🍩 Database of Original & NonTheoretical Uses of Topology
(found 24 matches in 0.002829s)


Loops Abound in the Cosmic Microwave Background: A \$4\sigma\$ Anomaly on SuperHorizon Scales (2021)
Pratyush PranavAbstract
We present a topological analysis of the temperature fluctuation maps from the \emph\Planck 2020\ Data release 4 (DR4) based on the \texttt\NPIPE\ data processing pipeline. For comparison, we also present the topological characteristics of the maps from \emph\Planck 2018\ Data release 3 (DR3). We perform our analysis in terms of the homology characteristics of the maps, invoking relative homology to account for analysis in the presence of masks. We perform our analysis for a range of smoothing scales spanning sub and superhorizon scales corresponding to \$FWHM = 5', 10', 20', 40', 80', 160', 320', 640'\$. Our main result indicates a significantly anomalous behavior of the loops in the observed maps compared to simulations that are modeled as isotopic and homogeneous Gaussian random fields. Specifically, we observe a \$4\sigma\$ deviation between the observation and simulations in the number of loops at \$FWHM = 320'\$ and \$FWHM = 640'\$, corresponding to superhorizon scales of \$5\$ degrees and larger. In addition, we also notice a mildly significant deviation at \$2\sigma\$ for all the topological descriptors for almost all the scales analyzed. Our results show a consistency across different data releases, and therefore, the anomalous behavior deserves a careful consideration regarding its origin and ramifications. Disregarding the unlikely source of the anomaly being instrumental systematics, the origin of the anomaly may be genuinely astrophysical  perhaps due to a yet unresolved foreground, or truly primordial in nature. Given the nature of the topological descriptors, that potentially encodes information of all orders, nonGaussianities, of either primordial or latetype nature, may be potential candidates. Alternate possibilities include the Universe admitting a nontrivial global topology, including effects induced by largescale topological defects. 
Finding Universal Structures in Quantum ManyBody Dynamics via Persistent Homology (2020)
Daniel Spitz, Jürgen Berges, Markus K. Oberthaler, Anna WienhardAbstract
Inspired by topological data analysis techniques, we introduce persistent homology observables and apply them in a geometric analysis of the dynamics of quantum field theories. As a prototype application, we consider simulated data of a twodimensional Bose gas far from equilibrium. We discover a continuous spectrum of dynamical scaling exponents, which provides a refined classification of nonequilibrium universal phenomena. A possible explanation of the underlying processes is provided in terms of mixing wave turbulence and vortex kinetics components in point clouds. We find that the persistent homology scaling exponents are inherently linked to the geometry of the system, as the derivation of a packing relation reveals. The approach opens new ways of analyzing quantum manybody dynamics in terms of robust topological structures beyond standard field theoretic techniques. 
Path Homology as a Stronger Analogue of Cyclomatic Complexity (2020)
Steve HuntsmanAbstract
Cyclomatic complexity is an incompletely specified but mathematically principled software metric that can be usefully applied to both source and binary code. We consider the application of path homology as a stronger analogue of cyclomatic complexity. We have implemented an algorithm to compute path homology in arbitrary dimension and applied it to several classes of relevant flow graphs, including randomly generated flow graphs representing structured and unstructured control flow. We also compared path homology and cyclomatic complexity on a set of disassembled binaries obtained from the grep utility. There exist control flow graphs realizable at the assembly level with nontrivial path homology in arbitrary dimension. We exhibit several classes of examples in this vein while also experimentally demonstrating that path homology gives identicial results to cyclomatic complexity for at least one detailed notion of structured control flow. We also experimentally demonstrate that the two notions differ on disassembled binaries, and we highlight an example of extreme disagreement. Path homology empirically generalizes cyclomatic complexity for an elementary notion of structured code and appears to identify more structurally relevant features of control flow in general. Path homology therefore has the potential to substantially improve upon cyclomatic complexity. 
