We generalize to the bidisc a theorem of Garnett and Jones relating the space BMO of functions of bounded mean oscillation to its martingale counterpart, dyadic BMO. Namely, translation-averages of suitable families of dyadic BMO functions belong to BMO. As a corollary, we deduce a biparameter version of a theorem of Burgess Davis connecting the Hardy space H¹ to martingale H¹. We also prove the analogues of the theorem of Garnett and Jones in the one-parameter and biparameter VMO spaces of functions of vanishing mean oscillation.

    We establish sufficient conditions under which the harmonic measure distribution functions h_n of a sequence of domains D_n converge pointwise to the distribution function h of the limiting domain D, at all points of continuity of h. In the case of a model example, we establish this convergence of the distribution functions. Here, the value of the function h(r) gives the harmonic measure of the part of the boundary of the domain that lies within distance r of a fixed basepoint in the domain, thus relating the geometry of the domain to the behavior of Brownian motion in the domain.

    Algorithms such as Kleinberg's HITS algorithm, the PageRank algorithm of Brin and Page, and the SALSA algorithm of Lempel and Moran use the link structure of a network of webpages to assign weights to each page in the network. The weights can then be used to rank the pages as authoritative sources. These algorithms share a common underpinning; they find a dominant eigenvector of a non-negative matrix that describes the link structure of the given network and use the entries of this eigenvector as the page weights. We use this commonality to give a unified treatment, proving the existence of the required eigenvector for the PageRank, HITS, and SALSA algorithms, the uniqueness of the PageRank eigenvector, and the convergence of the algorithms to these eigenvectors. However, we show that the HITS and SALSA eigenvectors need not be unique. We examine how the initialization of the algorithms affects the final weightings produced. We give examples of networks that lead the HITS and SALSA algorithms to return non-unique or non-intuitive rankings. We characterize all such networks, in terms of the connectivity of the related HITS authority graph. We propose a modification, Exponentiated Input to HITS, to the adjacency matrix input to the HITS algorithm. We prove that Exponentiated Input to HITS returns a unique ranking, so long as the network is weakly connected. Our examples also show that SALSA can give inconsistent hub and authority weights, due to non-uniqueness. We also mention a small modification to the SALSA initialization which makes the hub and authority weights consistent.

    The arithmetic-geometric mean of two positive numbers is defined as follows. Suppose $0 < a < b < \infty$. Recursively define two sequences $$ a_{n+1} = \sqrt{a_nb_n}, \qquad b_{n+1} = \frac{a_n + b_n}{2}, $$ with $a_0 = a$ and $b_0 = b$. Lagrange showed that $\{a_n\}$ and $\{b_n\}$ converge to a common limit. This limit is called the arithmetic-geometric mean $M(a,b)$ of $a$ and~$b$. Gauss developed an extensive theory of the arithmetic-geometric mean, involving elliptic integrals, Jacobian theta functions, and the lemniscate.
    In this paper we propose a new setting, involving harmonic measure in certain domains in the complex plane. We calculate the arithmetic-geometric mean $M(a,b)$ in terms of harmonic measure. We show how this setting allows a unified derivation of several well-known elliptic integral formulas for $M(a,b)$.

    The harmonic measure distribution function of a planar domain relates the geometry of the domain to the behaviour of Brownian motion in the domain. The value of the function h(r) specifies the harmonic measure of the part of the boundary of the domain which lies within any given distance r of a fixed basepoint in the domain. A longterm goal is to realize all suitable functions as distribution functions, by explicit construction of appropriate domains. We show here that increasing step functions can be realized as distribution functions of discs with concentric circular arcs deleted from their interiors. We also establish sufficient conditions under which the distribution functions h_n of a sequence of domains D_n converge pointwise to the distribution function h of the limiting domain D, at all points of continuity of h. In the case of a model example, we establish this convergence of the distribution functions.

    The process of translation averaging is known to improve dyadic BMO to the space BMO of functions of bounded mean oscillation, in the sense that the translation average of a family of dyadic BMO functions is necessarily a BMO function. The present work investigates the effect of translation averaging in other dyadic settings. We show that translation averages of dyadic doubling measures need not be doubling measures, translation averages of dyadic Muckenhoupt weights need not be Muckenhoupt weights, and translation averages of dyadic reverse Holder weights need not be reverse Holder weights. All three results are proved using the same construction.

