a solution to the continuous time rm h _ infty fixed interval smoother problem the minimum variance fixed interval smoother is a state space realization of the wiener solution generalized for time varying problems it involves forward and adjoint wiener hopf factor inverses in which the gains are obtained by solving a riccati equation this technical note introduces a continuous time h smoother having the structure of the minimum variance version in which the gains are obtained by solving a riccati equation that possesses an indefinite quadratic term it is shown that the smoother exhibits an increase in mean square error the error is bounded and the upper error bound is greater than that for the h filter

cox rings of cubic surfaces and fano threefolds abstract we determine the cox rings of the minimal resolutions of cubic surfaces with at most rational double points of blow ups of the projective plane at non general configurations of six points and of three dimensional smooth fano varieties of picard numbers one and two

computing cox rings we consider modifications for example blow ups of mori dream spaces and provide algorithms for investigating the effect on the cox ring e g testing finite generation or computing an explicit presentation in terms of generators and relations as a first application we compute the cox rings of all gorenstein log del pezzo surfaces of picard number one moreover we show computationally that all smooth rational surfaces of picard number at most six are mori dream surfaces and we provide explicit presentations of the cox ring for those not admitting a torus action finally we provide the cox rings of projective spaces blown up at a certain special point configurations

nombre de points de hauteur bornee sur les surfaces de del pezzo de degre 5 we state manin s conjecture in the particular case of the split del pezzo s surfaces of degree 5 over q we show that for an open set u v nu q b card p u q h p b c b log b 4 b the constant c is the one conjectured by e peyre sommaire

the cox ring of an algebraic variety with torus action abstract we investigate the cox ring of a normal complete variety x with algebraic torus action our first results relate the cox ring of x to that of a maximal geometric quotient of x as a consequence we obtain a complete description of the cox ring in terms of generators and relations for varieties with torus action of complexity one moreover we provide a combinatorial approach to the cox ring using the language of polyhedral divisors applied to smooth k surfaces our results give a description of the cox ring in terms of orlik wagreich graphs as examples we explicitly compute the cox rings of all gorenstein del pezzo k surfaces with picard number at most two and the cox rings of projectivizations of rank two vector bundles as well as cotangent bundles over toric varieties in terms of klyachko s description

optimum performance levels for minimax filters predictors and smoothers abstract for both discrete and continuous time linear time varying systems we obtain the achievable performance levels for minimax filters predictors and smoothers in terms of the finite escape times of some related discrete and continuous time riccati equations our game theoretic approach also yields an alternative derivation for the corresponding minimax estimators which were first obtained in 1 they are all bayes estimators with respect to particular gaussian distributions and admit recursive structures

grobner bases monomial group actions and the cox rings of del pezzo surfaces abstract we introduce the notion of monomial group action and study some of its consequences for grobner basis theory as an application we prove a conjecture of v batyrev and o popov describing the cox rings of del pezzo surfaces of degree 3 as quotients of a polynomial ring by an ideal generated by quadrics

non split toric bch codes on singular del pezzo surfaces in the article we construct low rate non split toric q ary codes on some singular surfaces more precisely we consider non split toric cubic and quartic del pezzo surfaces whose singular points are mathbb f q conjugate our codes turn out to be bch ones with sufficiently large minimum distance d indeed we prove that d d geqslant q lfloor 2 sqrt q rfloor 1 where d is the designed minimum distance in other words we significantly improve upon bch bound on the other hand the defect of the griesmer bound for the new codes is leqslant lfloor 2 sqrt q rfloor 1 which also seems to be quite good it is worth noting that to better estimate d we actively use the theory of elliptic curves over finite fields

spherical blow ups of grassmannians and mori dream spaces abstract in this paper we classify weak fano varieties that can be obtained by blowing up general points in prime fano varieties we also classify spherical blow ups of grassmannians in general points and we compute their effective cone these blow ups are in particular mori dream spaces furthermore we compute the stable base locus decomposition of the blow up of a grassmannian in one point and we show how it is determined by linear systems of hyperplanes containing the osculating spaces of the grassmannian at the blown up point and by the rational normal curves in the grassmannian passing through the blown up point

