Reconstructing fully-resolved trees from triplet cover distances

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset L of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in L. It is known that any set L with this property for a tree in which all interior vertices have degree 3 must form a cover for T {that is, for each interior vertex v of T, L must contain a pair of leaves from each pair of the three components of T ̶ v. Here we provide a partial converse of this result by showing that if a set L of leaf pairs forms a cover of a certain type for such a tree T then T and its edge weights can be uniquely determined from the distances between the pairs of leaves in L. Moreover, there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning `triplet covers', and is relevant to a problem arising in evolutionary genomics

Tensor structure for Nori motives

We construct a tensor product on Freyd's universal abelian category attached to an additive tensor category or a tensor quiver and establish a universal property. This is used to give an alternative construction for the tensor product on Nori motives.Comment: Revised & updated version, 23 page

SALT SENSITIVE HYPERTENSION AND OREXIN

Elevated plasma arginine vasopressin (AVP) levels have been found in human hypertension subjects and several salt dependent experimental animal models of hypertension including the Dahl salt sensitive hypertension (SSHTN) model. The orexin system is involved in AVP regulation and its over activation has been implicated in hypertension, however, the role of brain orexin in SSHTN is unknown. We hypothesized that increased activity of orexin in the paraventricular nucleus (PVN), a prominent region in AVP production, contributes to SSHTN via enhancing AVP signaling. Eight-week-old male adult Dahl salt sensitive (DS) and age and sex matched Sprague Dawley (SD) rats were placed on either a high salt (HS, 8% NaCl) or normal salt (NS, 0.4% NaCl) diet for 5 weeks. Five weeks HS intake did not increase mean arterial pressure (MAP) or alter PVN mRNA expression of chronic neuronal activation marker Fra1, orexin receptor 1 (OX1R), or orexin receptor 2 (OX2R) but increased PVN AVP mRNA expression in SD rats. HS diet induced significant increases in MAP and PVN mRNA levels of Fra1, AVP, OX1R, and prepro orexin in DS rats. Intracerebroventricular (ICV) infusion of orexin A (0.2 nmol) increased PVN AVP mRNA levels in SD rats. Incubation of cultured hypothalamus neurons from newborn SD rats with orexin A resulted in increases in AVP mRNA expression which were attenuated by OX1R blockade. In addition increased cerebrospinal fluid (CSF) sodium concentration through ICV infusion of NaCl salt solution (4μmol) increased PVN OX1R and AVP mRNA levels in SD rats. Furthermore, bilateral PVN microinjection of the OX1R antagonist SB408124 resulted in a greater reduction in MAP in HS intake (-16±5 mmHg) compared to NS fed (-4±4 mmHg) anesthetized DS rats. These results suggest that elevated PVN OX1R activation may be involved in SSHTN through enhancing AVP signaling

Teaching Differential Equations with Modeling and Visualization

In this article, I explain the history of using Interdisciplinary Lively Applications Projects (ILAPs) in an ordinary differential equations course. Students want to learn methods to solve real world problems, and incorporating ILAPs into the syllabus has been an effective way to apply solution methods to situations that students may encounter in other disciplines. Feedback has been positive and will be shared. Examples of ILAPs currently used will be referenced. For more information about how to develop ILAPs, see Huber and Myers (in Innovative Approaches to Undergraduate Mathematics Courses Beyond Calculus, 2005). Included with this document are three Interdisciplinary Lively Applications Projects (ILAPs). (1) Antaeus.pdf This ILAP uses the Laplace transform to model the struggle between Hercules and Antaeus, a tale from Greek mythology. It is appropriate for use in an introductory course on differential equations. (2) MechanicalResonance.pdf This ILAP discusses mechanical resonance in the context of a vibrating propeller on an airplane wing. It is suitable for use in an introductory course in differential equations. (3) Fever.pdf This ILAP uses Newton\u27s Law of Cooling to explore the question: How long should a thermometer be held in the mouth in order to get an accurate reading? The project is based on an article by Elmo Moore and Charles Biles in the UMAP journal, and is suitable for use in an introductory course in differential equations. In the project, students relate actual temperature measurements to derive the differential equations model, then use the model to answer the question

