a q analogue of a ramanujan type supercongruence involving central binomial coefficients abstract motivated by zudilin s work we give a q analogue of a ramanujan type supercongruence of van hamme and mortenson via the q wz method meanwhile we give a q analogue of a related congruence of sun in the same way we also propose several related conjectures on congruences involving central q binomial coefficients

embedded trees estimation of gaussian processes on graphs with cycles graphical models provide a powerful general framework for encoding the structure of large scale estimation problems however the graphs describing typical real world phenomena contain many cycles making direct estimation procedures prohibitively costly in this paper we develop an iterative inference algorithm for general gaussian graphical models it operates by exactly solving a series of modified estimation problems on spanning trees embedded within the original cyclic graph when these subproblems are suitably chosen the algorithm converges to the correct conditional means moreover and in contrast to many other iterative methods the tree based procedures we propose can also be used to calculate exact error variances although the conditional mean iteration is effective for quite densely connected graphical models the error variance computation is most efficient for sparser graphs in this context we present a modeling example suggesting that very sparsely connected graphs with cycles may provide significant advantages relative to their tree structured counterparts thanks both to the expressive power of these models and to the efficient inference algorithms developed herein the convergence properties of the proposed tree based iterations are characterized both analytically and experimentally in addition by using the basic tree based iteration to precondition the conjugate gradient method we develop an alternative accelerated iteration that is finitely convergent simulation results are presented that demonstrate this algorithm s effectiveness on several inference problems including a prototype distributed sensing application

a q analogue of the j 2 supercongruence of van hamme abstract the l 2 supercongruence of van hamme was proved by swisher recently in this paper we provide a conjectural q analogue of the l 2 supercongruence of van hamme and prove a weaker form of it by using the q wz method in the same way we prove a complete q analogue of the following congruence k 0 n 6 k 1 2 k k 3 512 n k 0 mod 4 2 n 1 2 n n which was conjectured by z w sun and confirmed by b he we also provide a conjectural q analogue of another congruence proved by swisher

factors of some truncated basic hypergeometric series abstract we prove that certain basic hypergeometric series truncated at k n 1 have the factor n q 2 where n q is the n th cyclotomic polynomial this confirms two recent conjectures of the author and zudilin we also put forward some conjectures on q congruences modulo n q 2

a generalized levinson algorithm for covariance extension with application to multiscale autoregressive modeling efficient computation of extensions of banded partially known covariance matrices is provided by the classical levinson algorithm one contribution of this paper is the introduction of a generalization of this algorithm that is applicable to a substantially broader class of extension problems this generalized algorithm can compute unknown covariance elements in any order that satisfies certain graph theoretic properties which we describe this flexibility which is not provided by the classical levinson algorithm is then harnessed in a second contribution of this paper the identification of a multiscale autoregressive mar model for the maximum entropy me extension of a banded partially known covariance matrix the computational complexity of mar model identification is an order of magnitude below that of explicitly computing a full covariance extension and is comparable to that required to build a standard autoregressive ar model using the classical levinson algorithm

properties of the solutions of the linearized monge ampere equation let r n r be a function strictly convex and smooth and det d 2 is the monge ampere generated by given x r n and t 0 let s x t y r n y x x y x t the purpose of this paper is to study the properties of the solutions of the linearized monge ampere equation given by a ij x d ij u 0 where the coefficients a ij x are the cofactors of the matrix d 2 x it is assumed that satisfies a doubling condition on the sets s x t and a uniform continuity condition at every scale with respect to lebesgue measure we establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant harnack s inequality on the sections s x t all the estimates are independent of the regularity of and depend only on the constants in the hypotheses made on the measure

