the mobius function and distal flows we prove that the mobius function is linearly disjoint from an analytic skew product on the 2 torus these flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points the previous cases for which such disjointness has been proved are all regular we also establish the linear disjointness of the mobius function from various distal homogeneous flows

random walk percolation based modeling of two phase flow in porous media breakthrough time and net to gross ratio estimation abstract fluid flow modeling in porous media has many applications in waste treatment hydrology and petroleum engineering in any geological model flow behavior is controlled by multiple properties these properties must be known in advance of common flow simulations when uncertainties are present deterministic modeling often produces poor results percolation and random walk rw methods have recently been used in flow modeling their stochastic basis is useful in dealing with uncertainty problems they are also useful in finding the relationship between porous media descriptions and flow behavior this paper employs a simple methodology based on random walk and percolation techniques the method is applied to a well defined model reservoir in which the breakthrough time distributions are estimated the results of this method and the conventional simulation are then compared the effect of the net to gross ratio on the breakthrough time distribution is studied in terms of shannon entropy use of the entropy plot allows one to assign the appropriate net to gross ratio to any porous medium

properties of nodal domains in a pseudointegrable barrier billiard we investigate experimentally the scaling properties of the number of nodal domains of the wave functions and the distribution of the nodal domain areas in a pseudointegrable barrier billiard the number of nodal domains is smaller than expected for billiards whose dynamics is chaotic or regular this reduction is explained by the appearance of superscars in the barrier billiard i e regular structures which are embedded into the underlying chaotic wave functions furthermore the area distribution of nodal domains follows the predictions of a percolation model for chaotic systems

mobius disjointness for topological models of ergodic measure preserving systems with quasi discrete spectrum abstract by using a decomposition result for ergodic measure preserving system with quasi discrete spectrum we prove that a generic point of an ergodic quasi discrete spectrum measure in a topological dynamical system satisfies the required disjointness condition in sarnak s mobius disjointness conjecture as a direct application we have that sarnak s mobius disjointness conjecture holds for any topological model of an ergodic measure preserving system with quasi discrete spectrum

connected floer homology of covering involutions using the covering involution on the double branched cover of s 3 branched along a knot and adapting ideas of hendricks manolescu and hendricks hom lidman we define new knot concordance invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots

effect of anisotropy on finite size scaling in percolation theory we investigate the effects of anisotropy on finite size scaling of site percolation in two dimensions we consider a lattice of size n x ifmmode times else texttimes fi n y we define an aspect ratio ensuremath omega n x n y and consider the mean connected fraction p averaged over the realizations as a function of the site occupancy probability p the system size n x and this aspect ratio it is clear that there is an easy direction for percolation which is in the short direction i e y if ensuremath omega g1 and a difficult direction which is along the long axis we define an apparent percolation threshold in each direction as the value of p when 50 of realizations connect in that direction we show that standard finite size scaling applies if we use this apparent threshold we also find a finite size scaling for the fluctuations about this mean connected fraction

polynomial sequences in groups abstract given a groupgwith lower central seriesg g1 g2 g3 we say that a sequenceg z gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference operatordg n g n 1g n 1 dtimes takes its values ingk we introduce the notion ofthe degree of a polynomial sequenceand we prove that polynomial sequences of degrees not exceeding a given one form a group as an application we obtain the following extension of the hall petresco theorem theorem letg g1 g2 g3 be the lower central series of a group g let x gk y gland letp qbe polynomials z z of degrees k and l respectively then there is a sequencez0 g zi gifori n such thatfor all n n

