su n quantum yang mills theory in two dimensions a complete solution a closed expression of the euclidean wilson loop functionals is derived for pure yang mills continuum theories with gauge groups su n and u 1 and space time topologies r1 r1 and r1 s1 for the u 1 theory we also consider the s1 s1 topology the treatment is rigorous manifestly gauge invariant manifestly invariant under area preserving diffeomorphisms and handles all piecewise analytic loops in one stroke equivalence between the resulting euclidean theory and and the hamiltonian framework is then established finally an extension of the osterwalder schrader axioms for gauge theories is proposed these axioms are satisfied in the present model

a relaxation method for constructing a beltrami flow in a bounded domain it is shown that a time dependent equation generates a beltrami flow or a force free field at time t from non beltrami initial data in an arbitrary bounded domain in three dimensions the generated beltrami flow has a zero normal component on the boundary and its helicity is equal to a prescribed value the existence and uniqueness of a solution to the equation are guaranteed globally in time and no finite time singularity occurs

projective techniques and functional integration for gauge theories a general framework for integration over certain infinite dimensional spaces is first developed using projective limits of a projective family of compact hausdorff spaces the procedure is then applied to gauge theories to carry out integration over the non linear infinite dimensional spaces of connections modulo gauge transformations this method of evaluating functional integrals can be used either in the euclidean path integral approach or the lorentzian canonical approach a number of measures discussed are diffeomorphism invariant and therefore of interest to the connection dynamics version of quantum general relativity the account is pedagogical in particular prior knowledge of projective techniques is not assumed

on the evolution of nearly circular vortex patches recently the classical problem of the evolution of patches of constant vorticity was reformulated as an evolution equation for the boundary of the patch we study this equation in the neighborhood of the circular vortex patch and introduce a hierarchy of area preserving nonlinear approximate equations the first of these equations is shown to have a rich rigid structure it possesses an exhaustive increasing sequence of linear invariant manifolds of arbitrarily large finite dimensions on each of these manifolds the equation can be written as an explicit finite system of ordinary differential equations solutions of these odes starting from arbitrarily small neighborhoods of the circular vortex patch are shown to blow up

a new description of force free magnetic fields the moses curl eigenfunctions are used to describe linear force free magnetic fields enabling a proof that such fields are defined entirely by the value of their curl transform on the unit hemisphere in transform space this change of viewpoint suggests an orderly approach to the exploration and classification of the properties of such fields which is briefly sketched the simplest force free fields defined on zero one and two dimensional sets on the transform sphere are exhibited and the possibility of fields with fractal support sets suggested connections with other mathematical descriptions are pointed out as well as several promising directions for further exploration all results apply equally well to the description of the trkalian subset of beltrami fields in fluid dynamics

instability of the cauchy kovalevskaya solution for a class of nonlinear systems we prove that in any c infty neighborhood of an analytic cauchy datum there exists a smooth function such that the corresponding initial value problem does not have any classical solution for a class of first order nonlinear systems we use a method initiated by g m etivier for elliptic systems based on the representation of solutions and on the fbi transform in our case the system can be hyperbolic at initial time but the characteristic roots leave the real line at positive times

spectral triples of holonomy loops the machinery of noncommutative geometry is applied to a space of connections a noncommutative function algebra of loops closely related to holonomy loops is investigated the space of connections is identified as a projective limit of lie groups composed of copies of the gauge group a spectral triple over the space of connections is obtained by factoring out the diffeomorphism group the triple consist of equivalence classes of loops acting on a hilbert space of sections in an infinite dimensional clifford bundle we find that the dirac operator acting on this hilbert space does not fully comply with the axioms of a spectral triple

on the helicity in 3d periodic navier stokes equations i the non statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

a class large solution of the 3d hall magnetohydrodynamic equations abstract in this paper we construct the global large solution to the three dimensional incompressible hall mhd equations with a class of initial data here the large solution means that the l norm can be arbitrarily large initially

on the helicity in 3d periodic navier stokes equations ii the statistical case we study the asymptotic behavior of the statistical solutions to the navier stokes equations using the normalization map 9 it is then applied to the study of mean energy mean dissipation rate of energy and mean helicity of the spatial periodic flows driven by potential body forces the statistical distribution of the asymptotic beltrami flows are also investigated we connect our mathematical analysis with the empirical theory of decaying turbulence with appropriate mathematically defined ensemble averages the kolmogorov universal features are shown to be transient in time we provide an estimate for the time interval in which those features may still be present our collaborator and friend basil nicolaenko passed away in september of 2007 after this work was completed honoring his contribution and friendship we dedicate this article to him

