a fast algorithm for subspace state space system identification via exploitation of the displacement structure two recent approaches van overschee de moor n4sid automatica 30 1 1994 75 verhaegen int j control 58 3 1993 555 in subspace identification problems require the computation of the r factor of the qr factorization of a block hankel matrix h which in general has a huge number of rows since the data are perturbed by noise the involved matrix h is in general full rank it is well known that from a theoretical point of view the r factor of the pr factorization of h is equivalent to the cholesky factor of the correlation matrix hth apart from a multiplication by a sign matrix in sima proceedings second niconet workshop paris versailles december 3 1999 p 75 a fast cholesky factorization of the correlation matrix exploiting the block hankel structure of h is described in this paper we consider a fast algorithm to compute the r factor based on the generalized schur algorithm the proposed algorithm allows to handle the rank deficient case c 2001 elsevier science b v all rights reserved

the magnetization energy scaling limit in high dimension in the single phase region including the critical point of a nearest neighbor ising ferromagnet with zero external field the block magnetization and energy within the infinite volume system are asymptotically for large block size independent gaussian variables when the dimensiond exceeds four for other models including ones with long range interactions a sufficient condition for such triviality of the scaling limit is finiteness of the bubble quantity

correlation inequalities for the truncated two point function of an ising ferromagnet we establish the following new correlation inequalities for the truncated twopoint function of an ising ferromagnet in a positive external field j l t j k t k l t and j l t k k j k t k l wherek is any set of sites which separatesj froml the inequalities are also valid for the pure phases with zero magnetic field at all temperatures above the critical temperature they reduce to known inequalities of griffiths and simon respectively

brief stability of extremum seeking feedback for general nonlinear dynamic systems while the mainstream methods of adaptive control both linear and nonlinear deal only with regulation to known set points or reference trajectories in many applications the set point should be selected to achieve a maximum of an uncertain reference to output equilibrium map the techniques of the so called extremum control or self optimizing control developed for this problem in the 1950 1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment in this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block we allow the plant to be a general nonlinear dynamic system possibly non affine in control and open loop unstable whose reference to output equilibrium map has a maximum and whose equilibria are locally exponentially stabilizable

fast extremum seeking for wiener hammerstein plants this paper considers an observer based extremum seeking es scheme acting on a wiener hammerstein wh plant the es scheme utilises a high frequency sinusoidal dither in order to achieve fast minimisation of the plant s static nonlinearity unlike the prevailing analyses of fast es schemes acting on wh plants the presented result is semi global and applicable to plants with fairly general unknown nonlinearities design of the es scheme requires knowledge of the relative orders of both the input and output dynamics of the plant two different tunings of the es parameters are considered one which forces the es dynamics to be fast and one which is less restrictive it is shown that approximate knowledge of the phase of the plant s input dynamics is required if the es scheme is to use the less restrictive tuning for both tunings it is shown how the es scheme may be designed in order to simultaneously achieve arbitrarily fast and accurate minimisation of the static nonlinearity from an arbitrarily large set of initial conditions moreover the result shows that a wh plant with unstable input dynamics can in fact be stabilised by the es scheme

sensitivity analysis for szegő polynomials szeg polynomials are orthogonal with respect to an inner product on the unit circle numerical methods for weighted least squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of szeg polynomials this paper discusses the conditioning of several mappings involving szeg polynomials and thereby sheds light on the sensitivity of some approximation problems involving trigonometric polynomials

a fast hankel solver based on an inversion formula for loewner matrices abstract we propose a new o n2 algorithm for solving complex n n linear systems that have hankel structure via ffts the hankel system is transformed into a loewner system an inversion formula enables us to calculate the inverse of the loewner matrix explicitely the parameters that occur in this inversion formula are calculated by solving two rational interpolation problems on the unit circle we present an o n2 algorithm to solve these interpolation problems one of the advantages of this algorithm is that it incorporates pivoting we have implemented our hankel solver in fortran 90 numerical examples are included they show the effectiveness of our pivoting strategy

