Ramification theory for degree pp extensions of arbitrary valuation rings in mixed characteristic (0,p)(0,p)

We previously obtained a generalization and refinement of results about the ramification theory of Artin-Schreier extensions of discretely valued fields in characteristic $p$ with perfect residue fields to the case of fields with more general valuations and residue fields. As seen in VT16, the "defect" case gives rise to many interesting complications. In this paper, we present analogous results for degree $p$ extensions of arbitrary valuation rings in mixed characteristic $(0,p)$ in a more general setting. More specifically, the only assumption here is that the base field $K$ is henselian. In particular, these results are true for defect extensions even if the rank of the valuation is greater than $1$. A similar method also works in equal characteristic, generalizing the results of VT16.Comment: 20 pages. VT16: Ramification Theory for Artin-Schreier Extensions of Valuation Rings, Journal of Algebra (2016), pp. 355-38

Invertibility of the TKF model of sequence evolution

We consider character sequences evolving on a phylogenetic tree under the TKF91 model. We show that as the sequence lengths tend to infinity the the topology of the phylogenetic tree and the edge lengths are determined by any one of (a) the alignment of sequences (b) the collection of sequence lengths. We also show that the probability of any homology structure on a collection of sequences related by a TKF91 process on a tree is independent of the root location. Keywords: phylogenetics, DNA sequence evolution models, identifiability, alignmentComment: 23 page

GG-reconstruction of graphs

Let $G$ be a group of permutations acting on an $n$-vertex set $V$, and $X$ and $Y$ be two simple graphs on $V$. We say that $X$ and $Y$ are $G$-isomorphic if $Y$ belongs to the orbit of $X$ under the action of $G$. One can naturally generalize the reconstruction problems so that when $G$ is $S_n$, the symmetric group, we have the usual reconstruction problems. In this paper, we study $G$-edge reconstructibility of graphs. We prove some old and new results on edge reconstruction and reconstruction from end vertex deleted subgraphs.Comment: 8 page

A reconstruction problem related to balance equations-II: the general case

A modified $k$-deck of a graph $G$ is obtained by removing $k$ edges of $G$ in all possible ways, and adding $k$ (not necessarily new) edges in all possible ways. Krasikov and Roditty asked if it was possible to construct the usual $k$-edge deck of a graph from its modified $k$-deck. Earlier I solved this problem for the case when $k=1$. In this paper, the problem is completely solved for arbitrary $k$. The proof makes use of the $k$-edge version of Lov\'asz's result and the eigenvalues of certain matrix related to the Johnson graph. This version differs from the published version. Lemma 2.3 in the published version had a typo in one equation. Also, a long manipulation of some combinatorial expressions was skipped in the original proof of Lemma 2.3, which made it difficult to follow the proof. Here a clearer proof is given.Comment: Improved version of Discrete Mathematics 194, no. 1-3(1999) 281-28

Kocay's lemma, Whitney's theorem, and some polynomial invariant reconstruction problems

Given a graph G, an incidence matrix N(G) is defined for the set of distinct isomorphism types of induced subgraphs of G. If Ulam's conjecture is true, then every graph invariant must be reconstructible from this matrix, even when the graphs indexing the rows and the columns of N(G) are unspecified. It is proved that the characteristic polynomial, the rank polynomial, and the number of spanning trees of a graph are reconstructible from its N-matrix. These results are stronger than the original results of Tutte in the sense that actual subgraphs are not used. It is also proved that the characteristic polynomial of a graph with minimum degree 1 can be computed from the characteristic polynomials of all its induced proper subgraphs. The ideas in Kocay's lemma play a crucial role in most proofs. Here Kocay's lemma is used to prove Whitney's subgraph expansion theorem in a simple manner. The reconstructibility of the characteristic polynomial is then demonstrated as a direct consequence of Whitney's theorem as formulated here.Comment: 31 page

