A class of identifiable birth-death models

In a striking result, Louca and Pennell (2020) recently proved that a large class of birth-death models are statistically unidentifiable from lineage-through-time (LTT) data. Specifically, they showed that any pair of sufficiently smooth birth and death rate functions is "congruent" to an infinite collection of other rate functions, all of which have the same likelihood for any LTT vector of any dimension. This fact has distressing implications for the thousands of studies which have utilized birth-death models to study evolution. In this paper, we qualify their finding by proving that an alternative and widely used class of birth-death models is indeed identifiable. Specifically, we show that piecewise constant birth-death models can, in principle, be consistently estimated and distinguished from one another, given a sufficiently large extant time tree and some knowledge of the present day population. Subject to mild regularity conditions, we further show that any unidentifiable birth-death model class can be arbitrarily closely approximated by a class of identifiable models. The sampling requirements needed for our results to hold are explicit, and are expected to be satisfied in many contexts such as the phylodynamic analysis of a global pandemic.

All-order quartic couplings in highly symmetric D-brane-anti-D-brane systems

We compute six-point string amplitudes for the scattering of one closed string Ramond-Ramond state, two tachyons and two gauge fields in the worldvolume of D-brane-anti-D-brane systems in the Type II superstring theories. From the resulting S-matrix elements, we read off the precise form of the couplings, including their exact numerical coefficients, of two tachyons and two gauge fields in the corresponding highly symmetric effective field eheory (EFT) Lagrangian in the worldvolume of D-brane-Anti-D-brane to all orders in α′, which modify and complete previous proposals. We verify that the EFT reproduces the infinite collection of stringy gauge field singularities in dual channels. Inspired by interesting similarities between the all-order highly symmetric EFTs and holographic duals of Vasiliev’s higher spin gravities à là Nilsson and Vasiliev, we make a proposal for tensionless limits of D-brane-anti-D-brane systems.

Generalized Catalan Numbers from Hypergraphs

The Catalan numbers $C_{n} \in \{1,1,2,5,14,42,\dots \}$ form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting rooted plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we actually define an infinite collection of generalizations $C_{n}^{(m)}$, $m\geq 1$, with $C_{n}^{(1)}$ equal to the usual Catalans $C_{n}$; the sequence $C_{n}^{(m)}$ comes from studying certain matrix models attached to hypergraphs. We also give some combinatorial interpretations of these numbers.

Solving a Paradox of Evidential Equivalence

David Builes presents a paradox concerning how confident you should be that any given member of an infinite collection of fair coins landed heads, conditional on the information that they were all flipped and only finitely many of them landed heads. We argue that if you should have any conditional credence at all, it should be 1/2.

INTERACTIVE TOOLS FOR LINEAR ALGEBRA: GEOUNIUD

GeoUniud is a user-friendly platform built-in interactive tutors which allow students to investigate specific tasks by selecting their own input values and working through a problem in a step-by-step fashion together with immediate feedback at each step. Lessons and exercises are stored and organized with a careful use of randomized controlled contents as exercises, geometrical pictures and abstract reasoning. The lessons are augmented by a virtually infinite collection of examples, and by interactive representations of concepts. As example we show the design of two interactive tools about linear transformation and change of basis in order to develop students’ sense-making in a dynamic geometry environment (DGE) within the perspective of semiotic mediation.

Chromatic homotopy theory is asymptotically algebraic

Inspired by the Ax–Kochen isomorphism theorem, we develop a notion of categorical ultraproducts to capture the generic behavior of an infinite collection of mathematical objects. We employ this theory to give an asymptotic solution to the approximation problem in chromatic homotopy theory. More precisely, we show that the ultraproduct of the E(n, p)-local categories over any non-principal ultrafilter on the set of prime numbers is equivalent to the ultraproduct of certain algebraic categories introduced by Franke. This shows that chromatic homotopy theory at a fixed height is asymptotically algebraic.

