a new characterization of invariant subspaces of h 2 and applications to the optimal sensitivity problem this paper gives a new equivalent characterization for invariant subspaces of h 2 when the underlying structure is specified by the so called pseudorational transfer functions this plays a fundamental role in computing the optimal sensitivity for a certain important class of infinite dimensional systems including delay systems a closed formula easier to compute than the well known zhou khargonekar formula is given for optimal sensitivity for such systems an example is given to illustrate the result 2004 elsevier b v all rights reserved

conditions for the equivalence of internal and external stability for distributed parameter systems conditions are given on a controlled observed distributed parameter system under which external stability also known as input output stability implies internal stability which is the exponential stability of the underlying semigroup generator it is shown that when the system satisfies a general definition of stabilizability and detectability external stability is equivalent to internal stability unlike the case previous definitions of stabilizability and detectability a system which satisfies these does not necessarily have a spectrum decomposition therefore in order to check internal stability it is often sufficient to check the boundedness of the transfer function in the right half complex plane this is illustrated with a simple example

practical output regulation for bounded linear infinite dimensional state space systems we develop the mathematical foundations of practical state space output regulation for bounded infinite dimensional linear systems by practical output regulation we mean asymptotic tracking of references and rejection of disturbances with a given accuracy our main results are general upper bounds for the norms perturbations to the parameters of the exosystem the plant and a controller which achieves exact output regulation these bounds depend explicitly on the desired tracking accuracy e 0 in this paper all perturbations are assumed to be bounded additive and linear our results apply for both feedforward and error feedback controllers and for arbitrary bounded uniformly continuous reference disturbance signals

author s reply to comments on nonlinear repetitive control our paper ghosh and paden nonlinear repetitive control ieee trans automat contr vol 45 p 949 54 2000 proves theorem 1 which establishes conditions for the stability of the nonlinear repetitive control system and proposition 2 which goes on to establish bounds on the steady state error signals these are correct as stated as they apply to repetitive controllers with a finite number of oscillators in their internal model an issue correctly raised in the commentary lucibello comments on nonlinear repetitive control ieee trans automat contr vol 48 p 1470 1 2003 is that the conditions of proposition 2 might be difficult to meet when the number of oscillators is large this observation does not challenge the veracity of the proposition but is certainly insightful in our introductory comments of section 11 we attempt and fail to make a qualitative connection to the pure time delay type repetitive control the following statement is incorrect and should be deleted from our paper this justifies the fact that in the limit n spl rarr spl infin the above model of the repetitive controller works as well as the ideal model in achieving perfect tracking for typical reference signals

ergodic averages of commuting transformations with distinct degree polynomial iterates we prove mean convergence as n to infty for the multiple ergodic averages frac 1 n sum n 1 n f 1 t 1 p 1 n x cdot ldots cdot f ell t ell p ell n x where p 1 ldots p ell are integer polynomials with distinct degrees and t 1 ldots t ell are invertible commuting measure preserving transformations acting on the same probability space this establishes several cases of a conjecture of bergelson and leibman that complement the case of linear polynomials recently established by tao furthermore we show that unlike the case of linear polynomials for polynomials of distinct degrees the corresponding characteristic factors are mixtures of inverse limits of nilsystems we use this particular structure together with some equidistribution results on nilmanifolds to give an application to multiple recurrence and a corresponding one to combinatorics

limiting behavior of the ginzburg landau functional we continue our study of the functional ee u u12 u 2 14e2 1 u 2 2dx for u h1 u r2 where u is a bounded open subset of r2 compactness results for the scaled jacobian of ue are proved under the assumption that ee ue is bounded uniformly by a function of e in addition the gamma limit of ee ue ln e 2 is shown to be e v 12 v 22 v m where v is the limit of j ue ln e j ue ue due and m is the total variation of a radon measure these results are applied to the ginzburg landau functionalfe u a hext u12 au 2 14e2 1 u 2 2 12 a hext dx with external magnetic field hext h ln e the gamma limit of fe ln e 2 is calculated to bef v a h 12 v a 22 v m a h 22 where v is as before and a is the limit of ae ln e

powers of sequences and recurrence we study recurrence and multiple recurrence properties along the k th powers of a given set of integers we show that the property of recurrence for some given values of k does not give any constraint on the recurrence for the other powers this is motivated by similar results in number theory concerning additive basis of natural numbers moreover motivated by a result of kamae and mendes france that links single recurrence with uniform distribution properties of sequences we look for an analogous result dealing with higher order recurrence and make a related conjecture

