saturation for valuations on two dimensional regular local rings we study valuations centered in a regualar local ring of dimension two we define the notion of saturation with respect to such a valuation extending the classical definitions the invariants associated in a natural way to the valuation are related with the saturated ring and som geometric properties are deduced

2 d exact model matching the exact model matching problem in single input single output 2 d systems is solved by applying polyaomial techniques all solutions are described in a parametric form natural solvability conditions for causal and or stable solutions are given and a simple design procedure is proposed it consists in solving a linear 2 d polynomial equation which directly yields a suitable compensator

skew prime polynomial matrices the polynomial model approach abstract we examine the concept of skew primeness of polynomial matrices in terms of the associated polynomial model it is shown that skew primeness can be characterized in terms of the property of decomposition of a vector space relative to an endomorphism this basic result is then applied to the special case of nonsingular polynomial matrices we investigate the nonuniqueness of skew complements of a skew prime pair it is shown that the space of equivalence classes of skew complements is in bijective correspondence with a finite dimensional linear space finally the equivalence of the solutions to the problem of output regulation with internal stability obtained via geometric methods and via polynomial matrix techniques is shown explicitly

extremal solutions of φ laplacian diffusion scalar problems with nonlinear functional boundary conditions in a unified way abstract the aim of this paper is to ensure the existence of extremal solutions lying between a pair of lower and upper solutions for a second order differential equation that includes both the laplacian and some classes diffusion equations the results hold from a suitable change of variables an example is given to expose the applicability of the obtained results

existence of solutions for weighted p r laplacian system boundary value problems this paper investigates the existence of solutions for weighted p r laplacian ordinary system boundary value problems via leray schauder degree the sufficient conditions for the existence of solutions be given as an application we obtain the existence of the radial solutions for the p x laplacian partial differential system dirichlet problems

local global double algebras for slow h sup infinity adaptation i inversion and stability for pt ii see ibid vol 36 no 2 p 143 51 1991 in this two part work a common algebraic framework is introduced for the frozen time analysis of stability and h sup infinity optimization in slowly time varying systems based on the notion of a normed algebra on which local and global products are defined relations between local stability local near optimality local coprime factorization and global versions of these properties are sought the framework is valid for time domain disturbances in l sup infinity h sup infinity behavior is related to l sup infinity input output behavior via the device of an approximate isometry between frequency and time domain norms the authors presently elaborate on the double algebra concept for volterra operators which approximately commute with the shift and summarize the main algebraic properties and norm inequalities local conditions for global invertibility are also obtained classical frozen time stability conditions are incorporated in relations between local and global spectra

regulation of split linear systems over rings coefficient assignment and observers a theory of regulators is developed for finite free split linear systems over a commutative ring k this is achieved by developing a theory of coefficient assignment and observers it is shown that the problem of coefficient assignment can be solved for reachable systems by using dynamic state feedback for strongly observable systems it is shown that observers with arbitrary dynamics can be constructed internal stability of observer and state feedback configuration is considered explicitly some examples are given to illustrate these techniques

two point higher order bvps with singularities in phase variables the existence of solutions for singular higher order differential equations with the lidstone or the n p boundary conditions is proved the right hand sides of differential equations can have singularities in the zero value of their phase variables and so higher derivatives of solutions changing their signs can pass through these singularities proofs are based on the method of a priori estimates the degree theory arguments and on vitali s convergence theorem

singular nonlinear boundary value problems for second order ordinary differential equations abstract the boundary value problem y f x y 0 y 0 y 0 0 y 1 y 1 0 is studied under conditions which are modeled on the problem y a x y p 0y 0 y 1 0 or y 0 y 1 0 which occurs in a number of applications the technical arguments involve properties of solutions i e concavity and the use of iterative techniques a new fixed point theorem for cones for decreasing mappings is proved and used it is of interest in its own right since most such theorems are for monotone increasing mappings

