computability and the morphological complexity of some dynamics on continuous domains the partially ordered set of compact intervals provides a convenient embedding space for the analysis of some dynamical systems crucial dynamical properties are transferred to it while allowing an investigation of stability and chaoticity in terms of computability in particular in the presence of singularities we will survey some results which display the connections between the geometric complexity of the dynamics and computability issues as well as new relations between dynamic predictability and effective decidability

local model network identification with gaussian processes a bayesian gaussian process gp modeling approach has recently been introduced to model based control strategies the estimate of the variance of the predicted output is the most useful advantage of gps in comparison to neural networks nns and fuzzy models however the gp model is computationally demanding and nontransparent to reduce the computation load and increase transparency a local linear gp model network is proposed in this paper the proposed methodology combines the local model network principle with the gp prior approach a novel algorithm for structure determination and optimization is introduced which is widely applicable to the training of local model networks the modeling procedure of the local linear gp lgp model network is demonstrated on an example of a nonlinear laboratory scale process rig

a clustering technique for the identification of piecewise affine systems we propose a new technique for the identification of discrete time hybrid systems in the piecewise affine pwa form this problem can be formulated as the reconstruction of a possibly discontinuous pwa map with a multi dimensional domain in order to achieve our goal we provide an algorithm that exploits the combined use of clustering linear identification and pattern recognition techniques this allows to identify both the affine submodels and the polyhedral partition of the domain on which each submodel is valid avoiding gridding procedures moreover the clustering step used for classifying the datapoints is performed in a suitably defined feature space which allows also to reconstruct different submodels that share the same coefficients but are defined on different regions measures of confidence on the samples are introduced and exploited in order to improve the performance of both the clustering and the final linear regression procedure

brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

toric degeneration of schubert varieties and gelfand tsetlin polytopes this note constructs the flat toric degeneration of the manifold fln of flags in cn due to gonciulea and lakshmibai transform groups 1 3 1996 215 as an explicit git quotient of the grobner degeneration due to knutson and miller grobner geometry of schubert polynomials ann math 2 to appear this implies that schubert varieties degenerate to reduced unions of toric varieties associated to faces indexed by rc graphs reduced pipe dreams in the gelfand tsetlin polytope our explicit description of the toric degeneration of fln provides a simple explanation of how gelfand tsetlin decompositions for irreducible polynomial representations of gln arise via geometric quantization

adaptation and parameter estimation in systems with unstable target dynamics and nonlinear parametrization in this paper we propose a solution to the problem of adaptive control and parameter estimation in systems with unstable target dynamics models of uncertainties are allowed to be nonlinearly parameterized and required to be smooth and monotonic functions of linear functionals of the parameters the mere assumption of existence of nonlinear operator gains for the target dynamics is sufficient to guarantee that system solutions are bounded reach a neighborhood of the target set and the mismatches between the modeled uncertainties and their compensator converge to zero with respect to parameter convergence a standard persistent excitation condition suffices to ensure that it is exponential when a weaker nonlinear version of persistent excitation is satisfied asymptotic convergence is guaranteed the spectrum of possible applications ranges from tyre road slip control to asynchronous message transmission in spiking neural oscillators

constructing narmax models using armax models this paper outlines how it is possible to decompose a complex non linear modelling problem into a set of simpler linear modelling problems local armax models valid within certain operating regimes are interpolated to construct a global narmax non linear narmax model knowledge of the system behaviour in terms of operating regimes is the primary basis for building such models hence it should not be considered as a pure black box approach but as an approach that utilizes a limited amount of a priori system knowledge it is shown that a large class of non linear systems can be modelled in this way and indicated how to decompose the systems range of operation into operating regimes standard system identification algorithms can be used to identify the narmax model and several aspects of the system identification problem are discussed and illustrated by a simulation example

fuzzy model based control stability robustness and performance issues a nonlinear controller based on a fuzzy model of mimo dynamical systems is described and analyzed the fuzzy model is based on a set of arx models that are combined using a fuzzy inference mechanism the controller is a discrete time nonlinear decoupler which is analyzed both for the adaptive and the fixed parameter cases a detailed stability analysis is carried out and the main result is that the closed loop is globally stable and robust with respect to unstructured uncertainty which may include modeling error and disturbances in addition bounds on the asymptotic and transient performance are given the main assumptions on the system and model are that they must not have strong nonminimum phase effects except time delay and the unstructured uncertainty must not be too large a simulation example illustrates some of the properties of the modeling method and model based control structure

