symmetries of the turbulent state the emphasis of this review is on fundamental properties degree of universality and symmetries of the turbulent state the central questions are which symmetries remain broken even when the symmetry breaking factor reaches zero and which symmetries in contrast emerge in the state of developed turbulence we shall see that time reversibility is broken in all cases since turbulence is a far from equilibrium state accompanied by dissipation as far as scale invariance is concerned we argue that it is always broken in direct cascades toward small scales no matter how far one goes away from the pumping scale in contrast inverse cascades become scale invariant as they go toward large scales moreover some properties of the inverse cascades seem to be conformal invariant and related to schramm loewner evolution a class of random curves that can be mapped to a 1d brownian walk

a blind approach to hammerstein model identification this paper discusses the hammerstein model identification using a blind approach by fast sampling at the output it is shown that identification of the linear part can be achieved based only on the output measurements that makes the hammerstein model identification possible without knowing the structure of the nonlinearity and the internal variables

the loewner equation maps and shapes an approach called schramm loewner evolution sle provides a new method for dealing with a wide variety of scale invariant problems in two dimensions this approach is based upon an older method called loewner evolution le which connects analytic and geometrical constructions in the complex plane in this paper the bases of le and sle are described and some simple applications are discussed in relatively non technical form a bibliography of the subject is presented

residual bounds of approximate solutions of the discrete time algebraic riccati equation let tilde x be a hermitian matrix which approximates the unique hermitian positive semi definite solution x to the discrete time algebraic riccati equation dare where f in cal c n times n c 2 in cal c m times m is hermitian positive definite g 1 in cal c n times m c in cal c r times n the pair f g 1 is stabilizable and the pair c f is detectable assume that i g tilde x is nonsingular and i g tilde x 1 f is stable let g g 1g 2 1 g 1 rm h h c rm h c and let be the residual of the dare with respect to tilde x define the linear operator vec l by the main result of this paper is if where denotes any unitarily invariant norm and then

on approximation of stable linear dynamical systems using laguerre and kautz functions approximation of stable linear dynamical systems by means of so called laguerre and kautz functions which are the laplace transforms of a class of orthonormal exponentials is studied since the impulse response of a stable finite dimensional linear dynamical system can be represented by a sum of exponentials times polynomials it seems reasonable to use basis functions of the same type assuming that the transfer function of a system is bounded and analytic outside a given disc it is shown that laguerre basis functions are optimal in a mini max sense this result is extended to the two parameter kautz functions which can have complex poles while the poles of laguerre functions are restricted to the real axis by conformai mapping techniques the two parameter kautz approximation problem is recast as two laguerre approximation problems thus the well developed theory of laguerre functions can be applied to analyze kautz approximations unilateral shifts are used to further develop the connection between laguerre functions and kautz functions results on 2 and approximation using kautz models are given furthermore the weighted l2 kautz approximation problem is shown to be equivalent to solving a block toeplitz matrix equation

dual mode predictive control algorithm for constrained hammerstein systems in the process industry there exist many systems which can be approximated by a hammerstein model moreover these systems are usually subjected to input magnitude constraints in this paper a multi channel identification algorithm mcia is proposed in which the coefficient parameters are identified by least squares estimation lse together with a singular value decomposition svd technique compared with traditional single channel identification algorithms the present method can enhance the approximation accuracy remarkably and provide consistent estimates even in the presence of coloured output noises under relatively weak assumptions on the persistent excitation pe condition of the inputs then to facilitate the following controller design this mcia is converted into a two stage single channel identification algorithm ts scia which preserves most of the advantages of mcia with this ts scia as the inner model a dual mode non linear model predictive control nmpc algorithm is developed in detail over a finite horizon an optimal input profile found by solving a open loop optimal control problem drives the non linear system state into the terminal invariant set afterwards a linear output feedback controller steers the state to the origin asymptotically in contrast to the traditional algorithms the present method has a maximal stable region a better steady state performance and a lower computational complexity finally simulation results on a heat exchanger are presented to show the efficiency of both the identification and the control algorithms

finger growth and selection in a poisson field solutions are found for the growth of infinitesimally thin two dimensional fingers governed by poisson s equation in a long strip the analytical results determine the asymptotic paths selected by the fingers which compare well with the recent numerical results of cohen and rothman j stat phys 167 703 712 2017 for the case of two and three fingers the generalisation of the method to an arbitrary number of fingers is presented and further results for four finger evolution given the relation to the analogous problem of finger growth in a laplacian field is also discussed