polynomial matrix fraction representations for linear time varying systems abstract this paper deals with the existence and associated realization theory of skew polynomial fraction representations for linear time varying discrete time systems it is shown that the time varying analog of the polynomial model always has a free state space necessary and sufficient conditions for the existence of bezout factorizations are obtained in terms system theoretic properties of state space realizatios

on the existence of approximately coprime factorizations for retarded systems and abstract this note establishes a result linking algebraically coprime factorizations of transfer matrices of delay systems to approximately coprime factorizations in the sense of distribu tions the latter have been employed by the second author in the study of function space controllability for such systems

first order representations of fliess models abstract fuhrmann s state space construction in its generalized form is used to obtain a general theory of first order representations of fliess models defined over an arbitrary noetherian commutative ring the case of arbitrary linear delay differential equations is involved

system theoretic and algebraic aspects of the rings of stable and proper stable rational functions abstract a new approach in transfer function methods for solving a variety of control theoretic problems is to work with fractional representations over the ring of stable or proper stable rational functions of various transfer matrices the rings of stable and proper stable rational functions are well known to be euclidean domains these rings differ critically from the polynomial ring by the nonuniqueness of the remainders obtained this major difficulty in extending the idea of polynomial models of fuhrmann to the rings of stable and causal stable rational functions is circumvented by choosing a remainder in a special form a natural realization theory is thus developed for matrix fraction representations of transfer function matrices over these rings as an application of the new theory developed linear matrix equations qx ry t and qx yr t over the rings of stable and causal stable rational functions are reduced to finite sets of linear equations over the base field

transfer matrices realization and control of continuous time linear time varying systems via polynomial fractional representations abstract an algebraic theory of transfer matrices fractional representations and control for linear continuous time time varying systems based on the realization theory of input output maps is given it is shown for the first time that the realization of such systems specified by an abstract input output map as a module homomorphism over noncommutative polynomial rings can be established using an abstract kalman input output map defined over a ring of skew polynomials with time varying coefficients it is shown that in fact transfer matrices can be defined as formal power

cones of curves and of line bundles at infinity abstract we consider pencils at infinity v f z d in the projective plane p 2 there exists a minimal composition of point blowing ups x v p 2 eliminating the indeterminacies of the rational map p 2 p 1 induced by v we systematically study the mordell weil group of v the cone of curves ne x v q of x v and their relationship in theorem 3 we explicitly compute both objects when the projective curve defined by f is a union of curves with one place at infinity we furthermore ask two questions on pencils in p 2 our question 2 is answered in the affirmative in a particular case theorem 4 which can be viewed as a version at infinity but for pencils of any degree of a classical result on cubic pencils

the abhyankar moh theorem for plane valuations at infinity abstract we introduce the class of plane valuations at infinity and prove an analogue to the abhyankar moh semigroup theorem for it

the total coordinate ring of a smooth projective surface in 5 cox introduced the homogeneous coordinate ring of a toric variety which is a polynomial ring that allows to show that such a variety behaves like a projective space in many ways an analogous to that ring can also be defined for a smooth projective variety x over an algebraically closed field k such that linear and numerical equivalence coincide for divisors on x condition which is assumed for all varieties considered in this paper indeed let us fix li i 1 a z basis of pic x and set n n1 n2 nr z and dn ri 1 nili by regarding the vector spaces h 0 x ox dn f k divx f dn 0 0 as k subvector spaces of the function field k of x the total coordinate ring of x or the cox ring of x is defined as the graded k subalgebra of k

on state models of linear time variant discrete time systems a decoupled full dimension form is introduced for state models in this form the dimension of the state vector is time invariant which is equal to the dimension of the state space at each time instant therefore the dimension of the state vector is minimized from the time variant point of view and all the state variables are independent moreover the state vector in a state model of the decoupled full dimension form is partitioned into two decoupled subvectors each of which alone forms a subsystem one of the subsystems is an fir system whose state vector is dependent only on inputs at finitely past instants in addition this subsystem is completely reachable the other subsystem is an iir system whose state space is equal to the force free solution set of the state vector of the subsystem it is proved that every state model in the conventional form can be mapped by a linear one to one map into an equivalent state model in the decoupled full dimension form in comparison with a state model in the

