ACDC: Analysis of Congruent Diversification Classes

Diversification rates inferred from phylogenies are not identifiable. There are infinitely many combinations of speciation and extinction rate functions that have the exact same likelihood score for a given phylogeny, building a congruence class. The specific shape and characteristics of such congruence classes have not yet been studied. Whether speciation and extinction rate functions within a congruence class share common features is also not known. Instead of striving to make the diversification rates identifiable, we can embrace their inherent non-identifiable nature. We use two different approaches to explore a congruence class: (i) testing of specific alternative hypotheses, and (ii) randomly sampling alternative rate function within the congruence class. Our methods are implemented in the open-source R package ACDC (https://github.com/afmagee/ACDC). ACDC provides a flexible approach to explore the congruence class and provides summaries of rate functions within a congruence class. The summaries can highlight common trends, i.e. increasing, flat or decreasing rates. Although there are infinitely many equally likely diversification rate functions, these can share common features. ACDC can be used to assess if diversification rate patterns are robust despite non-identifiability. In our example, we clearly identify three phases of diversification rate changes that are common among all models in the congruence class. Thus, congruence classes are not necessarily a problem for studying historical patterns of biodiversity from phylogenies.

The first families of highly symmetric Kirkman Triple Systems whose orders fill a congruence class

Kirkman triple systems (KTSs) are among the most popular combinatorial designs and their existence has been settled a long time ago. Yet, in comparison with Steiner triple systems, little is known about their automorphism groups. In particular, there is no known congruence class representing the orders of a KTS with a number of automorphisms at least close to the number of points. We partially fill this gap by proving that whenever $$v \equiv 39$$ v ≡ 39 (mod 72), or $$v \equiv 4^e48 + 3$$ v ≡ 4 e 48 + 3 (mod $$4^e96$$ 4 e 96 ) and $$e \ge 0$$ e ≥ 0 , there exists a KTS on v points having at least $$v-3$$ v - 3 automorphisms. This is only one of the consequences of an investigation on the KTSs with an automorphism group G acting sharply transitively on all but three points. Our methods are all constructive and yield KTSs which in many cases inherit some of the automorphisms of G, thus increasing the total number of symmetries. To obtain these results it was necessary to introduce new types of difference families (the doubly disjoint ones) and difference matrices (the splittable ones) which we believe are interesting by themselves.

Deformations of reducible Galois representations to Hida-families

The global deformation theory of residually reducible Galois representations with fixed auxiliary conditions is studied. We show that [Formula: see text] lifts to a Hida line for which the weights range over a congruence class modulo-[Formula: see text]. The advantage of the purely Galois theoretic approach is that it allows us to construct [Formula: see text]-adic families of Galois representations lifting the actual representation [Formula: see text], and not just the semisimplification.

Advance Care Planning—Complex and Working: Longitudinal Trajectory of Congruence in End-of-Life Treatment Preferences: An RCT

Context: The effect of advance care planning (ACP) interventions on the trajectory of end-of-life treatment preference congruence between patients and surrogate decision-makers is unstudied. Objective: To identify unobserved distinctive patterns of congruence trajectories and examine how the typology of outcome development differed between ACP and controls. Methods: Multisite, assessor-blinded, intent-to-treat, randomized clinical trial enrolled participants between October 2013 to March 2017 from 5 hospital-based HIV clinics. Persons living with HIV(PLWH)/surrogate dyads were randomized to 2 weekly 60-minute sessions: ACP (1) ACP facilitated conversation, (2) advance directive completion; or Control (1) Developmental/relationship history, (2) Nutrition/Exercise. Growth Mixed Modeling was used for 18-month post-intervention analysis. Findings: 223 dyads (N = 449 participants) were enrolled. PLWH were 56% male, aged 22 to 77 years, and 86% African American. Surrogates were 56% female, aged 18 to 82 years, and 84% African American. Two latent classes (High vs. Low) of congruence growth trajectory were identified. ACP influenced the trajectory of outcome growth (congruence in all 5 AIDS related situations) by latent class. ACP dyads had a significantly higher probability of being in the High Congruence latent class compared to controls (52%, 75/144 dyads versus 27%, 17/62 dyads, p = 0.001). The probabilities of perfect congruence diminished at 3-months post-intervention but was then sustained. ACP had a significant effect (β = 1.92, p = 0.006, OR = 7.10, 95%C.I.: 1.729, 26.897) on the odds of being in the High Congruence class. Conclusion: ACP had a significant effect on the trajectory of congruence growth over time. ACP dyads had 7 times the odds of congruence, compared to controls. Three-months post-intervention is optimal for booster sessions.

Fundamental identifiability limits in molecular epidemiology

Viral phylogenies provide crucial information on the spread of infectious diseases, and many studies fit mathematical models to phylogenetic data to estimate epidemiological parameters such as the effective reproduction ratio (Re) over time. Such phylodynamic inferences often complement or even substitute for conventional surveillance data, particularly when sampling is poor or delayed. It remains generally unknown, however, how robust phylodynamic epidemiological inferences are, especially when there is uncertainty regarding pathogen prevalence and sampling intensity. Here we use recently developed mathematical techniques to fully characterize the information that can possibly be extracted from serially collected viral phylogenetic data, in the context of the commonly used birth-death-sampling model. We show that for any candidate epidemiological scenario, there exist a myriad of alternative, markedly different and yet plausible “congruent” scenarios that cannot be distinguished using phylogenetic data alone, no matter how large the dataset. In the absence of strong constraints or rate priors across the entire study period, neither maximum-likelihood fitting nor Bayesian inference can reliably reconstruct the true epidemiological dynamics from phylogenetic data alone; rather, estimators can only converge to the “congruence class” of the true dynamics. We propose concrete and feasible strategies for making more robust epidemiological inferences from viral phylogenetic data.

Comparing nonorientable three genus and nonorientable four genus of torus knots

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let [Formula: see text] be any torus knot with [Formula: see text] even and [Formula: see text] odd. The difference between these two invariants on [Formula: see text] is at least [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text]. Hence, the difference between the two invariants on torus knots [Formula: see text] grows arbitrarily large for any fixed odd [Formula: see text], as [Formula: see text] ranges over values of a fixed congruence class modulo [Formula: see text]. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot [Formula: see text] is [Formula: see text], and Kronheimer and Mrowka later proved that the orientable four genus of [Formula: see text] is also this same value.

Congruence relations on almost distributive lattices-II

For a given ideal [Formula: see text] of an almost distributive lattice [Formula: see text], we study the smallest and the largest congruence relation on [Formula: see text] having [Formula: see text] as a congruence class.