on the geometric regularity conditions for the 3d navier stokes equations abstract we prove geometrically improved version of prodi serrin type blow up criterion let v and be the velocity and the vorticity of solutions to the 3d navier stokes equations and denote f max f 0 q t r 3 0 t if v v v l x t q t with 3 2 1 for some 3 and 1 2 then the local smooth solution v of the navier stokes equations on 0 t can be continued to 0 t for some 0 we also prove localized version of a special case of this let v be a suitable weak solution to the navier stokes equations in a space time domain containing z 0 x 0 t 0 let q z 0 r b x 0 r t 0 r 2 t 0 be a parabolic cylinder in the domain we show that if either v l x t q z 0 r with 3 2 1 or v v l x t q z 0 r with 3 2 2 2 2 then z 0 is a regular point for v this improves previous local regularity criteria for the suitable weak solutions

global regularity for some classes of large solutions to the navier stokes equations in previous works by the first two authors classes of initial data to the three dimensional incompressible navier stokes equations were presented generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large the main feature of the initial data considered in one of those studies is that it varies slowly in one direction though in some sense it is well prepared its norm is large but does not depend on the slow parameter the aim of this article is to generalize that setting to an ill prepared situation the norm blows up as the small parameter goes to zero as in those works the proof uses the special structure of the nonlinear term of the equation

dissipative continuous euler flows we show the existence of continuous periodic solutions of the 3d incompressible euler equations which dissipate the total kinetic energy

on the support of the ashtekar lewandowski measure we show that the ashtekar isham extension overline a g of the configuration space of yang mills theories a g is topologically and measure theoretically the projective limit of a family of finite dimensional spaces associated with arbitrary finite lattices

zero viscosity limit for analytic solutions of the navier stokes equation on a half space i existence for euler and prandtl equations this is the first of two papers on the zero viscosity limit for the incompressible navier stokes equations in a half space in this paper we prove short time existence theorems for the euler and prandtl equations with analytic initial data in either two or three spatial dimensions the main technical tool in this analysis is the abstract cauchy kowalewski theorem for the euler equations the projection method is used in the primitive variables to which the cauchy kowalewski theorem is directly applicable for the prandtl equations cauchy kowalewski is applicable once the diffusion operator in the vertical direction is inverted

beltrami fields with a nonconstant proportionality factor are rare we consider the existence of beltrami fields with a nonconstant proportionality factor f in an open subset u of mathbb r 3 by reformulating this problem as a constrained evolution equation on a surface we find an explicit differential equation that f must satisfy whenever there is a nontrivial beltrami field with this factor this ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the c k topology k geqq 7 in particular there are no nontrivial beltrami fields whenever f has a regular level set diffeomorphic to the sphere this provides an explanation of the helical flow paradox of morgulis et al commun pure appl math 48 571 582 1995

uniqueness of diffeomorphism invariant states on holonomy flux algebras loop quantum gravity is an approach to quantum gravity that starts from the hamiltonian formulation in terms of a connection and its canonical conjugate quantization proceeds in the spirit of dirac first one defines an algebra of basic kinematical observables and represents it through operators on a suitable hilbert space in a second step one implements the constraints the main result of the paper concerns the representation theory of the kinematical algebra we show that there is only one cyclic representation invariant under spatial diffeomorphisms while this result is particularly important for loop quantum gravity we are rather general the precise definition of the abstract algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group the variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection the uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory

extractor codes we study error correcting codes for highly noisy channels for example every received signal in the channel may originate from some half of the symbols in the alphabet our main conceptual contribution is an equivalence between error correcting codes for such channels and extractors our main technical contribution is a new explicit error correcting code based on trevisan s extractor that can handle such channels and even noisier ones our new code has polynomial time encoding and polynomial time soft decision decoding we note that reed solomon codes cannot handle such channels and our study exposes some limitations on list decoding of reed solomon codes another advantage of our equivalence is that when the johnson bound is restated in terms of extractors it becomes the well known leftover hash lemma this yields a new proof of the johnson bound which applies to large alphabets and soft decoding our explicit codes are useful in several applications first they yield algorithms to extract many hardcore bits using few auxiliary random bits second they are the key tool in a recent scheme to compactly store a set of elements in a way that membership in the set can be determined by looking at only one bit of the representation finally they are the basis for the recent construction of high noise almost optimal rate list decodable codes over large alphabets

