brief paper a dual optimization procedure for linear quadratic robust control problems this paper considers the problem of choosing a single constant linear state feedback control law which produces satisfactory performance for each of several operating points of a system the model for each operating point is assumed to be linear and the criterion for satisfactory performance is taken to be an infinite horizon quadratic cost functional this problem is reformulated as a finite dimensional optimization over the linear feedback gains which can be readily solved using standard nonlinear optimization techniques provided a stabilizing initial value of the gains can be found although the direct solution of this problem will be discussed briefly the major portion of the paper will be devoted to solution techniques when an initial stabilizing guess is not available

astala s conjecture on distortion of hausdorff measures under quasiconformal maps in the plane let e subset mathbb c be a compact set g mathbb c to mathbb c be a k quasiconformal map and let 0 t 2 let mathcal h t denote t dimensional hausdorff measure then mathcal h t e 0 quad rightarrow quad mathcal h t left ge right 0 quad t frac 2kt 2 left k 1 right t

uniform rectifiability and harmonic measure ii poisson kernels in l p imply uniform rectifiability the first author was supported by nsf grants dms 0801079 and dms 1101244 the second author was supported by micinn grant mtm2010 16518 and by csic pie 200850i015 the third author was partially supported by grants dms 0901524 career dms 1056965 us nsf sloan research foundation and mtm2010 16232 mtm2009 14694 c02 01 spain

the cauchy integral analytic capacity and uniform rectifiability several explanations concerning notation terminology and background are in order first notation by 7hi we have denoted the one dimensional hausdorff measure i e length and a z r stands for the closed disc with center z and radius r a curve f is called ad regular that is ahlfors david regular if it satisfies 1 2 with e f since the lower bound is automatic for curves this means that hl r n a z r 0 general sets satisfying 1 2 are called ad regular it is simplest to define the l2 boundedness of the cauchy singular integral operator via the truncated integrals we say that ce is bounded in l2 e without really defining the operator ce itself if there is m ox such that

calderon zygmund kernels and rectifiability in the plane abstract let e c be a borel set with finite length that is 0 h 1 e by a theorem of david and leger the l 2 h 1 e boundedness of the singular integral associated to the cauchy kernel or even to one of its coordinate parts x z 2 y z 2 z x y c implies that e is rectifiable we extend this result to any kernel of the form x 2 n 1 z 2 n z x y c n n we thus provide the first non trivial examples of operators not directly related with the cauchy transform whose l 2 boundedness implies rectifiability

a further note on the robustness of output feedback control methods to modeling errors in an earlier note it was shown that designing a nonstrictly proper output feedback compensator using a model that neglects parasitic elements might lead to an unstable closed loop system even when the neglected modes are asymptotically stable the purpose of this note is to show that this instability phenomenon might take place even when the reduced order design model is strictly proper

bergman type singular integral operators and the characterization of carleson measures for besov sobolev spaces on the complex ball the purposes of this paper are twofold first we extend the method of non homogeneous harmonic analysis of nazarov treil and volberg to handle bergman type singular integral operators the canonical example of such an operator is the beurling transform on the unit disc second we use the methods developed in this paper to settle the important open question about characterizing the carleson measures for the besov sobolev space of analytic functions b sigma 2 on the complex ball of bbb c d in particular we demonstrate that for any sigma 0 the carleson measures for the space are characterized by a t1 condition the method of proof of these results is an extension and another application of the work originated by nazarov treil and the first author

exact solutions of the harmonic oscillator plus non polynomial interaction the exact solutions to a one dimensional harmonic oscillator plus a non polynomial interaction ax2bx2 1cx2 a 0 c 0 are given by the confluent heun functions hc z the minimum value o

quasiconformal distortion of riesz capacities and hausdorff measures in the plane in this paper we prove the sharp distortion estimates for the quasi conformal mappings in the plane both in terms of the riesz capacities from non linear potential theory and in terms of the hausdorff measures

