9 s ay a g m c x realizing an arbitrary connected linear algebraic group as its n

exponential stability of a class of boundary control systems we study a class of partial differential equations with variable coefficients on a one dimensional spatial domain with control and observation at the boundary for this class of systems we provide simple tools to check exponential stability this class is general enough to include models of flexible structures traveling waves heat exchangers and bioreactors among others the result is based on the use of a generating function the energy for physical systems and an inequality condition at the boundary furthermore based on the port hamiltonian approach we give a constructive method to reduce this inequality to a simple matrix inequality

linear differential equations with finite differential galois group abstract for a finite irreducible subgroup h psl c n and an irreducible h invariant curve z p c n such that c z h c t a standard differential operator l s t c t d d t is constructed for n 2 this is essentially klein s work for n 2 an actual calculation of l s t is done by computing an evaluation of invariants c x 1 x n h c t and applying a scalar form of a theorem of e compoint in a procedure also in some cases where z is unknown evaluations are produced this new method is tested for n 2 and for three irreducible subgroups of sl 3 this supplements 18 the theory developed here relates to and continues classical work of h a schwarz g fano f klein and a hurwitz

on rigid tuples in linear groups of odd dimension abstract we construct examples of linearly rigid tuples which lead to regular galois realizations over q for linear and unitary groups gl2m 1 q and u2m 1 q2 where q is odd and m q 1 resp m q 1 the notion of linear rigidity was introduced by strambach and volklein j reine angew math to appear as an analogue of katz physically rigid systems see n m katz rigid local systems annals of mathematics studies vol 139 princeton univ press princeton nj 1996

image of the burau representation at d th roots of unity we show that the image of the braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of branch points is suciently large compared to the degree d this is deduced by proving the arithmeticity of the image of the braid group on n 1 letters under the burau representation evaluated at d th roots of unity when n 2d

boltzmann samplers for v balanced cycles this paper is devoted to the random generation of a new operator which we call v balanced colored cycle a cycle of which the components are assigned different colors with frequencies restricted by a vector v we propose an efficient approximate size sampler of linear expected time complexity following the boltzmann model introduced by duchon et al 2004 9 one of our motivations is to show that the absence of a decomposable specification which form the basis of the boltzmann model can be circumvented by mixing boltzmann samplers with other methods of random generation

on lame operators which are pull backs of hypergeometric ones we give a method that would allow one to calculate the number of lame operators 5n n e n with prescribed finite monodromy and do the calculation for the case n 1 we find a bound for the degree over q of the field of definition of the coefficients of a lame operator with prescribed finite monodromy and give examples of lame operators with finite monodromy finally we study lame operators with infinite monodromy and generic second order differential operators which are pull backs of hypergeometric ones under algebraic maps consider an ordinary differential operator 0 1 l a x an x where as x e k x and k is a number field over q what one can say about its solutions ly 0 and monodromy looking only at the coefficients as x in this paper we give some partial answers to this question for the special case of a lame operator one of the questions we will approach is the following for every n e n how many lame operators yn are there whose monodromy group is a dihedral group of order 2n n e n we will answer in the case n 1 2 14 2 15 the method we use for the proof works for general n for example the case n 2 can be found in co before summarizing our work we give a brief review of the known results since the work of schwarz on hypergeometric equations and the classification of the relative finite monodromy groups no basic improvements have been made recently the challenge of the grothendieck conjecture has stimulated some progress see ka ka1 ka2 ho be he dw1 we recall that this conjecture asserts that if in 0 1 almost all the reductions mod p prime ideal in the ring of integers of k have a complete set of solutions which is equivalent to l having zero p curvature for almost all p then equation 0 1 has only algebraic solutions on the other hand some improvements in the sense of klein and schwarz can be found in ba dw sing ba dw but we are still a long way from achieving the same level of understanding for general operators that we have attained in the hypergeometric case in received by the editors january 2 1991 1991 mathematics subject classification primary 34b30 secondary 14e20 the author was a visiting fellow at princeton university supported by c n r italy 1995 american mathematical society

