A distributionally robust perspective on uncertainty quantification and chance constrained programming

The objective of uncertainty quantification is to certify that a given physical, engineering or economic system satisfies multiple safety conditions with high probability. A more ambitious goal is to actively influence the system so as to guarantee and maintain its safety, a scenario which can be modeled through a chance constrained program. In this paper we assume that the parameters of the system are governed by an ambiguous distribution that is only known to belong to an ambiguity set characterized through generalized moment bounds and structural properties such as symmetry, unimodality or independence patterns. We delineate the watershed between tractability and intractability in ambiguity-averse uncertainty quantification and chance constrained programming. Using tools from distributionally robust optimization, we derive explicit conic reformulations for tractable problem classes and suggest efficiently computable conservative approximations for intractable ones

A note on the γ\gamma-coefficients of the "tree Eulerian polynomial"

We consider the generating polynomial of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. B. Drake proved that this polynomial factors completely over the integers. From his product formula it can be concluded that this polynomial has positive coefficients in the $\gamma$-basis and we show that a formula for these coefficients can also be derived. We discuss various combinatorial interpretations of these positive coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from previous results of the author and Wachs related to the poset of weighted partitions and the free multibracketed Lie algebra.Comment: 13 pages, 6 figures, Interpretations derived from results in arXiv:1309.5527 and arXiv:1408.541

The local hh-vector of the cluster subdivision of a simplex

The cluster complex $\Delta (\Phi)$ is an abstract simplicial complex, introduced by Fomin and Zelevinsky for a finite root system $\Phi$. The positive part of $\Delta (\Phi)$ naturally defines a simplicial subdivision of the simplex on the vertex set of simple roots of $\Phi$. The local $h$-vector of this subdivision, in the sense of Stanley, is computed and the corresponding $\gamma$-vector is shown to be nonnegative. Combinatorial interpretations to the entries of the local $h$-vector and the corresponding $\gamma$-vector are provided for the classical root systems, in terms of noncrossing partitions of types $A$ and $B$. An analogous result is given for the barycentric subdivision of a simplex.Comment: 21 pages, 4 figure

Symmetric chain decomposition for cyclic quotients of Boolean algebras and relation to cyclic crystals

The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the $\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.Comment: minor revisions; to appear in IMR

Unimodality Problems in Ehrhart Theory

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart $h^*$-vector. Ehrhart $h^*$-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart $h^*$-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This version updated October 2017 to correct an error in the original versio

Single cell measurement of telomerase expression and splicing using microfluidic emulsion cultures.

Telomerase is a reverse transcriptase that maintains telomeres on the ends of chromosomes, allowing rapidly dividing cells to proliferate while avoiding senescence and apoptosis. Understanding telomerase gene expression and splicing at the single cell level could yield insights into the roles of telomerase during normal cell growth as well as cancer development. Here we use droplet-based single cell culture followed by single cell or colony transcript abundance analysis to investigate the relationship between cell growth and transcript abundance of the telomerase genes encoding the RNA component (hTR) and protein component (hTERT) as well as hTERT splicing. Jurkat and K562 cells were examined under normal cell culture conditions and during exposure to curcumin, a natural compound with anti-carcinogenic and telomerase activity-reducing properties. Individual cells predominantly express single hTERT splice variants, with the α+/β- variant exhibiting significant transcript abundance bimodality that is sustained through cell division. Sub-lethal curcumin exposure results in reduced bimodality of all hTERT splice variants and significant upregulation of alpha splicing, suggesting a possible role in cellular stress response. The single cell culture and transcript abundance analysis method presented here provides the tools necessary for multiparameter single cell analysis which will be critical for understanding phenotypes of heterogeneous cell populations, disease cell populations and their drug response