961 research outputs found
Divisors on graphs, Connected flags, and Syzygies
We study the binomial and monomial ideals arising from linear equivalence of
divisors on graphs from the point of view of Gr\"obner theory. We give an
explicit description of a minimal Gr\"obner bases for each higher syzygy
module. In each case the given minimal Gr\"obner bases is also a minimal
generating set. The Betti numbers of the binomial ideal and its natural initial
ideal coincide and they correspond to the number of 'connected flags' in the
graph. In particular the Betti numbers are independent of the characteristic of
the base field. For complete graphs the problem was previously studied by
Postnikov and Shapiro and by Manjunath and Sturmfels. The case of a general
graph was stated as an open problem.Comment: to appear in International Mathematics Research Notices (IMRN
Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux
We study the combinatorics of Gr\"obner degenerations of Grassmannians and
the Schubert varieties inside them. We provide a family of binomial ideals
whose combinatorics is governed by tableaux induced by matching fields in the
sense of Sturmfels and Zelevinsky. We prove that these ideals are all
quadratically generated and they yield a SAGBI basis of the Pl\"ucker algebra.
This leads to a new family of toric degenerations of Grassmannians. Moreover,
we apply our results to construct a family of Gr\"obner degenerations of
Schubert varieties inside Grassmannians. We provide a complete characterization
of toric ideals among these degenerations in terms of the combinatorics of
matching fields, permutations, and semi-standard tableaux
Prime splittings of Determinantal Ideals
We consider determinantal ideals, where the generating minors are encoded in
a hypergraph. We study when the generating minors form a Gr\"obner basis. In
this case, the ideal is radical, and we can describe algebraic and numerical
invariants of these ideals in terms of combinatorial data of their hypergraphs,
such as the clique decomposition. In particular, we can construct a minimal
free resolution as a tensor product of the minimal free resolution of their
cliques. For several classes of hypergraphs we find a combinatorial description
of the minimal primes in terms of a prime splitting. That is, we write the
determinantal ideal as a sum of smaller determinantal ideals such that each
minimal prime is a sum of minimal primes of the summands.Comment: Final version to appear in Communications in Algebr
Toric degenerations of flag varieties from matching field tableaux
We present families of tableaux which interpolate between the classical
semi-standard Young tableaux and matching field tableaux. Algebraically, this
corresponds to SAGBI bases of Pl\"ucker algebras. We show that each such family
of tableaux leads to a toric ideal, that can be realized as initial of the
Pl\"ucker ideal, hence a toric degeneration for the flag variety
An Efficient Algorithm for Computing Network Reliability in Small Treewidth
We consider the classic problem of Network Reliability. A network is given
together with a source vertex, one or more target vertices, and probabilities
assigned to each of the edges. Each edge appears in the network with its
associated probability and the problem is to determine the probability of
having at least one source-to-target path. This problem is known to be NP-hard.
We present a linear-time fixed-parameter algorithm based on a parameter
called treewidth, which is a measure of tree-likeness of graphs. Network
Reliability was already known to be solvable in polynomial time for bounded
treewidth, but there were no concrete algorithms and the known methods used
complicated structures and were not easy to implement. We provide a
significantly simpler and more intuitive algorithm that is much easier to
implement.
We also report on an implementation of our algorithm and establish the
applicability of our approach by providing experimental results on the graphs
of subway and transit systems of several major cities, such as London and
Tokyo. To the best of our knowledge, this is the first exact algorithm for
Network Reliability that can scale to handle real-world instances of the
problem.Comment: 14 page
A Family of Quasisymmetry Models
We present a one-parameter family of models for square contingency tables
that interpolates between the classical quasisymmetry model and its Pearsonian
analogue. Algebraically, this corresponds to deformations of toric ideals
associated with graphs. Our discussion of the statistical issues centers around
maximum likelihood estimation.Comment: 17 pages, 10 figure
Tensors in statistics and rigidity theory
This is a short report on the discussions of appearance of tensors in
algebraic statistics and rigidity theory, during the semester ``AGATES:
Algebraic Geometry with Applications to TEnsors and Secants". We briefly survey
some of the existing results in the literature and further research directions.
We first provide an overview of algebraic and geometric techniques in the study
of conditional independence (CI) statistical models. We study different
families of algebraic varieties arising in statistics. This includes the
determinantal varieties related to CI statements with hidden random variables.
Such statements correspond to determinantal conditions on the tensor of joint
probabilities of events involving the observed random variables. We show how to
compute the irreducible decompositions of the corresponding CI varieties, which
leads to finding further conditional dependencies (or independencies) among the
involved random variables. As an example, we show how these methods can be
applied to extend the classical intersection axiom for CI statements. We then
give a brief overview about secant varieties and their appearance in the study
of mixture models. We focus on examples and briefly mention the connection to
rigidity theory which will appear in the forthcoming paper \cite{rigidity}.Comment: Comments are welcome! arXiv admin note: substantial text overlap with
arXiv:2103.1655
To Wear or Not to Wear the Hijab Online (a Study of the Identity Performances of Muslim Canadian Women on Facebook)
This paper looks at how Muslim women with an Iranian background and now living in Canada perform their identity through wearing the hijab. This was achieved by observing the behavior of six members of this community on Facebook using Erving Goffman’s stigmatization theory. The observation reveals that women who wear the hijab are more likely to identify themselves as Muslim-Canadian while those who have abandoned the hijab after immigration are more likely to identify themselves as Iranian-Canadian. Moreover, the results show that while Goffman’s theory is very useful in trying to understand the stigmatization of the veil after the 9/11 attacks as well as other extremists’ attacks, the pressures that this created on Muslim women, as well as the behavior of some women in dropping the veil in order to ‘pass’ such stigmatization, his theory is of limited use in understanding the more complicated performance of women who kept their hijab in spite of the challenges they faced
Construction of Adaptive Multistep Methods for Problems with Discontinuities, Invariants, and Constraints
Adaptive multistep methods have been widely used to solve initial value problems. These ordinary differential equations (ODEs) may arise from semi-discretization of time-dependent partial differential equations(PDEs) or may combine with some algebraic equations to represent a differential algebraic equations (DAEs).In this thesis we study the initialization of multistep methods and parametrize some well-known classesof multistep methods to obtain an adaptive formulation of those methods. The thesis is divided into three main parts; (re-)starting a multistep method, a polynomial formulation of strong stability preserving (SSP)multistep methods and parametric formulation of blocked multistep methods.Depending on the number of steps, a multistep method requires adequate number of initial values tostart the integration. In the view of first part, we look at the available initialization schemes and introduce two family of Runge--Kutta methods derived to start multistep methods with low computational cost and accurate initial values.The proposed starters estimate the error by embedded methods.The second part concerns the variable step-size blocked multistep methods. We use the polynomial formulation of multistep methods applied on ODEs to parametrize blocked multistep methods forthe solution of index-2 Euler-Lagrange DAEs. The performance of the adaptive formulation is verified by some numerical experiments. For the last part, we apply a polynomial formulation of multistep methods to formulate SSP multistep methods that are applied for the solution of semi-discretized hyperbolic PDEs. This formulationallows time adaptivity by construction
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