dirac structures on hilbert spaces and boundary control of distributed port hamiltonian systems abstract aim of this paper is to show how the dirac structure properties can be exploited in the development of energy based boundary control laws for distributed port hamiltonian systems usually stabilization of non zero equilibria has been achieved by looking at or generating a set of structural invariants namely casimir functions in closed loop since this approach fails when an infinite amount of energy is required at the equilibrium dissipation obstacle this paper illustrates a novel approach that enlarges the class of stabilizing controllers the starting point is the parametrization of the dynamics provided by the image representation of the dirac structure that is able to show the effects of the boundary inputs on the state evolution in this way energy balancing and control by state modulated source methodologies are extended to the distributed parameter scenario and a geometric interpretation of these control techniques is provided the theoretical results are discussed with the help of a simple but illustrative example i e a transmission line with an rlc load in both serial and parallel configurations in the latter case energy balancing controllers are not able to stabilize non zero equilibria because of the dissipation obstacle the problem is solved thanks to a boundary state modulated source

on boundary values for the laguerre operator in indefinite inner product spaces abstract it is shown that when ly xy 1 x y requires more than one boundary condition at x 0 in order to be self adjoint the more negative the more boundary conditions required

on representations and integrability of mathematical structures in energy conserving physical systems in the present paper we elaborate on the underlying hamiltonian structure of interconnected energy conserving physical systems it is shown that a power conserving interconnection of port controlled generalized hamiltonian systems leads to an implicit generalized hamiltonian system and a power conserving partial interconnection to an implicit port controlled hamiltonian system the crucial concept is the notion of a generalized dirac structure defined on the space of energy variables or on the product of the space of energy variables and the space of flow variables in the port controlled case three natural representations of generalized dirac structures are treated necessary and sufficient conditions for closedness or integrability of dirac structures in all three representations are obtained the theory is applied to implicit port controlled generalized hamiltonian systems and it is shown that the closedness condition for the dirac structure leads to strong conditions on the input vector fields

generalized basic hypergeometric equations this paper deals with regular singular generalized q hypergeometric equations with either large or small galois groups in particular we consider the fundamental problem of finding appropriate galoisian substitutes for the usual notion of local monodromy

achievable casimirs and its implications on control of port hamiltonian systems in this paper we extend results on interconnections of port hamiltonian systems to infinite dimensional port hamiltonian systems and to mixed finite and infinite dimensional port hamiltonian systems the problem of achievable dirac structures is now studied for systems with dissipation in the finite dimensional infinite dimensional and the mixed finite and infinite dimensional case we also characterize the set of achievable casimirs and study its application for the control of port hamiltonian systems

the uniformization of certain algebraic hypergeometric functions abstract the hypergeometric functions f n 1 n are higher transcendental functions but for certain parameter values they become algebraic because the monodromy of the defining hypergeometric differential equation becomes finite it is shown that many algebraic f n 1 n s for which the finite monodromy is irreducible but imprimitive can be represented as combinations of certain explicitly algebraic functions of a single variable namely the roots of trinomials this generalizes a result of birkeland and is derived as a corollary of a family of binomial coefficient identities that is of independent interest any tuple of roots of a trinomial traces out a projective algebraic curve and it is also determined when this so called schwarz curve is of genus zero and can be rationally parametrized any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic f n 1 n s many examples of such uniformizations are worked out explicitly even when the governing schwarz curve is of positive genus it is shown how it is sometimes possible to construct explicit single valued or multivalued parametrizations of individual algebraic f n 1 n s by parametrizing a quotiented schwarz curve the parametrization requires computations in rings of symmetric polynomials

calabi yau threefolds fibred by mirror quartic k3 surfaces abstract we study threefolds fibred by mirror quartic k3 surfaces we begin by showing that any family of such k3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic k3 surfaces this is then used to give a complete explicit description of all calabi yau threefolds fibred by mirror quartic k3 surfaces we conclude by studying the properties of such calabi yau threefolds including their hodge numbers and deformation theory

