1,689 research outputs found

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

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

### $G$-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

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