Title: Convexity, star-shapedness, and multiplicity of numerical range and its generalizations

Abstract:  Given an $n\times n$ complex matrix $A$, the classical numerical range (field of values) of $A$ is the following set associated with the quadratic form:
$$W(A) = \{x^*Ax: x*x=1, x\,\text{ is a complex }\, n\text{-tuple}\}$$We will start with the celebrated Toeplitz-Hausdorff (1918, 1919) convexity theorem for the classical numerical range. Then we will move on to introduce various generalizations and we will focus on those in the framework of semisimple Lie algebras and compact Lie groups. In our discussions, results on convexity, star-shapedness, and multiplicity will be reviewed, for example, the results of Embry (1970), Westwick (1975), Au-Yeung-Tsing (1983, 84), Cheung-Tsing (1996), Li-Tam (2000), Tam (2002), Dokovic-Tam (2003), Cheung-Tam (2008, 2011), Carden (2009), Cheung-Liu-Tam (2011) and Markus-Tam (2011). We will mention some unsolved problems.

Topic: Text as Data

Abstract: Professor Wedeking will give a summary of three projects that he has been involved in using text as data (1 is published, 1 is under review, and 1 is ongoing). Specifically, for each of the 3 projects, He will:  (1) describe the method he's using, what it generally is used for;  (2) the motivation for the project-e.g., the substantive research question and relevant background information;  (3) a brief description of the data; and  (4) the results of the method and the substantive conclusions.  The three projects are: (1) measuring how legal issues are framed (e.g., free speech vs. right to privacy, etc) and how that helps parties win; (2) uncovering the clarity of texts using readability formulas; and (3) scaling justices with texts- uncovering their ideological positions (how liberal or conservative they are) using their words.

Abstract:

We present a Multivariate Decomposition Method (MDM) for approximating integrals of functions with countably many variables. We assume that the integrands have mixed first order partial derivatives bounded in a γ = {γ_u }u⊂N+ -weighted Lp norm. We also assume that the integrands can be evaluated only at points with finitely many (d) coordinates different than zero and that the cost of such a sampling is equal to $(d) for a given cost function $. We show that MDM can approximate the integrals with the worst case error bounded by ε at cost proportional to −1+|O(ln(1/ε)/ ln(ln(1/ε)))| ε even if the cost function is exponential in d, i.e., $(d) = e^{O(d)}.  This is an almost optimal method since all algorithms for univariate functions (d = 1) from this space have the cost bounded from below by Ω(1/ε).