multiscale recursive estimation data fusion and regularization we describe a framework for modeling stochastic phenomena at multiple scales and for their efficient estimation or reconstruction given partial and or noisy measurements which may also be at several scales in particular multiscale signal representations lead naturally to pyramidal or tree like data structures in which each level in the tree corresponds to a particular scale of representation a class of multiscale dynamic models evolving on dyadic trees is introduced the main focus of this paper is on the description analysis and application of an extremely efficient optimal estimation algorithm for this class of models this algorithm consists of a fine to coarse filtering sweep followed by a coarse to fine smoothing step corresponding to the dyadic tree generalization of kalman filtering and rauch tung striebel smoothing the kalman filtering sweep consists of the recursive application of three steps a measurement update step a fine to coarse prediction step and a fusion step we illustrate the use of our methodology for the fusion of multiresolution data and for the efficient solution of fractal regularizations of ill posed signal and image processing problems encountered

multiscale methods for the segmentation and reconstruction of signals and images this paper addresses the problem of both segmenting and reconstructing a noisy signal or image the work is motivated by large problems arising in certain scientific applications such as medical imaging two objectives for a segmentation and denoising algorithm are laid out it should be computationally efficient and capable of generating statistics for the errors in the reconstruction and estimates of the boundary locations the starting point for the development of a suitable algorithm is a variational approach to segmentation shah 1992 this paper then develops a precise statistical interpretation of a one dimensional 1 d version of this variational approach to segmentation the 1 d algorithm that arises as a result of this analysis is computationally efficient and capable of generating error statistics a straightforward extension of this algorithm to two dimensions would incorporate recursive procedures for computing estimates of inhomogeneous gaussian markov random fields such procedures require an unacceptably large number of operations to meet the objective of developing a computationally efficient algorithm the use of previously developed multiscale statistical methods is investigated this results in the development of an algorithm for segmenting and denoising which is not only computationally efficient but also capable of generating error statistics as desired

multiscale representations of markov random fields recently a framework for multiscale stochastic modeling was introduced based on coarse to fine scale recursive dynamics defined on trees this model class has some attractive characteristics which lead to extremely efficient statistically optimal signal and image processing algorithms the authors show that this model class is also quite rich in particular they describe how 1 d markov processes and 2 d markov random fields mrfs can be represented within this framework the recursive structure of 1 d markov processes makes them simple to analyze and generally leads to computationally efficient algorithms for statistical inference on the other hand 2 d mrfs are well known to be very difficult to analyze due to their noncausal structure and thus their use typically leads to computationally intensive algorithms for smoothing and parameter identification in contrast their multiscale representations are based on scale recursive models and thus lead naturally to scale recursive algorithms which can be substantially more efficient computationally than those associated with mrf models in 1 d the multiscale representation is a generalization of the midpoint deflection construction of brownian motion the representation of 2 d mrfs is based on a further generalization to a midline deflection construction the exact representations of 2 d mrfs are used to motivate a class of multiscale approximate mrf models based on one dimensional wavelet transforms they demonstrate the use of these latter models in the context of texture representation and in particular they show how they can be used as approximations for or alternatives to well known mrf texture models

some supercongruences arising from symbolic summation abstract based on some combinatorial identities arising from symbolic summation we extend two supercongruences on partial sums of hypergeometric series which were originally conjectured by guo and schlosser and recently confirmed by jana and kalita

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

on the weak convergence in hf1 rn abstract let vmo f denote the closure of c c lip with respect to the seminorm bmo f where f is a family of sections and the space bmo f associated to the family f was introduced by caffarelli and gutierrez 1996 4 sections play an important role in the investigation of monge ampere equation and the linearized monge ampere equation we show that the dual of vmo f is the hardy space h f 1 defined in ding and lin 2005 8 as an application we prove that almost everywhere convergence of a sequence of functions bounded in h f 1 to a function in l 1 d implies the weak convergence