quantization of diffeomorphism invariant theories of connections with local degrees of freedom quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions this provides in particular a quantization of the husain kucha model the main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity in the riemannian case i e signature the approach appears to contain all the necessary ingredients already in the lorentzian case it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections

coherent state transforms for spaces of connections abstract the segal bargmann transform plays an important role in quantum theories of linear fields recently hall obtained a non linear analog of this transform for quantum mechanics on lie groups given a compact connected lie groupgwith its normalized haar measure h the hall transform is an isometric isomorphism froml2 g h to h g c l2 g c whereg c the complexification ofg h g c the space of holomorphic functions ong c and an appropriate heat kernel measure ong c we extend the hall transform to the infinite dimensional context of non abelian gauge theories by replacing the lie groupgby a certain extension of the space a g of connections modulo gauge transformations the resulting coherent state transform provides a holomorphic representation of the holonomyc algebra of real gauge fields this representation is expected to play a key role in a non perturbative canonical approach to quantum gravity in 4 dimensions

stratification of the generalized gauge orbit space different versions for defining ashtekar s generalized connections are investigated depending on the chosen smoothness category for the paths and graphs the label set for the projective limit our definition covers the analytic case as well as the case of webs

extractors for a constant number of polynomially small min entropy independent sources we consider the problem of randomness extraction from independent sources we construct an extractor that can extract from a constant number of independent sources of length n each of which have min entropy n gamma for an arbitrarily small constant gamma 0 our extractor is obtained by composing seeded extractors in simple ways we introduce a new technique to condense independent somewhere random sources which looks like a useful way to manipulate independent sources our techniques are different from those used in recent work b barak r impagliazzo and a wigderson siam j comput 36 2006 pp 1095 1118 b barak g kindler r shaltiel b sudakov and a wigderson simulating independence new constructions of condensers ramsey graphs dispersers and extractors in proceedings of the 37th annual acm symposium on theory of computing acm new york 2005 pp 1 10 r raz extractors with weak random seeds in proceedings of the 37th annual acm symposium on theory of computing acm new york 2005 pp 11 20 j bourgain int j number theory 1 2005 pp 1 32 for this problem in the sense that they do not rely on any results from arithmetic combinatorics using an extractor of bourgain s int j number theory 1 2005 pp 1 32 as a black box we obtain a new extractor for two independent block sources with few blocks even when the min entropy is as small as operatorname polylog n we also show how to modify the 2 source disperser for linear min entropy of barak et al simulating independence new constructions of condensers ramsey graphs dispersers and extractors in proceedings of the 37th annual acm symposium on theory of computing acm new york 2005 pp 1 10 and the three source extractor of raz extractors with weak random seeds in proceedings of the 37th annual acm symposium on theory of computing acm new york 2005 pp 11 20 to get dispersers extractors with exponentially small error and linear output length where previously both were constant

the second iterate for the navier stokes equation we consider the iterative resolution scheme for the navier stokes equation and focus on the second iterate more precisely on the map from the initial data to the second iterate at a given time t we investigate boundedness properties of this bilinear operator this new approach yields very interesting results a new perspective on koch tataru solutions a first step towards weak strong uniqueness for koch tataru solutions and finally an instability result in b q 1 for q 2

unbiased bits from sources of weak randomness and probabilistic communication complexity a new model for weak random physical sources is presented the new model strictly generalizes previous models e g the santha and vazirani model 27 the sources considered output strings according to probability distributions in which no single string is too probable the new model provides a fruitful viewpoint on problems studied previously such as extracting almost perfect bits from sources of weak randomness the question of possibility as well as the question of efficiency of such extraction schemes are addressed probabilistic communication complexity it is shown that most functions have linear communication complexity in a very strong probabilistic sense robustness of bpp with respect to sources of weak randomness generalizing a result of vazirani and vazirani 32 33

structure of helicity and global solutions of incompressible navier stokes equation in this paper we derive a new energy identity for the three dimensional incompressible navier stokes equations by a special structure of helicity the new energy functional is critical with respect to the natural scalings of the navier stokes equations moreover it is conditionally coercive as an application we construct a family of finite energy smooth solutions to the navier stokes equations whose critical norms can be arbitrarily large

