13. BMO from dyadic BMO on the bidisc

with Jill Pipher

In preparation. 31 pages.
PDF version.

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 *H*^{1} to
martingale *H*^{1}. 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.

12. Convergence properties of harmonic measure distributions for planar domains

with Marie A. Snipes*

Submitted, June 2005. 21 pages.

We establish
sufficient conditions under which the harmonic measure distribution functions
*h _{n}* of a sequence of domains

11. Authority rankings from HITS, PageRank, and SALSA:

Existence, Uniqueness, and Effect of Initialization

with Ayman Farahat, Thomas LoFaro, Joel C. Miller*, and Gregory Rae*

January 2005. To appear in SIAM Journal on Scientific
Computing (SISC). 20 pages.

PDF version.

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.

with Byron L. Walden

Submitted, June 2003. To appear in the American Mathematical
Monthly, subject to revisions. 9 pages.

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 $M(a,b)$.

9. Realizing step functions as harmonic measure distributions of planar domains

with Marie A. Snipes*

Ann. Acad. Sci. Fenn. Math., Vol 30 (2005), 353--360.

PDF version.

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

*Revista Matematica Iberoamericana*, Vol. 18 (2002), No. 2,
379--407.

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.

7. Modifications of Kleinberg's HITS algorithm using matrix exponentiation and web log records

with Joel C. Miller*, Gregory Rae*, Fred Schaefer*, Thomas LoFaro,
and Ayman Farahat
*Proceedings of the SIGIR 2001 Conference*, New Orleans,
September 2001, 444--445.

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.

6. Asymptotic behaviour of distributions of harmonic measure for planar domains

with Byron L. Walden
*Complex Variables: Theory and Applications*, Vol. 46 (2001),
No. 2, 157--177.

Review: MR 2002j:30034

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.

5. Fuchsian groups, quasiconformal groups, and conical limit sets

with Peter W. Jones
*Trans. Amer. Math. Soc.*, Vol. 352 (2000), 311--362.

(Also MSRI Preprint No. 1996-024.)

Review: CMP 1 458 326 (2000:02)

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.

4. Quasisymmetrically thick sets

with Susan G. Staples
*Ann. Acad. Sci. Fenn. Ser. A I Math.*, Vol. 23 (1998), 151--168.

Review: MR 98m:30031

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.

3. Paraexponentials, Muckenhoupt weights, and resolvents of paraproducts

with M. Cristina Pereyra
*Proc. Amer. Math. Soc.*, Vol. 126 (1998), 135--144.

(Also MSRI Preprint No. 1996-031.)

Review: MR 99c:42033

We analyze the stability of Muckenhoupt's
**RH _{p}^{d}** and

2. Distributions of harmonic measure for planar domains

with Byron L. Walden
*Proceedings of the 16th Nevanlinna Colloquium (Joensuu, 1995)*,

eds. Laine/Martio, de Gruyter & Co., Berlin, 1996, 289--299.

(Refereed conference proceedings.)

(Also UNSW Pure Mathematics Report PM95/21.)

Review: MR 98f:30026

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.

1. On the conical limit set of a complex hyperbolic group

with Gaven Martin

Mathematics Research Report No. MRR 046-95,

Centre for Mathematics and its Applications, The Australian
National University (1995). 26 pages.

Let $\Gamma$ be a discrete group of complex
hyperbolic isometries of the unit ball of **C ^{n}**. 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

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