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

symbolic summation methods and hypergeometric supercongruences abstract in this paper via the wz method and the summation package sigma we establish the following two supercongruences k 0 p 1 2 3 k 1 1 2 k 2 1 2 k 4 k k 3 p 6 p 3 1 p 2 p 3 1 p e p 3 mod p 4 k 0 p 1 3 k 1 1 2 k 2 1 2 k 4 k k 3 p 2 p 3 mod p 4 where p 3 is a prime e p 3 is the p 3 th euler number and 1 p 1 p 1 2 is the legendre symbol the first congruence modulo p 3 confirms a recent conjecture of guo and schlosser

dynamic trees for unsupervised segmentation and matching of image regions we present a probabilistic framework namely multiscale generative models known as dynamic trees dt for unsupervised image segmentation and subsequent matching of segmented regions in a given set of images beyond these novel applications of dts we propose important additions for this modeling paradigm first we introduce a novel dt architecture where multilayered observable data are incorporated at all scales of the model second we derive a novel probabilistic inference algorithm for dts structured variational approximation sva which explicitly accounts for the statistical dependence of node positions and model structure in the approximate posterior distribution thereby relaxing poorly justified independence assumptions in previous work finally we propose a similarity measure for matching dynamic tree models representing segmented image regions across images our results for several data sets show that dts are capable of capturing important component subcomponent relationships among objects and their parts and that dts perform well in segmenting images into plausible pixel clusters we demonstrate the significantly improved properties of the sva algorithm both in terms of substantially faster convergence rates and larger approximate posteriors for the inferred models when compared with competing inference algorithms furthermore results on unsupervised object recognition demonstrate the viability of the proposed similarity measure for matching dynamic structure statistical models

an overlapping tree approach to multiscale stochastic modeling and estimation recently a class of multiscale stochastic models has been introduced in which random processes and fields are described by scale recursive dynamic trees a major advantage of this framework is that it leads to an extremely efficient statistically optimal algorithm for least squares estimation in certain applications however estimates based on the types of multiscale models previously proposed may not be adequate as they have tended to exhibit a visually distracting blockiness we eliminate this blockiness by discarding the standard assumption that distinct nodes on a given level of the multiscale process correspond to disjoint portions of the image domain instead we allow a correspondence to overlapping portions of the image domain we use these so called overlapping tree models for both modeling and estimation in particular we develop an efficient multiscale algorithm for generating sample paths of a random field whose second order statistics match a prespecified covariance structure to any desired degree of fidelity furthermore we demonstrate that under easily satisfied conditions we can lift a random field estimation problem to one defined on an overlapped tree resulting in an estimation algorithm that is computationally efficient directly produces estimation error covariances and eliminates blockiness in the reconstructed imagery without any sacrifice in the resolution of fine scale detail

a q microscope for supercongruences abstract by examining asymptotic behavior of certain infinite basic q hypergeometric sums at roots of unity that is at a q microscopic level we prove polynomial congruences for their truncations the latter reduce to non trivial super congruences for truncated ordinary hypergeometric sums which have been observed numerically and proven rarely a typical example includes derivation from a q analogue of ramanujan s formula n 0 4 n 2 n 2 n n 2 2 8 n 3 2 n 8 n 1 2 3 of the two supercongruences s p 1 p 3 p mod p 3 and s p 1 2 p 3 p mod p 3 valid for all primes p 3 where s n denotes the truncation of the infinite sum at the n th place and 3 stands for the quadratic character modulo 3

approximation of functional depending on jumps by elliptic functional via t convergence we show how it is possible to approximate the mumford shah see 29 image segmentation functional by elliptic functionals defined on sobolev spaces the heuristic idea is to consider functionals h u z with z ranging between 0 and 1 and related to the set k the minimizing zh are near to 1 in a neighborhood of the set k and far from the neighborhood they are very small the neighborhood shrinks as h for a similar approach to the problem compare kulkarni see 25 the approximation of h to takes place in a variational sense the de giorgi f convergence

some congruences involving fourth powers of central q binomial coefficients we prove some congruences on sums involving fourth powers of central q binomial coefficients as a conclusion we confirm the following supercongruence observed by long pacific j math 249 2011 405 418 sum limits k 0 p r 1 2 displaystyle 4k 1 over 256 k left matrix 2k cr k right 4 equiv p r quad left bmod p r 3 right where p 5 is a prime and r is a positive integer our method is similar to but a little different from the wz method used by zudilin to prove ramanujan type supercongruences