application of the schwarz christoffel map to the laplacian growth of needles and fingers a numerical procedure based on the schwarz christoffel map suitable for the study of the laplacian growth of thin two dimensional protrusions is presented the protrusions take the form of either straight needles or curved fingers satisfying loewner s equation and are represented by slits in the complex plane particular use is made of driscoll s numerical procedure the sc toolbox for computing the schwarz christoffel map from a half plane to a slit half plane since the schwarz christoffel map applies only to polygonal regions the growth of curved fingers is approximated by an increasing number of short straight line segments the growth rate is given by a fixed power of the harmonic measure at the finger or needle tips and so includes the possibility of screening as the needles of fingers interact with themselves and with boundaries the method is illustrated with examples of multiple needle and finger growth in half plane and channel geometries the effect of on the trajectories of asymmetric bifurcating fingers is also studied

laplacian growth as one dimensional turbulence abstract a new model of laplacian stochastic growth is formulated using conformal mappings the model describes two growth regimes stable and turbulent separated by a sharp phase transition the first few fourier components of the mapping define the web an envelope of the cluster the web is used to study the transition and the dynamics of large scale features of the cluster characterized by evolution from macro to micro scales also we derive scaling laws for the cluster size

nonlinearity estimation in hammerstein systems based on ordered observations the nonlinear subsystem of a hammerstein system is identified i e its characteristic is recovered from input output ohservations of the whole system the input and disturbance are white stochastic processes the identified characteristic satisfies a piecewise lipschitz condition only algorithms presented in the paper are calculated from ordered input output observations i e from pairs of observations arranged in a sequence in which input measurements increase in value the mean integrated square error converges to zero as the number of observations tends to infinity convergence rates are insensitive to the shape of the probability density of the input signal results of numerical simulation are also shown

path selection in a poisson field a criterion for path selection for channels growing in a poisson field is presented we invoke a generalization of the principle of local symmetry we then use this criterion to grow channels in a confined geometry the channel trajectories reveal a self similar shape as they reach steady state analyzing their paths we identify a cause for branching that may result in a ramified structure in which the golden ratio appears

scaling of the discrete time algebraic riccati equation to enhance stability of the schur solution method a simple scaling procedure for discrete time riccati equations is introduced this procedure eliminates instabilities which can be associated with the linear equation solution step of the generalized schur method without changing the condition of the underlying problem a computable bound for the relative error of the solution of the riccati equation is also derived

brief paper perturbation analysis and condition numbers of symmetric algebraic riccati equations this paper is devoted to the perturbation analysis of symmetric algebraic riccati equations based on our perturbation analysis the upper bounds for the normwise mixed and componentwise condition numbers are presented the results are demonstrated by our preliminary numerical experiments

backward error for the discrete time algebraic riccati equation the normwise backward error of an approximate solution to the discrete time algebraic riccati equation is evaluated the results are illustrated by using simple numerical examples c elsevier sci

combined parametric nonparametric identification of hammerstein systems a novel parametric nonparametric methodology for hammerstein system identification is proposed assuming random input and correlated output noise the parameters of a nonlinear static characteristic and finite impulse response system dynamics are estimated separately each in two stages first the inner signal is recovered by a nonparametric regression function estimation method stage 1 and then system parameters are solved independently by the least squares stage 2 convergence properties of the scheme are established and rates of convergence are given

an optimal two stage identification algorithm for hammerstein wiener nonlinear systems abstract an optimal two stage identification algorithm is presented for hammerstein wiener systems where two static nonlinear elements surround a linear block the proposed algorithm consists of two steps the first one is the recursive least squares and the second one is the singular value decomposition of two matrices whose dimensions are fixed and do not increase as the number of the data point increases moreover the algorithm is shown to be convergent in the absence of noise and convergent with probability one in the presence of white noise

nonlinear black box models in system identification mathematical foundations we discuss several aspects of the mathematical foundations of the nonlinear black box identification problem we shall see that the quality of the identification procedure is always a result of a certain trade off between the expressive power of the model we try to identify the larger the number of parameters used to describe the model the more flexible is the approximation and the stochastic error which is proportional to the number of parameters a consequence of this trade off is the simple fact that a good approximation technique can be the basis of a good identification algorithm from this point of view we consider different approximation methods and pay special attention to spatially adaptive approximants we introduce wavelet and neuron approximations and show that they are spatially adaptive then we apply the acquired approximation experience to estimation problems finally we consider some implications of these theoretical developments for the practically implemented versions of the spatially adaptive algorithms