A reconstruction problem related to balance equations-I

A modified $k$-deck of a graph is obtained by removing $k$ edges in all possible ways and adding $k$ (not necessarily new) edges in all possible ways. Krasikov and Roditty used these decks to give an independent proof of M\"uller's result on the edge reconstructibility of graphs. They asked if a $k$-edge deck could be constructed from its modified $k$-deck. In this paper, we solve the problem when $k=1$. We also offer new proofs of Lov\'asz's result, one describing the constructed graph explicitly, (thus answering a question of Bondy), and another based on the eigenvalues of Johnson graph.Comment: 7 page

Revisiting an equivalence between maximum parsimony and maximum likelihood methods in phylogenetics

Tuffley and Steel (1997) proved that Maximum Likelihood and Maximum Parsimony methods in phylogenetics are equivalent for sequences of characters under a simple symmetric model of substitution with no common mechanism. This result has been widely cited ever since. We show that small changes to the model assumptions suffice to make the two methods inequivalent. In particular, we analyze the case of bounded substitution probabilities as well as the molecular clock assumption. We show that in these cases, even under no common mechanism, Maximum Parsimony and Maximum Likelihood might make conflicting choices. We also show that if there is an upper bound on the substitution probabilities which is `sufficiently small', every Maximum Likelihood tree is also a Maximum Parsimony tree (but not vice versa)

Maximum Parsimony on Subsets of Taxa

In this paper we investigate mathematical questions concerning the reliability (reconstruction accuracy) of Fitch's maximum parsimony algorithm for reconstructing the ancestral state given a phylogenetic tree and a character. In particular, we consider the question whether the maximum parsimony method applied to a subset of taxa can reconstruct the ancestral state of the root more accurately than when applied to all taxa, and we give an example showing that this indeed is possible. A surprising feature of our example is that ignoring a taxon closer to the root improves the reliability of the method. On the other hand, in the case of the two-state symmetric substitution model, we answer affirmatively a conjecture of Li, Steel and Zhang which states that under a molecular clock the probability that the state at a single taxon is a correct guess of the ancestral state is a lower bound on the reconstruction accuracy of Fitch's method applied to all taxa

Reconstructing pedigrees: a stochastic perspective

A pedigree is a directed graph that describes how individuals are related through ancestry in a sexually-reproducing population. In this paper we explore the question of whether one can reconstruct a pedigree by just observing sequence data for present day individuals. This is motivated by the increasing availability of genomic sequences, but in this paper we take a more theoretical approach and consider what models of sequence evolution might allow pedigree reconstruction (given sufficiently long sequences). Our results complement recent work that showed that pedigree reconstruction may be fundamentally impossible if one uses just the degrees of relatedness between different extant individuals. We find that for certain stochastic processes, pedigrees can be recovered up to isomorphism from sufficiently long sequences.Comment: 20 pages, 3 figure

A Smart Meter Data-driven Distribution Utility Rate Model for Networks with Prosumers

Distribution grids across the world are undergoing profound changes due to advances in energy technologies. Electrification of the transportation sector and the integration of Distributed Energy Resources (DERs), such as photo-voltaic panels and energy storage devices, have gained substantial momentum, especially at the grid edge. Transformation in the technological aspects of the grid could directly conflict with existing distribution utility retail tariff structures. We propose a smart meter data-driven rate model to recover distribution network-related charges, where the implementation of these grid-edge technologies is aligned with the interest of the various stakeholders in the electricity ecosystem. The model envisions a shift from charging end-users based on their KWh volumetric consumption, towards charging them a "grid access fee" that approximates the impact of end-users' time-varying demand on their local distribution network. The proposed rate incorporates two cost metrics affecting distribution utilities (DUs), namely 'magnitude' and 'variability' of customer demand. The proposed rate can be applied to prosumers and conventional consumers without DERs.Comment: Accepted to Utilities Policy Journal, to appear in 2021 (https://www.sciencedirect.com/journal/utilities-policy