near optimal output feedback regulation of ill conditioned linear systems a two time scale approximation for the linear quadratic optimal output feedback regulator program is examined necessary conditions for optimality as well as an algorithm for computing locally near optimal gains are derived if it is assumed that the slow and fast subsystem initial conditions are uniformly distributed optimal gains for the two time scale problem provide a second order approximation to optimal closed loop performance in the unperturbed system this is verified with a numerical example

riesz transforms of non integer homogeneity on uniformly disconnected sets in this paper we obtain precise estimates for the l norm of the s dimensional riesz transforms on very general measures supported on cantor sets in r with d 1 s d from these estimates we infer that for the so called uniformly disconnected compact sets the capacity s associated with the riesz kernel x x is comparable to the capacity 2 3 d s 3 2 from non linear potential theory

uniform rectifiability calderon zygmund operators with odd kernel and quasiorthogonality in this paper we study some questions in connection with uniform rectifiability and the l boundedness of calderon zygmund operators we show that uniform rectifiability can be characterized in terms of some new adimensional coefficients which are related to the jones numbers we also use these new coefficients to prove that n dimensional calderonzygmund operators with odd kernel of type c2 are bounded in l if is an n dimensional uniformly rectifiable measure

weighted morrey spaces on non homogeneous metric measure spaces abstract let x d be a metric measure space satisfying the so called upper doubling condition and the geometrically doubling condition in this setting the authors introduce the weighted morrey space and the weighted weak morrey space and show several properties of these spaces as applications some weighted weak type estimates for the hardy littlewood maximal operator and the calderon zygmund type operator are established

on the schrodinger equation for the interaction x2 λx2 1 gx2 by forming the 6 6 pade approximants to the energy perturbation series for the interaction x2 lambda x2 1 gx2 as obtained from the hypervirial relations the ground and the first three excited state energy levels are calculated for g or 2 the results of the present calculation are in good agreement with those of the variational calculations the authors also present another set of exact solutions to the schrodinger equation for the interaction x2 lambda x2 1 gx2

on the robustness of output feedback control methods to modeling errors singular perturbation analysis is employed to study the robustness of output feedback control schemes for linear time invariant systems the effect of designing the control strategy using a model that neglects unknown parasitic elements is considered it is shown that such a strategy may in general be destabilizing even when the neglected modes are asymptotically stable special structures that will not permit this situation and will guarantee asymptotic stability are illustrated

boundedness of calderon zygmund operators on non homogeneous metric measure spaces equivalent characterizations national natural science foundation of china 10871025 changjiang scholars and innovative research team in university of china fundamental research funds for the central universities of china 2011121040

near optimum regulators for stochastic linear singularly perturbed systems this paper presents a new approach to the decomposition and approximation of linear quadratic gaussian estimation and control problems for singularly perturbed systems the kalman filter is decomposed into separate slow mode and fast mode filters via the use of a decoupling transformation a near optimal control law is derived by approximating the coefficients of the optimal control law the order of approximation of the optimal performance is 0 mu n where n is the order of approximation of the coefficients

multilinear calderon zygmund operators on the product of lebesgue spaces with non doubling measures abstract under the assumption that is a non doubling measure on r d the author proves that for the multilinear calderon zygmund operator its boundedness from the product of hardy space h 1 h 1 into l 1 2 implies its boundedness from the product of lebesgue spaces l p 1 l p 2 into l p with 1 p 1 p 2 and p satisfying 1 p 1 p 1 1 p 2

principal values for riesz transforms and rectifiability let e rd with hn e where hn stands for the n dimensional hausdorff measure in this paper we prove that e is n rectifiable if and only if the limit lime 0 y e x y ex y x y n 1dhn y exists hn almost everywhere in e to prove this result we obtain precise estimates from above and from below for the l2 norm of the n dimensional riesz transforms on lipschitz graphs