singular boundary value problems for superlinear second order ordinary and delay differential equations abstract existence criteria are established for singular boundary value problems for nonlinear second order ordinary and delay differential equations the theorems obtained are very general and complement previous known results

dynamical compactifications of c 2 we find good dynamical compactifications for arbitrary polynomial mappings of c 2 and use them to show that the degree growth sequence satisfies a linear integral recursion formula for maps of low topological degree we prove that the green function is well behaved for maps of maximum topological degree we give normal forms

lidstone polynomials and boundary value problems abstract we define lidstone polynomials provide their explicit representations give their relations with bernoulli polynomials obtain best possible error estimates in lidstone interpolation and establish several best possible sharp inequalities which compare favourably with the known results next we use these results to study even order differential equations together with lidstone boundary conditions

δ sequences and evaluation codes defined by plane valuations at infinity we introduce the concept of sequence a sequence generates a well ordered semigroup s in z or r we show how to construct and compute parameters for the dual code of any evaluation code associated with a weight function defined by from the polynomial ring in two indeterminates to a semigroup s as above we prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces called plane valuations at infinity we also give algorithms to construct an unlimited number of sequences of the different existing types and so this paper provides the tools to know and use a new large set of codes

finite spectrum assignment and observer for multivariable systems with commensurate delays this paper is concerned with control of multivariable systems with commensurate delays the purpose of this paper is to enlarge the class of finite spectrum assignable systems to spectrally controllable multivariable systems with commensurate delays by overcoming an infinite number of eigenvalues which are contained in the system and make control very complex it is proved that if the system is spectrally controllable there is a delayed feedback matrix such that the closed loop system is spectrally controllable through a single input combining this result with previous ones about single input systems it is verified that spectral controllability is equivalent to finite spectrum assignability for multivariable systems with commensurate delays an observer for the system with delayed outputs is presented by using these results the multivariable system can be regulated as desired without preliminary knowledge of open loop eigenvalues and a stability test of transcendental functions whenever the system is spectrally controllable and spectrally observable

cones of curves and of line bundles on surfaces associated with curves having one place at infinity let v be a pencil of curves in bf p 2 with one place at infinity and x longrightarrow bf p 2 the minimal composition of point blow ups eliminating its base locus we study the cone of curves and the cones of numerically effective and globally generated line bundles on x it is proved that all of these cones are regular in particular this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves the surfaces in this class have non numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic an application is a global version on x of zariski s unique factorization theorem for complete ideals we also define invariants of the semigroup of globally generated line bundles on x depending only on the topology of v at infinity 2000 mathematical subject classification primary 14c20 secondary 14e05

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

local global double algebras for slow h sup infinity adaptation ii optimization of stable plants for pt i see ibid vol 36 no 2 p 130 42 1991 the authors presently establish an explicit formula linking global and local sensitivity for systems with stable plants in which local sensitivity is a lipschitz continuous function of data frequency domain estimates of time domain sensitivity norms which become accurate as rates of time variation approach zero are obtained notions of adaptive versus nonadaptive robust control are introduced it is shown that adaptive control can achieve better sensitivity than optimal nonadaptive control it is demonstrated that in general h sup infinity optimal interpolants do not depend lipschitz continuously on data however delta suboptimal interpolants of the aak central maximal entropy type are shown to satisfy a tractable lipschitz condition

singular dirichlet second order bvps with impulses abstract the paper deals with the existence of solutions to singular second order differential equations with impulse effects and with the dirichlet boundary conditions the right hand side of the differential equation can be singular in its phase variable