robust state estimation for uncertain systems with averaged integral quadratic constraints this paper considers a robust state estimation problem for a new class of uncertain systems the new uncertainty class introduced in the paper involves structured uncertainties which are required to satisfy a certain averaged integral quadratic constraint this uncertainty class combines aspects of norm bounded and stochastic uncertainty descriptions the solution to the robust state estimation problem is obtained by solving a parametrized riccati differential equation of the game type

h output feedback control for linear discrete time varying systems via the bounded real lemma in this paper we develop a solution to the discrete time output feedback control problem for linear time varying ltv systems the solution is developed along the strategy set up by doyle et al in 1989 and the main ingredient in its derivation is the extension of the well known bounded real lemma to a discrete time varying context developed by van der veen and verhaegen in 1995 this approach contributes to the conceptual simplicity and hence to the accessibility of the solution apart from that we treat the infinite horizon case for ltv system of non uniform state dimension and varying input and output dimension both situations can easily occur in practice for example in multirate sampled data control systems the algorithm that can be derived from the solution presented in then applied to the output feedback of a dynamical system changing from one operation point in its operation envelope to another

well posedness of stochastic kdv bo equation driven by fractional brownian motion the cauchy problem for the korteweg de vries benjamin ono equation driven by cylindrical fractional brownian motion is discussed in this paper fractional brownian motion is a family of processes b h it is known that the smaller the value of hurst parameter h is the worse of the regularity of fbm is using bourgain restriction method we obtain the lower bound of the hurst parameter h for the driving processes b h with h 38 we prove local existence results with initial value in classical sobolev spaces of negative indices i e h s with s 18

gibbsian dynamics and invariant measures for stochastic dissipative pdes we present a general strategy for proving ergodicity for stochastically forced nonlinear dissipative pdes it consists of two main steps the first step is the reduction to a finite dimensional gibbsian dynamics of the low modes the second step is to prove the equivalence between measures induced by different past histories using girsanov theorem as applications we prove ergodicity for ginzburg landau kuramoto sivashinsky and cahn hilliard equations with stochastic forcing

exemples de comptage de courbes sur les surfaces some examples of curves countings on surfaces let x be a surface whose cox ring has a single relation satisfying moreover a kind of linearity property we show that the geometric manin s conjectures hold for some degrees lying in the dual of the effective cone of x in particular for those degrees the moduli space of morphisms has the expected dimension the result applies to a class of generalized del pezzo surfaces which has been intensively studied in the context of the arithmetic manin s conjecture

the cox ring of overline _ 0 6 we prove that the cox ring of the moduli space v 0 6 of stable rational curves with 6 marked points is finitely generated by sections corresponding to the boundary divisors and divisors which are pull backs of the hyperelliptic locus in m 3 via morphisms p m 0 6 m 3 that send a 6 pointed rational curve to a curve with 3 nodes by identifying 3 pairs of points in particular this gives a self contained proof of hassett and tschinkel s result about the effective cone of m 0 6 being generated by the above mentioned divisors

numerical decomposition of the solution sets of polynomial systems into irreducible components in engineering and applied mathematics polynomial systems arise whose solution sets contain components of different dimensions and multiplicities in this article we present algorithms based on homotopy continuation that compute much of the geometric information contained in the primary decomposition of the solution set in particular ignoring multiplicities our algorithms lay out the decomposition of the set of solutions into irreducible components by finding at each dimension generic points on each component as by products the computation also determines the degree of each component and an upper bound on its multiplicity the bound is sharp i e equal to one for reduced components the algorithms make essential use of generic projection and interpolation and can if desired describe each irreducible component precisely as the common zeroes of a finite number of polynomials

h sub infinity minimum error state estimation of linear stationary processes a state estimator is derived which minimizes the h sub infinity norm of the estimation error power spectrum matrix two approaches are presented the first achieves the optimal estimator in the frequency domain by finding the filter transfer function matrix that leads to an equalizing solution the second approach establishes a duality between the problem of h sub infinity filtering and the problem of unconstrained input h sub infinity optimal regulation using this duality previously published results for the latter regulation problem are applied which lead to an optimal filter that possess the structure of the corresponding kalman filter the two approaches usually lead to different results they are compared by a simple example which also demonstrates a clear advantage of the h sub infinity estimate over the conventional l sub 2 estimate