Exact Topological Inference of the RestingState Brain Networks in Twins (2019)
Moo K. Chung, Hyekyoung Lee, Hernando Ombao, Victor Solo 
PhaseField Investigation of the Coarsening of Porous Structures by Surface Diffusion (2019)
PierreAntoine Geslin, Mickaël Buchet, Takeshi Wada, Hidemi KatoAbstract
Nano and microporous connected structures have attracted increasing attention in the past decades due to their high surface area, presenting interesting properties for a number of applications. These structures generally coarsen by surface diffusion, leading to an enlargement of the structure characteristic length scale. We propose to study this coarsening behavior using a phasefield model for surface diffusion. In addition to reproducing the expected scaling law, our simulations enable to investigate precisely the evolution of the topological and morphological characteristics along the coarsening process. In particular, we show that after a transient regime, the coarsening is selfsimilar as exhibited by the evolution of both morphological and topological features. In addition, the influence of surface anisotropy is discussed and comparisons with experimental tomographic observations are presented. 
Ghrist Barcoded Video Frames. Application in Detecting Persistent Visual Scene Surface Shapes Captured in Videos (2019)
Arjuna P. H. Don, James F. PetersAbstract
This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve complexes found in triangulations of video frames. A Ghrist barcode (also called a persistence barcode) is a topology of data pic tograph useful in representing the persistence of the features of changing shapes. The basic approach is to introduce a free Abelian group representation of intersecting filled polygons on the barycenters of the triangles of Alexandroff nerves. An Alexandroff nerve is a maximal collection of triangles of a common vertex in the triangulation of a finite, bounded planar region. In our case, the planar region is a video frame. A Betti number is a count of the number of generators is a finite Abelian group. The focus here is on the persistent Betti numbers across sequences of triangulated video frames. Each Betti number is mapped to an entry in a Ghrist barcode. Two main results are given, namely, vortex nerves are EdelsbrunnerHarer nerve complexes and the Betti number of a vortex nerve equals k + 2 for a vortex nerve containing k edges attached between a pair of vortex cycles in the nerve. 
A Classification of Topological Discrepancies in Additive Manufacturing (2019)
Morad Behandish, Amir M. Mirzendehdel, Saigopal NelaturiAbstract
Additive manufacturing (AM) enables enormous freedom for design of complex structures. However, the processdependent limitations that result in discrepancies between asdesigned and asmanufactured shapes are not fully understood. The tradeoffs between infinitely many different ways to approximate a design by a manufacturable replica are even harder to characterize. To support design for AM (DfAM), one has to quantify local discrepancies introduced by AM processes, identify the detrimental deviations (if any) to the original design intent, and prescribe modifications to the design and/or process parameters to countervail their effects. Our focus in this work will be on topological analysis. There is ample evidence in many applications that preserving local topology (e.g., connectivity of beams in a lattice) is important even when slight geometric deviations can be tolerated. We first present a generic method to characterize local topological discrepancies due to material underand overdeposition in AM, and show how it captures various types of defects in the asmanufactured structures. We use this information to systematically modify the asmanufactured outcomes within the limitations of available 3D printer resolution(s), which often comes at the expense of introducing more geometric deviations (e.g., thickening a beam to avoid disconnection). We validate the effectiveness of the method on 3D examples with nontrivial topologies such as lattice structures and foams. 
Fundamentals on Base Stations in Urban Cellular Networks: From the Perspective of Algebraic Topology (2018)
Ying Chen, Rongpeng Li, Zhifeng Zhao, Honggang Zhang 
Detecting Functional States of the Rat Brain With Topological Data Analysis (2018)
Nianqiao Ju, Ismar Volić, Michael Wiest 
Using Topological Data Analysis for Text Classification (2018)
Pratik Doshi 
Use of Topological Data Analysis in Motor Intention Based BrainComputer Interfaces (2018)
Fatih Altindis, Bulent Yilmaz, Sergey Borisenok, Kutay Icoz 
Topological Characteristics of Oil and Gas Reservoirs and Their Applications (2017)
V. A. Baikov, R. R. Gilmanov, I. A. Taimanov, A. A. YakovlevAbstract
We demonstrate applications of topological characteristics of oil and gas reservoirs considered as threedimensional bodies to geological modeling. 