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

a multidimensional szemeredi theorem for hardy sequences of different growth abstract we prove a variant of the multidimensional polynomial szemeredi theorem of bergelson and leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some hardy field and satisfy certain growth conditions we do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula a consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences another consequence is that every syndetic set of integers contains certain nonshift invariant patterns and every finite coloring of n with each color class a syndetic set contains certain polychromatic patterns results very particular to our non polynomial setup

a product estimate for ginzburg landau and corollaries abstract we prove a new inequality for the jacobian or vorticity associated to the ginzburg landau energy in any dimension it allows to retrieve existing lower bounds on the energy to extend them to the case of unbounded vorticity and to get a few other corollaries it also provides a new estimate on the time variation for time dependent families which has applications for the study of ginzburg landau dynamics

partial regularity for mass minimizing currents in hilbert spaces recently the theory of currents and the existence theory for plateau s problem have been extended to the case of finite dimensional currents in infinite dimensional manifolds or even metric spaces see acta math 185 2000 1 80 and also proc lond math soc 3 106 2013 1121 1142 adv calc var 7 2014 227 240 for the most recent developments in this paper in the case when the ambient space is hilbert we provide the first partial regularity result in a dense open set of the support for n dimensional integral currents which locally minimize the mass our proof follows with minor variants indiana univ math j 31 1982 415 434 implementing lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space

on the chern simons higgs vortex equation without gauge fields abstract this paper is concerned with the nonself dual chern simons higgs model on r 2 with vanishing gauge fields we prove the existence of radial solutions with the topological boundary condition and the nonexistence of radial solutions with the nontopological boundary condition we also establish the asymptotic properties of solutions and derive the quantization of the potential energy

some fine properties of currents and applications to distributional jacobians we study fine properties of currents in the framework of geometric measure theory on metric spaces developed by ambrosio and kirchheim and we prove a rectifiability criterion for flat currents of finite mass we apply these tools to study the structure of the distributional jacobians of functions in the space bnv defined by jerrard and soner we define the subspace of special functions of bounded higher variation and we prove a closure theorem

internal and external stability and robust stability condition for a class of infinite dimensional systems abstract in the current study of robust stability of infinite dimensional systems internal exponential stability is not necessarily guaranteed this paper introduces a new class of impulse responses called r in which the usual notion of l2 input output stability guarantees not only external but also internal exponential stability the result is applied to derive a closed loop stability condition and a version of the small gain theorem with internal exponential stability this leads to a robust stability condition that also assures internal stability an application to repetitive control systems is shown to illustrate the results

robust regulation theory for transfer functions with a coprime factorization classical frequency domain results of robust regulation are extended by requiring only a right or a left coprime factorization of a plant but not both the famous internal model principle is generalized first which leads to a necessary and sufficient solvability condition of the robust regulation problem and to a parametrization of all robustly regulating controllers in addition a procedure for constructing robustly regulating controllers is proposed

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

pseudo rational input output maps and their realizations a fractional representation approach to infinite dimensional systems this paper studies matrix fractional representation for impulse responses of a certain class of infinite dimensional linear systems which contains in particular delay differential systems such impulse responses are called pseudo rational in this paper this fractional representation is effectively used to derive concrete function space models from the abstract shift realizations given a fractional representation q 1 p a standard observable realization analogous to fuhrmann realizations for finite dimensional systems is associated to it a new notion of coprimeness called approximate left coprimeness is introduced and it is shown that the standard observable realization associated to the representation q 1 p is canonical if and only if q and p are approximately left coprime some examples are discussed to illustrate the relationships among various coprimeness concepts that have appeared in the literature