braid groups and hodge theory this paper gives an account of the unitary representations of the braid group that arise via the hodge theory of cyclic branched coverings of mathbb p 1 highlighting their connections with ergodic theory complex reflection groups moduli spaces of 1 forms and open problems in surface topology

modeling and control of the timoshenko beam the distributed port hamiltonian approach the purpose of this paper is to show how the timoshenko beam can be fruitfully described within the framework of distributed port hamiltonian dph systems so that rather simple and elegant considerations can be drawn regarding both the modeling and control of this mechanical system after the dph model of the beam is introduced the control problem is discussed in particular it is shown how control approaches already presented in the literature can be unified and a new control methodology is presented and discussed this control methodology relies on the generalization to infinite dimensions of the concept of structural invariant casimir function and on the extension to distributed systems of the so called control by interconnection methodology in this way finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy i e by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipative effect is introduced

mapping class group dynamics and the holonomy of branched affine structures we classify up to few exceptions the orbit closures of the mathrm mod sigma action on the affine character variety chi mathrm aff mathbb c we obtain from this classification that the only obstruction for a non abelian representation rho pi 1 sigma longrightarrow mathrm aff mathbb c to be the holonomy of a branched affine structure on sigma is to be euclidean and not to have positive volume where sigma is a closed oriented surface of genus g ge 2

transcendence in positive characteristic and special values of hypergeometric functions we prove a simple transcendence criterion suitable for function field arithmetic we apply it to show the transcendence of special values at non zero rational arguments or more generally at algebraic arguments which generate extension of the rational function field with less than q places at infinity of the entire hypergeometric functions in the function field over fq context and to obtain a new proof of the transcendence of special values at non natural p adic integers of the carlitz goss gamma function we also characterize in the balanced case the algebraicity of hypergeometric functions giving an analog of the result of f r villegas based on beukers heckman results in the classical hypergeometric case 0 introduction number theorists study the number fields and the function fields over finite fields together as global fields in view of the strong analogies that exist between them transcendental number theory originated with liouville s result of 1844 see 17 that an algebraic irrational cannot be well approximated by rationals k mahler 18 showed in 1949 that liouville s theorem holds for function fields over any field and also that the liouville s inequality is best possible if the ground field k is of positive characteristic p by considering t p i k 1 t which is algebraic of degree p as well as of the approximation exponent p in particular naive analog of roth s theorem fails see 30 cha 9 for a survey of general situation which is not even conjecturally fully understood yet so we need stronger approximations in function fields than in number fields counterparts to conclude transcendence by diophantine approximation on the other hand for many naturally occurring quantities in the function field arithmetic related to drinfeld modules and anderson s t motives such as periods special values of exponential logarithm gamma and zeta functions etc different techniques that exist algebraic group techniques more developed motivic machinery automata method strong diophantine approximation criterion of special kind etc in function fields have been used to prove very powerful results see 30 cha 10 11 for a survey of older results and 4 21 22 7 for some recent results many quantities of interest that occur naturally in the function field arithmetic have expansions that are fq linear or closely related to such so that they are date may 12 2010 1991 mathematics subject classification primary 11j81 11j93 secondary 11t55

torus equivariant d modules and hypergeometric systems abstract we formalize at the level of d modules the notion that a hypergeometric systems are equivariant versions of the classical hypergeometric equations for this purpose we construct a functor b a on a suitable category of torus equivariant d modules and show that it preserves key properties such as holonomicity regularity and reducibility of monodromy representation we also examine its effect on solutions characteristic varieties and singular loci by applying b a to suitable binomial d modules we shed new light on the d module theoretic properties of systems of classical hypergeometric differential equations