a generalized orthonormal basis for linear dynamical systems in many areas of signal system and control theory orthogonal functions play an important role in issues of analysis and design in this paper it is shown that there exist orthogonal functions that in a natural way are generated by stable linear dynamical systems and that compose an orthonormal basis for the signal space l sub 2sup n to this end use is made of balanced realizations of inner transfer functions the orthogonal functions can be considered as generalizations of for example the pulse functions laguerre functions and kautz functions and give rise to an alternative series expansion of rational transfer functions it is shown how we can exploit these generalized basis functions to increase the speed of convergence in a series expansion i e to obtain a good approximation by retaining only a finite number of expansion coefficients consequences for identification of expansion coefficients are analyzed and a bound is formulated on the error that is made when approximating a system by a finite number of expansion coefficients

needle crystal formation in two dimensions a simple model based on conformal mapping is constructed to simulate the two dimensional growth of needle crystals growing out regularly from a common point the resulting patterns are very similar to those which were obtained by other methods the radius of gyration exponent beta is measured and found to be in the range 0 58 0 65 independent of the number of needles n it is shown that for large patterns beta asymptotically approaches a value of 2 3 to compare the patterns with the dla aggregates the average slope alpha was measured for the density density correlation function plotted logarithmically log r to log c r the relation between alpha and beta which is characteristic of dla fractal aggregates is satisfied only in the case n 8

stochastic approximation in nonparametric identification of hammerstein systems derived from the idea of stochastic approximation recursive algorithms to identify a hammerstein system are presented two of them recover the characteristic of the nonlinear memoryless subsystem while the third one estimates the impulse response of the linear dynamic part the a priori information about both subsystems is nonparametric consistency in quadratic mean is shown and the convergence rate is examined results of numerical simulation are also presented

nonparametric identification of hammerstein systems a discrete time nonlinear hammerstein system is identified and the correlation and frequency domain methods for identification of its linear subsystem are presented the main results concern the estimation of the nonlinear memoryless subsystem no conditions concerning the functional form of the transform characteristic of the subsystem are made and an algorithm for estimation of the characteristic is given the algorithm is simply a nonparametric kernel estimate of the regression function calculated from dependent data it is shown that the algorithm converges to the characteristic of the subsystem regardless of the probability distribution of the input variable pointwise as well as global consistencies are established for lipschitz characteristics the rate of the convergence in probability is o n sup 1 3

higher conformal multifractality we derive from conformal invariance and quantum gravity the multifractal spectrum f of the harmonic measure i e electrostatic potential or diffusion field near any conformally invariant fractal in two dimensions it gives the hausdorff dimension of the set of points where the potential varies with distance r to the fractal frontier as r first examples are a random walk i e a brownian motion a self avoiding walk or a critical percolation cluster the generalized dimensions d n as well as the multifractal functions f are derived and are all identical for these three cases the external frontiers of a brownian motion and of a percolation cluster are thus identical to a self avoiding walk in the scaling limit the multifractal mf function f c of the electrostatic potential near any conformally invariant fractal boundary like a critical o n loop or a q state potts cluster is given as a function of the central charge c of the associated conformal field theory the dimensions dep of the external perimeter and dh of the hull of a critical scaling curve or cluster obey the superuniversal duality equation d text ep 1 d text d 1 tfrac 1 4 finally for a conformally invariant scaling curve which is simple i e without double points we derive higher multifractal functions like the universal function f2 which gives the hausdorff dimension of the points where the potential varies jointly with distance r as r on one side of the curve and as r on the other the general case of the potential distribution between the branches of a star made of an arbitrary number of scaling paths is also treated the results apply to critical o n loops potts clusters and to the sle process we present a duality between external perimeters of potts clusters and o n loops at their critical point as well as the corresponding duality in the sle process for 16

laplacian growth with separately controlled noise and anisotropy conformal mapping models are used to study the competition of noise and anisotropy in laplacian growth for this purpose a family of models is introduced with the noise level and directional anisotropy controlled independently fractalization is observed in both anisotropic growth and growth with varying noise the fractal dimension is determined from the cluster size scaling with cluster area for isotropic growth d 1 7 at both high and low noise for anisotropic growth with reduced noise the dimension can be as low as d 1 5 and apparently is not universal also we study the fluctuations of particle areas and observe in agreement with previous studies that exceptionally large particles may appear during growth leading to pathologically irregular clusters this difficulty is circumvented by using an acceptance window for particle areas

