torsion points on families of squares of elliptic curves in a recent paper we proved that there are at most finitely many complex numbers 0 1 such that the points 2 sqrt 2 2 lambda and 3 sqrt 6 3 lambda are both torsion on the elliptic curve defined by y 2 x x 1 x here we give a generalization to any two points with coordinates algebraic over the field q and even over c this implies a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes

existence of minimal nonsquare j symmetric factorizations for self adjoint rational matrix functions in this paper we show that any rational matrix function having hermitian values on the imaginary axis and with constant signature and constant pole signature admits a minimal symmetric factorization with possibly nonsquare factors our proof is based on a construction which shows that any such function can be extended preserving its mcmillan degree to a function that admits j symmetric factorization with square factors also we consider other properties of the factors in j symmetric factorizations particular attention is given to the study of the common invariant zero structure of these factors

minimal nonsquare spectral factors we consider the problem of parametrizing the set of all nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis

j inner outer factorization j spectral factorization and robust control for nonlinear systems the problem of expressing a given nonlinear state space system as the cascade connection of a lossless system and a stable minimum phase system inner outer factorization is solved for the case of a stable system having state space equations affine in the inputs the solution is given in terms of the stabilizing solution of a certain hamilton jacobi equation the stable minimum phase factor is obtained as the solution of an associated nonlinear spectral factorization problem as an application one can arrive at the solution of the nonlinear h sub spl infin control problem for the disturbance feedforward case

nonsquare spectral factors via factorizations of a unitary function abstract we consider the problem of parametrizing the set of nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis the tool here is the use of factorizations of a certain type of the function w 1w where w resp w is the minimal square spectral factor that has all its poles and zeros in the right resp left halfplane

inner outer factorization for nonlinear noninvertible systems this paper considers inner outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor an application of the main result to control of nonminimum phase nonlinear systems is indicated

nonsquare spectral factors of nonlinear control systems in terms of inner inner factorizations this paper considers nonsquare spectral factors of nonlinear input affine state space control systems in continuous time more specifically we obtain a parametrization of nonsquare spectral factors in terms of a special class of inner inner factorizations explicit formulas for different classes of spectral factors and related inner systems in terms of solutions of hamilton jacobi equations are provided furthermore the important notion of a minimal inner embedding for nonlinear control systems is introduced and used to derive several relationships between inner systems

soliton surfaces in the mechanical equilibrium of closed membranes for a closed material membrane in equilibrium in a force field we investigate whether the external observables membrane geometry and force field determine the internal membrane response the stress tensor t when the mean stress open image in new window tr t is known for membranes with boundary the indeterminacy of the response is classical for closed membranes the geometry decides the question we show uniqueness for all but a class of soliton surfaces the globally isothermic surfaces 1 2 the physical phenomenon exhibited by any closed globally isothermic membrane in equilibrium is that with all observables static there is in total a 1 parameter family of responses with the same mean stress all canonically determined by membrane geometry theorem 1 there exist closed embedded globally isothermic surfaces of every genus the recognition of the role of these soliton surfaces settles the old classification problem for the space of static shears in any closed membrane explicitly identifying all static shears where before only a genus dependent dimension bound was known theorem 2

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

interconnection of nonlinear causal systems for nonlinear systems it is not a mathematically obvious fact that the simplest feedback system yields equations which are solvable even when the gain in the loop is

zero sets of solutions to semilinear elliptic systems of first order consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over an n dimensional manifold assume the operator has the strong unique continuation property we show that the zero set of the solution is contained in a countable union of smooth n 2 dimensional submanifolds hence it is countably n 2 rectifiable and its hausdorff dimension is at most n 2 moreover it has locally finite n 2 dimensional hausdorff measure we show by example that every real number between 0 and n 2 actually occurs as the hausdorff dimension for a suitable choice of operator we also derive results for scalar elliptic equations of second order