look ahead levinson and schur algorithms for non hermitian toeplitz systems we present generalizations of the nonsymmetric levinson and schur algorithms for non hermitian toeplitz matrices with some singular or ill conditioned leading principal submatrices the underlying recurrences allow us to go from any pair of successive well conditioned leading principal submatrices to any such pair of larger order if the look ahead step size between these pairs is bounded our generalized levinson and schur recurrences require o n 2 operations and the schur recurrences can be combined with recursive doubling so that an o n log 2 n algorithm results the overhead in operations and storage of look ahead steps is very small there are various options for applying these algorithms to solving linear systems with toeplitz matrix

formally biorthogonal polynomials and a look ahead levinson algorithm for general teoplitz systems abstract systems of linear equations with toeplitz coefficient matrices arise in many important applications the classical levinson algorithm computes solutions of toeplitz systems with only o n 2 arithmetic operations as compared to o n 3 operations that are needed for solving general linear systems however the levinson algorithm in its original form requires that all leading principal submatrices be nonsingular in this paper an extension of the levinson algorithm to general toeplitz systems is presented the algorithm uses look ahead to skip over exactly singular as well as ill conditioned leading submatrices and at the same time it still fully exploits the toeplitz structure in our derivation of this algorithm we make use of the intimate connection of toeplitz matrices with formally biorthogonal polynomials in particular the occurence of singular or ill conditioned submatrices corresponds to a breakdown or near breakdown in the standard recurrence relations for biorthogonal polynomials we present new general recurrence relations that connect successive pairs in any given subsequence of all existing formally biorthogonal polynomials these recurrences then immediately lead to the proposed look ahead levinson algorithm for solving toeplitz systems implementation details for this algorithm and operation counts are given numerical experiments for toeplitz systems with ill conditioned submatrices are reported

more inequalities for critical exponents a variety of rigorous inequalities for critical exponents is proved most notable is the low temperature josephson inequalitydv 2 2 others are 1 1 v 1 1 1 d 1 1 for d dv 3 for d 4 and 2m 2m 2 form 2 the hypotheses vary all inequalities are true for the spin 1 2 ising model with nearest neighbor ferromagnetic pair interactions

monte carlo test of a hyperscaling relation for the two dimensional self avoiding walk the authors simulated self avoiding walks on the square lattice with fixed endpoints by means of a dynamic monte carlo algorithm from these data they obtain an evaluation of the effective coordination number mu and the critical exponents alpha and v they can therefore test the hyperscaling relation 2 alpha dv with a careful estimate of systematic and statistical errors

a stabilized superfast solver for indefinite hankel systems abstract we present a stabilized superfast solver for indefinite hankel systems whose size is a power of 2 the hankel system is transformed into a loewner system which is solved by using an inversion formula for loewner matrices this explicit formula for the inverse of a loewner matrix contains certain parameters that are computed by solving two linearized rational interpolation problems on the unit circle the heart of our hankel solver is a superfast algorithm to solve these interpolation problems this algorithm is stabilized via pivoting iterative improvement and by giving the so called difficult interpolation points an adequate treatment we have implemented our algorithm in fortran 90 numerical examples illustrate the effectiveness of our approach

stochastic averaging in continuous time and its applications to extremum seeking we investigate stochastic averaging theory in continuous time for locally lipschitz systems and the applications of this theory to stability analysis of stochastic extremum seeking algorithms first we establish a general stochastic averaging principle and some related stability theorems for a class of continuous time nonlinear systems with stochastic perturbations and remove or weaken several significant restrictions present in existing results global lipschitzness of the nonlinear vector field equilibrium preservation under the stochastic perturbation global exponential stability of the average system and compactness of the state space of the perturbation process then we propose a continuous time extremum seeking algorithm with stochastic excitation signals instead of deterministic periodic signals we analyze the stability of stochastic extremum seeking for static maps and for general nonlinear dynamic systems