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

supervised hierarchical clustering in fuzzy model identification this paper presents a new supervised hierarchical clustering algorithm suhiclust for fuzzy model identification the presented algorithm solves the problem of global model accuracy together with the interpretability of local models as valid linearizations of the modeled nonlinear system the algorithm combines the advantages of supervised hierarchical algorithms which are based on heuristic tree construction algorithms together with the advantages of fuzzy product space clustering the high flexibility of the validity functions that is obtained by fuzzy clustering combined with supervised learning results in an efficient partitioning algorithm which is independent of initialization and results in a parsimonious fuzzy model furthermore the usability of suhiclust is very undemanding because it delivers in contrast with many other methods reproducible results in order to get reasonable results the user only has to set either a threshold for the maximum number of local models or a value for the maximum allowed global model error as a termination criterion for fine tuning the interpolation smoothness controls the degree of regularization the performance is illustrated on both analytical examples and benchmark problems from the literature

okounkov bodies and restricted volumes along very general curves abstract given a big divisor d on a normal complex projective variety x we show that the restricted volume of d along a very general complete intersection curve c x can be read off from the okounkov body of d with respect to an admissible flag containing c from this we deduce that if two big divisors d 1 and d 2 on x have the same okounkov body with respect to every admissible flag then d 1 and d 2 are numerically equivalent

computability noncomputability and undecidability of maximal intervals of ivps let c r denote the maximal interval of existence of solutions for the initial value problem dx dt f t x x t 0 x 0 where e is an open subset of r m 1 f is continuous in e and t 0 x 0 e we show that under the natural definition of computability from the point of view of applications there exist initial value problems with computable f and t 0 x 0 whose maximal interval of existence is noncomputable the fact that f may be taken to be analytic shows that this is not a lack of the regularity phenomenon moreover we get upper bounds for the degree of noncomputability by showing that is r e recursively enumerable open under very mild hypotheses we also show that the problem of determining whether the maximal interval is bounded or unbounded is in general undecidable

model structure selection by cross validation two criteria for choosing between different model structures are proposed their derivation is within a natural cross validatory assessment context and is fairly assumption free in particular the two criteria can be used for discriminating between non nested model structures and more importantly the true system is not required to belong to the considered set of models should the true system belong to the model set the two proposed criteria will asymptotically reduce to some well known structure selection criteria this is believed to be a desirable feature of our proposals on the other hand it provides a nice cross validation interpretation of some well known model structure selection rules also the cross validation interpretation helps one to choose which criteria to use in a given application the paper also has a second purpose somewhat decoupled from that mentioned above it contains an extensive survey of the literature that is useful in its own right

non computable impressions of computable external rays of quadratic polynomials we discuss computability of impressions of prime ends of compact sets in particular we construct quadratic julia sets which possess explicitly described non computable impressions

geometric lifting of the canonical basis and semitoric degenerations of richardson varieties in the sl n case a berenstein and a zelevinsky studied the schutzenberger involution in terms of lusztig s canonical basis 3 we generalize their construction and formulas for any semisimple lie algebra we use for this the geometric lifting of the canonical basis on which an analogue of the schutzenberger involution can be given as an application we construct semitoric degenerations of richardson varieties following a method of p caldero 6

normality of torus orbit closures in g p abstract the purpose of this note is to classify the torus orbit closures in an arbitrary algebraic homogeneous space g p that are toric varieties