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

deformation of hypersurfaces preserving the mobius metric and a reduction theorem abstract a hypersurface without umbilics in the n 1 dimensional euclidean space f m n r n 1 is known to be determined by the mobius metric g and the mobius second fundamental form b up to a mobius transformation when n 3 in this paper we consider mobius rigidity for hypersurfaces and deformations of a hypersurface preserving the mobius metric in the high dimensional case n 4 when the highest multiplicity of principal curvatures is less than n 2 the hypersurface is mobius rigid when the multiplicities of all principal curvatures are constant deformable hypersurfaces and the possible deformations are also classified completely in addition we establish a reduction theorem characterizing the classical construction of cylinders cones and rotational hypersurfaces which helps to find all the non trivial deformable examples in our classification with wider application in the future

torsion hypersurfaces on abelian schemes and betti coordinates in this paper we extend to arbitrary complex coefficients certain finiteness results on unlikely intersections linked to torsion in abelian surface schemes over a curve which have been recently proved for the case of algebraic coefficients in this way we complete the solution of zilber pink conjecture for abelian surface schemes over a curve as experience has shown also in previous cases the extension from algebraic to complex coefficients often requires entirely new arguments whereas simple specialization arguments fail the outcome gives as a byproduct new finiteness results when the base of the scheme has arbitrary dimension another consequence is a proof of an expectation of mazur concerning the structure of the locus in the base when a given section is torsion finally we show the link with an old work of griffiths and harris on a higher dimensional extension of a theorem of poncelet

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a characterization of minimal square spectral factors two different well known approaches to the spectral factorization problem phi s w s w s are connected together by relating the geometric properties of the solution set of the underlying algebraic riccati equation to the structure of the all pass factor of each minimal solution w s

on cancellation in h optimal controllers abstract this note clarifies the mechanism of pole zero cancellations that appeared in the derivation of the h optimal controller via a solution of the corresponding model matching problem the cancellable factor is explicitly expressed using chain scattering matrix descriptions a simple state space formula of the sub optimal controller is also presented based on the chain scattering forms and we show that there is a mixed sensitivity problem that can be solved by calculating a solution of only one riccati equation in each iteration

minimal square spectral factors abstract recently there has been renewed interest in the problem of spectral factorization and in particular the problem of parametrizing all square minimal spectral factors of a given spectrum for instance let us mention two recent papers in this journal one of which dealing with more computational aspects of this problem clements 1993 the other giving a parametrization under additional constraints ferrante et al 1993 this paper is motivated in a large part by ferrante et al 1993 in fact we shall show that a parametrization similar to the one given there can be achieved without one of the additional constraints imposed in ferrante et al 1993 a secondary aim of this paper is to give an overview of several parametrizations available in the literature

a normal form approach to approximate input output linearization for maximum phase nonlinear siso systems a control synthesis scheme is presented for nonlinear single input single output systems which have completely unstable antistable zero dynamics the approach is similar in spirit to linear approaches for nonminimum phase systems and involves the derivation of an input output linearizing controller for a suitably defined nonlinear minimum phase approximation to the original system the linearizing controller achieves an approximately linear input output response and internal stability

inner outer factorization of nonlinear operators abstract a fundamental result in the theory of hardy spaces of analytic matrix and operator valued functions on the unit disk is inner outer factorization for the case of rational matrix functions explicit formulas for the inner and outer factors have recently been obtained in these formulas the functions are presented in terms of a collection of finite matrices as the transfer function of a linear system the key ingredient in the construction is the solution of a linear matrix equation called a stein equation in this paper we obtain extensions of these results to the setting of input output maps of nonlinear discrete time systems a particaular computable class of examples is presented for illustration

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

non minimal factorization of nonlinear systems the observable case minimal factorizations of nonlinear systems have been studied earlier here non minimal factorizations of nonlinear systems are studied for the case where the product contains no extra non observable states observable factorizations a description for all such factorizations is given in terms of invariant foliations of the system and its inverse the results are applied to the special cases of stable factorizations coprime