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

newton like extremum seeking for the control of thermoacoustic instability in practice the convergence rate and stability of perturbation based extremum seeking schemes can be very sensitive to the curvature of the plant map an example of this can be seen in the use of extremum seeking to reduce the amplitude of thermoacoustic oscillations in premixed gas turbine combustors this sensitivity to the plant map curvature arises from the use of a gradient descent adaptation algorithm such extremum seeking schemes may need to be conservatively tuned in order to maintain stability over a wide range of operating conditions resulting in slower optimization than could be achieved for a fixed operating condition this can severely reduce the effectiveness of perturbation based extremum seeking schemes in some applications in this paper a sinusoidally perturbed extremum seeking scheme using a newton like step is developed non local stability results for the scheme are formulated using a lyapunov analysis a local analysis of the scheme is given to investigate the influence of plant dynamics and to show that the local rate of convergence is independent of the plant map curvature the benefit of this plant map curvature independence is then experimentally demonstrated in minimizing the thermoacoustic oscillations in a model premixed combustor

hyperbolic householder transformations a class of transformation matrices analogous to the householder matrices is developed with a nonorthogonal property designed to permit the efficient deletion of data from least squares problems these matrices which we term hyperbolic householder are shown to effect deletion or simultaneous addition and deletion of data with much less sensitivity to rounding errors than for techniques based on normal equations when the addition deletion sets are large this numerical robustness is obtained at the expense of only a modest increase in computations and when only a relatively small fraction of the data set is modified there is a decrease in required computations two applications to signal processing problems are considered first these transformations are used to obtain a square root algorithm for windowed recursive least squares filtering second the transformations are employed to implement the rejection of spurious data from the weight vector estimator in an adaptive array

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

a unified framework for input to state stability in systems with two time scales this paper develops a unified framework for studying robustness of the input to state stability iss property and presents new results on robustness of iss to slowly varying parameters to rapidly varying signals and to generalized singular perturbations the common feature in these problems is a time scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory to address various robustness problems simultaneously the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables the main results establish that if the boundary layer and averaged systems are iss then the iss bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time scales the generality of the framework is illustrated by making connection to various classical two time scale problems and suggesting extensions

rigorous studies of critical behavior the task addressed in the talk is the analysis of the critical behavior in statistical mechanical systems and related problems in the theory of random fields while it would have been very useful to have mathematical means of approaching this subject at the level of generality presented here in professor dobrushin s analysis of the high temperature phases we find that the richness of the behavior exhibited at low temperatures calls for more restricted treatments of special models nevertheless it was found that the somewhat delicate issues of the phase structure and the critical behavior may be approached by methods which apply to a number of systems in particular the picture offered by random walk representations led to very useful insights for the study of ferromagnetic systems 4 d field theory and percolation models it is hoped that random surface expansions might play a somewhat similar role for gauge fields and some models of spin glasses

a note on the ising model in high dimensions we consider thed dimensional ising model with a nearest neighbor ferromagnetic interactionj d 1 4d we show that asd the phase and the phase approaches a product measure with density given by the mean field approximation in particular the spontaneous magnetization converges to its mean field value a similar result holds for the unique gibbs measure of the system subject to an external fieldh 0

strict inequality for critical values of potts models and random cluster processes we prove that the critical value c of a ferromagnetic potts model is astrictly decreasing function of the strengths of interaction of the process this is achieved in the more general context of the random cluster representation of fortuin and kasteleyn by deriving and utilizing a formula which generalizes the technique known in percolation theory as russo s formula as a byproduct of the method we present a general argument for showing that at any given point on the critical surface of a multiparameter process the values of a certain critical exponent do not depend on the direction of approach of that point our results apply to all random cluster processes satisfying the fkg inequality

high performance algorithms for toeplitz and block toeplitz matrices in this paper we present several high performance variants of the classical schur algorithm to factor various toeplitz matrices for positive definite block toeplitz matrices we show how hyperbolic householder transformations may be blocked to yield a block schur algorithm this algorithm uses blas3 primitives and makes efficient use of a memory hierarchy we present three algorithms for indefinite toeplitz matrices two of these are based on look ahead strategies and produce an exact factorization of the toeplitz matrix the third produces an inexact factorization via perturbations of singular principal miners we also present an analysis of the numerical behavior of the third algorithm and derive a bound for the number of iterations to improve the accuracy of the solution for rank deficient toeplitz least squares problems we present a variant of the generalized schur algorithm that avoids breakdown due to an exact rank deficiency in the presence of a near rank deficiency an approximate rank factorization of the toeplitz matrix is produced finally we suggest an algorithm to solve the normal equations resulting from a real toeplitz least squares problem based on transforming to cauchy like matrices this algorithm exploits both realness and symmetry in the normal equations

extremum seeking in singularly perturbed hybrid systems this technical note considers the stability of a class of singularly perturbed hybrid systems with a continuous slow subsystem this type of system encompasses many slowly varying adaptation schemes such as extremum seeking acting on hybrid systems such as urban traffic control and switching control conditions are given for semi global practical asymptotic stability of the closed loop system when the dynamics of the reduced system are also semi globally practically asymptotically stable this extends previous results which require global asymptotic stability of the reduced system furthermore the stability result is used to show convergence of an extremum seeking scheme acting upon a hybrid system