smooth maps null sets for integralgeometric measure and analytic capacity here ifi is the one dimensional lebesgue measure of a set f lying on a line and p0 denotes the orthogonal projection onto the line lo through the origin having normal cos 0 sin 0 we call a map segmental if it maps every line segment onto a line segment note that there are nonaffine segmental diffeomorphisms for example x y 1 x y x the converse of the above theorem also holds if f is a segmental diffeomorphism then i p0e i 0 for almost all o e 0 st implies ijp je i 0 for almost all 0 e 0 st in fact an application of fubini s theorem shows that ipoe i 0 for almost all 0 if and only if almost all lines through almost any point of a do not meet e this latter condition is easily seen to be preserved under segmental diffeomorphisms we can restate the above remarks by saying that a c2 diffeomorphism preserves the null sets for the integralgeometric favard measure if and only if it is segmental for a borel set e c r2 the integralgeometric measure is given by

output feedback control of linear two time scale systems output feedback control of linear time invariant singularly perturbed systems is studied the set of all compensators that stabilize a singularly perturbed system while preserving its two time scale structure is parameterized the parameterization is used to show that any two frequency scale stabilizing compensator can be asymptotically approximated by a compensator designed via a sequential procedure in this procedure a fast high frequency compensator is designed first to stabilize the fast model of the system then a strictly proper slow low frequency compensator is designed to stabilize a modified slow model the parallel connection of the two compensators forms a two frequency scale stabilizing compensator for the singularly perturbed system

finitely randomized dyadic systems and bmo on metric measure spaces abstract we study the connection between bmo and dyadic bmo in metric measure spaces using finitely randomized dyadic systems and give a garnett jones type proof for a theorem of uchiyama on a construction of certain bmo functions we obtain a relation between the bmo norm of a suitable expectation over dyadic systems and the dyadic bmo norms of the original functions in different systems the expectation is taken over only finitely randomized dyadic systems to overcome certain measurability questions applying our result we derive uchiyama s theorem from its dyadic counterpart which we also prove

new integral solutions of the non polynomial oscillator v x x2 λx2 1 gx2 when λ 2g 2 3g the authors find exact solutions of the integral form and the corresponding eigenvalues for the non polynomial oscillator v x x2 x2 1 gx2 the solutions are shown to exist when the parameter is constrained to satisfy a relation of supersymmetric origin involving the coupling constant g

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

numerical fixed point solution for near optimum regulators of linear quadratic gaussian control problems for singularly perturbed systems the fixed point method is developed for obtaining an efficient numerical solution of the linear quadratic gaussian problem for singularly perturbed systems it is shown that each iteration step improves the accuracy by an order of magnitude that is the accuracy 0 ek can be obtained by performing only k 1 iterations in addition only low order systems are involved in algebraic manipulations and no analyticity requirements are imposed on the system coefficients

the tb theorem on non homogeneous spaces 0 introduction main objects and results 3 0 1 main results 4 0 2 an application of t1 heorem electric intensity capacity 7 0 3 how to interpret calderon zygmund operator t 9 0 3 1 bilinear form is defined on lipschitz functions 10 0 3 2 bilinear form is defined for smooth functions 11 0 3 3 apriori boundedness 12 0 4 plan of the paper 15

menger curvature and rectifiability where hi is the 1 dimensional hausdorff measure in rn c x y z is the inverse of the radius of the circumcircle of the triangle x y z that is following the terminology of 6 the menger curvature of the triple x y z a borel set e c rn is said to be purely unrectifiable if for any lipschitz function y r t r 1h l e n r 0 whereas it is said to be rectifiable if there exists a countable family of lipschitz functions yi r e rn such that hi1 e ui yi ir 0 it may be seen from this definition that any 1 set e that is e borel and 0 7 t e oo can be decomposed into two subsets

a theory of state covariance assignment for discrete systems in many regulation control problems it is desired to design the controller so that various system states have acceptable root mean squared values this is a multiple objective problem one way to satisfy these objectives is to assign a specified state covariance to the system this paper introduces and solves the following problem 1 characterize the entire set of state covariances which may be assigned to a linear discrete time system by state feedback and 2 find the set of all state feedback gains which will assign an admissible state covariance to the system extensions are also presented for state estimate feedback