some new results for singular problems with sign changing nonlinearities we present some general existence theorems for second order boundary value problems with sign changing nonlinearities both the upper and lower solution approach and the growth restriction approach are discussed

comments on 2 d exact model matching two examples are given to indicate that in 2 d coefficient assignment model matching some cancellations are unavoidable in the author s belief and the split condition not only the zero coprimeness is necessary

big rational surfaces we prove that the cox ring of a smooth rational surface with big anticanonical class is finitely generated we classify surfaces of this type that are blow ups of mathbb p 2 at distinct points lying on a possibly reducible cubic

on existence and multiplicity of positive solutions to singular multi point boundary value problems the existence and multiplicity of positive solutions are established for the multi point boundary value problem p t u t f t u t 0 0 t 1 u 0 j 1maju xj w 1 j 1mbjw xj where w t p t u t aj bj 0 with 0 j 1maj 1 and j 1mbj 1 xj 0 1 with 0 x1 x2 xm 1 under certain conditions on p and f the arguments are based upon the positivity of the green s function and the krasnosel skii fixed point theorem

the cone conjecture for calabi yau pairs in dimension 2 a central idea of minimal model theory as formulated by mori is to study algebraic varieties using convex geometry the cone of curves of a projective variety is defined as the convex cone spanned by the numerical equivalence classes of algebraic curves the dual cone is the cone of nef line bundles for fano varieties varieties with ample anticanonical bundle these cones are rational polyhedral by the cone theorem 21 theorem 3 7 for more general varieties these cones are not well understood they can have infinitely many isolated extremal rays or they can be round both phenomena occur among calabi yau varieties such as k3 surfaces 22 which can be considered the next simplest varieties after fano varieties the morrison kawamata cone conjecture would give a clear picture of the nef cone for calabi yau varieties 29 30 19 the conjecture says that the action of the automorphism group of the variety on the nef cone has a rational polyhedral fundamental domain the conjecture includes an analogous statement about the movable cone see section 2 for details thus for calabi yau varieties the failure of the nef cone to be rational polyhedral is always explained by an infinite discrete group of automorphisms of the variety it is not clear where these automorphisms should come from nonetheless the conjecture has been proved for calabi yau surfaces by sterk looijenga and namikawa 41 33 19 the heart of the proof being the torelli theorem for k3 surfaces of piatetski shapiro and shafarevich 3 theorem 11 1 kawamata proved the cone conjecture for all 3 dimensional calabi yau fiber spaces over a positive dimensional base 19 the conjecture is wide open for calabiyau 3 folds despite significant results by oguiso and peternell 38 szendroi 43 uehara 46 and wilson 48 the conjecture was generalized from calabi yau varieties to klt calabi yau pairs x in 45 here is a divisor on x and calabi yau means that kx rather than kx is numerically trivial in this paper we prove the cone conjecture for all klt calabi yau pairs of dimension 2 theorem 4 1 using the geometry of groups acting on convex cones and reduction to the case of k3 surfaces this is enough to show that the conjecture is reasonable in the greater generality of pairs more concretely the theorem gives control over the nef cone and the automorphism group for a large class of rational surfaces including the fano surfaces as well as many others in particular we get a good description of when the cox ring or total coordinate ring is finitely generated in this class of surfaces corollary 5 1 thanks to caucher birkar igor dolgachev brian harbourne artie prendergastsmith chenyang xu and the referees for their comments

minimum variance prediction for linear time varying systems in this paper we study the problem of minimum variance prediction for linear time varying systems we consider the standard time varying autoregression moving average arma model and develop a predictor which guarantees minimum variance prediction for a large class of linear time varying systems the predictor is developed based on a pseudocommutation technique for dealing with noncommutativity of linear time varying operators in a transfer operator framework we also show connections between this input output predictor and the kalman predictor via an example

cox rings of surfaces and the anticanonical iitaka dimension abstract in this paper we investigate the relation between the finite generation of the cox ring r x of a smooth projective surface x and its anticanonical iitaka dimension k x