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

a least squares approach to reduce stable discrete linear systems preserving their stability abstract a new stability preserving model reduction algorithm for discrete linear siso systems based on a least squares approach is proposed similar to the pade approximation an equation system for the markov parameters involving a high dimensional hankel matrix is considered it is proved that approximate solutions computed via the moore penrose pseudo inverse give rise to a stability preserving reduction scheme furthermore the proposed algorithm is compared to the balanced truncation method showing comparable performance of the reduced systems

intersection properties of simple random walks a renormalization group approach we study estimates for the intersection probability g m of two simple random walks on lattices of dimensiond 4 4 e as a problem in euclidean field theory we rigorously establish a renormalization group flow equation forg m and bounds on the function which show that ind 4 g m tends to zero logarithmically as the killing rate mass m tends to zero and that the fixed point g ind 4 e is bounded by const e g conste our methods also yield estimates on the intersection probability of three random walks ind 3 3 e for e 0 these results were first obtained by lawler 1

semi global stability analysis of observer based extremum seeking for hammerstein plants this paper considers semi global stability of an observer based extremum seeking es scheme acting on a hammerstein plant unlike previous semi global analyses no restriction is placed on the speed of the es dynamics relative to the plant dynamics for any set of frequencies of a probing sinusoidal dither it is shown how the plant output can be made to converge to an arbitrarily small neighborhood of its minimum from an arbitrarily large set of initial conditions furthermore in the absence of noise it is shown how increasing the dither frequency can allow arbitrarily fast convergence of the plant input to a small neighborhood of its optimum practical application of the results requires no a priori knowledge of the plant nonlinearity however in order to design the es scheme to operate over a wide range of dither frequencies some knowledge of the plant dynamics is required simulation examples are used to demonstrate these results and to highlight practical considerations when using a high frequency dither

dither re use in nash equilibrium seeking nash equilibrium seeking nes scheme consists of a number of decentralized extremum seeking es agents each controlling an input such that an associated selfish cost is regulated to its steady state nash equilibrium a non local stability result for the nes scheme is provided which allows two agents to use the same dither signal if the effect of each agent on the other s steady state cost function is sufficiently weak its application to plants with quadratic cost functions is presented as an example it is then demonstrated in simulation that by reducing the number of distinct dither signals the proposed scheme still has acceptable convergence properties while design effort is reduced

minres qlp a krylov subspace method for indefinite or singular symmetric systems cg symmlq and minres are krylov subspace methods for solving symmetric systems of linear equations when these methods are applied to an incompatible system that is a singular symmetric least squares problem cg could break down and symmlq s solution could explode while minres would give a least squares solution but not necessarily the minimum length pseudoinverse solution this understanding motivates us to design a minres like algorithm to compute minimum length solutions to singular symmetric systems minres uses qr factors of the tridiagonal matrix from the lanczos process where r is upper tridiagonal minres qlp uses a qlp decomposition where rotations on the right reduce r to lower tridiagonal form on ill conditioned systems singular or not minres qlp can give more accurate solutions than minres we derive preconditioned minres qlp new stopping rules and better estimates of the solution and residual norms the matrix norm and the condition number

unified frameworks for sampled data extremum seeking control global optimisation and multi unit systems two frameworks are proposed for extremum seeking of general nonlinear plants based on a sampled data control law within which a broad class of nonlinear programming methods is accommodated it is established that under some generic assumptions semi global practical convergence to a global extremum can be achieved in the case where the extremum seeking algorithm satisfies a stronger asymptotic stability property the converging sequence is also shown to be stable using a trajectory based proof as opposed to a lyapunov function type approach the former is more straightforward and insightful this allows for more general optimisation algorithms than considered in existing literature such as those which do not admit a state update realisation and or lyapunov functions lying at the heart of the analysis throughout is robustness of the optimisation algorithms to additive perturbations of the objective function multi unit extremum seeking is also investigated with the objective of accelerating the speed of convergence