   Kleinberg's HITS algorithm, a method of link analysis, uses the link structure of a network of web pages to assign authority and hub weights to each page in the network. These weights can then be used to rank authoritative sources on a particular topic. We have found that certain tree-like web structures can lead the HITS algorithm to return either arbitrary or non-intuitive results; we characterize these web structures. We also present two modifications to the adjacency matrix input to the HITS algorithm. Exponentiated Input, our first modification, includes in the modified matrix information not only on direct links between pages, but also on paths of arbitrary length between pages in the network. It resolves both the limitations mentioned above. Usage Weighted Input, our second modification, weights links according to how often they were followed by users in a given time period; it incorporates user feedback without requiring direct querying of users.

    We resolve several questions about the harmonic measure distribution function of a planar domain. This function h(r) specifies the harmonic measure of the part of the boundary of the domain which lies within any given distance r of a fixed basepoint in the domain. We focus on the asymptotic behaviour of the function as r decreases towards the distance from the basepoint to the boundary of the domain. We show that for each $\beta$ between zero and one, there is a domain whose distribution is asymptotically exponential with exponent $\beta$, proving our earlier conjecture. We prove that if the basepoint in any fixed domain is moved directly towards the closest boundary point, then the value of $\beta$ cannot decrease. Finally we construct a domain whose distribution function is not asymptotically exponential.

    We construct examples showing that the normalized Lebesgue measure of the conical limit set of a uniformly quasiconformal group acting discontinuously on the disc may take any value between zero and one. This is in contrast to the cases of Fuchsian groups acting on the disc, conformal groups acting discontinuously on the ball in dimension three or higher, uniformly quasiconformal groups acting discontinuously on the ball in dimension three or higher, and discrete groups of biholomorphic mappings acting on the ball in several complex dimensions. In these cases the normalized Lebesgue measure is either zero or one.

    A subset of the real line is called quasisymmetrically thick if all its images under quasisymmetric self-mappings of the real line have positive Lebesgue measure. We establish two sufficient conditions for a set to be quasisymmetrically thick, give an example distinguishing the conditions, and show that one of these conditions, which applies to sets with a Cantor-type structure, is sharp. We give the analogues of these conditions for sets all of whose K-quasisymmetric images have positive measure, for fixed K. These results are related to Wu's work on sets all of whose quasisymmetric images have measure zero. We also prove a result about when a Cantor set of positive measure cannot be mapped quasisymmetrically to a set of zero measure; for instance, a middle-interval Cantor set of positive measure, constructed in the usual way, cannot be mapped quasisymmetrically to the ternary Cantor set.

    We analyze the stability of Muckenhoupt's RH_p^d and A_p^d classes of weights under a non-linear operation, the $\lambda$-operation. We prove that the dyadic doubling reverse Holder classes RH_p^d are not preserved under the $\lambda$-operation, but that the dyadic doubling A_p classes A_p^d are preserved for $\lambda$ in [0,1]. We give an application to the structure of resolvent sets of dyadic paraproduct operators.

    We answer some questions, based on a problem posed by K. Stephenson, concerning how much geometric information about a planar domain can be distilled from knowledge of its harmonic measure distribution. The harmonic measure distribution specifies the harmonic measure of the part of the boundary of the domain which lies within any given distance from a fixed basepoint in the domain. Specifically, we give examples where the distribution does not determine the domain. Also, we consider the problem of deciding whether a particular function may be realized as a harmonic measure distribution. The paper includes several conjectures and poses some questions for further inquiry.

    Let $\Gamma$ be a discrete group of complex hyperbolic isometries of the unit ball of Cⁿ. We define and discuss the conical limit set and the exponent of convergence of the Poincare series for such a group. We show that the exponent of convergence is at most n, and that this bound can be achieved; that the conical limit set has either zero or full Lebesgue measure; and the the conical limit set has measure zero if and only if the Poincare series converges at the exponent n. These results, while similar to their analogues in real hyperbolic geometry, are not the same. The first of two principal differences between the complex hyperbolic and real hyperbolic settings is that, in the complex hyperbolic case, the cones used to define the conical limit set allow tangential approach in the directions of the complex tangent space. Second, the highly anisotropic nature of the action of a discrete complex hyperbolic group on the boundary of the ball distinguishes this setting from the classical one.

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