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

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

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

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

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 $h^\ast$-polynomials, thereby making progress towards conjectures and questions of Stanley, Hibi-Ohsugi, and others regarding the unimodality of their $h^\ast$-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 $h^\ast$-unimodal

Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras

The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope for strength one flows, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. We characterize the DAGs that admit ample framings, and we enumerate the number of ample framings for a fixed DAG. We establish a connection between maximal simplices in DKK triangulations and $\tau$-tilting posets for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that for full DAGs, i.e., those DAGs with inner vertices having in-degree and out-degree equal to two, the flow polytopes are Gorenstein and have unimodal Ehrhart $h^\ast$-polynomials.Comment: Minor revisions. Added Proposition 6.15, which contains a characterization of DAGs with Gorenstein flow polytope

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