translation invariant gibbs states for the ising model we prove that all the translation invariant gibbs states of the ising model are a linear combination of the pure phases open image in new window for any open image in new window this implies that the average magnetization is continuous for open image in new window furthermore combined with previous results on the slab percolation threshold b2 this shows the validity of pisztora s coarse graining pi up to the critical temperature

on the double random current nesting field we relate the planar random current representation introduced by griffiths hurst and sherman to the dimer model more precisely we provide a measure preserving map between double random currents obtained as the sum of two independent random currents on a planar graph and dimers on an associated bipartite graph we also define a nesting field for the double random current which under this map corresponds to the height function of the dimer model as applications we provide an alternative derivation of some of the bozonization rules obtained recently by dubedat and show that the spontaneous magnetization of the ising model on a planar biperiodic graph vanishes at criticality

the random walk representation of classical spin systems and correlation inequalities ii the skeleton inequalities on utilise la representation marche aleatoire pour prouver les premieres d une nouvelle famille d inegalites de correlation pour des modeles reticules ferromagnetiques 4 ces inegalites etablissent que les sommes partielles finies du developpement en perturbation resomme du propagateur pour la fonction a 4 points forment un ensemble alterne de limites superieures et inferieures rigoureuses pour la fonction a 4 points exactes

a unified least squares approach to identify and to reduce continuous asymptotically stable systems abstract the stability preserving model reduction algorithm for the discrete time case s feldmann p lang a least squares approach to reduce stable discrete linear systems preserving their stability laa 381 2004 141 163 is extended to both the identification and reduction of continuous time systems the obtained reduction algorithm turns out to be c stability preserving

multiplicity of phase transitions and mean field criticality on highly non amenable graphs we consider independent percolation ising and potts models and the contact process on infinite locally finite connected graphs

on the computation of the rank of block bidiagonal toeplitz matrices in the present paper we study the computation of the rank of a block bidiagonal toeplitz bbt sequence of matrices we propose matrix based numerical and symbolical updating and direct methods computing the rank of bbt matrices and comparing them with classical procedures the methods deploy the special form of the bbt sequence significantly reducing the required flops and leading to fast and efficient algorithms the numerical implementation of the algorithms computes the numerical rank in contrast with the symbolical implementation which guarantees the computation of the exact rank of the matrix the combination of numerical and symbolical operations suggests a new approach in software mathematical computations denoted as hybrid computations

a superfast method for solving toeplitz linear least squares problems in this paper we develop a superfast o m n log 2 m n complexity algorithm to solve a linear least squares problem with an m n toeplitz coefficient matrix the algorithm is based on the augmented matrix approach the augmented matrix is further extended to a block circulant matrix and dft is applied this leads to an equivalent tangential interpolation problem where the nodes are roots of unity this interpolation problem can be solved by a divide and conquer strategy in a superfast way to avoid breakdowns and to stabilize the algorithm pivoting is used and a technique is applied that selects difficult points and treats them separately the effectiveness of the approach is demonstrated by several numerical examples 2003 elsevier science inc all rights reserved

fast direct solvers for some complex symmetric block toeplitz linear systems we consider the solution of a class of complex symmetric block toeplitz linear systems arising from integral equation problems algorithms that exploit the toeplitz structure provide considerable savings on the number of arithmetic operations compared to the classical cholesky factorization we propose a generalized schur algorithm adapted to the complex symmetric case we detail blocked variants that perform better by using blas 3 primitives we also propose a solver based on an augmented matrix approach that allows a substantial decrease in the use of memory by avoiding an explicit assembly of the cholesky factor all algorithms have been implemented and numerical results are included to illustrate the effectiveness of our approach