stable controllers for robust stabilization of systems with infinitely many unstable poles abstract this paper studies the problem of robust stabilization by a stable controller for a linear time invariant single input single output infinite dimensional system we consider a class of plants having finitely many simple unstable zeros but possibly infinitely many unstable poles first we show that the problem can be reduced to an interpolation minimization by a unit element next by the modified nevanlinna pick interpolation we obtain both lower and upper bounds on the multiplicative perturbation under which the plant can be stabilized by a stable controller in addition we find stable controllers to provide robust stability we also present a numerical example to illustrate the results and apply the proposed method to a repetitive control system

asymptotics for the ginzburg landau equation in arbitrary dimensions let be a bounded simply connected regular domain of rn n 2 for 0 e 1 let ue c be a smooth solution of the ginzburg landau equation in with dirichlet boundary condition ge i e formula we are interested in the asymptotic behavior of ue as e goes to zero under the assumption that ee ue m0 log e and some conditions on ge which allow singularities of dimension n 3 on

multiple recurrence and nilsequences aiming at a simultaneous extension of khintchine s and furstenberg s recurrence theorems we address the question if for a measure preserving system x mathcal x mu t and a set a in mathcal x of positive measure the set of integers n such that mu a cap t n a cap t 2n a cap ldots cap t kn a mu a k 1 epsilon is syndetic the size of this set surprisingly enough depends on the length k 1 of the arithmetic progression under consideration in an ergodic system for k 2 and k 3 this set is syndetic while for k 4 it is not the main tool is a decomposition result for the multicorrelation sequence int f x f t n x f t 2n x ldots f t kn x d mu x where k and n are positive integers and f is a bounded measurable function we also derive combinatorial consequences of these results for example showing that for a set of integers e with upper banach density d e 0 and for all e 0 the set big n in mathbb z colon d big e cap e n cap e 2n cap e 3n big d e 4 epsilon big is syndetic

relationships between internal and external stability for infinite dimensional systems with applications to a servo problem a study is made of the relationships among various stability motions for a class of infinite dimensional systems which contains a class of systems not covered by existing methods e g those having infinitely many unstable poles it is proved that internal l sup 2 stability and exponential stability are equivalent and internal stability implies h sup infinity stability several necessary and sufficient conditions for internal stability are derived in particular under certain conditions a canonical realization is internally stable if it is externally stable these results are applied to the servo problem involving this class of systems it is shown that i an internal model is necessary for tracking ii an internal model along with closed loop stability implies tracking a typical example called a repetitive control system is discussed to illustrate the results

topological multivortex solutions of the self dual maxwell chern simons higgs system abstract we study the existence and various behaviors of topological multivortex solutions of the relativistic self dual maxwell chern simons higgs system we first prove the existence of general topological solutions by applying variational methods to the newly discovered minimizing functional then by an iteration method we prove the existence of topological solutions satisfying some extra conditions which we call admissible solutions we establish asymptotic exponential decay estimates for these topological solutions we also investigate the limiting behavior of the admissible solutions as parameters in our system go to some limits for the abelian higgs limit we obtain strong convergence result while for the chern simons limit we only obtained that our admissible solutions weakly approximate one of the chern simons solutions

the global existence in the cauchy problem of the maxwell chern simons higgs system in this article we prove the global existence of solution of the classical maxwell chern simons higgs equations in 2 1 dimensional minkowski space time in the lorentz gauge we also prove that the topological solution of the maxwell chern simons higgs system converges to that of maxwell higgs system as the chern simons constant goes to zero reproducing the classical result by moncrief j math phys 21 2291 1980 on the global existence of the maxwell klein gordon system in 2 1 dimension

a modica mortola approximation for branched transport and applications the m energy which is usually minimized in branched transport problems among singular one dimensional rectifiable vector measures is approximated by means of a sequence of elliptic energies defined on more regular vector fields the procedure recalls the one of modica mortola related to the approximation of the perimeter in our context the double well potential is replaced by a concave term the paper contains a proof of convergence and numerical simulations of optimal networks based on that previous result