A matroid associated with a phylogenetic tree

A (pseudo-)metric D on a finite set X is said to be a `tree metric' if there is a finite tree with leaf set X and non-negative edge weights so that, for all x,y ∈X, D(x,y) is the path distance in the tree between x and y. It is well known that not every metric is a tree metric. However, when some such tree exists, one can always find one whose interior edges have strictly positive edge weights and that has no vertices of degree 2, any such tree is 13; up to canonical isomorphism 13; uniquely determined by D, and one does not even need all of the distances in order to fully (re-)construct the tree's edge weights in this case. Thus, it seems of some interest to investigate which subsets of X, 2 suffice to determine (`lasso') these edge weights. In this paper, we use the results of a previous paper to discuss the structure of a matroid that can be associated with an (unweighted) X-tree T defined by the requirement that its bases are exactly the `tight edge-weight lassos' for T, i.e, the minimal subsets of X, 2 that lasso the edge weights of T

Phylogenetic Flexibility via Hall-Type Inequalities and Submodularity

Given a collection τ of subsets of a finite set X, we say that τ is phylogenetically flexible if, for any collection R of rooted phylogenetic trees whose leaf sets comprise the collection τ , R is compatible (i.e. there is a rooted phylogenetic X-tree that displays each tree in R). We show that τ is phylogenetically flexible if and only if it satisfies a Hall-type inequality condition of being ‘slim’. Using submodularity arguments, we show that there is a polynomial-time algorithm for determining whether or not τ is slim. This ‘slim’ condition reduces to a simpler inequality in the case where all of the sets in τ have size 3, a property we call ‘thin’. Thin sets were recently shown to be equivalent to the existence of an (unrooted) tree for which the median function provides an injective mapping to its vertex set; we show here that the unrooted tree in this representation can always be chosen to be a caterpillar tree. We also characterise when a collection τ of subsets of size 2 is thin (in terms of the flexibility of total orders rather than phylogenies) and show that this holds if and only if an associated bipartite graph is a forest. The significance of our results for phylogenetics is in providing precise and efficiently verifiable conditions under which supertree methods that require consistent inputs of trees can be applied to any input trees on given subsets of species

Combinatorial properties of triplet covers for binary trees

It is a classical result that an unrooted tree $T$ having positive real-valued edge lengths and no vertices of degree two can be reconstructed from the induced distance between each pair of leaves. Moreover, if each non-leaf vertex of $T$ has degree 3 then the number of distance values required is linear in the number of leaves. A canonical candidate for such a set of pairs of leaves in $T$ is the following: for each non-leaf vertex $v$, choose a leaf in each of the three components of $T-v$, group these three leaves into three pairs, and take the union of this set over all choices of $v$. This forms a so-called `triplet cover' for $T$. In the first part of this paper we answer an open question (from 2012) by showing that the induced leaf-to-leaf distances for any triplet cover for $T$ uniquely determine $T$ and its edge lengths. We then investigate the finer combinatorial properties of triplet covers. In particular, we describe the structure of triplet covers that satisfy one or more of the following properties of being minimal, `sparse', and `shellable'

Measuring DNS over TCP in the Era of Increasing DNS Response Sizes: A View from the Edge

The Domain Name System (DNS) is one of the most crucial parts of the Internet. Although the original standard defined the usage of DNS over UDP (DoUDP) as well as DNS over TCP (DoTCP), UDP has become the predominant protocol used in the DNS. With the introduction of new Resource Records (RRs), the sizes of DNS responses have increased considerably. Since this can lead to truncation or IP fragmentation, the fallback to DoTCP as required by the standard ensures successful DNS responses by overcoming the size limitations of DoUDP. However, the effects of the usage of DoTCP by stub resolvers are not extensively studied to this date. We close this gap by presenting a view at DoTCP from the Edge, issuing 12.1M DNS requests from 2,500 probes toward Public as well as Probe DNS recursive resolvers. In our measurement study, we observe that DoTCP is generally slower than DoUDP, where the relative increase in Response Time is less than 37% for most resolvers. While optimizations to DoTCP can be leveraged to further reduce the response times, we show that support on Public resolvers is still missing, hence leaving room for optimizations in the future. Moreover, we also find that Public resolvers generally have comparable reliability for DoTCP and DoUDP. However, Probe resolvers show a significantly different behavior: DoTCP queries targeting Probe resolvers fail in 3 out of 4 cases, and, therefore, do not comply with the standard. This problem will only aggravate in the future: As DNS response sizes will continue to grow, the need for DoTCP will solidify.Comment: Published in ACM SIGCOMM Computer Communication Review Volume 52 Issue 2, April 202

The Gromov Norm of the Product of Two Surfaces

We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors have genus 2. The case of arbitrary genera is easy to deduce form this one.Comment: The journal version contains an error that invalidates one direction of the main theorem. The present version contains an erratum, at the end, explaining thi