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© 2018 Elsevier B.V. DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G. Such models are used to model complex cause–effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence. Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e., skeleton) and the same set of the induced subDAGs i→j←k, known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, we introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and we studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some well-studied families of graphs including paths, stars, cycles, spider graphs, caterpillars, and balanced binary trees. In doing so, we recover connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictate the number and size of MECs.

© 2019 Elsevier Inc. We consider the construction of insurance premiums that are monotonically increasing with respect to a loading parameter. By introducing weight functions that are totally positive of higher order, we derive higher monotonicity properties of generalized weighted premiums; in particular, we deduce for weight functions that are totally positive of order three a monotonicity property of the variance-to-mean ratio, or index of dispersion, of the loss variable. We derive the higher order total positivity properties of some ratios that arise in actuarial and insurance analysis of combined risks. Further, we examine seven classes of weight functions that have appeared in the literature and we ascertain the higher order total positivity properties of those functions.

© 2018 Elsevier B.V. DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G. Such models are used to model complex cause–effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence. Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e., skeleton) and the same set of the induced subDAGs i→j←k, known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, we introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and we studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some well-studied families of graphs including paths, stars, cycles, spider graphs, caterpillars, and balanced binary trees. In doing so, we recover connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictate the number and size of MECs.

Cells sense physical and chemical signals from their local microenvironment and transduce them to the nucleus to regulate genomic programs. In this review, we first discuss different modes of mechanotransduction to the nucleus. Then we highlight the role of the spatial organization of chromosomes for integrating these signals. In particular, we emphasize the importance of chromosome intermingling for gene regulation. We also discuss various geometric models and recent advances in microscopy and genomics that have allowed accessing these nanoscale chromosome intermingling regions. Taken together, the recent work summarized in this review culminates in the hypothesis that the chromosome intermingling regions are mechanical hotspots for genome regulation. Maintenance of such mechanical hotspots is crucial for cellular homeostasis, and alterations in them could be precursors for various cellular reprogramming events including diseases., DARPA (Contract W911NF-16-1-0551), NSF (Grant DMS-1651995), ONR (Grant N00014-17-1-2147)

Abnormalities in nuclear and chromatin organization are hallmarks of many diseases including cancer. This review provides our understanding of how the cellular microenvironment regulates nuclear morphology and with it the spatial organization of chromosomes and genes, resulting in cell-type specific genomic programs. We also discuss the molecular basis for maintaining nuclear and genomic integrity and how alterations in nuclear mechanotransduction pathways result in various diseases. Finally, we highlight the importance of digital pathology based on nuclear morphometric features combined with single-cell genomics for early cancer diagnostics., NSF (Grant DMS-1651995), ONR (Grant N00014-17-1-2147)

In this Current Opinion, we highlight the importance of the material properties of tissues and how alterations therein, which influence epithelial-to-mesenchymal transitions, represent an important layer of regulation in a number of diseases and potentially also play a critical role in host-pathogen interactions. In light of the current SARS-CoV-2 pandemic, we here highlight the possible role of lung tissue stiffening with ageing and how this might facilitate increased SARS-CoV-2 replication through matrix-stiffness dependent epithelial-to-mesenchymal transitions of the lung epithelium. This emphasizes the need for integrating material properties of tissues in drug discovery programs.

The space of discrete copulas admits a representation as a convex polytope, and this has been exploited in entropy-copula methods used in hydrology and climatology. In this paper, we focus on the class of component-wise convex copulas, i.e., ultramodular copulas, which describe the joint behavior of stochastically decreasing random vectors. We show that the family of ultramodular discrete copulas and its generalization to component-wise convex discrete quasi-copulas also admit representations as polytopes. In doing so, we draw connections to the Birkhoff polytope, the alternating sign matrix polytope, and their generalizations, thereby unifying and extending results on these polytopes from both the statistics and the discrete geometry literature., NSF (Grant DMS-1651995), ONR (Grant N00014-17-1-2147)