diffusion limited aggregation and its response to anisotropy the stability of diffusion limited growths with n equivalent major fingers is investigated in two dimensions using a conformal mapping the results imply that square lattice but not hexagonal lattice bias are relevant in simple diffusion limited aggregation dla in general it is found that the maximum number of fingers stable with respect to finger loss by competition is given by nmax 2 2 d 1 where d is the apparent fractal dimension of the fingers at least when n is even computer simulations of dla with n fold symmetric growth rules are shown for n 5 8 12 finally it is argued that unbiased isotropic fractal dla should lie at the stability limit when combined with cone angle arguments independently relating d to the effective numbers of fingers this determines the value d 1 12 2 1 707 for dla in two dimensions this corresponds to a characteristic exterior half angle at the leading tips given by 2

chordal loewner families and univalent cauchy transforms we study chordal loewner families in the upper half plane and show that they have a parametric representation we show one that to every chordal loewner family there corresponds a unique measurable family of probability measures on the real line and two that to every measurable family of probability measures on the real line there corresponds a unique chordal loewner family in both cases the correspondence is being given by solving the chordal loewner equation we use this to show that any probability measure on the real line with finite variance and mean zero has univalent cauchy transform if and only if it belongs to some chordal loewner family if the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the cauchy transform the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal cauchy transform and the second in terms of moment inequalities corresponding to the grunsky inequalities

technical communique decentralized adaptive control of linear interconnected systems based on laguerre series representation this paper investigates the problem of completely decentralized adaptive control of large scale systems when the subsystems are stable a new approach which eliminates the assumptions about the interconnection terms the time delay the order etc is proposed each subsystem is modeled by an orthonormal laguerre network put in state space form a simple decentralized predictive control law is proposed the potential of such a scheme is illustrated by simulation results

residual bounds of approximate solutions of the algebraic riccati equation let tilde x geq 0 approximate the unique hermitian positive semi definite solution x to the algebraic riccati equation are where f g geq 0 a f is stabilizable and a g is detectable let be the residual of the are with respect to tilde x and define the linear operator vec l by by applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution tilde x we obtained the following result if a f tilde x is stable and if 4 vec l 1 vec l 1 hat r f 1 for any unitarily invariant norm then

sensitivity analysis of the discrete time algebraic riccati equation abstract consider the discrete time algebraic riccati equation dare a t xa x a t xb s r b t xb 1 b t xa s t q 0 where a r n n b s r n m r rt r m m q qt r n n the available perturbation theory for the dare can only be applied to the case r 0 however in some control problems the matrix r can be singular in this paper we study perturbation properties of the dare without the restriction r 0 perturbation bounds and a relative condition number for the stabilizing solution of the dare are derived computable residual bounds for an approximate solution are also derived the theoretical results are illustrated by numerical examples

fingered growth in channel geometry a loewner equation approach a simple model of laplacian growth is considered in which the growth takes place only at the tips of long thin fingers following carleson and makarov l carleson and n makarov j anal math 87 103 2002 the evolution of the fingers is studied with use of the deterministic loewner equation the method is then extended to study the growth in a linear channel with reflecting sidewalls one and two finger solutions are found and analyzed it turns out that the presence of the walls has a significant influence on the shapes of the fingers and the dynamics of the screening process in which longer fingers suppress the growth of the shorter ones

behaviour of the harmonic measure at the bottom of fjords diffusion limited aggregates are among many important fractal shapes that involve deep indentations usually called fjords to estimate the harmonic measure at the bottom of a fjord seems a prohibitive task but the authors find that a new mathematical equality due to beurling carleson and jones makes it easy they find that the harmonic measure at the bottom of a fjord as a function of its euclidean depth can exhibit a wide range of behaviours they introduce an infinite family of model fjords for which the equality takes a very simple form in this family the decay of the harmonic measure at their bottoms can be for example power law semi exponential stretched exponential and exponentially stretched exponential they show that self affinity or randomness can lead to faster than power law decays of the minimal growth probability on boundaries

semi global exponential stabilization of linear systems subject to input saturation via linear feedbacks abstract it is known that a linear time invariant system subject to input saturation can be globally asymptotically stabilized if it has no eigenvalues with positive real parts it is also shown by fuller 1997 and sussmann and yang 1991 that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws in this paper we show the existence of linear state feedback and or output feedback control laws for semi global exponential stabilization rather than global asymptotic stabilization of such systems we explicitly construct linear static state feedback laws and or linear dynamic output feedback laws that semi globally exponentially stabilize the given system our results complement the negative result of fuller 1977 and sussmann and yang 1991