ising n fold integrals as diagonals of rational functions and integrality of series expansions we show that the n fold integrals n of the magnetic susceptibility of the ising model as well as various other n fold integrals of the ising class or n fold integrals from enumerative combinatorics like lattice green functions correspond to a distinguished class of functions generalizing algebraic functions they are actually diagonals of rational functions as a consequence the power series expansions of the analytic at x 0 solutions of these linear differential equations derived from geometry are globally bounded which means that after just one rescaling of the expansion variable they can be cast into series expansions with integer coefficients we also give several results showing that the unique analytical solution of calabi yau odes and more generally picard fuchs linear odes with solutions of maximal weights are always diagonals of rational functions besides in a more enumerative combinatorics context generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions we finally address the question of the relations between the notion of integrality series with integer coefficients or more generally globally bounded series and the modularity of odes

energy shaping of distributed parameter port hamiltonian systems based on finite element approximation the main contribution of this paper is a procedure for the control by energy shaping via casimir generation of infinite dimensional port hamiltonian systems based on a particular finite element approximation the proposed approach is justified by the fact that the adopted spatial discretization technique is able to preserve casimir functions in the closed loop system when going from the distributed to the approximated lumped parameter system besides the intrinsic difficulties related to the large number of state variables the finite element model is generally given in terms of a dirac structure and is completely a causal which implies that the plant dynamics is not given in standard input state output form but as a set of daes consequently the classical energy casimir method has to be extended in order to deal with dynamical systems with constraints usually appearing in the form of lagrangian multipliers the general methodology is illustrated with the help of an example in which the distributed parameter system is a lossless transmission line

arithmeticity of some hypergeometric monodromy groups in sp 4 abstract the article 14 gives a list of 51 symplectic hypergeometric monodromy groups corresponding to primitive pairs of degree four polynomials which are products of cyclotomic polynomials and for which the absolute value of the leading coefficient of the difference polynomial is greater than 2 it follows from 12 and 14 that 12 of the 51 monodromy groups are arithmetic cf table 1 and the thinness of 13 of the remaining 39 monodromy groups follows from 3 cf table 2 in this article we show that 15 of the remaining 26 monodromy groups are arithmetic cf table 3

algebraic a hypergeometric functions we formulate and prove a combinatorial criterion to decide if an a hypergeometric system of differential equations has a full set of algebraic solutions or not this criterion generalises the so called interlacing criterion in the case of hypergeometric functions of one variable

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

laguerre polynomial expansions abstract the differential operator generated by the laguerre differential equation xy 1 x y y 0 is considered for all both in the classical setting l2 0 x e x and in the sobolev space h1 0 x 1e x x e x in l2 a shifted operator exists for 1 in h1 we prove that for 0 the laguerre operator self adjoint in l2 remains self adjoint with a restricted domain for 0 a shifted operator corresponds to the case 0 just as in the l2 case

hamiltonian discretization of boundary control systems a fundamental problem in the simulation and control of complex physical systems containing distributed parameter components concerns finite dimensional approximation numerical methods for partial differential equations pdes usually assume the boundary conditions to be given while more often than not the interaction of the distributed parameter components with the other components takes place precisely via the boundary on the other hand finite dimensional approximation methods for infinite dimensional input output systems e g in semi group format are not easily relatable to numerical techniques for solving pdes and are mainly confined to linear pdes in this paper we take a new view on this problem by proposing a method for spatial discretization of boundary control systems based on a particular type of mixed finite elements resulting in a finite dimensional input output system the approach is based on formulating the distributed parameter component as an infinite dimensional port hamiltonian system and exploiting the geometric structure of this representation for the choice of appropriate mixed finite elements the spatially discretized system is again a port hamiltonian system which can be treated as an approximating lumped parameter physical system of the same type in the current paper this program is carried out for the case of an ideal transmission line described by the telegrapher s equations and for the two dimensional wave equation