The Topology of the Cosmic Web in Terms of Persistent Betti Numbers (2017)
Pratyush Pranav, Herbert Edelsbrunner, Rien van de Weygaert, Gert Vegter, Michael Kerber, Bernard J. T. Jones, Mathijs WintraeckenAbstract
Abstract. We introduce a multiscale topological description of the Megaparsec weblike cosmic matter distribution. Betti numbers and topological persistence of 
Topology of Force Networks in Granular Media Under Impact (2017)
M. X. Lim, R. P. BehringerAbstract
We investigate the evolution of the force network in experimental systems of twodimensional granular materials under impact. We use the first Betti number, , and persistence diagrams, as measures of the topological properties of the force network. We show that the structure of the network has a complex, hysteretic dependence on both the intruder acceleration and the total force response of the granular material. can also distinguish between the nonlinear formation and relaxation of the force network. In addition, using the persistence diagram of the force network, we show that the size of the loops in the force network has a Poissonlike distribution, the characteristic size of which changes over the course of the impact. 
Cliques of Neurons Bound Into Cavities Provide a Missing Link Between Structure and Function (2017)
Michael W. Reimann, Max Nolte, Martina Scolamiero, Katharine Turner, Rodrigo Perin, Giuseppe Chindemi, Paweł Dłotko, Ran Levi, Kathryn Hess, Henry MarkramAbstract
The lack of a formal link between neural network structure and its emergent function has hampered our understanding of how the brain processes information. We have now come closer to describing such a link by taking the direction of synaptic transmission into account, constructing graphs of a network that reflect the direction of information flow, and analyzing these directed graphs using algebraic topology. Applying this approach to a local network of neurons in the neocortex revealed a remarkably intricate and previously unseen topology of synaptic connectivity. The synaptic network contains an abundance of cliques of neurons bound into cavities that guide the emergence of correlated activity. In response to stimuli, correlated activity binds synaptically connected neurons into functional cliques and cavities that evolve in a stereotypical sequence towards peak complexity. We propose that the brain processes stimuli by forming increasingly complex functional cliques and cavities. 
Topological Data Analysis of Biological Aggregation Models (2015)
Chad M. Topaz, Lori Ziegelmeier, Tom HalversonAbstract
We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in positionvelocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters. 
Multidimensional Persistence in Biomolecular Data (2015)
Kelin Xia, GuoWei WeiAbstract
Persistent homology has emerged as a popular technique for the topological simplification of big data, including biomolecular data. Multidimensional persistence bears considerable promise to bridge the gap between geometry and topology. However, its practical and robust construction has been a challenge. We introduce two families of multidimensional persistence, namely pseudomultidimensional persistence and multiscale multidimensional persistence. The former is generated via the repeated applications of persistent homology filtration to high dimensional data, such as results from molecular dynamics or partial differential equations. The latter is constructed via isotropic and anisotropic scales that create new simiplicial complexes and associated topological spaces. The utility, robustness and efficiency of the proposed topological methods are demonstrated via protein folding, protein flexibility analysis, the topological denoising of cryoelectron microscopy data, and the scale dependence of nano particles. Topological transition between partial folded and unfolded proteins has been observed in multidimensional persistence. The separation between noise topological signatures and molecular topological fingerprints is achieved by the LaplaceBeltrami flow. The multiscale multidimensional persistent homology reveals relative local features in Betti0 invariants and the relatively global characteristics of Betti1 and Betti2 invariants. 
