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.

On the existence of four or more curved foldings with common creases and crease patterns

Consider an oriented curve $$\Gamma $$ Γ in a domain D in the plane $${\varvec{R}}^2$$ R 2 . Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space $${\varvec{R}}^3$$ R 3 . This can be expressed as the image of an “origami map” $$\Phi :D\rightarrow {\varvec{R}}^3$$ Φ : D → R 3 such that $$\Gamma $$ Γ is the singular set of $$\Phi $$ Φ , the word “origami” coming from the Japanese term for paper folding. We call the singular set image $$C:=\Phi (\Gamma )$$ C : = Φ ( Γ ) the crease of $$\Phi $$ Φ and the singular set $$\Gamma $$ Γ the crease pattern of $$\Phi $$ Φ . We are interested in the number of origami maps whose creases and crease patterns are C and $$\Gamma $$ Γ , respectively. Two such possibilities have been known. In the authors’ previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possible values for the number N of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if C is a non-closed simple arc, then $$N=4$$ N = 4 if and only if both $$\Gamma $$ Γ and C do not admit any symmetries. On the other hand, when C is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.

FACTORIZATION LENGTH DISTRIBUTION FOR AFFINE SEMIGROUPS III: MODULAR EQUIDISTRIBUTION FOR NUMERICAL SEMIGROUPS WITH ARBITRARILY MANY GENERATORS

For numerical semigroups with a specified list of (not necessarily minimal) generators, we describe the asymptotic distribution of factorization lengths with respect to an arbitrary modulus. In particular, we prove that the factorization lengths are equidistributed across all congruence classes that are not trivially ruled out by modular considerations.

On the number of factorizations of $ t $ mod $ N $ and the probability distribution of Diffie-Hellman secret keys for many users

<p style='text-indent:20px;'>We study the number <inline-formula><tex-math id="M3">\begin{document}$ R_n(t,N) $\end{document}</tex-math></inline-formula> of tuplets <inline-formula><tex-math id="M4">\begin{document}$ (x_1,\ldots, x_n) $\end{document}</tex-math></inline-formula> of congruence classes modulo <inline-formula><tex-math id="M5">\begin{document}$ N $\end{document}</tex-math></inline-formula> such that</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} x_1\cdots x_n \equiv t \pmod{N}. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>As a result, we derive a recurrence for <inline-formula><tex-math id="M6">\begin{document}$ R_n(t,N) $\end{document}</tex-math></inline-formula> and prove some multiplicative properties of <inline-formula><tex-math id="M7">\begin{document}$ R_n(t,N) $\end{document}</tex-math></inline-formula>. Furthermore, we apply the result to study the probability distribution of Diffie-Hellman keys used in multiparty communication. We show that this probability distribution is not uniform.</p>

Metric Diophantine approximation with congruence conditions

We prove a version of the Khinchin–Groshev theorem for Diophantine approximation of matrices subject to a congruence condition. The proof relies on an extension of the Dani correspondence to the quotient by a congruence subgroup. This correspondence together with a multiple ergodic theorem are used to study rational approximations in several congruence classes simultaneously. The result in this part holds in the generality of weighted approximation but is restricted to simple approximation functions.

Brick polytopes, lattices and Hopf algebras

International audience Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope