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# Posters

## On the dimension of a certain measure in the plane

In this poster, I will discuss the Hausdorff dimension of a measure related to a positive weak solution of a certain partial differential equation in a simply connected domain in the plane.

Our work generalizes work of Lewis and coauthors when the measure is p-harmonic and also for p=2, the well known theorem of Makarov regarding the Hausdorff dimension of harmonic measure relative to a point in a simply connected domain.

## Asymptotic behaviors of higher Mahler Measure

We study behaviors of higher Mahler measure which is a generalization of classical Mahler measure. Apart from giving a quick broad overview of classical Mahler measure and higher Mahler mea- sure, in this talk we shall show several new asymptotic properties of zeta Mahler measure (of a specific class of x − a) involving partitions of integers, the Riemann zeta function, the Dirichlet η-function and π.

## Disease Simulation on Circle Packings

Circle packings and stochastic processes have seen several connections including results concerning the Dirichlet problem and harmonic measure. We will describe and analyze simulated results of disease outbreaks using random walks and circle packings. This method of disease study can be applied to real infectious diseases as well some fictitious diseases such as a zombie virus.

## Julia Variation Formula for Circle Packing

Circle packing is a configuration, P, of circles with specified patterns of tangency. This was first introduced by William Thurston, at the Biebarbach Conference in 1985, where he explained the geometric approach to "the Finite Riemann Mapping Theorem". In our poster, we shall describe a version of the classical Julia variation formula for Circle Packing. Let D be a region in plane bounded by a piecewise analytic curve. Let P be an Euclidean packing of the region D with "n" circles. Let R be the radius function for packing P, which maps the vertices of the circles to their corresponding radii. Now, we perturb P by incrementing the radius r to (r+k \epsilon), \epsilon >0, where r is the radius of a boundary circle. The main idea is to find the distribution function for the new radius function. In the case of a packing consisting of three circles, the result that we obtain is that the new radius function is deformed by the constant multiple of \epsilon. The modeling of the dynamics of the circles in a packing can be done by using the concept of random walks or Markov chains. So, using the idea of random walks, in general (for a packing with more than 3 circles), we may expect the new radius function would be deformed by the factor of order of \epsilon. The classical Julia variation has proven to be a very effective tool for solving extremal problems. The discrete version incorporates the combinatorial flexibility of circle packing and is a promising line of attack on several outstanding open problems. In addition, it is expected to lead to improved error and convergence estimates for circle packing.

## Composition Operators in Sobolev Spaces on a Carnot Group

Mainly we study mappings inducing composition operators on Sobolev spaces. In this poster we are going to present the basic notions regarding the problem we investigate. Moreover, we formulate our main result in the direction when the composition operator is an isomorphism of Sobolev spaces. The research involves a diversity of analytic and geometric techniques.

## Nevanlinna functions with fast growth of the spherical derivative

The purpose of this presentation is to give an example of a function with bounded Nevanlinna characteristic and arbitrary growth of the spherical derivative. Similar results were obtained by M.Tsuzuki (1967), S.Yamashita (1980), R.Aulaskari and J.Rattya (2009). Construction of our example is based on the property of mutual arrangement of a-points for meromorphic functions with given growth of the spherical derivative and technique of P-points.

## Blaschke's rolling ball property and conformal metric ratios

This poster presents my recent work submitted for publication, collaborated with Dr. David Herron from the University of Cincinnati, about Blaschke’s Rolling Ball Property. This property has been used to study various problems from mathematical morphology, image analysis, and smoothing. The purpose of this research was to characterize the closed sets in Euclidean space that satisfy a two-sided rolling ball property and to show that certain conformal metric ratios have a boundary value of one as an application. The main theorem proved by a geometric approach can be summarized as follows:
A non-empty and closed set has the two-sided rolling ball property if and only if it is an orientable C^{1,1} smooth embedded submanifold, and there is a globally defined Lipschitz continuous unit normal vector field along it.

## On the coefficients of the normalized conformal mapping onto the exterior of the Mandelbrot set

It is well known that the Mandelbort set is connected, but its local connectivity is still unknown. Douady and Hubbard demonstrated the connectedness of the Mandelbrot set by constructing a conformal homeomorphism Φ from the exterior of the Mandelbrot set onto the exterior of the closed unit disk. If the inverse map Ψ := Φ^{−1} extends continuously to the unit circle, then the Mandelbrot set is locally connected, according to Carathéodory’s continuity theorem. This is the motivation of our study.
Jungreis has presented an algorithm to compute the coefficients b_m of the Laurent series of Ψ(z) in [3]. Several detailed studies of b_m were given in [1], and others. There are also several empirical observations by Zagier mentioned in [1]. Furthermore Ewing and Schober studied the coefficients a_m of the Taylor series of the function f(z) := 1/Ψ(1/z) in [2]. After that, Yamashita studied a generalization of b_m in [5].
We will present several properties of a generalization of the coefficients a_m and b_m, with a focus on Zagier’s observations. Specifically a formula for these coefficients are given. Infinitely many coefficients are zero and also an estimate for the growth of the denominator is given.

References

[1] B. Bielefeld, Y. Fisher and F. V. Haeseler, Computing the Laurent series of the map Ψ: C−D → C−M, Adv. in Appl. Math. 14 (1993), 25-38.

[2] J. Ewing and G. Schober, Coefficients associated with the reciprocal of the Mandelbrot set, J. Math. Anal. Appl. 170 (1992), no. 1, 104-114.

[3] I. Jungreis, The uniformization of the complement of the Mandelbrot set, Duke Math. J. 52 (1985), no. 4, 935-938.

[4] H. Shimauchi, On the coefficients of the Riemann mapping function for the exterior of the Multibrot set, Topics in Finite or Infinite Dimensional Complex Analysis, Tohoku University Press (2013), 237-248.

[5] O. Yamashita, On the Coefficients of the Mapping to the exterior of the Mandelbrot set (in Japanese), Master thesis of Graduate School of Information Science, Nagoya University, (1998). http://www.math.human.nagoya-u.ac.jp/master.thesis/1997.html

## Convex functions and barycenters on CAT(1)-spaces of small radii

We establish a new local convex property of CAT(1)-spaces by proving the convexity of a certain function dened on any small metric ball in a CAT(1)-space. Our function was found by W. Kendall and has its origin in the study of harmonic maps. As an application, we also prove that any probability measure on a complete CAT(1)-space of small radius admits a unique barycenter.

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