a globally convergent algorithm for the optimal constant output feedback problem a dual version of the levine athans algorithm for solving the optimal constant output feedback gain for continuous time systems is studied global convergence of the procedure to a stationary point of the loss function is proved a global convergence property of the levine athans algorithm is also obtained

bounds on the optimum quadratic cost of structure constrained controllers in this note lower and upper bounds are derived on the quadratic cost of optimal output feedback controllers subject to constraints on the controller structure the use of these bounds in forming a comparative assessment of alternative controller confugurations is described and an illustrative example is given the principal advantage of these bounds arises from the fact that they can save considerable computational effort

singular integrals on self similar sets and removability for lipschitz harmonic functions in heisenberg groups in this paper we study singular integrals on small that is measure zero and lower than full dimensional subsets of metric groups the main examples of the groups we have in mind are euclidean spaces and heisenberg groups in addition to obtaining results in a very general setting the purpose of this work is twofold we shall extend some results in euclidean spaces to more general kernels than previously considered and we shall obtain in heisenberg groups some applications to harmonic in the heisenberg sense functions of some results known earlier in euclidean spaces

parametric lq control the parametric linear quadratic plq control problem is considered for discrete time stochastic systems the control problem consists of a non linear optimization problem in the parameters of a linear regulator results on global convergence and on rate of convergence are given for the linear descent mapping method for solving the optimal regulator parameter values furthermore a non linear mapping method based on the solution of a generalized discrete riccati equation is given the plq control problem is a useful extension of the optimal output feedback problem with applications to the design of low order regulators decentralized control and adaptive control

calderon zygmund operators with non doubling measures in this survey the authors review some basic results on calderon zygmund theory on r d with the non doubling radon measure instead of the usual doubling condition the measure is only assumed to satisfy some growth condition that is there exist positive con stants n 0 d a ndc such that b x r cr n for all x r d and r 0 in this setting the results that we review are the following the hardy space h 1 and the regularized bmo space rbmo some interpolation results related to h 1 a nd rbmo the calderon zygmund decomposition the t 1 theorem and the properties of the calderon zygmund op erator on lebesgue spaces and hardy spaces cotlar s inequality and the boundedness of the maximal calderon zygmund operator the behaviors of the multilinear calderon zygmund op erator on lebesgue spaces weighted norm inequalities for the calderon zygmund operator the boundedness of the commutator generated by the calderon zygmund operator and rbmo function moreover some new progress obtained recently is also included

the recursive algorithm for the optimal static output feedback control problem of linear singularity perturbed systems a recursive algorithm is developed for solving the algebraic equations comprising the solution of the optimal static output feedback control problem of singularly perturbed linear systems the algorithm is very efficient from the numerical point of view since only low order systems are involved in algebraic calculations and the required solution can be easily obtained up to an arbitrary order of accuracy that is o epsilon sup k where epsilon is a small perturbation parameter the real world example demonstrates the failure of o epsilon theory used so far in the study of this problem and the necessity for the existence of o epsilon sup k theory

the planar cantor sets of zero analytic capacity and the local t b theorem in this paper we obtain rather precise estimates for the analytic capacity of a big class of planar cantors sets in fact we show that analytic capacity and positive analytic capacity are comparable for these sets the main tool for the proof is an appropriate version of the t b theorem

characterization and semiadditivity of the c 1 harmonic capacity the c 1 harmonic capacity c plays a central role in problems of approximation by harmonic functions in the c 1 norm in n 1 in this paper we prove the comparability between the capacity c and its positive version c as a corollary we deduce the semiadditivity of c this capacity can be considered as a generalization in n 1 of the continuous analytic capacity in c moreover we also show that the so called inner boundary conjecture fails for dimensions n 1 unlike in the case n 1