a quasi infinite horizon nonlinear model predictive control scheme with guaranteed stability we present in this paper a novel nonlinear model predictive control scheme that guarantees asymptotic closed loop stability the scheme can be applied to both stable and unstable systems with input constraints the objective functional to be minimized consists of an integral square error ise part over a finite time horizon plus a quadratic terminal cost the terminal state penalty matrix of the terminal cost term has to be chosen as the solution of an appropriate lyapunov equation furthermore the setup includes a terminal inequality constraint that forces the states at the end of the finite prediction horizon to lie within a prescribed terminal region if the jacobian linearization of the nonlinear system to be controlled is stabilizable we prove that feasibility of the open loop optimal control problem at time t 0 implies asymptotic stability of the closed loop system the size of the region of attraction is only restricted by the requirement for feasibility of the optimization problem due to the input and terminal inequality constraints and is thus maximal in some sense

crossing formulae for critical percolation in an annulus an exact formula is given for the probability that there exists a spanning cluster between opposite boundaries of an annulus in the scaling limit of critical percolation the entire distribution function for the number of distinct spanning clusters is also derived these results are found using coulomb gas methods their forms are compared with the expectations of conformal field theory

sensitivity of the stable discrete time lyapunov equation the sensitivity of the stable discrete time lyapunov equation is analyzed through the spectral norm of the inverse lyapunov operator this leads to a directly computable easy to interpret sensitivity measure which involves only the open loop state matrix and also provides insight into the connection between sensitivity stability radius and conditioning of the eigenproblem of the open loop state matrix these results are an extension to the discrete time case of analogous results for the continuous time lyapunov equation

singular laplacian growth the general equations of motion for two dimensional laplacian growth are derived using the conformal mapping method in the singular case all singularities of the conformal map are on the unit circle and the map is a degenerate schwarz christoffel map the equations of motion describe the motions of these singularities despite the typical fractal like outcomes of laplacian growth processes the equations of motion are shown to be not particularly sensitive to initial conditions it is argued that the sensitivity of this system derives from a novel cause the non uniqueness of solutions to the differential system by a mechanism of singularity creation every solution can become more complex even in the absence of noise without violating the growth law these processes are permitted but are not required meaning the equation of motion does not determine the motion even in the small

monte carlo study of linear diffusion limited aggregation linear diffusion limited aggregation is studied on the square lattice the radius of gyration exponent nu 0 79 or 0 01 is estimated using the monte carlo method and a scaling analysis

model based predictive control for hammerstein wiener systems in this paper a model based predictive control mpc algorithm is presented for hammerstein wiener systems this type of system consists of a linear dynamic block preceded and followed by a static non linear block these systems appear useful in modelling several non linear processes encountered in industry directly using such a model in a mpc algorithm will in general lead to a non linear optimization problem due to the static non linearities this can be avoided by exploiting the structure of these models in this paper the non linearities are transformed into polytopic descriptions this procedure enables one to use robust linear mpc techniques for controlling these systems in such a way a convex optimization problem is retained for the presented mpc algorithm which is stated as an optimization problem subject to linear matrix inequalities nominal closed loop stability is proven in two examples it is shown that by means of transforming the non linearities into polytopic descriptions as done in t

online optimization of the time scale in adaptive laguerre based filters a new online method to optimize the free parameter in adaptive laguerre based filters is presented it is based on the minimization of a criterion that is equivalent to an upper bound for the quadratic approximation error the proposed technique presents a fast convergence and a good robustness

an optimum time scale for discrete laguerre network any discrete time stable transfer function can be expressed by a discrete time laguerre series with a chosen time scale an optimum time scale such that an index is minimized is derived this index ensures that the coefficients of higher order laguerre functions go toward zero quickly the solution derived requires the knowledge of the impulse response of the discrete plant cases of first order plants second order underdamped plants and plants with multiunit delay are also discussed

efficient robust predictive control predictive constrained control of time varying and or uncertain linear systems has been effected through the use of ellipsoidal invariant sets kothare et al 1996 linear matrix inequalities lmis have been used to design a state dependent state feedback law that maintains the state vector inside invariant feasible sets for the purposes of prediction however at each time instant the state feedback law is assumed constant in addition due to the large number of lmis involved online computation becomes intractable for anything other than small dimensional systems here we propose an approach that deploys a fixed state feedback law but introduces extra degrees of freedom through the use of perturbations on the fixed state feedback law the problem is so formulated that all demanding computations can be performed offline leaving only a simple optimization problem to be solved online over and above the very significant reduction in computational cost the extra degrees of freedom allow for better performance and wider applicability