rectifiable and flat g chains in a metric space chains in a complete metric space which have coefficients in a normed abelian group g are studied an m dimensional rectifiable chain is the lipschitz push forward of a region in bbb r m equipped with a measurable g valued density flat chains are obtained by completion using a certain flat norm on polyhedral or lipschitz chain approximations numerous basic results of geometric measure theory for these chains are derived including the rectifiability of finite mass flat chains provided that g contains no nonconstant lipschitz curves the work here generalizes and uses many ideas from the 1999 paper of b white on at g chains in bbb r n and the 2000 paper of l ambrosio and b kirchheim on currents in a metric space the use of flat and rectifiable chains in geometric variational problems or in defining geometric homology theories may reveal geometric properties of spaces

a note on linear input output maps of bounded type this note studies a class of constant linear continuous time input output maps which is called input output maps of bounded type an algebraic condition for an input output map to be of bounded type is given its application to realization theory especially for delay differential systems is considered an example is given to illustrate how it gives rise to a concrete realization procedure

design of multivariable feedback systems with simple unstable plant this paper proposes a design methodology for distributed linear multivariable feedback systems with simple unstable plants a simple unstable plant has either first or second order unstable poles the methodology developed provides a global characterization of all realizable compensators which stabilize a given simple unstable plant a design example is given to show that this methodology can be used to generate in an appropriate computer aided design environment controllers which are optimal with respect to designer specified criteria additionally it is shown that the nature of the design methodology gives geometric insight into the dynamics of the process whereby an unstable plant is stabilized

repetitive control system a new type servo system for periodic exogenous signals a control scheme called repetitive control is proposed in which the controlled variables follow periodic reference commands a high accuracy asymptotic tracking property is achieved by implementing a model that generates the periodic signals of period l into the closed loop system sufficient conditions for the stability of repetitive control systems and modified repetitive control systems are derived by applying the small gain theorem and the stability theorem for time lag systems synthesis algorithms are presented by both the state space approach and the factorization approach in the former approach the technique of the kalman filter and perfect regulation is utilized while coprime factorization over the matrix ring of proper stable rational functions and the solution of the hankel norm approximation are used in the latter one

set polynomials and polynomial extension of the hales jewett theorem an abstract hales jewett type extension of the polynomial van der waerden theorem bl is established theorem let r d q n there exists n n such that for any r coloring of the set of subsets of v 1 n d 1 q there exist a set a v and a nonempty set 1 n such that a d 1 q and the subsets a a d 1 a d 2 a d q are all of the same color this polynomial hales jewett theorem contains refinements of many combinatorial facts as special cases the proof is achieved by introducing and developing the apparatus of set polynomials polynomials whose coefficients are finite sets and applying the methods of topological dynamics

on the lower semicontinuous envelope of functionals defined on polyhedral chains abstract in this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains more precisely denoting by h r 0 an even subadditive and lower semicontinuous function with h 0 0 and by h the functional induced by h on polyhedral m chains namely h p i 1 n h i h m i for every p i 1 n i i p m r n we prove that the lower semicontinuous envelope of h coincides on rectifiable m currents with the h mass m h r e h x d h m x for every r e r m r n

improved partial regularity for manifold constrained minimisers of subquadratic energies we consider minimising p harmonic maps from three dimensional domains to the real projective plane for 1 p 2 these maps arise as least energy configurations in variational models for nematic liquid crystals we show that the singular set of such a map decomposes into a 1 dimensional set which can be physically interpreted as a non orientable line defect and a locally finite set i e a collection of point defects