identification of non linear system structure and parameters using regime decomposition abstract an off line algorithm for empirical modeling and identification of non linear dynamic systems is presented the minimal input to the algorithm is a sequence of empirical data and the model order using this information the algorithm searches for an optimal model structure and parameters within a rich non linear model set the model representation is based on the interpolation of a number of simple local models where each local model has a limited range of validity but the local models yield a complete global model when interpolated the method is illustrated using simulated data

nonlinear internal model control application of inverse model based fuzzy control this paper investigates the possible applications of dynamical fuzzy systems to control nonlinear plants with asymptotically stable zero dynamics using a fuzzy nonlinear internal model control strategy the developed strategy consists in including a dynamical takagi sugeno fuzzy model of the plant within the control structure in this way the controller design simply results in a fuzzy model inversion in this framework the originality of the presented work lies in the use of a dynamical fuzzy model and its inversion in order to be able to implement the control structure two crucial points have to be addressed in the considered fuzzy context on the one hand the model representation and identification on the other the model inversion as the fuzzy system can be viewed as a collection of elementary subsystems its inversion is approached here in a local way i e on the elementary subsystems capable to provide an inverse solution in this case the inversion of the global fuzzy system is thus tackled by inversion of some of its components by doing so exact inversion is obtained and offset free performances are ensured in order to guarantee a desired regulation behavior and robustness of stability of the control system the fuzzy controller is connected in series with a robustness filter the potential of the proposed method is demonstrated with simulation examples

okounkov bodies and toric degenerations let vardelta be the okounkov body of a divisor d on a projective variety x we describe a geometric criterion for vardelta to be a lattice polytope and show that in this situation x admits a flat degeneration to the corresponding toric variety this degeneration is functorial in an appropriate sense

multiplicities of points on a schubert variety in a minuscule gp in this paper we prove the results announced in 13 let g be a semi simple simply connected algebraic group defined over an algebraically closed field k let t be a maximal torus b a bore1 subgroup b 3 t let w be the weyl group of g let r resp r be the set of roots resp positive roots relative to t resp b let s be the set of simple roots in r let p be a maximal parabolic subgroup in g with associated fundamental weight w let w be the weyl group of p and wp be the set of minimal representatives of w w for w e wp let e w be the point and x w the schubert variety in g p associated to w in this paper we deter mine the multiplicity m w of x w at e z where e z e x w for all minuscule p s and also for p pgn g being of type c here pun denotes the maximal parabolic subgroup obtained by omitting a the determina tion of m w is done as follows let l be the ample generator of pic g p a basis has been constructed for x w l in terms of standard monomials on x w cf 16 11 i let u be the unipotent subgroup of g generated by u et r rp here r denotes the set of roots of p and u denotes the unipotent subgroup of g associated to tl e r then u e r gives an affme neighborhood of e z in g p let a be the affine algebra of u e z and a a where is the ideal of elements of a that vanish on x w n u e z let m be the maximal ideal in a h corresponding to e r then using the results of 16 111 we obtain a basis of m f this enables us to obtain an inductive formula for f the hilbert polynomial of x w at e t cf corollaries 3 8 and 4 11 and also express m w in terms of m w s x w s being the schubert divisors in x w such that e t e x w cf theorems 3 7 and 4 10 using this we

vanishing theorem for the cohomology of line bundles on bott samelson varieties abstract we use the grossberg karshon s degeneration of bott samelson varieties to toric varieties and the description of cohomology of line bundles on toric varieties to deduce vanishing results for the cohomology of lines bundles on bott samelson varieties

