72 research outputs found
Geometric Properties of Weighted Projective Space Simplices
A long-standing conjecture in geometric combinatorics entails the interplay between three properties of lattice polytopes: reflexivity, the integer decomposition property (IDP), and the unimodality of Ehrhart h*-vectors. Motivated by this conjecture, this dissertation explores geometric properties of weighted projective space simplices, a class of lattice simplices related to weighted projective spaces.
In the first part of this dissertation, we are interested in which IDP reflexive lattice polytopes admit regular unimodular triangulations. Let Delta(1,q)denote the simplex corresponding to the associated weighted projective space whose weights are given by the vector (1,q). Focusing on the case where Delta(1,q) is IDP reflexive such that q has two distinct parts, we establish a characterization of the lattice points contained in Delta(1,q) and verify the existence of a regular unimodular triangulation of its lattice points by constructing a Grobner basis with particular properties.
In the second part of this dissertation, we explore how a necessary condition for IDP that relaxes the IDP characterization of Braun, Davis, and Solus yields a natural parameterization of an infinite class of reflexive simplices associated to weighted projective spaces. This parametrization allows us to check the IDP condition for reflexive simplices having high dimension and large volume, and to investigate h* unimodality for the resulting IDP reflexives in the case that Delta(1,q) is 3-supported
Projective Normality and Ehrhart Unimodality for Weighted Projective Space Simplices
Within the intersection of Ehrhart theory, commutative algebra, and algebraic
geometry lie lattice polytopes. Ehrhart theory is concerned with lattice point
enumeration in dilates of polytopes; lattice polytopes provide a sandbox in
which to test many conjectures in commutative algebra; and many properties of
projectively normal toric varieties in algebraic geometry are encoded through
corresponding lattice polytopes. In this article we focus on reflexive
simplices and work to identify when these have the integer decomposition
property (IDP), or equivalently, when certain weighted projective spaces are
projectively normal. We characterize the reflexive, IDP simplices whose
associated weighted projective spaces have one projective coordinate with
weight fixed to unity and for which the remaining coordinates can assume one of
three distinct weights. We show that several subfamilies of such reflexive
simplices have unimodal -polynomials, thereby making progress towards
conjectures and questions of Stanley, Hibi-Ohsugi, and others regarding the
unimodality of their -polynomials. We also provide computational
results and introduce the notion of reflexive stabilizations to explore the
(non-)ubiquity of reflexive simplices that are simultaneously IDP and
-unimodal
Link Prediction Based on Local Random Walk
The problem of missing link prediction in complex networks has attracted much
attention recently. Two difficulties in link prediction are the sparsity and
huge size of the target networks. Therefore, the design of an efficient and
effective method is of both theoretical interests and practical significance.
In this Letter, we proposed a method based on local random walk, which can give
competitively good prediction or even better prediction than other
random-walk-based methods while has a lower computational complexity.Comment: 6 pages, 2 figure
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
Ground Testing of the EMCS Seed Cassette for Biocompatibility with the Cellular Slime Mold, Dictyostelium Discoideum
The European Modular Cultivation System, EMCS, was developed by ESA for plant experiments. To expand the use of flight verified hardware for various model organisms, we performed ground experiments to determine whether ARC EMCS Seed Cassettes could be adapted for use with cellular slime mold for future space flight experiments. Dictyostelium is a cellular slime mold that can exist both as a single-celled independent organism and as a part of a multicellular colony which functions as a unit (pseudoplasmodium). Under certain stress conditions, individual amoebae will aggregate to form multicellular structures. Developmental pathways are very similar to those found in Eukaryotic organisms, making this a uniquely interesting organism for use in genetic studies. Dictyostelium has been used as a genetic model organism for prior space flight experiments. Due to the formation of spores that are resistant to unfavorable conditions such as desiccation, Dictyostelium is also a good candidate for use in the EMCS Seed Cassettes. The growth substratum in the cassettes is a gridded polyether sulfone (PES) membrane. A blotter beneath the PES membranes contains dried growth medium. The goals of this study were to (1) verify that Dictyostelium are capable of normal growth and development on PES membranes, (2) develop a method for dehydration of Dictyostelium spores with successful recovery and development after rehydration, and (3) successful mock rehydration experiments in cassettes. Our results show normal developmental progression in two strains of Dictyostelium discoideum on PES membranes with a bacterial food source. We have successfully performed a mock rehydration of spores with developmental progression from aggregation to slug formation, and production of morphologically normal spores within 9 days of rehydration. Our results indicate that experiments on the ISS using the slime mold, Dictyostelium discoideum could potentially be performed in the flight verified hardware of the EMCS ARC Seed Cassettes
