strict monotonicity for critical points in percolation and ferromagnetic models when is the numerical value of the critical point changed by an enhancement of the process or of the interaction ferromagnetic spin models independent percolation and the contact process are known to be endowed with monotonicity properties in that certain enhancements are capable of shifting the corresponding phase transition in only an obvious direction e g the addition of ferromagnetic couplings can only increase the transition temperature the question explored here is whether enhancements do indeed change the value of the critical point we present a generally applicable approach to this issue for ferromagnetic ising spin systems with pair interactions of finite range ind 2 dimensions it is shown that the critical temperaturetc is strictly monotone increasing in each coupling with the first order derivatives bounded by positive functions which are continuous on the set of fullyd dimensional interactions for independent percolation with 0 pc 1 we prove that any essential enhancement of the process has an effect on the critical probability a result with applications to the question of the existence of entanglements and to invasion percolation with trapping

entanglement in percolation we study the existence of finite and infinite entangled graphs in the bond percolation process in three dimensions with density p after a discussion of the relevant definitions the entanglement critical probabilities are defined the size of the maximal entangled graph at the origin is studied for small p and it is shown that this graph has radius whose tail decays at least as fast as exp n log n indeed the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant we explore the question of almost sure uniqueness of the infinite maximal open entangled graph when p is large and we establish two relevant theorems we make several conjectures concerning the properties of entangled graphs in percolation

towards a universal theory for natural patterns abstract our goal is to find a macroscopic description of patterns that both unifies and simplifies classes of externally stressed dissipative pattern forming systems such as convecting fluids liquid crystals wideband lasers that are seemingly unrelated at the microscopic level we construct an order parameter equation which provides a controlled approximation of the original microscopic field in the limit of large aspect ratios it is built from and is a regularization of the cross newell phase diffusion equation obtained by averaging over the local periodicity of the pattern unlike the latter it is valid for all wavenumbers and can correctly capture the nucleation shape and nontrivial properties of the far fields of disclinations dislocations and grain boundaries it reduces to the cross newell equation away from pattern singularities and to the newell whitehead segel equation near onset as a consequence it correctly determines all the long wave instability boundaries zig zag eckhaus skew varicose of the busse balloon far from onset the order parameter is a real variable but its equation involves a functional corresponding to its local amplitude the local amplitude and phase required for the order parameter equation and the reconstruction of the approximation to the original field respectively are extracted from the order parameter field by wavelet analysis numerical comparisons between solutions of the original equation and the regularized equation are carried out we also explore a new class of singular and weak solutions of the cross newell equation which take account of the energetics of defects as well as their topologies these solutions correspond to convex and concave disclinations and their composites including saddles vortices targets dislocations and two new objects handles and bridges finally we show that phase grain boundaries lines across which the wavevector is discontinuous but the phase is continuous are captured by shock solutions of the phase diffusion equation

defects are weak and self dual solutions of the cross newell phase diffusion equation for natural patterns abstract we show that defects are weak solutions of the phase diffusion equation for the macroscopic order parameter for natural patterns further by exploring a new class of nontrivial solutions for which the graph of the phase function has vanishing gaussian curvature in 3d all sectional curvatures excetp at points we are able to derive explicit expressions which capture the anatomies of point and line and surface defects in two and three dimensional patterns together with their topological characters and energetic constraints

knotted paths in percolation we study the topology of doubly infinite paths in the bond percolation model on the three dimensional cubic lattice we propose a natural definition of a knotted doubly infinite path we prove the existence of a critical probability pk satisfying pc pk there are unknotted doubly infinite paths

the spatio temporal structure of spiral defect chaos abstract we present a study of the recently discovered spatially extended chaotic state known as spiral defect chaos which occurs in low prandtl number large aspect ratio rayleigh benard convection we employ the modulus squared of the space time fourier transform of time series of two dimensional shadowgraph images to construct the structure factor s k this analysis is used to characterize the average spatial and temporal scales of the chaotic state we find that the correlation length and time can be described by power law dependences on the reduced rayleigh number these power laws have as yet no theoretical explanation