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

a fast learning algorithm for parismonious fuzzy neural systems in this paper a novel learning algorithm for dynamic fuzzy neural networks based on extended radial basis function neural networks which are functionally equivalent to takagi sugeno kang fuzzy systems is proposed the algorithm comprises 4 parts 1 criteria of rules generation 2 allocation of premise parameters 3 determination of consequent parameters and 4 pruning technology the salient characteristics of the approach are 1 a hierarchical on line self organizing learning paradigm is employed so that not only parameters can be adjusted but also the determination of structure can be self adaptive without partitioning the input space a priori 2 fast learning speed can be achieved so that the system can be implemented in real time simulation studies and comprehensive comparisons with some other learning algorithms demonstrate that the proposed algorithm is superior in terms of simplicity of structure learning efficiency and performance

generic singularities of certain schubert varieties let g be a connected semisimple algebraic group b a borel subgroup t a maximal torus in b with weyl group w and q a subgroup containing b for w in w let x wq denote the schubert variety overline bwq q for y in w such that x yq subseteq x wq one knows that byq q admits a t stable transversal in x wq which we denote by cal n yq wq we prove that under certain hypotheses cal n yq wq is isomorphic to the orbit closure of a highest weight vector in a certain weyl module we also obtain a generalisation of this result under slightly weaker hypotheses further we prove that our hypotheses are satisfied when q is a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight and x yq is an irreducible component of the boundary of x wq that is the complement of the open orbit of the stabiliser in g of x wq as a consequence we describe the singularity of x wq along byq q and obtain that the boundary of x wq equals its singular locus

computable caratheodory theory abstract the conformal riemann mapping of the unit disk onto a simply connected domain w is a central object of study in classical complex analysis the first complete proof of the riemann mapping theorem given by p koebe in 1912 is constructive and theoretical aspects of computing the riemann map have been extensively studied since caratheodory theory describes the boundary extension of the riemann map in this paper we develop its constructive version with explicit complexity bounds

multiobjective identification of takagi sugeno fuzzy models the problem of identifying the parameters of the constituent local linear models of takagi sugeno fuzzy models is considered in order to address the tradeoff between global model accuracy and interpretability of the local models as linearizations of a nonlinear system two multiobjective identification algorithms are studied particular attention is paid to the analysis of conflicts between objectives and we show that such information can be easily computed from the solution of the multiobjective optimization this information is useful to diagnose the model and tune the weighting priorities of the multiobjective optimization moreover the result of the conflict analysis can be used as a constructive tool to modify the fuzzy model structure including membership functions in order to meet the multiple objectives simple illustrative examples as well as experimental results show the usefulness of the method

nonlinear gray box identification using local models applied to industrial robots in this paper we study the problem of estimating unknown parameters in nonlinear gray box models that may be multivariable nonlinear unstable and resonant at the same time a straightforward use of time domain predication error methods for this type of problem easily ends up in a large and numerically stiff optimization problem we therefore propose an identification procedure that uses intermediate local models that allow for data compression and a less complex optimization problem the procedure is based on the estimation of the nonparametric frequency response function frf in a number of operating points the nonlinear gray box model is linearized in the same operating points resulting in parametric frfs the optimal parameters are finally obtained by minimizing the discrepancy between the nonparametric and parametric frfs the procedure is illustrated by estimating elasticity parameters in a six axis industrial robot different parameter estimators are compared and experimental results show the usefulness of the proposed identification procedure the weighted logarithmic least squares estimator achieves the best result and the identified model gives a good global description of the dynamics in the frequency range of interest for robot control

application of a general multi model approach for identification of highly nonlinear processes a case study an identification method for highly nonlinear processes is proposed based on a multi model approach and kolmogorov gabor polynomials owing to the large number of possible terms in this general model structure the significant terms are selected by several statistical test procedures leading automatically to a minimal order model realization the performance of this method is evaluated in an in depth case study using a simulated ph neutralization process the effects of important variables such as range of operating conditions signal noise ratio and data length are discussed