well posedness for the navier stokes equations where u is the velocity and p is the pressure it is well known that the navierstokes equations are locally well posed for smooth enough initial data as long as one imposes appropriate boundary conditions on the pressure at for instance it is easy to see see 9 for much more general results that if s n 2 then for any h initial data there exists a unique c 0 t h r local solution with a pressure p c 0 t h r in the sequel we consider solutions for less regular initial data this has to be understood in the sense that the map from the initial data to the solution extends continuously to rougher function spaces the question we are interested in is the global well posedness for small data and local well posedness for large data with respect to a certain space of

complex trkalian fields and solutions to euler s equations for the ideal fluid we consider solutions to the complex trkalian equation vec nabla times vc vc where vc is a 3 component vector function with each component in the complex field and may be expressed in the form vc e ig vec nabla f with g real and f complex we find there are precisely two classes of solutions one where g is a cartesian variable and one where g is the spherical radial coordinate we consider these flows to be the simplest of all exact 3 d solutions to the euler s equation for the ideal incompressible fluid the novel approach we use in solving for these classes of solutions to these 3 dimensional vector pdes involves differential geometric techniques one may employ the method to generate solutions to other classes of vector pdes

physical properties of quantum field theory measures well known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory the difference of typical configurations of free massive scalar field theories with different masses is studied we apply the same methods to study the ashtekar lewandowski al measure on spaces of connections in particular we prove that the diffeomorphism group acts ergodically with respect to the al measure on the ashtekar isham space of quantum connections modulo gauge transformations we also prove that a typical with respect to the al measure quantum connection restricted to a piecewise analytic curve leads to a parallel transport discontinuous at every point of the curve

global smooth solutions of mhd equations with large data abstract in this paper we establish the global existence of smooth solutions of the three dimensional mhd system for a class of large initial data both the initial velocity and magnetic field can be arbitrarily large in the critical norm

the inverse loop transform the loop transform in quantum gauge field theory can be recognized as the fourier transform or characteristic functional of a measure on the space of generalized connections modulo gauge transformations since this space is a compact hausdorff space conversely we know from the riesz markov theorem that every positive linear functional on the space of continuous functions thereon qualifies as the loop transform of a regular borel measure on the moduli space in the present article we show how one can compute the finite joint distributions of a given characteristic functional that is we derive the inverse loop transform

projective loop quantum gravity i state space instead of formulating the state space of a quantum field theory over one big hilbert space it has been proposed by kijowski to describe quantum states as projective families of density matrices over a collection of smaller simpler hilbert spaces beside the physical motivations for this approach it could help designing a quantum state space holding the states we need in a latter work by okolow the description of a theory of abelian connections within this framework was developed an important insight being to use building blocks labeled by combinations of edges and surfaces the present work generalizes this construction to an arbitrary gauge group g in particular g is neither assumed to be abelian nor compact this involves refining the definition of the label set as well as deriving explicit formulas to relate the hilbert spaces attached to different labels if the gauge group happens to be compact we also have at our disposal the well established ashtekar lewandowski hilbert space which is

notes on the geometric regularity criterion of 3d navier stokes system in this note we show that if the volume of parallelepiped which is defined by three unit vectors direction of the velocity direction of the vorticity and the direction of the curl of the vorticity is small enough then the solution to the 3d navier stokes equations is classical

the beltrami spectrum for incompressible fluid flows recently v yakhot s orszag and their co workers have suggested that turbulent flows in various regions of space organize into a coherent hierarchy of weakly interacting superimposed approximate beltrami flows a mathematical framework is developed here to study organized beltrami hierarchies in a systematic fashion this framework is applied to several important classes of examples with universal beltrami hierarchies an analysis of the persistence of such beltrami hierarchies is also presented for general solutions of the navier stokes equations

an efficient reduction from two source to nonmalleable extractors achieving near logarithmic min entropy the breakthrough result of chattopadhyay and zuckerman explicit two source extractors and resilient functions in proceedings of the 48th annual acm sigact symposium on theory of computing stoc

stratification of the orbit space in gauge theories the role of nongeneric strata gauge theory is a theory with constraints and for that reason the space of physical states is not a manifold but a stratified space orbifold with singularities the classification of strata for smooth and generalized connections as well as the formulation of the physical space as the zero set of a momentum map is reviewed several important features of nongeneric strata are discussed and new results are presented suggesting an important role for these strata as concentrators of the measure in ground state functionals and as a source of multiple structures in low lying excitations