on contractible curves on normal surfaces we give characterizations of contractible curves on proper normal algebraic surfaces in terms of complementary weil divisors from this we obtain some generalizations of the classical criteria for contractibility of castelnuovo and artin furthermore we will derive a finiteness result on homogeneous spectra defined by weil divisors on proper normal algebraic surfaces

fractional representations for systems over a p i d a constructive technique abstract in this note we present a constructive technique for obtaining polynomial matrix fraction representations for reachable linear systems over a principal ideal domain an example is given to illustrate the algorithm

pole placement for linear systems over bezout domains the problem of pole placement for linear systems over rings is considered for a class of rings our results show that one can construct reduced order dynamic output feedback compensators for pole placement our results also provide an alternative proof of some previously known results on the problem of pole placement by state feedback for a class of reachable pairs f g over bezout domains

positive solutions of singular dirichlet and periodic boundary value problems let a and t be positive numbers the singular differential equation r x x q t f t x is considered here r 0 on 0 a may be singular at x 0 and f t x 0 may be singular at x 0 and x a effective sufficient conditions imposed on r q and f are given for the existence of a solution x to the above equation satisfying either the dirichlet conditions x 0 x t 0 or the periodic conditions x 0 x t x 0 x t and in addition 0 x a on 0 t

on the control of linear systems whose coefficients are functions of parameters the stabilization and regulation of linear discrete time systems whose coefficients depend on one or more parameters is studied for linear systems whose coefficients are continuous functions of real or complex parameters respectively analytic or rational functions of real parameters it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous analytic rational functions of the parameters closed form expressions for a collection of stabilizing gains are given in terms of the reachability gramian for systems which are stabilizable for all values of the parameters it is shown that continuous analytic rational stabilizing gains can be computed from a finite time solution to a riccati difference equation whose coefficients are functions of the parameters these results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters

on the computation of darboux first integrals of a class of planar polynomial vector fields abstract we study the class of planar polynomial vector fields admitting darboux first integrals of the type i 1 r f i i where the i s are positive real numbers and the f i s are polynomials defining curves with only one place at infinity we show that these vector fields have an extended reduction procedure and give an algorithm which from a part of the extended reduction of the vector field computes a darboux first integral for generic exponents

singular lidstone boundary value problem with given maximal values for solutions abstract the singular problem 1 nx 2n f t x x 2n 2 x 2j 0 x 2j t 0 0 j n 1 max x t 0 t t a depending on the parameter is considered here the positive caratheodory function f may be singular at the zero value of all its phase variables the paper presents conditions which guarantee that for any a 0 there exists a 0 such that the above problem with a has a solution x ac2n 1 0 t which is positive on 0 t the proofs are based on the regularization and sequential techniques and use the leray schauder degree and vitali s convergence theorem

existence of solutions for some nonlinear problems with p laplacian like operators abstract the purpose of this paper is to obtain some existence results of solutions for the nonlinear boundary value problems with p laplacian like operators

generalized model matching and f g invariant submodules for linear systems over rings abstract a generalized version of the exact model matching problem gemmp is considered for linear multivariable systems over an arbitrary commutative ring k with identity reduced forms of this problem are introduced and a characterization of all solutions and minimal order solutions is given both with and without the properness constraint on the solutions in terms of linear equations over k and k modules an approach to the characterization of all stable solutions is presented which under a certain bezout condition and a freeness condition provides a parametrization of all stable solutions the results provide an explicit parametrization of all solutions and all stable solutions in case k is a field without the bezout condition this is achieved through a very simple characterization and a generalization to an arbitrary field k of the fixed poles of the model matching problem in terms of invariant factors of a certain polynomial matrix the results also show that whenever the gemmp has a solution there exist solutions whose poles can be chosen arbitrarily as far as they contain the fixed poles with the right multiplicities in the algebraic closure of k implications of these results in regard to inverse systems are shown equivalent simpler forms in state space form of the problem are shown to be obtainable a theory of finitely generated f g invariant submodules for linear systems over rings is developed and the geometric equivalent of the model matching problem the dynamic cover problem is formulated to which the results of the previous sections provide a solution in the reduced case