h_ infty fixed interval smoothing estimation for time delay systems this paper is concerned with the h fixed interval smoothing estimation for time delay systems which include continuous time case and discrete time case in the case of discrete time systems the problem can be solved by using the conventional state augmentation approach however this approach is not suitable for the continuous time case in this paper we will propose a unified approach to study the h fixed interval smoothing problem for both continuous time and discrete time systems with l output delays by introducing a suitable stochastic linear time delay models in an indefinite space it is shown that the h fixed interval smoother can be obtained by calculating h2 fixed interval smoother for time delay systems in an indefinite space therefore based on the orthogonal projection theory in an indefinite space the h fixed interval smoothers for time delay systems are designed by performing l 1 riccati equations with the same dimension as the original systems moreover a necessary and sufficient conditions for the existence of the h fixed interval smoother will also be given

on images of mori dream spaces the purpose of this paper is to study the geometry of images of morphisms from mori dream spaces first we prove that a variety which admits a surjective morphism from a mori dream space is again a mori dream space secondly we introduce a natural fan structure on the effective cone of mori dream spaces we show that it encodes the information of zariski decompositions which in turn is equivalent to the information of the variation of git quotients of their cox rings finally we show that under a surjective morphism between mori dream spaces the fan of the target space coincides with the restriction of the fan of the source

sampled data h optimal control of time varying systems the sampled data h optimal control problem is studied for time varying systems the problem consists of finding all sampled data controllers with a piece wise constant control signal for which the l2 induced norm from the continuous time disturbance to the continuous time output is less than a specified bound a game theoretic approach is used to obtain a complete solution to the problem in the finite horizon time varying case the solution involves a mixed discrete continuous riccati equation and a variable transformation which takes the problem to a standard discrete finite dimensional h filtering problem

a game theoretic approach to a finite time disturbance attenuation problem a disturbance attenuation problem over a finite time interval is considered by a game theoretic approach where the control restricted to a function of the measurement history plays against adversaries composed of the process and measurement disturbances and the initial state a zero sum game formulated as a quadratic cost criterion subject to linear time varying dynamics and measurements is solved by a calculus of variation technique by first maximizing the quadratic cost criterion with respect to the process disturbance and initial state a full information game between the control and the measurement residual subject to the estimator dynamics results the resulting solution produces an n dimensional compensator which expresses the controller as a linear combination of the measurement history a disturbance attenuation problem is solved based on the results of the game problem for time invariant systems it is shown that under certain conditions the time varying controller becomes time invariant on the infinite time interval the resulting controller satisfies an h sub infinity norm bound

cox ring of the generic fiber abstract given a surjective morphism x y of normal varieties satisfying some regularity hypotheses we prove how to recover a cox ring of the generic fiber of from the cox ring of x as a corollary we show that in some cases it is also possible to recover the cox ring of a very general fiber and finally we give an application in the case of the blowing up of a toric fiber space

picard graded betti numbers and the defining ideals of cox rings abstract we introduce a complex of vector spaces whose homology determines the picard graded betti numbers of the cox ring of a mori dream space as an application of this construction we prove a conjecture of batyrev and popov which describes presentations for the cox rings of del pezzo surfaces of degree at least two

brief state estimation of continuous time radial basis function networks this paper presents state estimators for previously trained rbf networks based on a minimax approach the uncertainty class is characterized in terms of an approximation error vector minimizing the objective function over this uncertainty class is used to obtain state estimates the initial portion of the paper presents the derivation and properties of such a minimax state estimator for rbf networks in the later portion this is extended to the problem of minimax adaptive state estimation this approach is useful in the cases where the pretrained rbf network is to be used in state estimation since the latest information is used in improving estimates of the rbf weights a special class of estimators for the appended state estimation problem is also considered where a constant gain estimator matrix can be obtained this section gives sufficient conditions for the existence of such an observer and follows directly from the structure of the rbf networks finally example applications of the minimax and adaptive minimax estimation for nonlinear problems are presented the results show the accuracy of state estimation using the proposed minimax estimator