Identification of Copy Number Aberrations in Breast Cancer Subtypes Using Persistence Topology (2015)
Javier Arsuaga, Tyler Borrman, Raymond Cavalcante, Georgina Gonzalez, Catherine ParkAbstract
DNA copy number aberrations (CNAs) are of biological and medical interest because they help identify regulatory mechanisms underlying tumor initiation and evolution. Identification of tumordriving CNAs (driver CNAs) however remains a challenging task, because they are frequently hidden by CNAs that are the product of random events that take place during tumor evolution. Experimental detection of CNAs is commonly accomplished through array comparative genomic hybridization (aCGH) assays followed by supervised and/or unsupervised statistical methods that combine the segmented profiles of all patients to identify driver CNAs. Here, we extend a previouslypresented supervised algorithm for the identification of CNAs that is based on a topological representation of the data. Our method associates a twodimensional (2D) point cloud with each aCGH profile and generates a sequence of simplicial complexes, mathematical objects that generalize the concept of a graph. This representation of the data permits segmenting the data at different resolutions and identifying CNAs by interrogating the topological properties of these simplicial complexes. We tested our approach on a published dataset with the goal of identifying specific breast cancer CNAs associated with specific molecular subtypes. Identification of CNAs associated with each subtype was performed by analyzing each subtype separately from the others and by taking the rest of the subtypes as the control. Our results found a new amplification in 11q at the location of the progesterone receptor in the Luminal A subtype. Aberrations in the Luminal B subtype were found only upon removal of the basallike subtype from the control set. Under those conditions, all regions found in the original publication, except for 17q, were confirmed; all aberrations, except those in chromosome arms 8q and 12q were confirmed in the basallike subtype. These two chromosome arms, however, were detected only upon removal of three patients with exceedingly large copy number values. More importantly, we detected 10 and 21 additional regions in the Luminal B and basallike subtypes, respectively. Most of the additional regions were either validated on an independent dataset and/or using GISTIC. Furthermore, we found three new CNAs in the basallike subtype: a combination of gains and losses in 1p, a gain in 2p and a loss in 14q. Based on these results, we suggest that topological approaches that incorporate multiresolution analyses and that interrogate topological properties of the data can help in the identification of copy number changes in cancer. 
Multiphase Mixing Quantification by Computational Homology and Imaging Analysis (2011)
Jianxin Xu, Hua Wang, Hui FangAbstract
The purpose of this study is to introduce a new technique for quantifying the efficiency of multiphase mixing. This technique based on algebraic topology is illustrated by using the hydraulic modeling of gas agitated reactors stirred by top lance gas injection and image analysis. The zeroth Betti numbers are used to estimate the numbers of pieces in the patterns, leading to a useful parameter to characterize the mixture homogeneity. The first Betti numbers are introduced to characterize the nonhomogeneity of the mixture. The mixing efficiency can be characterized by the Betti numbers for binary images of the patterns. This novel method may be applied for studying a variety of multiphase mixing problems in which multiphase components or tracers are visually distinguishable. 
Alpha, Betti and the Megaparsec Universe: On the Topology of the Cosmic Web (2011)
Rien Van De Weygaert, Gert Vegter, Herbert Edelsbrunner, Bernard J. T. Jones, Pratyush Pranav, Changbom Park, Wojciech A. Hellwing, Bob Eldering, Nico Kruithof, E. G. P. Bos, Johan Hidding, Job Feldbrugge, Eline Ten Have, Matti Van Engelen, Manuel Caroli, Monique TeillaudAbstract
We study the topology of the Megaparsec Cosmic Web in terms of the scaledependent Betti numbers, which formalize the topological information content of... 
The Persistent Cosmic Web and Its Filamentary Structure – II. Illustrations (2011)
T. Sousbie, C. Pichon, H. KawaharaAbstract
Abstract. The recently introduced discrete persistent structure extractor (DisPerSE, Sousbie, Paper I) is implemented on realistic 3D cosmological simulations 
Persistent Betti Numbers for a Noise Tolerant ShapeBased Approach to Image Retrieval (2011)
Patrizio Frosini, Claudia LandiAbstract
In contentbased image retrieval a major problem is the presence of noisy shapes. It is well known that persistent Betti numbers are a shape descriptor that admits a dissimilarity distance, the matching distance, stable under continuous shape deformations. In this paper we focus on the problem of dealing with noise that changes the topology of the studied objects. We present a general method to turn persistent Betti numbers into stable descriptors also in the presence of topological changes. Retrieval tests on the Kimia99 database show the effectiveness of the method. 
TopologyBased Signal Separation (2004)
V. Robins, N. Rooney, E. Bradley