a length operator for canonical quantum gravity we construct an operator that measures the length of a curve in four dimensional lorentzian vacuum quantum gravity we work in a representation in which an su 2 connection is diagonal and it is therefore surprising that the operator obtained after regularization is densely defined does not suffer from factor ordering singularities and does not require any renormalization we show that the length operator admits self adjoint extensions and compute part of its spectrum which like its companions the volume and area operators already constructed in the literature is purely discrete and roughly quantized in units of the planck length the length operator contains full and direct information about all the components of the metric tensor which facilitates the construction of so called weave states which approximate a given classical three geometry

on the global regularity of a helical decimated version of the 3d navier stokes equations we study the global regularity for all time and all initial data in h 1 2 o f ar e cently introduced decimated version of the incompressible 3d navier stokes dns equa tions the model is based on a projection of the dynamical evolution of navier stokes ns equations into the subspace where helicity the l 2 scalar product of velocity and vorticity is sign definite the presence of a second beside energy sign definite inviscid conserved quadratic quantity which is equivalent to the h 1 2 sobolev norm allows us to demonstrate global existence and uniqueness of space periodic solutions together with continuity with respect to the initial conditions for this decimated 3d model this is achieved thanks to the establishment of two new estimates for this 3d model which show that the h 1 2 and the time average of the square of the h 3 2 norms of the velocity field remain finite such two additional bounds are known in the spirit of the work of h fujita and t kato arch ration mech anal 16 269 315 1964 rend semin mat univ padova 32 243 260 1962 to be sufficient for showing well posedness for the 3d ns equations furthermore they are di rectly linked to the helicity evolution for the dns model and therefore with a clear physical meaning and consequences

closed formula for the matrix elements of the volume operator in canonical quantum gravity we derive a closed formula for the matrix elements of the volume operator for canonical lorentzian quantum gravity in four space time dimensions in the continuum in a spin network basis we also display a new technique of regularization which is state dependent but we are forced to it in order to maintain diffeomorphism covariance and in that sense it is natural we arrive naturally at the expression for the volume operator as defined by ashtekar and lewandowski up to a state independent factor

some notes on periodic beltrami fields in cartesian geometry some mathematical facts about beltrami fields which are time independent solutions of the three dimensional incompressible euler equations with nontrivial helicity are assembled the linearization about an arbitrary fixed beltrami field is studied in a hamiltonian framework a factorization of this linearization is introduced and used to characterize null spaces for beltrami fields with ergodic streamlines an interesting property of unstable modes is noticed and its consequences concerning the nature of potential instabilities are discussed

on global regularity of incompressible mhd equations in r3 abstract this article considers the existence of global smooth solutions to the cauchy problem in 3d incompressible magnetohydrodynamic mhd flows and prove the global regularity of classical solutions for a family of large initial data with finite energy

construction of three dimensional stationary euler flows from pseudo advected vorticity equations some vorticity equations with pseudoadvection terms are proposed they yield threedimensional stationary euler flows at t in particular one of them generates the beltrami flow

extracting randomness using few independent sources in this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1 2 specifically for every delta 0 we give an explicit construction for extracting randomness from a constant depending polynomially on 1 delta number of distributions over bits n each having min entropy delta n these extractors output n bits that are 2 n close to uniform this construction uses several results from additive number theory and in particular a recent result of bourgain et al we also consider the related problem of constructing randomness dispersers for any constant output length m our dispersers use a constant number of identical distributions each with requires min entropy omega log n and outputs every possible m bit string with positive probability the main tool we use is a variant of the stepping up lemma of erdods and hajnal used in establishing a lower bound on the ramsey number for hypergraphs

on the nonlinear instability of euler and prandtl equations in this paper we give examples of nonlinearly unstable solutions of euler equations in the whole space 2 the half space the periodic strip the strip 1 1 and the periodic torus 2 with an application to vortex sheets using the same methods we prove an instability result for prandtl type boundary layers that appear in and 2000 john wiley sons inc

an infinite dimensional calculus for generalized connections on hypercubic lattices a space for gauge theories is defined using projective limits as subsets of cartesian products of homomorphisms from a lattice on the structure group in this space non interacting and interacting measures are defined as well as functions and operators from projective limits of test functions and distributions on products of compact groups a projective gauge triplet is obtained which provides a framework for the infinite dimensional calculus in gauge theories the gauge measure behavior on non generic strata is also obtained