nonlinear control of hammerstein systems with passive nonlinear dynamics a nonlinear dynamic compensator framework for hammerstein systems with passive nonlinear dynamics is proposed for this class of systems controlled by passive nonlinear dynamic compensators we prove the global closed loop stability by modifying the dynamic compensator to include a suitable input nonlinearity the proof of this result is based on the dissipativity theory and shows that the nonlinear controller modification counteracts the effects of the input nonlinearity by recovering the passivity of the plant and the compensator

laplacian networks growth local symmetry and shape optimization inspired by river networks and other structures formed by laplacian growth we use the loewner equation to investigate the growth of a network of thin fingers in a diffusion field we first review previous contributions to illustrate how this formalism reduces the network s expansion to three rules which respectively govern the velocity the direction and the nucleation of its growing branches this framework allows us to establish the mathematical equivalence between three formulations of the direction rule namely geodesic growth growth that maintains local symmetry and growth that maximizes flux into tips for a given amount of growth surprisingly we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips

stabilizing effect of tip splitting on the interface motion pattern forming processes such as electrodeposition dielectric breakdown or viscous fingering are often driven by instabilities accordingly the resulting growth patterns are usually highly branched fractal structures however in some of the unstable growth processes the envelope of the structure grows in a highly regular manner with the perturbations smoothed out over the course of time in this paper we show that the regularity of the envelope growth can be connected to small scale instabilities leading to the tip splitting of the fingers at the advancing front of the structure whenever the growth velocity becomes too large the finger splits into two branches in this way it can absorb an increased flux and thus damp the instability hence somewhat counterintuitively the instability at a small scale results in a stability at a larger scale the quantitative analysis of these effects is provided by means of the loewner equation which one can use to reduce the problem of the interface motion to that of the evolution of the conformal mapping onto the complex plane this allows an effective analysis of the multifingered growth in a variety of different geometries we show how the geometry impacts the shape of the envelope of the growing pattern and compare the results with those observed in natural systems

dynamics of finger arrays in a diffusion limited growth model with a drift abstract we propose a phenomenological model of fingering dynamics in the presence of an external drift motivated by recent experiments on quasi two dimensional electrodeposition with hydrodynamic convection we study the dynamical transition between a laplacian growth regime and a weak finger competition regime the model proposed defines a wavelength selection problem of the finger array which is coupled in a nontrivial way to the single finger selection problem in a channel the wavelength selection mechanism proposed is associated to the crossover from a diffusive to a ballistic behavior of the aggregating particles as the coarsening spatial scale of the pattern matches the appropriate diffusion length this yields specific predictions for the selected wavelength the dynamics of the model are studied numerically with boundary integral methods in the quasistatic approximation and complementarily using a biased random walk monte carlo scheme on a lattice results are in qualitative agreement with the theoretical predictions

fingering in a channel and tripolar loewner evolutions a class of laplacian growth models in the channel geometry is studied using the formalism of tripolar loewner evolutions in which three points namely the channel corners and the point at infinity are kept fixed initially the problem of fingered growth where growth takes place only at the tips of slitlike fingers is revisited and a class of exact solutions of the corresponding loewner equation is presented for the case of stationary driving functions a model for interface growth is then formulated in terms of a generalized tripolar loewner equation and several examples are presented it is shown that the growing interface evolves into a steadily moving finger and that tip competition arises for nonsymmetric initial configurations with multiple tips

optimum choice of free parameter in orthonormal approximations this correspondence investigates the choice of a free parameter usually related to time scale that minimizes the error energy when approximating a given signal with some widely used orthonormal basis functions the proposed solution is the best that can be achieved with the limited required knowledge of the signal it is appealing for experimental data for which no exact mathematical expression is available

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

interface growth in two dimensions a loewner equation approach the problem of laplacian growth in two dimensions is considered within the loewner equation framework initially the problem of fingered growth is revisited and an exact solution for a three finger configuration is reported then a general class of growth models for an interface growing in the upper half plane is introduced and the corresponding loewner equation for the problem is derived several examples are given including interfaces with one or more tips as well as multiple growing interfaces a generalization of our interface growth model in terms of loewner domains where the growth rule is specified by a time evolving measure is briefly discussed