the stochastic korteweg de vries equation inl2 r abstract we present here two results of global existence and uniqueness of the cauchy problem concerning the korteweg de vries equation forced by a random term of white noise type in the functional framework of weighted spaces on l 2 r this work is motivated by both physical experiments where a forced generation of weakly nonlinear waves by localized disturbances was observed 1 15 and a result of uniqueness in weighted space in the homogeneous case 8 14

continuous time tracking problems in an h sub spl infin setting a game theory approach this paper investigates the problem of finite horizon h sub spl infin tracking for linear time varying systems three tracking problems are considered depending on whether the reference signal is perfectly known in advance measured on line or previewed in fixed interval of time ahead no a priori knowledge of a dynamic model for the reference signal is assumed a game theory approach to the latter tracking problems is presented where given a specific reference signal the controllers play against nature which can choose any initial state and any bounded energy disturbance input and measurement noise necessary and sufficient conditions for the existence of a saddle point equilibrium are determined and h sub spl infin tracking controllers for both the cases of state and output feedback are derived tracking problems for time invariant systems on infinite horizon are also analyzed and time invariant tracking controllers are obtained for zero initial conditions

automorphism groups on normal singular cubic surfaces with no parameters the classification of normal singular cubic surfaces in p 3 over a complex number field c was given by j w bruce and c t c wall in this paper first we prove their results by a different way second we provide normal forms of normal singular cubic surfaces according to the type of singularities and finally we determine automorphism groups on normal singular cubic surfaces with no parameters

game theory approach to optimal linear state estimation and its relation to the minimum h sub infinity norm estimation a possible game theory approach to optimal state estimation is presented it is found that in a certain differential game the minimizer s policy is identical to the one obtained by optimal estimation in the minimum h sub infinity norm sense this interpretation of h sub infinity optimal state estimation provides better insight into the mechanism of h sub infinity optimal filtering especially in the case where the exogenous signals are not energy bounded

a game theoretic approach to h control for time varying systems a representation formula for all controllers that satisfy an mathcal l infty type constraint is derived for time varying systems it is now known that a formula based on two indefinite algebraic riccati equations may be found for time invariant systems over an infinite time support see j c doyle et al ieee trans automat control ac 34 1989 pp 831 847 k glover and j c doyle systems control lett 11 1988 pp 167 172 k glover et al siam j control optim 29 1991 pp 283 324 m green et al siam j control optim 28 1990 pp 1350 1371 d j n limebeer et al in proc ieee conf on decision and control austin tx 1988 g tadmor math control systems signal processing 3 1990 pp 301 324 in the time varying case two indefinite riccati differential equations are required a solution to the design problem exists if these equations have a solution on the optimization interval the derivation of the representation formula illustrated in this paper mak

smoothing properties and existence of solutions for the generalized benjamin ono equation abstract we study the cauchy problem for the generalized benjamin ono equation tu d d 2 u dv u where d d dx u 0 and v e 1 r r with v 0 v 0 0 using commutator identities and estimates we prove that the equation exhibits smoothing properties similar to those of the generalized korteweg de vries equation gkdv which is the special case 1 of if v satisfies suitable estimates at zero and at infinity we prove that the cauchy problem for the equation with initial data u 0 u0 has a solution u l r l2 l2loc r h loc if u0 l2 and has a solution u l r h l2loc r h2 loc if u0 h finite energy solution in addition we prove that the usual benjamin ono equation 1 2 v u u 2 has a solution u l r h1 l2loc r h 3 2 loc for u0 h1 those results are easily extended to related equations such as the intermediate long wave equation or the smith equation

asymptotic behavior of stochastic discrete wave equations with nonlinear noise and damping in this article we study the asymptotic behavior for a class of discrete wave equations with nonlinear noise and damping defined on a k dimensional integer set the well posedness of the system is established when the nonlinear drift function and the nonlinear diffusion term are only locally lipschitz continuous the mean random dynamical system associated with the non autonomous system is shown to possess a unique tempered weak pullback mean random attractor in l2 f l2 l2 the existence of invariant measures for the autonomous system is also derived by using the krylov bogolyubov method the difficulty in proving the tightness of a family of distribution laws of the solutions is overcome by using the idea of uniform estimates on the tails of solutions