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

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

dynamical approximation entropies and topological entropy in operator algebras dynamical entropy invariants based on a general approximation approach are introduced for c and w algebra automorphisms this includes a noncommutative extension of topological entropy

holomorphic disks and three manifold invariants properties and applications in 27 we introduced floer homology theories hf y s hf y s hf y t hf y s and hf red y s associated to closed oriented three manifolds y equipped with a spiny structures s spin c y in the present paper we give calculations and study the properties of these invariants the calculations suggest a conjectured relationship with seiberg witten theory the properties include a relationship between the euler characteristics of hf and turaev s torsion a relationship with the minimal genus problem thurston norm and surgery exact sequences we also include some applications of these techniques to three manifold topology

the slice ribbon conjecture for 3 stranded pretzel knots we determine the smooth concordance order of the 3 stranded pretzel knots p p q r with p q r odd we show that each one of finite order is in fact ribbon thereby proving the slice ribbon conjecture for this family of knots as corollaries we give new proofs of results first obtained by fintushel stern and casson gordon

embedding seifert manifolds in s 4 using an obstruction based on donaldson s theorem on the intersection forms of definite 4 manifolds we determine which connected sums of lens spaces smoothly embed in s4 we also find constraints on the seifert invariants of seifert 3 manifolds which embed in s4 when either the base orbifold is non orientable or the first betti number is odd in addition we construct some new embeddings and use these along with the d and invariants to examine the question of when the double branched cover of a 3 or 4 strand pretzel link embeds

sequences from zero entropy noncommutative toral automorphisms and sarnak conjecture abstract in this paper we study c algebra version of sarnak conjecture for noncommutative toral automorphisms let a be a noncommutative torus and be the noncommutative toral automorphism arising from a matrix s g l d z we show that if the voiculescu brown entropy of is zero then the sequence n u n z is a sum of a nilsequence and a zero density sequence where u a and is any state on a then by a result of green and tao 9 this sequence is linearly disjoint from the mobius function

the lens space realization problem we determine the lens spaces that arise by integer dehn surgery along a knot in the three sphere specically if surgery along a knot produces a lens space then there exists an equivalent surgery along a berge knot with the same knot floer homology groups this leads to sharp information about the genus of such a knot the arguments rely on tools from floer homology and lattice theory they are primarily combinatorial in nature

dynamical entropy in banach spaces we introduce a version of voiculescu brown approximation entropy for isometric automorphisms of banach spaces and develop within this framework the connection between dynamics and the local theory of banach spaces as discovered by glasner and weiss our fundamental result concerning this contractive approximation entropy or ca entropy characterizes the occurrence of positive values both geometrically and topologically this leads to various applications for example we obtain a geometric description of the topological pinsker factor and show that a c algebra is type i if and only if every multiplier inner automorphism has zero ca entropy we also examine the behaviour of ca entropy under various product constructions and determine its value in many examples including isometric automorphisms of l p for 1 p and noncommutative tensor product shifts

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

measure complexity and mobius disjointness abstract in this paper the notion of measure complexity is introduced for a topological dynamical system and it is shown that sarnak s mobius disjointness conjecture holds for any system for which every invariant borel probability measure has sub polynomial measure complexity we then apply this result to a number of situations including certain systems whose invariant measure don t all have discrete spectrum

theoretical investigation of finite size effects at dna melting we investigated how the finiteness of the length of a sequence affects the phase transition that takes place at the dna melting temperature for this purpose we modified the transfer integral method to adapt it to the calculation of both extensive partition function entropy specific heat etc and nonextensive order parameter and average separation between paired bases thermodynamic quantities of finite sequences with open boundary conditions and applied the modified procedure to two different dynamical models we characterized in some detail the three effects that take place when the length of the sequence is decreased namely i the decrease of the critical temperature ii the decrease of the peak values of all quantities that diverge at the thermodynamic limit but remain finite for finite sequences like the specific heat and the correlation length and iii the broadening of the temperature range over which the transition affects the dynamics of the system we also performed a finite size scaling analysis of the two models and showed that the singular part of the free energy can indeed be expressed in terms of a homogeneous function however josephson s identity is satisfied for none of the investigated models so that the derivation of the characteristic exponents which appear for example in the expression of the specific heat requires some care