some explicit solutions of the additive deligne simpson problem and their applications abstract in this paper we construct three infinite series and two extra triples e8 and e 8 of complex matrices b c and a b c of special spectral types associated to simpson s classification in amer math soc proc 1 1992 157 and magyar et al classification in adv math 141 1999 97 this enables us to construct fuchsian systems of differential equations which generalize the hypergeometric equation of gauss riemann in a sense they are the closest relatives of the famous equation because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions in all the cases except for e8 we also explicitly construct scalar products such that a b and c are self adjoint with respect to them in the context of fuchsian systems these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions when the eigenvalues of a b and c are real the matrices and the scalar products become real as well we find inequalities on the eigenvalues of a b and c which make the scalar products positive definite as proved by klyachko spectra of three hermitian or real symmetric matrices b c and a b c form a polyhedral convex cone in the space of triple spectra he also gave a recursive algorithm to generate inequalities describing the cone the inequalities we obtain describe non recursively some faces of the klyachko cone

isomonodromy aspects of the tt equations of cecotti and vafa ii riemann hilbert problem in guest et al arxiv 1209 2045 part i we computed the stokes data for the smooth solutions of the tt toda equations whose existence we had previously established by p d e methods here we formulate the existence problem as a riemann hilbert problem based on this stokes data and solve it under certain conditions theorem 5 4 in the process we compute the connection matrix for all smooth solutions thus completing the computation of the monodromy data theorem 5 5 we also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions theorem 4 1 clarifying the region of validity of the formulae established earlier by tracy and widom finally we resolve some conjectures of cecotti and vafa concerning the positivity of s st where s is the stokes matrix and the unimodularity of the eigenvalues of the monodromy matrix theorem 5 6 in particular we show that unitarity implies regularity for the tt toda equations

middle convolution of fuchsian systems and the construction of rigid differential systems abstract in m dettweiler s reiter an algorithm of katz and its application to the inverse galois problem j symbolic comput 30 2000 761 798 a purely algebraic analogon of katz middle convolution functor see n m katz rigid local systems ann of math stud vol 139 princeton university press 1997 is given in this paper we find an explicit riemann hilbert correspondence for this functor this leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the riemann hilbert correspondence

control by interconnection of mixed port hamiltonian systems in this note the regulation problem for mixed finite and infinite dimensional port hamiltonian systems m ph systems is discussed a m ph system results from the power conserving interconnection of finite and infinite dimensional systems in port hamiltonian form in particular the system given by the interconnection of two finite dimensional systems one of which is the controller by means of an infinite dimensional connection is studied the proposed control methodology is a generalization to the infinite dimensional case of a well established passivity based control technique for finite dimensional port hamiltonian systems the control by interconnection and energy shaping according to which the open loop energy function is shaped so that a minimum in the desired configuration is introduced this procedure is possible once the state variable of the controller is related to the state variable of the plant by constraining the state of the closed loop system on a structural invariant defined by a set of casimir functions in this way the energy function of the controller which is freely assignable becomes a function of the configuration of the plant and then it can be easily shaped in order to solve the regulation problem

a differential equation with monodromy group 2 j2 abstract we construct a sixth order differential equation having the central extension of c 2 by the hall janko group j 2 as monodromy group moreover it arises from an iterated application of tensor products and convolution operations from a first order differential equation

rank one perturbations in a pontryagin space with one negative square let n 1 denote the class of generalized nevanfinna functions with one negative square and let n 1 n 0 be the subclass of functions q z is an element of n with the additional properties lim y infinity q i y y 0 and lim sup y infinity y im q i y i these classes form an analytic framework for studying generalized rank one perturbations a a in a pontryagin space setting many functions appearing in quantum mechanical models of point interactions either belong to the subclass n 1 n 0 or can be associated with the corresponding generalized friedrichs extension in this paper a spectral theoretical analysis of the perturbations a and the associated friedrichs extension is carried out many results such as the explicit characterizations for the critical eigenvalues of the perturbations a are based on a recent factorization result for generalized nevanlinna functions c 2002 elsevier science usa