SMOOTHED QUANTILE REGRESSION PROCESSES FOR BINARY RESPONSE MODELS

In this article, we consider binary response models with linear quantile restrictions. Considerably generalizing previous research on this topic, our analysis focuses on an infinite collection of quantile estimators. We derive a uniform linearization for the properly standardized empirical quantile process and discover some surprising differences with the setting of continuously observed responses. Moreover, we show that considering quantile processes provides an effective way of estimating binary choice probabilities without restrictive assumptions on the form of the link function, heteroskedasticity, or the need for high dimensional nonparametric smoothing necessary for approaches available so far. A uniform linear representation and results on asymptotic normality are provided, and the connection to rearrangements is discussed.

Zeno of Elea (fl. c.450 BC)

The Greek philosopher Zeno of Elea was celebrated for his paradoxes. Aristotle called him the ‘founder of dialectic’. He wrote in order to defend the Eleatic metaphysics of his fellow citizen and friend Parmenides, according to whom reality is single, changeless and homogeneous. Zeno’s strength was the production of intriguing arguments which seem to show that apparently straightforward features of the world – most notably plurality and motion – are riddled with contradiction. At the very least he succeeded in establishing that hard thought is required to make sense of plurality and motion. His paradoxes stimulated the atomists, Aristotle and numerous philosophers since to reflect on unity, infinity, continuity and the structure of space and time. Although Zeno wrote a book full of arguments, very few of his actual words have survived. Secondary reports (some from Plato and Aristotle) probably preserve accurately the essence of Zeno’s arguments. Even so, we know only a fraction of the total. According to Plato the arguments in Zeno’s book were of this form: if there are many things, then the same things are both F and not-F; since the same things cannot be both F and not-F, there cannot be many things. Two instances of this form have been preserved: if there were many things, then the same things would be both limited and unlimited; and the same things would be both large (that is, of infinite size) and small (that is, of no size). Quite how the components of these arguments work is not clear. Things are limited (in number), Zeno says, because they are just so many, rather than more or less, while they are unlimited (in number) because any two of them must have a third between them, which separates them and makes them two. Things are of infinite size because anything that exists must have some size: yet anything that has size is divisible into parts which themselves have some size, so that each and every thing will contain an infinite number of extended parts. On the other hand, each thing has no size: for if there are to be many things there have to be some things which are single, unitary things, and these will have no size since anything with size would be a collection of parts. Zeno’s arguments concerning motion have a different form. Aristotle reports four arguments. According to the Dichotomy, motion is impossible because in order to cover any distance it is necessary first to cover half the distance, then half the remainder, and so on without limit. The Achilles is a variant of this: the speedy Achilles will never overtake a tortoise once he has allowed it a head start because Achilles has an endless series of tasks to perform, and each time Achilles sets off to catch up with the tortoise it will turn out that, by the time Achilles arrives at where the tortoise was when he set off, the tortoise has moved on slightly. Another argument, the Arrow, purports to show that an arrow apparently in motion is in fact stationary at each instant of its ‘flight’, since at each instant it occupies a region of space equal in size to itself. The Moving Rows describes three rows (or streams) of equal-sized bodies, one stationary and the other two moving at equal speeds in opposite directions. If each body is one metre long, then the time taken for a body to cover two metres equals the time taken for it to cover four metres (since a moving body will pass two stationary bodies while passing four bodies moving in the opposite direction), and that might be thought impossible. Zeno’s arguments must be resolvable, since the world obviously does contain a plurality of things in motion. There is little agreement, however, on how they should be resolved. Some points can be identified which may have misled Zeno. It is not true, for example, that the sum of an infinite collection of parts, each of which has size, must itself be of an infinite size (it will be false if the parts are of proportionally decreasing size); and something in motion will pass stationary bodies and moving bodies at different velocities. In many other cases, however, there is no general agreement as to the fallacy, if any exists, of Zeno’s argument.