# Mathematics Colloquium

## Some Sylvester-type Matrix Equations and Tensor Equations over the Quaternion Algebra

### When

April 6, 2018 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

Zhuo-Heng He, Auburn University

### Abstract

Sylvester-type equations have many applications in neural network, robust control, output feedback control, the almost noninteracting control by measurement feedback problem, graph theory, and so on. In this talk, we consider some Sylvester-type matrix equations and tensor equations over the quaternion algebra. We present some necessary and sufficient conditions for the solvability to these Sylvester-type matrix equations and tensor equations over the quaternion algebra. Moreover, the general solutions to these quaternion matrix equations and tensor equations are explicitly given when they are solvable. We also provide some numerical examples to illustrate our results.

## Elementary and Advanced Perspectives of Measurement and Ratio

### When

March 30, 2018 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

James Madden, Louisiana State University

### Abstract

Measurement, ratio, and proportion are topics in elementary school mathematics, yet there are profound connections to current research in algebra and analysis, e.g., the theorem of Hölder on archimedean totally-ordered groups, the Yosida Representation Theorem for archimedean vector lattices, and my own work interpreting the Yosida Theorem in point-free topology. In this talk, I will trace the history of ratio from Eudoxus to "point-free Yosida", with stops along the way to examine interactions between academic mathematics and the mathematics taught in school.

## Decomposition Towers and their Forcing

### When

March 23, 2018 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

Alexander Blokh, UAB and Michal Misiurewicz, IUPUI, Indianapolis

### Abstract

We define the *decomposition tower*, a new characteristic of cyclic permutations. A cyclic permutation π of the set N = {1,…,*n*} has a *block structure* if N can be divided into consecutive blocks permuted by π. The set N might be partitioned into blocks in a few ways; then those partitions get finer and finer. Decomposition towers reflect the variety of sizes of blocks of such partitions. Set

4 >> 6 >> 3 >> … >> 4n >> 4n + 2 >> … >> 2 >> 1,

define the lexicographic extension of >> onto towers, and denote it >> too. We prove that if *N* >> *M* and an interval map *f* has a cycle with decomposition tower *N* then *f* must have a cycle with decomposition tower *M*. The results are joint with Michal Misiurewicz (IUPUI, Indianapolis), inspired by the Sharkovsky Theorem, and based upon our (M – B) recent results.

## Optimal Quantization

### When

February 9, 2018 | 2:30 - 3:30 p.m.

### Where

Campbell Hall 443

### Speaker

Mrinal Kanti Roychowdhury, The University of Texas Rio Grande Valley

### Abstract

The basic goal of quantization for probability distribution is to reduce the number of values, which is typically uncountable, describing a probability distribution to some finite set and thus approximation of a continuous probability distribution by a discrete distribution. Though the term 'quantization' is known to electrical engineers for the last several decades, it is still a new area of research to the mathematical community. In my presentation, first I will give the basic definitions that one needs to know to work in this area. Then, I will give some examples, and talk about the quantization on mixed distributions. Mixed distributions are an exciting new area for optimal quantization. I will also tell some open problems relating to mixed distributions.