poincare recurrence and number theory thirty years later hillel furstenberg s 1981 article in the bulletin gives an elegant introduction to the interplay between dynamics and number theory summarizing the major developments that occurred in the few years after his landmark paper 21 the field has evolved over the past thirty years with major advances on the structural analysis of dynamical systems and new results in combinatorics and number theory furstenberg s article continues to be a beautiful introduction to the subject drawing together ideas from seemingly distant fields furstenberg s article 21 gave a general correspondence between regularity properties of subsets of the integers and recurrence properties in dynamical systems now dubbed the furstenberg correspondence principle he then showed that such recurrence properties always hold proving what is now referred to as the multiple recurrence theorem combined these results gave a new proof of szemeredi s theorem 45 if s z has positive upper density then s contains arbitrarily long arithmetic progressions this proof lead to an explosion of activity in ergodic theory and topological dynamics beginning with new proofs of classic results of ramsey theory and ultimately leading to significant new combinatorial and number theoretic results the full implications of these connections have yet to be understood the approach harks back to the earliest results on recurrence in the measurable setting and in the topological setting a measure preserving system is a quadruple x b t where x denotes a set b is a algebra on x is a probability measure on x b and t x x is a measurable transformation such that t 1 a a for all a b poincare recurrence states that if x b t is a measure preserving system and a b with a 0 there exists n n such that a t na 0 a topological dynamical system is a pair x t where x is a compact metric space and t x x is a continuous map one can show that any such topological space admits a borel probability measure that preserves t in particular poincare recurrence implies recurrence in the topological setting birkhoff 13 gave a direct proof of

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

analysis of nematic liquid crystals with disclination lines we investigate the structure of nematic liquid crystal thin films described by the landau de gennes tensor valued order parameter model with dirichlet boundary conditions on the sides of nonzero degree we prove that as the elasticity constant goes to zero in the energy a limiting uniaxial nematic texture forms with a finite number of defects all of degree frac 1 2 or all of degree frac 1 2 corresponding to vertical disclination lines at those locations we also state a result on the limiting behavior of minimizers of the chern simons higgs model without magnetic field that follows from a similar proof

distributed system transfer functions of exponential order o is a real number we construct a transfer function algebra of fractions viz f o for modelling possibly unstable distributed systems such that i fcirc in f o is holomorphic in re s o i e is o stable iff fcirc is o exponentially stable and ii we allow delay in the direct input output transmission of the system this algebra is a a restriction of the algebra b o developed by callier and desoer 1978 1980 a b an extension of the algebra of proper rational functions such that the exponential order properties of the latter transfer functions of lumped systems are maintained the algebra f o can be used for modelling and feedback system design it is shown that standard semigroup systems are better modelled by a transfer function in f o rather than b o

sensitivity reduction by stable controllers for mimo infinite dimensional systems via the tangential nevanlinna pick interpolation we study the problem of finding a stable stabilizing controller that satisfies a desired sensitivity level for an mimo infinite dimensional system the systems we consider have finitely many simple transmission zeros in c but they are allowed to possess infinitely many poles in c we compute both upper and lower bounds of the minimum sensitivity achievable by a stable controller via the tangential nevanlinna pick interpolation we also obtain stable controllers attaining such an upper bound to illustrate the results we discuss a repetitive control system as an application of the proposed method

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

a nilpotent roth theorem let t and s be invertible measure preserving transformations of a probability measure space x we prove that if the group generated by t and s is nilpotent then lim n ras infty frac 1 n sum n 1 n u t n x v s n x exists in l 2 norm for any u v l x we also show that for a with a 0 one has lim n ras infty frac 1 n sum n 1 n mu a cap t n a cap s n a 0 by the way of contrast we bring examples showing that if measure preserving transformations t s generate a solvable group then i the above limits do not have to exist ii the double recurrence property fails that is for some a a 0 one may have a t n a s n a 0 for all n finally we show that when t and s generate a nilpotent group of class c lim n ras infty frac 1 n sum n 1 n u t n x v s n x int ud mu int vd mu in l 2 x for all u v l x if and only if t s is ergodic on x x and the group generated by t 1 s t 2 s 2 t c s c acts ergodically on x

sensitivity reduction by strongly stabilizing controllers for mimo distributed parameter systems this note investigates a sensitivity reduction problem by stable stabilizing controllers for a linear time invariant multi input multioutput distributed parameter system the plant we consider has finitely many unstable zeros which are simple and blocking but can possess infinitely many unstable poles we obtain a necessary condition and a sufficient condition for the solvability of the problem using the matrix nevanlinna pick interpolation with boundary conditions we also develop a necessary and sufficient condition for the solvability of the interpolation problem and show an algorithm to obtain the solutions our method to solve the interpolation problem is based on the schur nevanlinna algorithm

