“Iterated ultrapowers for the masses”, part 2
Comparing Near-linearity Notions in Open Induction
There have been works in number theory on characterizing the class of Beatty sequences (integer parts of natural multiples of a fixed nonnegative real slope). The same is true for the inhomogeneous case when a fixed intercept is added before taking the integer part. We consider some notions of multiplicative or additive near-linearity and elaborate on the extent to which they charecterize various such sequences. We show some implications from standard number theory carry over to Open Induction and some do not. [In a second talk we could relate this to the weak fragment allowing the standard integers as a direct summand of a model. That second talk would include two more multiplicative vs. additive topics, details to follow.]
Some additive vs. multiplicative issues in subrecursivity, maximality, and near-linearity
We deal with three topics around addition without or with multiplication.
We first present algorithms to compute a certain real, generating its Beatty sequence or base 2 expansion. The former calculates in integers with addition, in conjunction with the counting operator. The latter calculates in integers with addition and multiplication. Motivation comes from subrecursive reals.
Next, let F be an ordered field, D a maximal discrete subring of F, and G a maximal discrete additive subgroup of F. We point out that although there are examples where F has elements of infinite distance to D, it can never realize any gaps of G. If F is countable, then G can be constructed Delta^0_2 relative to F.
Finally we finish and extend the talk of last week by considering some nonstandard models M of weak arithmetic which have the integers as an additive direct summand. We present functions f and g from M to M whose value at a sum minus sum of values is always 0 or 1 yet for some x,y,u,v ≥ 1in M, we have f(xy) < xf(y) and g(uv) > ug(v) + u – 1.
Assessing the Reverse Mathematical Strength of Gratzer-Schmidt Theorem
Gratzer-Schmidt theorem in lattice theory states that all complete and compactly generated lattices are isomorphic to the congruence lattice of an algebra. There has been an effort to assess the strength of this theorem in the reverse mathematical setting. I will discuss my recent progress on this topic and its potential implications.
Title: Analysis of partisan gerrymandering tools in advance of the US 2020 census
Abstract: Over the last decade, mapmakers have precisely gerrymandered political districts for the benefit of their party. In response, political scientists and mathematicians have more extensively investigated tools to quantify and understand the mathematical structure of redistricting problems. Two primary tools for determining whether a particular redistricting plan is fair are partisan-gerrymandering metrics and stochastic sampling algorithms. In this work we explore the Declination, a new metric intended to detect partisan gerrymandering. Within out analyses, we show that Declination cannot detect all forms of packing and cracking, and we compare the Declination to the Efficiency Gap. We show that these two metrics can behave quite differently, and give explicit examples where that occurs.
Title: Structural Identifiability of Biological Models
Abstract: Parameter identifiability analysis addresses the problem of which unknown parameters of a model can be determined from given input/output data. If all of the parameters of a model can be determined from data, the parameters and the model are called identifiable. However, if some subset of the parameters can not be determined from data, the model is called unidentifiable. We examine this problem for the case of perfect input/output data, i.e. absent of any experimental noise. This is called the structural identifiability problem. We show that, even in the ideal case of perfect input/output data, many biological models are structurally unidentifiable, meaning some subset of the parameters can take on an infinite number of values, yet yield the same input/output data. In this case, one attempts to reparametrize the model in terms of new parameters that can be determined from data. In this talk, we discuss the problem of finding an identifiable reparametrization and give necessary and sufficient conditions for a certain class of linear compartmental models to have an identifiable reparametrization. We also discuss finding classes of identifiable models and finding identifiable submodels of identifiable models. Our work uses graph theory and tools from computational algebra. This is joint work with Elizabeth Gross and Anne Shiu.