automorphism groups of cubic surfaces abstract the automorphism group of a generic cubic surface is trivial k in this article we determine the automorphism group of each nonsingular cubic surface

on the stochastic benjamin ono equation abstract we discuss the cauchy problem for the stochastic benjamin ono equation in the function class h s r s 3 2 when there is a zero order dissipation we also establish the existence of an invariant measure with support in h 2 r many authors have discussed the cauchy problem for the deterministic benjamin ono equation but our results are new for the stochastic benjamin ono equation our goal is to extend known results for the deterministic equation to the stochastic equation

homogeneous coordinates for algebraic varieties abstract we associate to every divisorial e g smooth variety x with only constant invertible global functions and finitely generated picard group a pic x graded homogeneous coordinate ring this generalizes the usual homogeneous coordinate ring of the projective space and constructions of cox and kajiwara for smooth and divisorial toric varieties we show that the homogeneous coordinate ring defines in fact a fully faithful functor for normal complex varieties x with only constant global functions we even obtain an equivalence of categories finally the homogeneous coordinate ring of a locally factorial complete irreducible variety with free finitely generated picard group turns out to be a krull ring admitting unique factorization

nonlocal models for nonlinear dispersive waves abstract considered herein are model equations for the unidirectional propagation of small amplitude nonlinear dispersive long waves such as those governed by the classical korteweg de vries equation of special interest are physical situations in which the linear dispersion relation is not appropriately approximated by a polynomial so that the operator modelling dispersive effect is nonlocal particular cases in view here are the benjamin ono equation and the intermediate long wave equation which arise in internal wave theory and smith s equation which governs certain types of continental shelf waves the initial value problem for these equations is shown to be globally well posed in the classical sense including continuous dependence upon the initial data and in certain cases upon the modelling of nonlinear and dispersive effects whilst the results are stated for the specific equations listed above the techniques utilized are seen to have a considerable range of generality as regards application to nonlinear dispersive evolution equations particularly worthy of note is our theorem implying that solutions of the intermediate long wave equation converge strongly to solutions of the korteweg de vries equation or to solutions of the benjamin ono equation in appropriate asymptotic limits

a new parametrization of h suboptimal controllers elementary state space concepts are used to derive a transparent solution to the h control problem on an infinite horizon the main contribution of this solution is a novel representation of suboptimal controllers in terms of a pair of parametric riccati equations with a coupling constraint unlike the classical parametrization in terms of linear fractional transformation this state space representation has a homogeneous riccati formulation which should help to make the most of the suboptimal controller diversity potential applications include the design of reduced order controllers and more generally the selection of suboptimal controllers to meet or optimize additional constraints

on blowing up the weighted projective plane we investigate the blow up of a weighted projective plane at a general point we provide criteria and algorithms for testing if the result is a mori dream surface and we compute the cox ring in several cases moreover applications to the study of overline m 0 n are discussed

filtering and smoothing in an h sup infinity setting the problems of filtering and smoothing are considered for linear systems in an h sup infinity setting i e the plant and measurement noises have bounded energies are in l sub 2 but are otherwise arbitrary two distinct situations for the initial condition of the system are considered the initial condition is assumed known in one case while in the other the initial condition is not known but the initial condition the plant and measurement noise are in some weighted ball of r sup n xl sub 2 finite horizon and infinite horizon cases are considered necessary and sufficient conditions are presented for the existence of estimators both filters and smoothers that achieve a prescribed performance bound and algorithms that result in performance within the bounds are developed in case of smoothers the optimal smoother is also presented the approach uses basic quadratic optimization theory in a time domain setting as a consequence of which both linear time varying and time invariant systems can be considered with equal ease in the smoothing problem for linear time varying systems one considers only the finite horizon case

factorially graded rings and cox rings abstract cox rings of normal prevarieties are factorially graded i e homogeneous elements allow a unique decomposition into homogeneous factors we study this property from an algebraic point of view and give a criterion which in a sense reduces it to factoriality this will allow us to detect and construct cox rings in a purely algebraic manner