convergence of a numerical algorithm for calculating optimal output feedback gains a sequential numerical algorithm is described which obtains gains minimizing a broad class of performance indexes including the standard lq case the primary contribution is a proof that the algorithm converges to a local minimum under nonrestrictive assumptions numerical examples illustrate the theory the second example demonstrates an important lq design technique which permits the designer to prespecify the feedback structure subject to the requirement of output feedback stabilizability

a geometric approach to stabilization by output feedback in this paper we present a geometric approach to the stabilizability of linear systems by output feedback the geometric approach is completely analysed for both discrete time and continuous time single input single output systems two examples are worked out in detail to illustrate the theoretical development we also show how our approach can be extended to dyadic systems the simplicity of the geometric approach in the special case when the number of inputs or outputs is one less than the order of the system is illustrated with the help of examples present address bell laboratories south plainfield new jersey 07080 u s a

multilinear commutators of calderon zygmund operators on hardy type spaces with non doubling measures abstract under the assumption that is a non negative radon measure on r d which only satisfies some growth condition the authors obtain the boundedness in some hardy type spaces of multilinear commutators generated by calderon zygmund operators or fractional integrals with rbmo functions where the hardy type spaces are some appropriate subspaces associated to the considered rbmo functions of the hardy space h 1 of tolsa

double exponential sinc collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function we show that the double exponential sinc collocation method provides an efficient uniformly accurate solution to the one dimensional time independent schrodinger equation for a general class of rational potentials of the form v x p x q x the derived algorithm is based on the discretization of the hamiltonian of the schrodinger equation using sinc expansions this discretization results in a generalized eigenvalue problem the eigenvalues of which correspond to approximations of the energy values of the starting hamiltonian a systematic numerical study is conducted beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree

paper optimal output feedback design of systems with ill conditioned dynamics singular perturbation concepts are exploited to develop a procedure for designing a constant gain output feedback control system it is assumed that the original system is ill conditioned in the sense that the plant contains widely separated dynamics and that an accurate description for the high frequency behavior may not be available the design procedure attempts to stabilize the system by minimizing a quadratic cost function made up of the control and states associated with a reduced order low frequency model for the plant and a measure of stability for the neglected fast dynamics the resulting design procedure does not require knowledge of the fast dynamics

bounded point evaluations for rationally multicyclic subnormal operators abstract let s be a pure bounded rationally multicyclic subnormal operator on a separable complex hilbert space h and let m z be the minimal normal extension on a separable complex hilbert space k containing h let b p e s be the set of bounded point evaluations and let a b p e s be the set of analytic bounded point evaluations we show a b p e s b p e s i n t s the result affirmatively answers a question asked by j b conway concerning the equality of the interior of b p e s and a b p e s for a rationally multicyclic subnormal operator s as a result if 0 i n t s and d i m k e r s 0 n where n is the minimal number of cyclic vectors for s then the range of s 0 is closed hence 0 s e s

holder curves and parameterizations in the analyst s traveling salesman theorem abstract we investigate the geometry of sets in euclidean and infinite dimensional hilbert spaces we establish sufficient conditions that ensure a set of points is contained in the image of a 1 s holder continuous map f 0 1 l 2 with s 1 our results are motivated by and generalize the sufficient half of the analyst s traveling salesman theorem which characterizes subsets of rectifiable curves in r n or l 2 in terms of a quadratic sum of linear approximation numbers called jones beta numbers the original proof of the analyst s traveling salesman theorem depends on a well known metric characterization of rectifiable curves from the 1920s which is not available for higher dimensional curves such as holder curves to overcome this obstacle we reimagine jones non parametric proof and show how to construct parameterizations of the intermediate approximating curves f k 0 1 we then find conditions in terms of tube approximations that ensure the approximating curves converge to a holder curve as an application to the geometry of measures we identify conditions that guarantee fractional rectifiability of pointwise doubling measures in r n

the design of optimal output regulators for linear multivariable systems with constant disturbances a method recently developed for solving the output regulator problem is extended to linear systems with constant disturbances the design of static and low order dynamic regulators for linear systems based on the use of projective controls is reviewed first the proposed method determines through a unified procedure the order and the parameters of the compensator together with the feedback parameters this methodology is then applied to systems with external disturbances and various control structures incorporating integral action are investigated in the light of the proposed methodology