the orientation of yang mills moduli spaces and 4 manifold topology si x est une 4 variete lisse orientee fermee dont la forme intersection q h 2 x z torsion z est definie negative alors la forme est equivalente sur les entiers a la forme standard 1 1 1 soit x une 4 variete lisse orientee fermee avec les trois proprietes suivantes 1 h 1 x z n a pas de 2 torsion 2 la forme intersection q sur h 2 x torsion a une partie positive de rang 1 ou 2 3 la forme intersection est paire alors q est equivalente sur les entiers a une des formes 01 10 01 10 01 10

orientability for gauge theories on calabi yau manifolds abstract we study orientability issues of moduli spaces from gauge theories on calabi yau manifolds our results generalize and strengthen those for donaldson thomas theory on calabi yau manifolds of dimensions 3 and 4 we also prove a corresponding result in the relative situation which is relevant to the gluing formula in dt theory

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

rational homology spheres and the four ball genus of knots using the heegaard floer homology of ozsvath and szabo we investigate obstructions to a rational homology sphere bounding a four manifold with a definite intersection pairing as an application we obtain new lower bounds for the four ball genus of montesinos links

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

effect of anisotropy on the scaling of connectivity and conductivity in continuum percolation theory we investigate the effects of anisotropy on the finite size scaling of connectivity and conductivity of continuum percolation in three dimensions we consider a system of size x ifmmode times else texttimes fi y ifmmode times else texttimes fi z in which cubic bodies of size a ifmmode times else texttimes fi b ifmmode times else texttimes fi c are placed randomly we define two aspect ratios to request anisotropy then we expect that the displacement of average connected fraction p averaged over the realizations about the isotropic universal curves will be a function of the two aspect ratios this is accounted by considering an apparent percolation threshold in each direction which leads to 50 of realizations connecting in that direction we find the aspect ratios dependency of the apparent threshold and investigate the finite size scaling transformations for the mean connected fraction and its associated fluctuations moreover we apply a single phase pressure solver to determine the conductivity of various realizations of the system finally we apply the same idea to account for the effect of anisotropy on the conductivity scaling

absolutely graded floer homologies and intersection forms for four manifolds with boundary abstract in ozsvath and szabo holomorphic triangles and invariants for smooth four manifolds math sg 0110169 2001 we introduced absolute gradings on the three manifold invariants developed in ozsvath and szabo holomorphic disks and topological invariants for closed three manifolds math sg 0101206 ann of math 2001 to appear coupled with the surgery long exact sequences we obtain a number of three and four dimensional applications of this absolute grading including strengthenings of the complexity bounds derived in ozsvath and szabo holomorphic disks and three manifold invariants properties and applications math sg 0105202 ann of math 2001 to appear restrictions on knots whose surgeries give rise to lens spaces and calculations of hf for a variety of three manifolds moreover we show how the structure of hf constrains the exoticness of definite intersection forms for smooth four manifolds which bound a given three manifold in addition to these new applications the techniques also provide alternate proofs of donaldson s diagonalizability theorem and the thom conjecture for cp 2

donaldson s theorem heegaard floer homology and knots with unknotting number one abstract we establish an obstruction to unknotting an alternating knot by a single crossing change the obstruction is lattice theoretic in nature and combines donaldson s diagonalization theorem with an obstruction developed by ozsvath and szabo using heegaard floer homology as an application we enumerate the alternating 3 braid knots with unknotting number one and show that each has an unknotting crossing in its standard alternating diagram

on the heegaard floer homology of branched double covers abstract let l s 3 be a link we study the heegaard floer homology of the branched double cover l of s 3 branched along l when l is an alternating link hf of its branched double cover has a particularly simple form determined entirely by the determinant of the link for the general case we derive a spectral sequence whose e 2 term is a suitable variant of khovanov s homology for the link l converging to the heegaard floer homology of l