Computing Exponentials of Essentially Non-negative Matrices with Entry-wise Accuracy
Speaker: Qiang Ye, University of Kentucky
Abstract:

A real square matrix is said to be essentially non-negative if all of its off-diagonal entries are non-negative. In this talk, I will present new perturbation results and algorithms that demonstrate that the exponential of an essentially non-negative matrix can be computed with entrywise relative accuracy.

Learning Algorithms for Restricted Boltzmann Machines
Speaker: Devin Willmott, University of Kentucky
Abstract: Restricted Boltzmann machines (RBMs) have played a central role in the development of deep learning. In this talk, we will introduce the theoretical framework behind stochastic binary RBMs, give motivation and a derivation for the most commonly used RBM learning algorithm (contrastive divergence), and prove some analytic results related to its convergence properties.

Abstract: As biology has become a data-rich science, more biological phenomena have become amenable to modeling and analysis using mathematical and statistical methods. At the same time, more mathematical areas have developed applications in the biosciences, in particular algebra, discrete mathematics, topology, and geometry. This talk will present some case studies from algebra and discrete mathematics applied to the construction and analysis of dynamic models of biological networks. Some emerging themes will be highlighted, outlining a broader research agenda at the interface of biology and algebra and discrete mathematics. No special knowledge in any of these fields is required to follow the presentation.

Title: "A Matrix Analysis of Centrality Measures"

Abstract:  When analyzing a network, one of the most basic concerns is identifying the "important" nodes in the network. What defines "important" can vary from network to network, depending on what one is trying to analyze about the network. In this paper by Benzi and Klymko several different centrality measures, methods of computing node importance, are introduced and compared. We will see that some centrality measures give more information about the network on a local scale, while others help to analyze on a more global scale. In particular, the paper analyzes the behavior of these measures as we let the parameters defining them approach certain limits that appear to be problematic.

Title: Optimality of the Neighbor Joining Algorithm and Faces of the Balanced Minimum Evolution Polytope

Abstract:  Balanced minimum evolution (BME) is a statistically consistent distance-based method to reconstruct a phylogenetic tree from an alignment of molecular data. In 2008, Eickmeyer, Huggins, Pachter, and myself developed a notion of the BME polytope, the convex hull of the BME vectors obtained from Pauplin's formula applied to all binary trees. We also showed that the BME can be formulated as a linear programming problem over the BME polytope.  The BME is related to the Neighbor Joining (NJ) algorithm, now known to be a greedy optimization of the BME principle. Further, the NJ and BME algorithms have been studied previously to understand when the NJ algorithm returns a BME tree for small numbers of taxa. In this talk we aim to elucidate the structure of the BME polytope and strengthen knowledge of the connection between the BME method and NJ algorithm. We first show that any subtree-prune-regraft move from a binary tree to another binary tree corresponds to an edge of the BME polytope. Moreover, we describe an entire family of faces parametrized by disjoint clades. We show that these clade-faces are smaller-dimensional BME polytopes themselves. Finally, we show that for any order of joining nodes to form a tree, there exists an associated distance matrix (i.e., dissimilarity map) for which the NJ algorithm returns the BME tree. More strongly, we show that the BME cone and every NJ cone associated to a tree T have an intersection of positive measure.  We end this talk with the current and future projects on phylogenomics with biologists in University of Kentucky and Eastern Kentucky University.  This work is supported by NIH.

 

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).

As an active research area in biomechanics, growth mechanics studies mechanical aspects of growth (and resorption) of biological tissues [1]-[5]. While the classical theory of mechanics has provided useful tools in such studies, the physiological process of growth in turn presents special challenges to the classical theory of continuum mechanics. For solids, the kinematical theory in continuum mechanics is based on a deformation function which maps a fixed reference configuration of the body to the current configuration, and the constitutive equation relates the stress in the current configuration to certain kinematical quantities defined through the reference configuration. These and some other fundamental concepts of continuum mechanics are not adequate to describe growth, especially surface growth where a growth body may undergo changes in topology.

In this talk, we address some of these issues, and develop a growth theory using consistent and clean kinematical and constitutive equations. The kinematics of a growing body is described by the velocity field and a growth rate function defined on the region occupied by the body at the current time. Also defined on this region is the mass density field. An equation for the balance of mass is derived with these functions. To describe surface growths as well as volume growths, the growth rate function may be singular on a surface in the current configuration, while the velocity field and mass density field may suffer jump discontinuities across the surface. They are weak solutions of the equation of mass balance.

For constitutive equations, we discard the standard practice of using a fixed reference configuration, and use the current configuration as the reference. In contrast to the constitutive equations for an elastic body in the classical continuum mechanics, the response function of a growing elastic body in the present theory gives the Cauchy stress when the body in the current grown/deformed configuration is subjected to an imaginary further elastic deformation.  In particular, it gives the Cauchy stress at the current configuration when this imaginary deformation is the identity deformation. The form of such a response function changes constantly with growth and deformation even when the intrinsic mechanical properties of the material remain unchanged. The evolution equation for the response function is derived by considering a sequence of growth/deformation processes. Again, this evolution equation may admit weak solutions for surface growths.

The balance of mass, the evolution equation for the response function, and the usual balance of linear momentum equation, along with appropriate initial and boundary conditions, can be solved to yield the velocity field, the mass density, the stress field, and the response function.  In particular, one can find residual stresses due to growth when the body is subjected to zero applied load.

Material tensors pertaining to polycrystalline aggregates should manifest the influence of crystallographic texture on the material properties. A representation theorem on material tensors of weakly-textured polycrystals has been established by Man and Huang (2011) under the classical assumption in quantitative texture analysis that the orientation distribution function (ODF) is defined on the rotation group. By this theorem, a given material tensor can be expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters.  In this talk, an extension of this representation theorem where the ODF is defined on the orthogonal group will be discussed. The extended theorem now covers all the 32 cases of point-group crystal symmetry.