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

stable invariant lagrangian subspaces factorization of symmetric rational matrix functions and other applications several applications of earlier results by the authors concerning various notions of stability of invariant lagrangian subspaces are studied the applications include stability of symmetric minimal factorizations of real symmetric rational matrix functions stability of factorizations of certain classes of symmetric matrix polynomials and stably well posed matricial boundary value problems with symmetries an interpretation of many stability results in terms of equilibria of lie group actions on subspaces is given as well

the rational points of a definable set let x r n be a set that is definable in an o minimal structure over r this article shows that in a suitable sense there are very few rational points of x which do not lie on some connected semialgebraic subset of x of positive dimension

parametrization of all minimal square spectral factors abstract two different parametrizations of minimal square spectral factors are given these parametrizations extend recent results by picci and pinzoni since they characterize spectral factors without specification on the zero or pole structure

torsion points on families of products of elliptic curves abstract in a recent paper we proved a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non simple surfaces this involves some entirely new and crucial issues

curved flats and isothermic surfaces we show how pairs of isothermic surfaces are given by curved flats in a pseudo riemannian symmetric space and vice versa calapso s fourth order partial differential equation is derived and using a solution of this equation a mobius invariant frame for an isothermic surface is built

isothermic triangulated surfaces we found a class of triangulated surfaces in euclidean space which have similar properties as isothermic surfaces in differential geometry we call a surface isothermic if it admits an infinitesimal isometric deformation preserving the mean curvature integrand locally we show that this class is mobius invariant isothermic triangulated surfaces can be characterized either in terms of circle patterns or based on conformal equivalence of triangle meshes this definition generalizes isothermic quadrilateral meshes a consequence is a discrete analog of minimal surfaces here the weierstrass data needed to construct a discrete minimal surface consist of a triangulated plane domain and a discrete harmonic function

zeros of spectral factors the geometry of splitting subspaces and the algebraic riccati inequality in this paper it is shown how the zero dynamics of not necessarily square spectral factors relate to the splitting subspace geometry of stationary stochastic models and to the corresponding algebraic riccati inequality the notion of output induced subspace of a minimal markovian splitting subspace which is the stochastic analogue of the supremal output nulling subspace in geometric control theory is introduced through this concept the analysis can be made coordinate free and straightforward geometric methods can be applied it is shown how the zero structure of the family of spectral factors relates to the geometric structure of the family of minimal markovian splitting subspaces in the sense that the relationship between the zeros of different spectral factors is reflected in the partial ordering of minimal splitting subspaces finally the well known characterization of the solutions of the algebraic riccati equation is generalized in terms of lagrangian subspaces invariant under the corresponding hamiltonian to the larger solution set of the algebraic riccati inequality

nonsquare spectral factorization for nonlinear control systems this paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time more specifically we obtain a parametrization of nonsquare spectral factors in terms of invariant lagrangian submanifolds and associated solutions of hamilton jacobi inequalities this inequality is a nonlinear analogue of the bounded real lemma and the control algebraic riccati inequality by way of an application we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system

factorization and feedback stabilization for nonlinear systems we establish the equivalence of internal input out stability for two feedback configurations of a nonlinear time varying plant p for which a related plant g is assumed to have a factorization g r with both r and r 1 incrementally stable this extends a factorization principle for stabilizability previously given only for the linear time invariant case as an application of a special case we recover a version of the youla parametrization of stabilizing compensators for the nonlinear case previously presented in the literature we use degree theory to parametrize a collection of solutions of the h control problem for the case of a 1 gain stable or lossless plant in the case of a plant g having a j inner outer factorization this last result combined with the above mentioned factorization principle leads to results on the h control problem for p

linear quadratic optimization for systems in the behavioral approach in this paper the following formulation of the linear quadratic optimal control problem for dynamical systems in the behavioral setting is proposed given a linear time invariant and complete system find the set of trajectories of the system that optimize a quadratic type cost function and that satisfy some linear static constraints this formulation provides a unifying framework where several generalized versions of the classical lq optimal control can be stated and solved the existence of solutions is first discussed it is shown that a necessary and sufficient condition for the existence of solutions may be obtained as a by product of a reduction procedure translating the problem into an equivalent one of minimum complexity such a procedure is based on the theory of ell 2 systems in the behavioral setting once the complexity is reduced a parametrization of the set of optimal solutions is obtained by employing a behavioral realization technique and a two step optimization procedure

non linear minimal square spectral factorization recent work has identified a correspondence between minimal square spectral factors of a given spectral density invariant subspaces of an associated hamiltonian matrix solutions of an appropriate algebraic riccati equation and minimal unitary left divisors of a certain unitary function on the imaginary line the purpose of this report is to obtain a natural extension of these ideas to a non linear setting at least for the special case where only minimal square stable spectral factors are considered