multivariable fuzzy inference system for fingerprinting indoor localization the emergence of wireless sensor network has raised the need for cheap wireless indoor localization technique this paper considers the problem of fingerprinting indoor localization based on signal strength measurements rss a new approach based on fuzzy logic has been put forward the proposal makes use of k nearest neighbor classification in signal space the localization of target node is then determined as a weighted combination of nearest fingerprints the weights are determined using takagi sugeno fuzzy controller with two inputs a new enhancement to the knn is proposed to enhance the accuracy of location estimation this enhancement allows the knn to outlier some miss elected neighbors based on triangular area measurements the performance of the developed estimation algorithm has been evaluated using both monte carlo simulations and real testbed scenarios while compared to other alternative approaches

rational curves on minuscule schubert varieties introduction let us denote by c the variety of lines in p3 meeting a fixed line it is a grassmannian and hence minuscule schubert variety in 16 we described the irreducible components of the scheme of morphisms from p1 to c and the general morphism in each irreducible component in this text we study the scheme of morphisms from p1 to any minuscule schubert variety x let us recall that we studied in 15 the scheme of morphisms from p1 to any homogeneous variety the main idea in the case of a minuscule schubert variety x is to restrict ourselves to the dense orbit under the stabilizer stab x of x and apply the results of 15 more precisely let u be the dense orbit under stab x in x and let y be the complement because x is a minuscule schubert variety the closed subset y of x is of codimension at least 2 see section 2 2 this fact and the stratification of x by schubert subvarieties gives us a surjective morphism see section 1 s pic u a1 x

non computable mandelbrot like sets for a one parameter complex family abstract we show the existence of computable complex numbers for which the bifurcation locus of the one parameter complex family f b z z b z 2 z 3 is not turing computable

non computable julia sets polynomial julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system apart from the beautiful mathematics one of the reasons for their popularity is the beauty of the computer generated images of such sets the algorithms used to draw these pictures vary the most naive work by iterating the center of a pixel to determine if it lies in the julia set milnor s distance estimator algorithm mil uses classical complex analysis to give a one pixel estimate of the julia set this algorithm and its modifications work quite well for many examples but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution moreover there are examples even in the family of quadratic polynomials when no satisfactory pictures of the julia set exist in this paper we study computability properties of julia sets of quadratic polynomials under the definition we use a set is computable if roughly speaking its image can be generated by a computer with an arbitrary precision under this notion of computability we show

convex bodies appearing as okounkov bodies of divisors abstract based on the work of okounkov okounkov 1996 15 2003 16 lazarsfeld and musta 2009 13 and kaveh and khovanskii preprint 10 have independently associated a convex body called the okounkov body to a big divisor on a smooth projective variety with respect to a complete flag in this paper we consider the following question what can be said about the set of convex bodies that appear as okounkov bodies we show first that the set of convex bodies appearing as okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable we then give a complete characterisation of the set of convex bodies that arise as okounkov bodies of r divisors on smooth projective surfaces such okounkov bodies are always polygons satisfying certain combinatorial criteria finally we construct two examples of non polyhedral okounkov bodies in the first one the variety we deal with is fano and the line bundle is ample in the second one we find a mori dream space variety such that under small perturbations of the flag the okounkov body remains non polyhedral

identification of non linear parametrically varying models using separable least squares in this paper a novel identification algorithm for a class of non linear possibly parameter varying models is proposed the algorithm is based on separable least squares ideas these models are given in the form of a linear fractional transformation lft where the forward part is represented by a conventional linear regression and the feedback part is given by a non linear map which can take into account scheduling variables available for measurement the non linear part of the model can be parameterized according to various paradigms like e g neural network nn or general nonlinear autoregressive exogenous narx models the estimation algorithm exploits the separability of the criterion used to estimate the parameters when using a nn it is possible the explicit computation of the frechet derivative needed to implement a separable least square algorithm