Effective and Efficient Similarity Index for Link Prediction of Complex Networks
Predictions of missing links of incomplete networks like protein-protein
interaction networks or very likely but not yet existent links in evolutionary
networks like friendship networks in web society can be considered as a
guideline for further experiments or valuable information for web users. In
this paper, we introduce a local path index to estimate the likelihood of the
existence of a link between two nodes. We propose a network model with
controllable density and noise strength in generating links, as well as collect
data of six real networks. Extensive numerical simulations on both modeled
networks and real networks demonstrated the high effectiveness and efficiency
of the local path index compared with two well-known and widely used indices,
the common neighbors and the Katz index. Indeed, the local path index provides
competitively accurate predictions as the Katz index while requires much less
CPU time and memory space, which is therefore a strong candidate for potential
practical applications in data mining of huge-size networks.Comment: 8 pages, 5 figures, 3 table
Uncovering missing links with cold ends
To evaluate the performance of prediction of missing links, the known data
are randomly divided into two parts, the training set and the probe set. We
argue that this straightforward and standard method may lead to terrible bias,
since in real biological and information networks, missing links are more
likely to be links connecting low-degree nodes. We therefore study how to
uncover missing links with low-degree nodes, namely links in the probe set are
of lower degree products than a random sampling. Experimental analysis on ten
local similarity indices and four disparate real networks reveals a surprising
result that the Leicht-Holme-Newman index [E. A. Leicht, P. Holme, and M. E. J.
Newman, Phys. Rev. E 73, 026120 (2006)] performs the best, although it was
known to be one of the worst indices if the probe set is a random sampling of
all links. We further propose an parameter-dependent index, which considerably
improves the prediction accuracy. Finally, we show the relevance of the
proposed index on three real sampling methods.Comment: 16 pages, 5 figures, 6 table
Link Prediction in Complex Networks: A Survey
Link prediction in complex networks has attracted increasing attention from
both physical and computer science communities. The algorithms can be used to
extract missing information, identify spurious interactions, evaluate network
evolving mechanisms, and so on. This article summaries recent progress about
link prediction algorithms, emphasizing on the contributions from physical
perspectives and approaches, such as the random-walk-based methods and the
maximum likelihood methods. We also introduce three typical applications:
reconstruction of networks, evaluation of network evolving mechanism and
classification of partially labelled networks. Finally, we introduce some
applications and outline future challenges of link prediction algorithms.Comment: 44 pages, 5 figure
Recommender Systems
The ongoing rapid expansion of the Internet greatly increases the necessity
of effective recommender systems for filtering the abundant information.
Extensive research for recommender systems is conducted by a broad range of
communities including social and computer scientists, physicists, and
interdisciplinary researchers. Despite substantial theoretical and practical
achievements, unification and comparison of different approaches are lacking,
which impedes further advances. In this article, we review recent developments
in recommender systems and discuss the major challenges. We compare and
evaluate available algorithms and examine their roles in the future
developments. In addition to algorithms, physical aspects are described to
illustrate macroscopic behavior of recommender systems. Potential impacts and
future directions are discussed. We emphasize that recommendation has a great
scientific depth and combines diverse research fields which makes it of
interests for physicists as well as interdisciplinary researchers.Comment: 97 pages, 20 figures (To appear in Physics Reports
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