automorphism groups of quartic del pezzo surfaces abstract the automorphism group of a generic quartic del pezzo surface is isomorphic to the group 2 23 4 m koitabashi j algebra116 1988 130 142 in this article we determine the automorphism group of each quartic del pezzo surface

computing resolutions of quotient singularities abstract let g gl n be a finite group without pseudo reflections we present an algorithm to compute and verify a candidate for the cox ring of a resolution x c n g which is based just on the geometry of the singularity c n g without further knowledge of its resolutions we explain the use of our implementation of the algorithms in singular as an application we determine the cox rings of resolutions x c 3 g for all g gl 3 with the aforementioned property and of order g 12 we also provide examples in dimension 4

game theory approach to finite time horizon optimal estimation in this game a measurement record is given and the first player looks for the best estimate of a prespecified combination of the system states in the presence of a hostile process noise signal and system initial condition that are applied by his adversary it turns out that the game possesses a saddle point solution which leads to an optimal smoothed estimate that is identical to the corresponding l sub 2 optimal estimate a similar game in which the estimate is restricted to be causal is formulated and solved this game provides for the first time a saddle point equilibrium interpretation to finite time h sub infinity optimal filtered estimation the two games are very closely related it is shown that in the first game the first player s strategy which is the optimal smoothed estimate is a linear fractional transformation of the h sub infinity optimal filter which applies a nonzero free contracting q parameter it therefore achieves a unity h sub infinity norm bound for the operator that relates the exogeneous signals to the estimation error

nondefinite least squares and its relation to h sub infinity minimum error state estimation the problem of recursive nondefinite least squares state estimation of continuous time stationary processes is solved by applying pontryagin s maximum principle a comparison of the derived solution to the result that is obtained for the h sub infinity minimum error estimation suggests a new interpretation for the h sub infinity optimal estimation mechanism according to this interpretation the estimator tries to optimally estimate the required combination of the states in the l sub 2 norm sense against the worst disturbance signal that stems from a fictitious measurement of this combination

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

white noise driven korteweg de vries equation abstract we consider a stochastic korteweg de vries equation on the real line the noise is additive we use function spaces similar to those introduced by bourgain to prove well posedness results for the korteweg de vries equation in l 2 r we are able to handle a noise which is locally white in space and time more precisely it is a space time white noise multiplied by an l 2 function of the space variable due to the lack of a priori estimates we can only get a local existence result in time however we obtain the global existence of l 2 r solutions when the covariance operator of the noise is hilbert schmidt in l 2 r

cox rings and combinatorics given a variety x with a finitely generated total coordinate ring we describe basic geometric properties of x in terms of certain combinatorial structures living in the divisor class group of x for example we describe the singularities we calculate the ample cone and we give simple fano criteria as we show by means of several examples the results allow explicit computations as immediate applications we obtain an effective version of the kleiman chevalley quasiprojectivity criterion and the following observation on surfaces a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non factorial singularities admit a common affine neighbourhood

dynamics of fractional stochastic reaction diffusion equations on unbounded domains driven by nonlinear noise abstract this paper is concerned with the asymptotic behavior of the solutions of the fractional reaction diffusion equations with polynomial drift terms of arbitrary order driven by locally lipschitz nonlinear diffusion terms defined on r n we first prove the well posedness of the equation based on pathwise uniform estimates as well as uniform estimates on average we then define a mean random dynamical system via the solution operators and prove the existence and uniqueness of weak pullback mean random attractors we finally establish the existence of invariant measures when the diffusion terms are globally lipschitz continuous the uniform estimates on the tails of solutions are employed to prove the tightness of a family of probability distributions of solutions in order to overcome the non compactness of usual sobolev embeddings on unbounded domains

on complementary models and fixed interval smoothing a new algorithm is derived for the standard fixed interval linear smoothing problem in which the signal is generated by a state model the structure of this new algorithm allows the smoothed estimate to be easily updated in response to a change in the initial state covariance matrix pi 0 since unlike in existing algorifiuns the relevant riccati equation is entirely independent of pi 0 the derivation of the algorithm is based on properties of complementary models