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

monotonicity of uniqueness for percolation on cayley graphs all infinite clusters are born simultaneously consider site or bond percolation with retention parameter p on an infinite cayley graph in response to questions raised by grimmett and newman 1990 and benjamini and schramm 1996 we show that the property of having almost surely a unique infinite open cluster is increasing in p moreover in the standard coupling of the percolation models for all parameters a s for all p2 p1 pc each infinite p2 cluster contains an infinite p1 cluster this yields an extension of alexander s 1995 simultaneous uniqueness theorem as a corollary we obtain that the probability v p that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p pc all our results extend to quasi transitive infinite graphs with a unimodular automorphism group

the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

on the analytic signal the teager kaiser energy algorithm and other methods for defining amplitude and frequency this paper compares the teager kaiser algorithm tka and other similar local methods to the analytic signal as procedure the general concepts of the instantaneous amplitude and frequency are discussed it is shown that only as meets certain physical conditions for the amplitude phase and frequency apf the advantage of accuracy and simplicity of the as is also demonstrated

modular generalized springer correspondence iii exceptional groups we complete the construction of the modular generalized springer correspondence for an arbitrary connected reductive group with a uniform proof of the disjointness of induction series that avoids the case by case arguments for classical groups used in previous papers in the series we show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the sylow subgroups of the weyl group under some assumptions we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character we also provide tables and other information on the modular generalized springer correspondence for quasi simple groups of exceptional type including a complete classification of cuspidal pairs in the case of good characteristic and a full determination of the correspondence in type g 2

uniqueness of the infinite entangled component in three dimensional bond percolation we prove uniqueness of the infinite entangled component for bond percolation on the three dimensional cubic lattice above the entanglement critical probability this improves earlier results by grimmett and holroyd

stereoscopic ranging by matching image modulations we apply an am fm surface albedo model to analyze the projection of surface patterns viewed through a binocular camera system this is used to support the use of modulation based stereo matching where local image phase is used to compute stereo disparities the local image phase is an advantageous feature for image matching since the problem of computing disparities reduces to identifying local phase shifts between the stereoscopic image data local phase shifts however are problematic at high frequencies due to phase wrapping when disparities exceed spl plusmn spl pi we meld powerful multichannel gabor image demodulation techniques for multiscale coarse to fine computation of local image phase with a disparity channel model for depth computation the resulting framework unifies phase based matching approaches with am fm surface image models we demonstrate the concepts in a stereo algorithm that generates a dense accurate disparity map without the problems associated with phase wrapping

the supercritical phase of percolation is well behaved we prove a general result concerning the critical probabilities of subsets of a lattice l it is a consequence of this result that the critical probability of a percolation process on l equals the limit of the critical probability of a slice of l as the thickness of the slice tends to infinity this verification of one of the standard hypotheses of the subject settles many questions concerning supercritical percolation

localized measurement of emergent image frequencies by gabor wavelets the authors derive implement and demonstrate a computational approach for the measurement of emergent image frequencies measuring emergent signal frequencies requires spectral measurements accurate in both frequency and time or space conflicting requirements that are shown to be balanced by a generalized uncertainty relationship such spectral measurements can be obtained from the responses of multiple wavelet like channel filters that sample the signal spectrum and that yield a locus of possible solutions for each locally emergent frequency it is shown analytically that this locus of solutions is maximally localized in both space and frequency if the channel filters used are gabor wavelets a constrained solution is obtained by imposing a stabilizing term that develops naturally from the assumptions on the signal the measurement of frequencies is then cast as an ill posed extremum problem regularized by the stabilizing term leading to an iterative constraint propagation algorithm the technique is demonstrated by application to a variety of 2 d textured images

patterns defects and integrability abstract in this paper recent results on the behavior of roll patterns in a class of problems typified by high prandtl number convection are presented a key finding is that the gaussian curvature of the crumpled phase surface which consists of patches with an almost constant wave number line defects on which most of the free energy is stored and point defects with nontrivial topologies condenses onto line and point defects this property allows considerable mathematical simplification in that the fourth order nonlinear diffusion equation governing stationary states can be effectively reduced to the linear helmholtz equation the observed patterns have much is common with the deformation of thin elastic sheets

convection patterns in large aspect ratio systems abstract a theory which should have widespread application is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio rayleigh benard boxes for case of presentation the theory is developed using model equations wavenumber selection the shape of patterns stability and the time dependence resulting from long wavelength instabilities are discussed the effects of adding the local mean drift important in low prandtl number situations are investigated our theory includes the notion of the busse stability balloon reduces near critical values of the stress parameter to the newell whitehead segel equations and contains the pomeau manneville phase equation it also gives a detailed description of the way in which the field amplitude which is slaved to the phase gradient away from threshold becomes an independent order parameter near the critical point

a wavelet based algorithm for pattern analysis abstract patterns with a local roll structure arise in many diverse physical systems which have little in common at the microscopic level in this paper we construct an algorithm based on the wavelet transform that can be used as a diagnostic tool to extract from such patterns macroscopic information like the local director field the local amplitude away from defects and slowly varying fields such as mean drift which may be soft modes of pattern it allows a precise detection of phase grain boundaries and point defects several tests are conducted on numerically generated signals to demonstrate the applicability and precision of the algorithm finally the algorithm is applied to actual experimental convection patterns allowing us to draw several conclusions about the nature of the wave director field in such patterns