eigenvalues of lidstone boundary value problems we consider the following boundary value problem 1 ny 2 n lf t y n 1 t 0 1 y 2 i 0 y 2 i 1 0 0 0 the values of l are characterized so that the boundary value problem has a positive solution in addition we derive explicit intervals of l such that for any l in the interval existence of a positive solution of the boundary value problem is guaranteed several examples are also included to dwell upon the importance of the results obtained

existence of multiple solutions for weighted p r laplacian equation dirichlet problems abstract this paper deals with the existence of multiple solutions for weighted p r laplacian equation dirichlet problems several sub super solution type results have been given via leray schauder s degree method in particular we give a counterexample which means that 2 in paper j henderson h b thompson existence of multiple solutions for second order boundary value problems j differential equations 166 2000 443 454 is not an open subset of c 0 1

a transfer function approach to linear time varying discrete time systems in the first part of the paper a transfer function approach is developed for the class of linear time varying discrete time systems the theory is specified in terms of skew noncommutative rings of polynomials and formal power series both with coefficients in a ring of time functions the transfer function matrix is defined to be a matrix whose entries belong to a skew ring of formal power series it is shown that various system properties such as asymptotic stability can be characterized in terms of the skew ring framework in the last part of the paper the transfer function framework is applied to the study of feedback control new results are obtained on assignability of system dynamics by using dynamic output feedback and dynamic state feedback the results are applied to the control of an armature controlled do motor with a variable loading

the cone of curves and the cox ring of rational surfaces given by divisorial valuations abstract we consider surfaces x defined by plane divisorial valuations of the quotient field of the local ring r at a closed point p of the projective plane p 2 over an arbitrary algebraically closed field k and centered at r we prove that the regularity of the cone of curves of x is equivalent to the fact that is non positive on o p 2 p 2 l where l is a certain line containing p under these conditions we characterize when the characteristic cone of x is closed and its cox ring finitely generated equivalent conditions to the fact that is negative on o p 2 p 2 l k are also given

decomposition of linear automata over residue rings into shift registers abstract a linear automaton of a over a residue ring n n e is in general not decomposable into a parallel connection of shift registers we are able to formulate necessary and sufficient conditions for such a decomposition by using the theory of artin local rings r and the r x module structure of a

trajectory tracking and disturbance rejection for linear time varying systems input output representation this paper considers the problem of asymptotic trajectory tracking and or disturbance rejection for linear time varying systems via input output representation inspired by the structure of repetitive control a novel feedback mechanism is proposed based on which the necessary and sufficient condition for the problem is derived this condition is a generalization to the time varying case of the result in time invariant setting based on the complete knowledge of the plant model a design is provided and it is applicable to repetitive control for time varying plants an important feature of the proposed approach is that a systematic way of constructing time varying internal model unit is provided

on matrix fraction representations for linear systems over commutative rings this paper deals with the problem of existence of polynomial matrix fraction representations for transfer matrices of linear systems over rings as well as the related realization theory these representations are then used in establishing new results for various classes of systems including split systems the relevance of these results to the regulation of various nonclassical classes of linear systems for example delay differential systems two dimensional systems etc is also discussed

stabilizability and stable proper factorizations for linear time varying systems the objective of this paper is to study in detail stabilizability and stable proper factorizations for linear time varying discrete time systems our main results are i if a linear time varying system can be stabilized by dynamic state feedback then it can also be stabilized by memoryless state feedback ii a complete characterization of the existence of stable proper factorizations for linear time varying input output operators this characterization is nontrivial there exist input output operators that do not admit stable proper factorizations