gamma convergence of maxwell chern simons higgs energy in this article we derive the limit of the maxwell chern simons higgs energy with log 2 energy bound our interests focus on the case with external electro magnetic potentials similarly to the abelian ginzburg landau model the corresponding obstacle problems are derived

rectifiability of flat chains we prove without using federer s structure theorem that a finite mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0 dimensional slices are rectifiable this implies that every flat chain of finite mass and finite size is rectifiable it also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite mass flat

nilsequences and a structure theorem for topological dynamical systems abstract we characterize inverse limits of nilsystems in topological dynamics via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems we provide two applications of the structure the first is to nilsequences which have played an important role in recent developments in ergodic theory and additive combinatorics we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally meaning on finite intervals the second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system we show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1

the green tao theorem on arithmetic progressions in the primes an ergodic point of view a long standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions until recently the only progress on this conjecture was due to van der corput who showed in 1939 that there are infinitely many triples of primes in arithmetic progression in an amazing fusion of methods from analytic number theory and ergodic theory ben green and terence tao showed that for any positive integer k there exist infinitely many arithmetic progressions of length k consisting only of prime numbers this is an introduction to some of the ideas in the proof concentrating on the connections to ergodic theory

brief robust learning control for a class of nonlinear systems with periodic and aperiodic uncertainties this paper addresses the robust learning control problem for a class of nonlinear systems with structured periodic and unstructured aperiodic uncertainties a recursive technique is proposed which extends the backstepping idea to the robust repetitive learning control systems a learning evaluation function instead of a lyapunov function is formulated as a guideline for derivation of the control strategy which guarantees the asymptotic stability of the tracking system a design example is given

lower bounds for the energy of unit vector fields and applications we prove lower bounds for the dirichlet energy of a unit vector field defined in a perforated domain of r2with nonzero degree on the outer boundary in terms of the total diameter of the holes we use this to derive lower bounds and then compactness results for sequences ue of minimizers or almost minimizers of ginzburg landau functionals with coupling constant 1 e2tending to

multiple recurrence and convergence for sequences related to the prime numbers for any measure preserving system x mathcal x mu t and a in mathcal x with mu a 0 we show that there exist infinitely many primes p such that mu bigl a cap t p 1 a cap t 2 p 1 a bigr 0 furthermore we show the existence of the limit in l 2 mu of the associated double average over the primes a key ingredient is a recent result of green and tao on the von mangoldt function a combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p 1 for some prime p

algebraic theory of linear multivariable feedback systems this paper presents an algebraic theory for analysis and design of linear multivariable feedback systems the theory is developed in an algebraic setting sufficiently general to include as special cases continuous and discrete time systems both lumped and distributed designs are implemented by construction of a controller with two vector inputs and one vector output use of controllers of this type is shown to generate convenient stability results and convenient global parametrizations of all i o maps and all disturbance to output maps achievable for a given plant by a stabilizing compensator these parametrizations are then used to show that any such i o map and any such disturbance to output map may be simultaneously realized by choice of an appropriate controller in the special case of lumped systems it is shown that the design theory can be reduced to manipulations involving polynomial matrices only the resulting design procedure is thus shown to be more efficient computationally finally the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting it is shown that the classical results on asymptotic tracking can be generalized to this setting additionally sufficient conditions for robustness of asymptotic tracking and robustness of stability are developed

