635 research outputs found
On Flow Polytopes, nu-Associahedra, and the Subdivision Algebra
This dissertation studies the geometry and combinatorics related to a flow polytope Fcar(ν) constructed from a lattice path ν, whose volume is given by the ν-Catalan numbers. It begins with a study of the ν-associahedron introduced by Ceballos, Padrol, and Sarmiento in 2019, but from the perspective of Schröder combinatorics. Some classical results for Schröder paths are extended to the ν-setting, and insights into the geometry of the ν-associahedron are obtained by describing its face poset with two ν-Schröder objects. The ν-associahedron is then shown to be dual to a framed triangulation of Fcar(ν), which is a geometric realization of the ν-Tamari complex. The dual graph of this triangulation is the Hasse diagram of the ν-Tamari lattice due to Préville-Ratelle and Viennot. The dual graph of a second framed triangulation of Fcar(ν) is shown to be the Hasse diagram of a principal order ideal of Young’s lattice generated by ν, and is used to show that the h∗-vector of Fcar(ν) is given by ν-Narayana numbers. This perspective serves to unify these two important lattices associated with ν-Dyck paths through framed triangulations of a flow polytope. Via an integral equivalence between Fcar(ν) and a subpolytope UI,J of a product of two simplices subdivisions of UI,J are shown to be obtainable with Mészáros’ subdivision algebra, which answers a question of Ceballos, Padrol, and Sarmiento. Building on this result, the subdivision algebra is extended to encode subdivisions of a product of two simplices, giving a new tool for their future study
Triangulations, order polytopes, and generalized snake posets
This work regards the order polytopes arising from the class of generalized
snake posets and their posets of meet-irreducible elements. Among generalized
snake posets of the same rank, we characterize those whose order polytopes have
minimal and maximal volume. We give a combinatorial characterization of the
circuits in these order polytopes and then conclude that every regular
triangulation is unimodular. For a generalized snake word, we count the number
of flips for the canonical triangulation of these order polytopes. We determine
that the flip graph of the order polytope of the poset whose lattice of filters
comes from a ladder is the Cayley graph of a symmetric group. Lastly, we
introduce an operation on triangulations called twists and prove that twists
preserve regular triangulations.Comment: 39 pages, 26 figures, comments welcomed
A Next-Generation Liquid Xenon Observatory for Dark Matter and Neutrino Physics
The nature of dark matter and properties of neutrinos are among the mostpressing issues in contemporary particle physics. The dual-phase xenontime-projection chamber is the leading technology to cover the availableparameter space for Weakly Interacting Massive Particles (WIMPs), whilefeaturing extensive sensitivity to many alternative dark matter candidates.These detectors can also study neutrinos through neutrinoless double-beta decayand through a variety of astrophysical sources. A next-generation xenon-baseddetector will therefore be a true multi-purpose observatory to significantlyadvance particle physics, nuclear physics, astrophysics, solar physics, andcosmology. This review article presents the science cases for such a detector.<br
A next-generation liquid xenon observatory for dark matter and neutrino physics
The nature of dark matter and properties of neutrinos are among the most pressing issues in contemporary particle physics. The dual-phase xenon time-projection chamber is the leading technology to cover the available parameter space for weakly interacting massive particles, while featuring extensive sensitivity to many alternative dark matter candidates. These detectors can also study neutrinos through neutrinoless double-beta decay and through a variety of astrophysical sources. A next-generation xenon-based detector will therefore be a true multi-purpose observatory to significantly advance particle physics, nuclear physics, astrophysics, solar physics, and cosmology. This review article presents the science cases for such a detector
A subdivision algebra for a product of simplices via flow polytopes
For any lattice path from the origin to a point , we construct
an associated flow polytope arising from
an acyclic graph where bidirectional edges are permitted. We show that the flow
polytope admits a subdivision whose dual
is a -simplex, where is the number of valleys in the path .
Refinements of this subdivision can be obtained by reductions of a polynomial
in a generalization of M\'esz\'aros' subdivision algebra for acyclic
root polytopes that allows negative roots. Via an integral equivalence between
and the product of simplices
, we obtain a subdivision algebra for a product of two
simplices. As a special case, we give a reduction order for reducing
which encodes the cyclic -Tamari complex of Ceballos, Padrol, and
Sarmiento.Comment: 19 pages, 10 figure
Pólya's Enumeration Theorem and Its Applications
This thesis presents and proves Pólya's enumeration theorem (PET) along with the necessary background knowledge. Also, applications are presented in coloring problems, graph theory, number theory and chemistry. The statement and proof of PET is preceded by detailed discussions on Burnside's lemma, the cycle index, weight functions, configurations and the configuration generating function. After the proof of PET, it is applied to the enumerations of colorings of polytopes of dimension 2 and 3, including necklaces, the cube, and the truncated icosahedron. The general formulas for the number of n-colorings of the latter two are also derived. In number theory, work by Chong-Yun Chao is presented, which uses PET to derive generalized versions of Fermat's Little Theorem and Gauss' Theorem. In graph theory, some classic graphical enumeration results of Pólya, Harary and Palmer are presented, particularly the enumeration of the isomorphism classes of unlabeled trees and (v,e)-graphs. The enumeration of all (5,e)-graphs is given as an example. The thesis is concluded with a presentation of how Pólya applied his enumeration technique to the enumeration of chemical compounds
On the subdivision algebra for the polytope
The polytopes were introduced by Ceballos, Padrol,
and Sarmiento to provide a geometric approach to the study of
-Tamari lattices. They observed a connection between certain
and acyclic root polytopes, and wondered if
M\'esz\'aros' subdivision algebra can be used to subdivide all
. We answer this in the affirmative from two
perspectives, one using flow polytopes and the other using root polytopes. We
show that is integrally equivalent to a flow polytope
that can be subdivided using the subdivision algebra. Alternatively, we find a
suitable projection of to an acyclic root polytope
which allows subdivisions of the root polytope to be lifted back to
. As a consequence, this implies that subdivisions of
can be obtained with the algebraic interpretation of
using reduced forms of monomials in the subdivision algebra. In addition, we
show that the -Tamari complex can be obtained as a triangulated
flow polytope.Comment: 19 pages, 3 figure
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Triangulations, Order Polytopes, and Generalized Snake Posets
This work regards the order polytopes arising from the class of generalized snake posets and their posets of meet-irreducible elements. Among generalized snake posets of the same rank, we characterize those whose order polytopes have minimal and maximal volume. We give a combinatorial characterization of the circuits in related order polytopes and then conclude that all of their triangulations are unimodular. For a generalized snake word, we count the number of flips for the canonical triangulation of these order polytopes. We determine that the flip graph of the order polytope of the poset whose lattice of upper order ideals comes from a ladder is the Cayley graph of a symmetric group. Lastly, we introduce an operation on triangulations called twists and prove that twists preserve regular triangulations.Mathematics Subject Classifications: 52B20, 52B05, 52B12, 06A07Keywords: Order polytopes, triangulations, flow polytopes, circuit
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