We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A 1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees.The authors would like to thank ML degree of special complete intersections. Moreover, a geometric characterisation of the ML degree of a smooth variety in the case when the divisor corresponding to the rational function is a normal crossings divisor is given in [6]. In the same paper an explicit combinatorial formula for the ML degree of a toric variety is derived by relaxing the restrictive smoothness assumption and allowing some mild singularities. For an introduction to the geometry behind the MLE for algebraic statistical models for discrete data the interested reader is refered to [24], which includes most of the current results on the MLE problem from the perspective of algebraic geometry as well as statistical motivation.This article is concerned with the problem of MLE on toric Fano varieties. Toric varieties correpond to log-linear models in statistics. Since the seminal papers by L.A. Goodman in the 70s [16,17], log-linear models have been widely used in statistics and areas like natural language processing when analyzing crossclassified data in multidimensional contingency tables [4]. The ML degree of a toric variety is bounded above by its degree. Toric Fano varieties provide several interesting classes of toric varieties for investigating the ML degree drop. We focus on studying the maximum likelihood estimation for 2-dimensional Gorenstein toric Fano varieties, Veronese(2, 2) with different scalings and toric Fano varieties associated to 3-valent phylogenetic trees.Two-dimensional Gorenstein toric Fano varieties correspond to reflexive polygons and by the classification results there are exactly 16 isomorphism classes of such polygons, see for example [27]. Out of these 16 isomorphism classes five correspond to smooth del Pezzo surfaces and 11 correspond to del Pezzo surfaces with singularities. Our first main result Theorem 3.1 states that the ML degree is equal to the degree of the surface in all cases except for the quintic del Pezzo surface with two singular points of type A 1 . Furthermore, in Table 2, we provide explicit expressions that all...

The fixing number of a graph G is the smallest cardinality of a set of vertices S such that only the trivial automorphism of G fixes every vertex in S. The fixing set of a group Γ is the set of all fixing numbers of finite graphs with automorphism group Γ. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label G so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.

ImportanceCancer screening deficits during the first year of the COVID-19 pandemic were found to persist into 2021. Cancer-related deaths over the next decade are projected to increase if these deficits are not addressed.ObjectiveTo assess whether participation in a nationwide quality improvement (QI) collaborative, Return-to-Screening, was associated with restoration of cancer screening.Design, Setting, and ParticipantsAccredited cancer programs electively enrolled in this QI study. Project-specific targets were established on the basis of differences in mean monthly screening test volumes (MTVs) between representative prepandemic (September 2019 and January 2020) and pandemic (September 2020 and January 2021) periods to restore prepandemic volumes and achieve a minimum of 10% increase in MTV. Local QI teams implemented evidence-based screening interventions from June to November 2021 (intervention period), iteratively adjusting interventions according to their MTVs and target. Interrupted time series analyses was used to identify the intervention effect. Data analysis was performed from January to April 2022.ExposuresCollaborative QI support included provision of a Return-to-Screening plan-do-study-act protocol, evidence-based screening interventions, QI education, programmatic coordination, and calculation of screening deficits and targets.Main Outcomes and MeasuresThe primary outcome was the proportion of QI projects reaching target MTV and counterfactual differences in the aggregate number of screening tests across time periods.ResultsOf 859 cancer screening QI projects (452 for breast cancer, 134 for colorectal cancer, 244 for lung cancer, and 29 for cervical cancer) conducted by 786 accredited cancer programs, 676 projects (79%) reached their target MTV. There were no hospital characteristics associated with increased likelihood of reaching target MTV except for disease site (lung vs breast, odds ratio, 2.8; 95% CI, 1.7 to 4.7). During the preintervention period (April to May 2021), there was a decrease in the mean MTV (slope, −13.1 tests per month; 95% CI, −23.1 to −3.2 tests per month). Interventions were associated with a significant immediate (slope, 101.0 tests per month; 95% CI, 49.1 to 153.0 tests per month) and sustained (slope, 36.3 tests per month; 95% CI, 5.3 to 67.3 tests per month) increase in MTVs relative to the preintervention trends. Additional screening tests were performed during the intervention period compared with the prepandemic period (170 748 tests), the pandemic period (210 450 tests), and the preintervention period (722 427 tests).Conclusions and RelevanceIn this QI study, participation in a national Return-to-Screening collaborative with a multifaceted QI intervention was associated with improvements in cancer screening. Future collaborative QI endeavors leveraging accreditation infrastructure may help address other gaps in cancer care.

We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x, y]/ q , where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including diagrams of infinite projective dimension, into pure diagrams. Boij-Söderberg theory completely describes the cone of Betti diagrams over a standard graded polynomial ring; our result provides the first example of another graded ring for which the cone of Betti diagrams is entirely understood. * β R 0,0 (M ) β R

We investigate decompositions of Betti diagrams over a polynomial ring within the framework of Boij-Söderberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the requirement that the degree sequences in such pure diagrams be totally ordered, we are able to define a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from [ES09]. In higher codimension, obstructions arise that inspire our work on an alternative algorithm.