refined jacobian estimates and gross pitaevsky vortex dynamics we study the dynamics of vortices in solutions of the gross pitaevsky equation i u t delta u 1 over varepsilon 2 u left 1 u 2 right in a bounded simply connected domain omega subset mathbb r 2 with natural boundary conditions on previous rigorous results have shown that for sequences of solutions u varepsilon with suitable well prepared initial data one can determine limiting vortex trajectories and moreover that these trajectories satisfy the classical ode for point vortices in an ideal incompressible fluid we prove that the same motion law holds for a small but fixed varepsilon and we give estimates of the rate of convergence and the time interval for which the result remains valid the refined jacobian estimates mentioned in the title relate the jacobian j u of an arbitrary function u in h 1 omega mathbb c to its ginzburg landau energy in the analysis of the gross pitaevsky equation they allow us to use the jacobian to locate vortices with great precision and they also provide a sort of dynamic stability of the set of multi vortex configurations

a simple robust controller for the approximate regulation of almost periodic signals for linear systems by using a delay line as a signal generator we design a simple robust controller for the approximate regulation i e approximate asymptotic tracking and rejection of almost periodic signals for linear systems as opposed to the related existing results our controller design does not require online parameter tuning besides the usual assumptions about stabilizability detectability and nonexistence of transmission zeros for the plant the controller design algorithm of this paper only requires knowledge of the desired tracking accuracy based on this information alone we can only design a robust controller achieving in the presence of unknown trigonometric polynomial disturbances and additive perturbations to the plant and the controller here is the output of the plant and is an arbitrary almost periodic reference signal taken from an infinite dimensional generalized sobolev space of almost periodic functions in this paper we also study robust output regulation in the limiting case where it turns out that due to the general loss of exponential closed loop stability asymptotic tracking rejection can also be lost as unless the exogenous signals are smooth enough this last result extends some recent theorems on nonrobust open loop output regulation for infinite dimensional exosystems the results of this paper are new and potentially useful for finite dimensional plants but the main results are also true for infinite dimensional systems with bounded control and observation operators

an inverse theorem for the gowers u s 1 n norm we prove the inverse conjecture for the gowers u s 1 n norm for all s 1 this is new for s 4 more precisely we establish that if f n 1 1 is a function with f u s 1 n d then there is a bounded complexity s step nilsequence f g n g that correlates with f where the bounds on the complexity and correlation depend only on s and d from previous results this conjecture implies the hardy littlewood prime tuples conjecture for any linear system of finite complexity

comments on relationships between internal and external stability for infinite dimensional systems with application to a servo problem with reply the author takes exception to two statements in the above referred paper ibid vol 33 p 1044 52 nov 1988 one is that servo systems for infinite dimensional systems have only been studied in two papers the commenter points out two more the other comment regards exponential decay of the error in l sup 2 stability the authors agree with the commenter s point and add some clarification

gamma limit of the nonself dual chern simons higgs energy abstract we consider the gamma limit of the abelian chern simons higgs energy g csh 1 2 u a e u e 2 2 4 curl a e h ex 2 u e 2 1 e 2 u e 2 1 u e 2 2 d x on a bounded simply connected two dimensional domain under the e 0 limit as a first step we study the gamma limit of e csh 1 2 u u e 2 1 e 2 u e 2 1 u e 2 2 d x under two different scalings e csh log e and e csh log e 2 we apply the log e 2 scaling result to the full chern simons higgs energy g csh and as a consequence we are able to compute the first critical field h 1 h 1 u for the nucleation of a vortex the method entails estimating in certain weak topologies the jacobian j u e det u e in terms of the chern simons higgs energy e csh

rectifiability of flat chains in banach spaces with coefficients in zp in this paper we study the behaviour of a class of currents with coefficients in zp in metric spaces obtained by taking a suitable quotient of currents with coefficients in z we prove under fairly general assumption on the ambient space the rectifiability of the so called slice mass and under more restrictive assumptions the rectifiability of the mass measure

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

nonself dual chern simons and maxwell chern simons vortices on bounded domains abstract we first show the existence of the multivortex solutions of maxwell chern simons higgs mcsh model on bounded domains and prove that the solutions of the euler lagrange equations of the mcsh energy functional converge to those of the abelian higgs ah model and the chern simons higgs csh model in suitable limits respectively we also show the existence of the multivortex solutions of the nonself dual csh model on bounded domains besides we study asymptotics for the minimizers of mcsh energy functional when the gauge field vanishes