inference of disjoint linear and nonlinear sub domains of a nonlinear mapping this paper investigates new ways of inferring nonlinear dependence from measured data the existence of unique linear and nonlinear sub spaces which are structural invariants of general nonlinear mappings is established and necessary and sufficient conditions determining these sub spaces are derived the importance of these invariants in an identification context is that they provide a tractable framework for minimising the dimensionality of the nonlinear modelling task specifically once the linear nonlinear sub spaces are known by definition the explanatory variables may be transformed to form two disjoint sub sets spanning respectively the linear and nonlinear sub spaces the nonlinear modelling task is confined to the latter sub set which will typically have a smaller number of elements than the original set of explanatory variables constructive algorithms are proposed for inferring the linear and nonlinear sub spaces from noisy data

julia sets in parameter spaces given a complex number of modulus 1 we show that the bifurcation locus of the one parameter family fb z z bz2 z3 b contains quasi conformal copies of the quadratic julia set j z z2 as a corollary we show that when the julia set j z z2 is not locally connected for example when z z z2 has a cremer point at 0 the bifurcation locus is not locally connected to our knowledge this is the first example of complex analytic parameter space of dimension 1 with connected but non locally connected bifurcation locus we also show that the set of complex numbers of modulus 1 for which at least one of the parameter rays has a non trivial accumulation set contains a dense g subset of s1

asmo dan algorithm for adaptive spline modelling of observation data nonlinear system identification by modelling the underlying relationships in observation data is an important application area for artificial neural networks and other learning paradigms splines have been used for scattered data interpolation but the applications have mainly been restricted to low dimensional input spaces this paper describes asmod a new learning paradigm for higher dimensional data 3 based on b spline interpolation the models can be trained online and a method for step wise model refinement is applied during model training for gradually increasing the modelling capability until the desired or best possible accuracy is obtained for every refinement step a number of possible refinement actions are evaluated and the one that gives the highest improvement of the model accuracy is chosen the model structure is hence adapted to the modelling problem giving a model of small size and high accuracy asmod has very efficient implementations on serial computers the scheme has been

an effective riemann mapping theorem the main results of the paper are two effective versions of the riemann mapping theorem the first uniform version is based on the constructive proof of the riemann mapping theorem by cheng and by bishop and bridges and formulated in the computability framework developed by kreitz and weihrauch it states which topological information precisely one needs about a nonempty proper open connected and simply connected subset of the complex plane in order to compute a description of a holomorphic bijection from the unit disk onto this set and vice versa which topological information about the set can be obtained from a description of a holomorphic bijection the second version which is derived from the first by considering the sets and the functions with computable descriptions characterizes the subsets of the complex plane for which there exists a computable holomorphic bijection from the unit disk this gives a partial answer to a problem posed by pour el and richards we also show that this class of sets is strictly larger than a class of sets considered by zhou which solves an open problem posed by him in preparation recursively enumerable open subsets and closed subsets of euclidean spaces are considered and several effective results in complex analysis are proved

genetic programming for model selection of tsk fuzzy systems this paper compares a genetic programming gp approach with a greedy partition algorithm lolimot for structure identification of local linear neuro fuzzy models the crisp linear conclusion part of a takagi sugeno kang tsk fuzzy rule describes the underlying model in the local region specified in the premise the objective of structure identification is to identify an optimal partition of the input space into gaussian axis orthogonal fuzzy sets the linear parameters in the rule consequent are then estimated by means of a local weighted least squares algorithm lolimot is an incremental tree construction algorithm that partitions the input space by axis orthogonal splits in each iteration it greedily adds the new model that minimizes the classification error gp performs a global search for the optimal partition tree and is therefore able to backtrack in case of sub optimal intermediate split decisions we compare the performance of both methods for function approximation of a highly non linear two dimensional test function and an engine characteristic map

construction of fuzzy models through clustering techniques abstract in this note we will discuss fuzzy models designed with the aid of fuzzy clustering the clustering method utilizes a generalized objective function involving a collection of linear varieties in this way the model is distributed and consists of a series of local linear type models based on the revealed clusters in comparison to the algorithms existing in the literature and producing function like models the proposed one is of a relational character allowing for multidirectional accessibility a detailed identification algorithm is given and illustrated with the aid of numerical examples