some calderon zygmund kernels and their relations to wolff capacities and rectifiability we consider the calderon zygmund kernels k alpha n x x i 2n 1 x 2n 1 alpha i 1 d in mathbb r d for 0 alpha le 1 and n in mathbb n we show that on the plane for 0 alpha 1 the capacity associated to the kernels k alpha n is comparable to the riesz capacity c frac 2 3 2 alpha frac 3 2 of non linear potential theory as consequences we deduce the semiadditivity and bilipschitz invariance of this capacity furthermore we show that for any borel set e subset mathbb r d with finite length the l 2 mathcal h 1 lfloor e boundedness of the singular integral associated to k 1 n implies the rectifiability of the set e we thus extend to any ambient dimension results previously known only in the plane

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

the semiadditivity of continuous analytic capacity and the inner boundary conjecture let a e be the continuous analytic capacity of a compact set e c c in this paper we obtain a characterization of a in terms of curvature of measures with zero linear density and we deduce that a is countably semiadditive this result has important consequences for the theory of uniform rational approximation on compact sets in particular it implies the so called inner boundary conjecture 1 introduction the continuous analytic capacity a was introduced by erokhin and vitushkin see vi in the 1950s in order to study problems of uniform rational approximation in compact subsets of the complex plane up to now the geometric properties of the capacity a have not been well understood it has been an open question if a is semiadditive as a set function that is to say if

the molecular characterization of the hardy space h 1 on non homogeneous metric measure spaces and its application national natural science foundation of china 11171027 11101339 specialized research fund for the doctoral program of higher education of china 20120003110003 natural science foundation of fujian province of china 2013j01020 fundamental research funds for central universities of china 2013121004

on the anderson moore method for solving the optimal output feedback problem a descent anderson moore method for solving the optimal constant output feedback gains for the stochastic discrete time optimal output feedback problem is discussed an efficient descent mapping algorithm is given in detail the algorithm involves a partial line search mapping implemented as a finite search process to determine a step length parameter so as to guarantee global convergence of the algorithm to a stationary point of the loss function under very mild assumptions furthermore the algorithm involves a scheme to bound the condition numbers of certain critical matrix inverses in the anderson moore method so that the algorithm can be applied to a wider class of control problems

riesz transforms and harmonic lip1 capacity in cantor sets we estimate the l norm of the s dimensional riesz transforms on some cantor sets in rd to this end we show that the riesz transforms truncated at different scales behave in quasiorthogonal way as an application we obtain some precise numerical estimates for the lipschitz harmonic capacity of these sets

fast suboptimal solution to the static output control problem of linear singularly perturbed systems abstract the reduced order suboptimal solution to the static output control problem of linear singularly perturbed system is obtained in terms of the fast variables only assuming the special structure of initial conditions for the slow and fast variables the problem of big initial conditions of the fast variables is considered

computational methods for parametric lq problems a survey iterative methods for finding the optimal constant feedback gains for parametric lq problems notably for optimal constant output feedback problems are surveyed the connections of several methods to loss function expansions are discussed with important implications to the understanding of their convergence properties especially the descent anderson moore method levine athans like methods and the newton method are considered convergence results are also included the initialization problem and the output feedback stabilization problem are also discussed furthermore it is shown that the concepts and methods surveyed in this paper are useful in solving many realistic generalized parametric lq problems as well notably so called robust parametric lq problems

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have

bilipschitz maps analytic capacity and the cauchy integral let c c be a bilipschitz map we prove that if e c is compact and e e stand for its analytic and continuous analytic capacity respectively then c 1 e e c e and c 1 e e c e where c depends only on the bilipschitz constant of further we show that if is a radon measure on c and the cauchy transform is bounded on l 2 then the cauchy transform is also bounded on l 2 where is the image measure of by to obtain these results we estimate the curvature of by means of a corona type decomposition