travelling wave like solutions of the navier stokes and the related equations abstract we present a new family of travelling wave like solutions to the navier stokes equations of incompressible fluid flows and other regularized equations of the euler equations obtain their trend to the solutions of the euler equations as the viscosity tends to zero and estimate the rate of convergence we also find a singularizing effect of the viscosity term in the navier stokes equations i e we have a local moving blow up of unbounded solutions with the blow up s speed depending on viscosity we demonstrate that if the initial function is the beltrami flow then the solution of the navier stokes equations conserves the beltrami flow property for all time

quantum group connections the ahtekar isham c algebra known from loop quantum gravity is the algebra of continuous functions on the space of generalized connections with a compact structure lie group the algebra can be constructed by some inductive techniques from the c algebra of continuous functions on the group and a family of graphs embedded in the manifold underlying the connections we generalize the latter construction replacing the commutative c algebra of continuous functions on the group by a noncommutative c algebra defining a compact quantum group

a new type of loop independence and su n quantum yang mills theory in two dimensions the expectation values of wilson loop products for the pure euclidean yang mills theory on r r given by ashtekar et al in the article su n quantum yang mills theory in two dimensions a complete solution j math phys 38 5453 1997 are determined directly for all piecewise analytic loops for that purpose we enlarge their calculations from quadratic lattices to general floating lattices introducing a new kind of loop independence and slightly modifying the regularization scheme

two source dispersers for polylogarithmic entropy and improved ramsey graphs in his 1947 paper that inaugurated the probabilistic method erdos proved the existence of 2 o 1 log n ramsey graphs on n vertices matching erdos s result with a constructive proof is cons

2 source dispersers for n o 1 entropy and ramsey graphs beating the frankl wilson construction the main result of this paper is an explicit disperser for two independent sources on n bits each of min entropy k 2 1 0 for some small absolute constant 0 0 put differently setting n 2 and k 2 we construct an explicit n n boolean matrix for which no k k sub matrix is monochromatic viewed as the adjacency matrix of a bipartite graph this gives an explicit construction of a bipartite k ramsey graph of 2n vertices this improves the previous the previous bound of k o n of barak kindler shaltiel sudakov and wigderson bks05 as a corollary we get a construction of a 2 log1 0 n non bipartite ramsey graph of 2 vertices significantly improving the previous bound of 2o n due to frankl and wilson fw81 we also give a construction of a new independent sources extractor that can extract from a constant number of sources of polynomially small min entropy with exponentially small error this improves independent sources extractor of rao rao06 which only achieved polynomially small error our dispersers combine ideas and constructions from several previous works in the area together with some new ideas in particular we rely on the extractors of raz raz05 and bourgain bou05 as well as an improved version of the extractor of rao rao06 a key ingredient that allows us to beat the barrier of k n is a new and more complicated variant of the challenge response mechanism of barak et al bks05 that allows us to locate the min entropy concentrations in a source of low min entropy

simulating independence new constructions of condensers ramsey graphs dispersers and extractors we present new explicit constructions of deterministic randomness extractors dispersers and related objects we say that a distribution x on binary strings of length n is a source if x assigns probability at most 2 n to any string of length n for every 0 we construct the following poly n time computable functions 2 source disperser d 0 1 n 2 0 1 such that for any two independent sources x1 x2 we have that the support of d x1 x2 is 0 1 bipartite ramsey graph let ne2n a corollary is that the function d is a 2 coloring of the edges of kn n the complete bipartite graph over two sets of n vertices such that any induced subgraph of size n by n is not monochromatic 3 source extractor e 0 1 n 3 0 1 such that for any three independent sources x1 x2 x3 we have that e x1 x2 x3 is o 1 close to being an unbiased random bit no previous explicit construction was known for either of these for any a component in these results is a new construction of condensers that may be of independent interest this is a function c 0 1 n 0 1 n c d where c and d are constants that depend only on such that for every source x one of the output blocks of c x is exponentially close to a 0 9 source this result was obtained independently by ran raz the constructions are quite involved and use as building blocks other new and known objects a recurring theme in these constructions is that objects that were designed to work with independent inputs sometimes perform well enough with correlated high entropy inputs the construction of the disperser is based on a new technique which we call the challenge response mechanism that in some sense allows identifying high entropy regions in a given pair of sources using only one sample from the two sources