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

critical exponents for the contact process under the triangle condition we show that a continuous time version of the triangle condition for percolation implies mean field values for several contact process critical exponents our results support the belief that the upper critical spatial dimension for the contact process is four

analysis for a class of singularly perturbed hybrid systems via averaging a class of singularly perturbed hybrid dynamical systems is analyzed the fast states are restricted to a compact set a priori the continuous time boundary layer dynamics produce solutions that are assumed to generate a well defined average vector field for the slow dynamics this average the projection of the jump map in the direction of the slow states and flow and jump sets from the original dynamics define the reduced or average hybrid dynamical system assumptions about the average system lead to conclusions about the original higher dimensional system for example forward pre completeness for the average system leads to a result on closeness of solutions between the original and average system on compact time domains in addition global asymptotic stability for the average system implies semiglobal practical asymptotic stability for the original system we give examples to illustrate the averaging concept and to relate it to classical singular perturbation results as well as to other singular perturbation results that have appeared recently for hybrid systems we also use an example to show that our results can be used as an analysis tool to design hybrid feedbacks for continuous time plants implemented by fast but continuous actuators

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

fast qr factorization of cauchy like matrices abstract in this paper we present two fast numerical methods for computing the qr factorization of an n n cauchy like matrix c c qr with data points lying on the real axis or on the unit circle in the complex plane it is shown that the rows of the q factor of c are the eigenvectors of a rank structured matrix partially determined by some prescribed spectral data this property establishes a basic connection between the computation of q and the solution of an inverse eigenvalue problem for a rank structured matrix exploiting the structure of this problem enables us to develop quadratic time i e o n 2 qr factorization algorithms

look ahead methods for block hankel systems in this paper the updating formulas used by three look ahead methods for solving hankel systems are generalized to the square block case each of the original methods was described in the literature using the terminology of different but strongly related fields formal orthogonal polynomials pade approximants structured matrices this paper gives several of these connections generalized to the block case and shows that each viewpoint has its own merits

lace expansion for the ising model the lace expansion has been a powerful tool for investigating mean field behavior for various stochastic geometrical models such as self avoiding walk and percolation above their respective upper critical dimension in this paper we prove the lace expansion for the ising model that is valid for any spin spin coupling for the ferromagnetic case we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace expansion coefficients for self avoiding walk as a result we obtain gaussian asymptotics of the critical two point function for the nearest neighbor model with d gg 4 and for the spread out model with d 4 and l gg 1 without assuming reflection positivity

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

non intersection of transient branching random walks let g be a cayley graph of a nonamenable group with spectral radius rho 1 it is known that branching random walk on g with offspring distribution mu is transient i e visits the origin at most finitely often almost surely if and only if the expected number of offspring overline mu satisfies overline mu le rho 1 benjamini and muller groups geom dyn 6 231 247 2012 conjectured that throughout the transient supercritical phase 1 overline mu le rho 1 and in particular at the recurrence threshold overline mu rho 1 the trace of the branching random walk is tree like in the sense that it is infinitely ended almost surely on the event that the walk survives forever this is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely we prove this conjecture along with several other related conjectures made by the same authors a central contribution of this work is the introduction of the notion of local unimodularity which we expect to have several further applications in the future

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

on singular perturbations due to fast actuators in hybrid control systems a stability result is given for hybrid control systems singularly perturbed by fast but continuous actuators if a hybrid control system has a compact set globally asymptotically stable when the actuator dynamics are omitted or equivalently are infinitely fast then the same compact set is semiglobally practically asymptotically stable in the finite speed of the actuator dynamics this result which generalizes classical results for differential equations justifies using a simplified plant model that ignores fast but continuous actuator dynamics even when using a hybrid feedback control algorithm

a lookahead algorithm for the solution of block toeplitz systems abstract we give a weakly stable algorithm to solve a block toeplitz system of linear equations if the lookahead steps taken to compute the parameters of the inversion formula for the block toeplitz matrix are small compared to the order n of the matrix the algorithm requires o n2 floating point operations the parameters of the inversion formula are interpreted and computed in a recursive way as solutions of certain interpolation problems given the formal laurent series based on the data of the block toeplitz matrix

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems

emergent planarity in two dimensional ising models with finite range interactions the known pfaffian structure of the boundary spin correlations and more generally order disorder correlation functions is given a new explanation through simple topological considerations within the model s random current representation this perspective is then employed in the proof that the pfaffian structure of boundary correlations emerges asymptotically at criticality in ising models on mathbb z 2 with finite range interactions the analysis is enabled by new results on the stochastic geometry of the corresponding random currents the proven statement establishes an aspect of universality seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases