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Method for computing topological features of a space at different spatial resolutions
analysis, persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are
Persistent_homology
Multiscale analog of homology group
In persistent homology, a persistent homology group is a multiscale analog of a homology group that captures information about the evolution of topological
Persistent_homology_group
Analysis of datasets using techniques from topology
study of "shape". The main tool is persistent homology, an adaptation of homology to point cloud data. Persistent homology has been applied to many types
Topological_data_analysis
A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological
Persistence_module
Concept in algebraic topology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of
Simplicial_homology
Algebraic structure associated with a topological space
Borel–Moore homology Cellular homology Cyclic homology Hochschild homology Floer homology Intersection homology K-homology Khovanov homology Morse homology Persistent
Homology_(mathematics)
In persistent homology, a persistent Betti number is a multiscale analog of a Betti number that tracks the number of topological features that persist
Persistent_Betti_number
Research field in deep learning
based on extended persistent homology descriptors, persistence landscapes, or coordinate functions. In parallel, persistent homology also found applications
Topological_deep_learning
Subfield of mathematical topology
reductions for pre-processing homology computations, as in the Perseus software package. Algorithms to compute persistent homology of filtered complexes, as
Computational_topology
Branch of mathematics
algebraic topology – specifically, via the theory of persistent homology. Encode the persistent homology of a data set in the form of a parameterized version
Topology
Topological data analysis tool
topological features, over a range of parameters, by way of computing the persistent homology of the entire filtration. It is named after Leopold Vietoris and
Vietoris–Rips_filtration
Topological data
the implementation of many basic operations useful to computing persistent homology. This data structure was invented by Jean-Daniel Boissonnat and Clément
Simplex_tree
Distance function defined between probability distributions
"Determining clinically relevant features in cytometry data using persistent homology". PLOS Computational Biology. 18 (3) e1009931. arXiv:2203.06263.
Wasserstein_metric
Combinatorial approach of studying the topology of a manifold
the Persistent Homology software. Mischaikow, Konstantin; Nanda, Vidit (2013). "Morse Theory for Filtrations and Efficient computation of Persistent Homology"
Discrete_Morse_theory
bifiltration has been an object of study within multidimensional persistent homology and topological data analysis. Following the notation of Corbet et
Multicover_bifiltration
Hybrid sorting algorithm based on insertion sort and merge sort
PersiSort, an algorithm that extends on the merge criterion with persistent homology. In 2015, Dutch and German researchers in the EU FP7 ENVISAGE project
Timsort
is a parameter-free and density-sensitive vehicle for performing persistent homology computations on point cloud data. It is standard practice in topological
Degree-Rips_bifiltration
Topological space formed from distances
all Vietoris–Rips complexes is a commonly applied construction in persistent homology and topological data analysis, and is known as the Rips filtration
Vietoris–Rips_complex
features of a data set. The offset filtration commonly arises in persistent homology and the field of topological data analysis. Utilizing a union of
Offset_filtration
Statistical method in data analysis
algorithm Numerical taxonomy OPTICS algorithm Statistical distance Persistent homology Nielsen, Frank (2016). "8. Hierarchical Clustering". Introduction
Hierarchical_clustering
Shape descriptions in a geometrical/topological sense
functor are strictly related to the concept of persistent homology group studied in persistent homology. It is worth to point out that the size function
Size_function
Spanish chess grandmaster (born 1987)
in applied mathematics to engineering and Ultrasonic NDT: Persistent and Tangential homology for defect classification in Time of Flight Diffraction (TOFD)
Pepe_Cuenca
Complex recording the pattern of intersections between a topological family's sets
Günter M. Ziegler , Section 4.3 Virk, Žiga (2022). Introduction to Persistent Homology (PDF). Faculty of Computer and Information Science, University of
Nerve_complex
Geological record of biological activity
space-filling trajectories, such as those preserved in Parapsammichnites. Persistent homology analyses of these trace-fossil patterns have been interpreted as
Trace_fossil
concept of persistent homology group, studied in persistent homology. It is worth to point out that the i {\displaystyle i\ } -th persistent homology group
Size_functor
Technique in topological data analysis
S2CID 456712. Zomorodian, Afra; Carlsson, Gunnar (2005). "Computing Persistent Homology". Discrete & Computational Geometry. 33 (2): 249–274. doi:10.1007/s00454-004-1146-y
Persistence_barcode
functor are strictly related to the concept of persistent homology group studied in persistent homology. It is worth to point out that the size function
Size_theory
Homologies are "structural 'resonances'...between the different elements making up a socio-cultural whole." (Middleton 1990, p. 9) Examples include Alan
Homology_(sociology)
Measure of distance between persistence modules
modules, a common object of study in topological data analysis and persistent homology. The interleaving distance was first introduced by Frédéric Chazal
Interleaving_distance
by means of G {\displaystyle G} -invariant persistent homology and by combining classical persistent homology with the use of G-equivariant non-expansive
Natural_pseudodistance
Australian mathematician
from 1999, is one of the three works that independently introduced persistent homology in topological data analysis. As well as working on mathematical
Vanessa_Robins
Technique in topological data analysis
the multicover bifiltration, implying that they have isomorphic persistent homology. A combinatorial proof of this statement was given in Sheehy's original
Subdivision_bifiltration
Russian mathematician
S2CID 119133707. "UC Berkeley Mathematics Department Colloquium: Persistent homology and applications from PDE to symplectic topology". events.berkeley
Serguei_Barannikov
Mathematician
collaboration with others, he has demonstrated the utility of both persistent homology and the Mapper methodology in a series of papers. This work is central
Gunnar_Carlsson
analysis, where they serve as alternatives to simplicial complexes in persistent homology; geometric group theory, where CAT(0) cube complexes provide spaces
Cubical_complex
Professor of Applied Mathematics
Helen M.; Harrington, Heather A. (2021-10-12). "Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors". Proceedings
Helen_Byrne
Monograph
to recover useful information about the space, such as its homology and persistent homology. Although the book is primarily aimed at specialists in these
Geometric and Topological Inference
Geometric_and_Topological_Inference
Applied mathematics researcher
ASC program from 2002-2007. He researched informatics and applied persistent homology from 2008-2011. Since 2011, he researches mesh generation, sampling
Scott_A._Mitchell
Broad-spectrum herbicide
pat. The two genes and their proteins have 80% homology on the DNA level and 86% amino acid homology and are each 158 amino acids long. Glufosinate is
Glufosinate
Species of virus
acid homology with WHV than HBV, with 82% nucleotide and 78% amino acid homology between GSHV and WHV and 55% nucleotide and 46% amino acid homology between
Ground squirrel hepatitis virus
Ground_squirrel_hepatitis_virus
Roofing bones of the skull
the novel evolution of the frontal bones in tetrapods and their lack of homology with the "frontals" of fish. Further tetrapodomorph discoveries, such as
Skull_roof
Species of bacterium
involved with bacterial pathogenesis and at least three of these ORFs show homology to staphylococcal hemolysins. The S. haemolyticus genome also contains
Staphylococcus_haemolyticus
Sea anemone toxin
sodium channels, they belong to distinct families and share no sequence homology. The toxins AFT-II (from Anthopleura fuscoviridis) and ATX-II differ by
ATX-II
Protein-coding gene in the species Homo sapiens
alterations in response to prenatal exposure of maternal cigarette smoking: A persistent epigenetic impact on health from maternal lifestyle?". Archives of Toxicology
C11orf52
Artery of the inner ear
structure is still present after birth in which case it is referred to as a persistent stapedial artery (PSA). While the prevalence of this anomaly is unknown
Stapedial branch of posterior auricular artery
Stapedial_branch_of_posterior_auricular_artery
Medical condition
hyperplasia Disorders of sex development Fraser syndrome Homology (biology) Intersex Persistent Müllerian duct syndrome Sex chromosome anomalies XX male
XY_gonadal_dysgenesis
Species of bacterium
different serovars. The antigenic relatedness of the serovars reflects the homology levels of DNA between MOMP genes, especially within these segments. Furthermore
Chlamydia_trachomatis
Unicellular organism lacking a membrane-bound nucleus
since eukaryotes may be part of the archaean clade and have multiple homologies with other Archaea. The cellular components of prokaryotes are not enclosed
Prokaryote
Protein
not a single molecule but rather a family of proteins sharing structural homology and permeability-regulating activity. Members of the zonulin family belong
Zonulin
Proposed cause of autoimmunity
or TCR (T cell receptor). This highlights the importance of structural homology in the theory of molecular mimicry. Upon activation, these "peptide mimic"
Molecular_mimicry
Neuroscientific theory
increase in synaptic efficacy arises from a presynaptic cell's repeated and persistent stimulation of a postsynaptic cell. It is an attempt to explain synaptic
Hebbian_theory
mimicry, disease associated with immune complexes, and the idea that persistent viruses can alter cellular functioning in significant ways. His research
Michael_Oldstone
Brock, G. A. (2026). "Sclerite fusion, ultrastructure, micro-ornament and homology in early Cambrian eccentrothecimorph tommotiids". Palaeontology. 69 (3)
2026_in_paleontology
Attempt to explain evolution by forces other than natural selection
plentiful evidence both that natural selection is effective and, from deep homology, that genes have been involved in shaping organisms throughout evolutionary
Structuralism_(biology)
Order of insects often called true bugs
Malpighian Tubules, and Related Behaviors in Juvenile Cicadas: Evidence of Homology with Spittlebugs (Hemiptera: Cicadoidea & Cercopoidea)". Zoologischer Anzeiger
Hemiptera
Neurodevelopmental disorder chiefly affecting motor skills
1007/s40474-018-0126-5. Scharff C, Petri J (July 2011). "Evo-devo, deep homology and FoxP2: implications for the evolution of speech and language". Philosophical
Developmental coordination disorder
Developmental_coordination_disorder
T cell which has not yet encountered its cognate antigen
phosphorylates the membrane adaptor Lat, which subsequently recruits several Src homology domain–containing proteins, including phospholipase C-γ1 (PLC-γ1) activation
Naive_T_cell
Species of virus
when found in the brain, and maedi when infecting the lungs. Lifelong, persistent infections in sheep occur in the lungs, lymph nodes, spleen, joints, central
Visna-maedi_virus
Family of DNA sequences found in prokaryotic organisms
cell's natural DNA repair systems, including non-homologous end joining, homology-directed repair, or mismatch repair, to modify, insert, or delete genetic
CRISPR
Second period of the Mesozoic Era
Jurassic of Patagonia: Implications for conifer megasporangiate cone homologies". Review of Palaeobotany and Palynology. 151 (3–4): 110–122. Bibcode:2008RPaPa
Jurassic
Erectile female sexual organ
writers of the Middle Ages, felt that it was important to highlight "homologies between male and female structures and function" by adding "a psychology
Clitoris
Clade of reptiles
Pittman, M.; Saitta, E.; Kaye, T.G.; Vinther, J. (2016). "Structure and homology of Psittacosaurus tail bristles". Palaeontology. 59 (6): 793–802. Bibcode:2016Palgy
Dinosaur
Sexually transmitted virus of corn earworm moths
HzNV-1; it is likely that HzNV-1 is a variant of HzNV-2 which exists in a persistent state in infected moths. HzNV-2 was discovered in 1995. Originally, symptomatic
Helicoverpa_zea_nudivirus_2
Primary organ of the respiratory system
original (PDF) on 11 June 2010. Longo S, Riccio M, McCune AR (June 2013). "Homology of lungs and gas bladders: Insights from arterial vasculature". Journal
Lung
Organ central to the nervous system
back end of the nerve cord, known as a "tail brain". There are coarse homologies in the anterior-posterior patterning as well as axial patterning of the
Brain
Medical condition
social context. Such disturbances may occur as single episodes, may be persistent, or may be relapsing–remitting. There are many different types of mental
Mental_disorder
Species of virus
host range, different serological properties, and no nucleotide sequence homology with WMV. WMV also has different cytoplasmic inclusion bodies that can
Papaya_ringspot_virus
Extinct genus of tetrapodomorphs
ISSN 0028-0836. PMID 17522683. S2CID 4410652. Davis, Marcus C. (2013). "The Deep Homology of the Autopod: Insights from Hox Gene Regulation". Integrative and Comparative
Tiktaalik
Species of fruit fly
Multiple elements of the Drosophila JAK-STAT signalling pathway bear direct homology to human JAK-STAT pathway genes. JAK-STAT signalling is induced upon various
Drosophila_melanogaster
Contagious disease caused by SARS-CoV-2
risk of developing severe symptoms and dying. Some people experience persistent symptoms (long COVID), for months or years after infection, including
COVID-19
Organisms that live in salt water
Mooi R, David B (December 1998). "Evolution Within a Bizarre Phylum: Homologies of the First Echinoderms". American Zoologist. 38 (6): 965–974. doi:10
Marine_life
Mouse used for scientific research
clade, which includes humans. This close relationship, the associated high homology with humans, their ease of maintenance and handling, and their high reproduction
Laboratory_mouse
ISBN 978-0-521-62971-3.[page needed] Söding J (April 2005). "Protein homology detection by HMM-HMM comparison". Bioinformatics. 21 (7): 951–60. doi:10
List of sequence alignment software
List_of_sequence_alignment_software
Fine hair-like growth on plants
cold desert climates.[citation needed] The filamentous sheaths form a persistent sticky network that helps maintain soil structure. Plant trichomes have
Trichome
Interplay between observation, experiment, and theory in science
tautologically true forms (viz. the Euler characteristic) into or out of forms from homology, or more abstractly, from homological algebra. Lakatos proposed an account
Scientific_method
Extinct genus of plant
seed cone (sometimes described as a "flower" though this does not imply homology with angiosperm flowers) consists of an ovulate receptacle enclosed by
Williamsonia_(plant)
Techniques to train attention and awareness
University Press. Sandilands O (August 2019). L'espace contemplatif. Étude des homologies entre expérience esthétique et méditation (Master's Thesis). École des
Meditation
Family of transport proteins
be persistently (or tonically) active. Some kinds of ion channels are naturally persistently active. However, genetic mutations that cause persistent activity
Voltage-gated_sodium_channel
Clade of flowering plants
ISBN 9783662035337., In Kubitzki & Huber (1998) Tomlinson, P. B. (1995). Non-homology of vascular organisation in monocotyledons and dicotyledons. pp. 589–622
Monocotyledon
Parvorder of cetaceans
; Amundin, Mats; Norris, Kenneth S. (1996). "Functional morphology and homology in the odontocete nasal complex: Implications for sound generation". Journal
Toothed_whale
Species of shrub
spines are longer (6-15mm), the 2 adaxial spines usually 1-2mm long. Persistent hair bases cause the base of the spine creates a rough texture; seed is
Sclerolaena_birchii
Protein(s) forming a major part of an organism's immune system
optimize the relative orientation of the light and heavy chains, as well as homology models that predict successful docking of antibodies with their unique
Antibody
Class of artificial neural network
Grammar learning Handwriting recognition Human action recognition Protein homology detection Predicting subcellular localization of proteins Several prediction
Recurrent_neural_network
Species of virus
infection are much less well conserved. Even in cases where genes have low homology, the RLV infection genes are functionally interchangeable with EBV genes
Rhesus_lymphocryptovirus
Gene
electrochemical gradient. The protein consists of four homology domains, HD1, HD2, HD3 and HD4. Each homology domain has six alpha helical segments, S1-S6. The
Paralytic_(gene)
Species of fungus
"Trichophyton tonsurans strains from Brazil: phenotypic heterogeneity, genetic homology, and detection of virulence genes". Canadian Journal of Microbiology. 59
Trichophyton_tonsurans
Statistics and machine learning technique
classifier based prediction of G-protein-coupled receptor classes in low homology". Neurocomputing. 154: 110–118. doi:10.1016/j.neucom.2014.12.013. Xue,
Ensemble_learning
Acetylcholine receptors named for their selective binding of muscarine
of M1 and M2 receptor proteins. The others were found by searching for homology, using bioinformatic techniques. G proteins contain an alpha-subunit that
Muscarinic acetylcholine receptor
Muscarinic_acetylcholine_receptor
Vertebrate brain region
are not warped into an S shape or enfolded by the dentate gyrus, but the homology is indicated by strong chemical and functional affinities. There is evidence
Hippocampus
Protein found in humans
Of note, ISG15 shows substantial sequence variation among species, with homology as low as 30% between orthologs. After induction by type I interferon,
ISG15
N. R. (2026). "The Bonaparte hypothesis and the evolution of dental homologies in monotremes". Australian Zoologist. 45 (2) AZ26025. doi:10.1071/AZ26025
2026_in_paleomammalogy
Part of the brain responsible for personality, decision-making, and social behavior
cortex in primates and nonprimates. This allowed the establishment of homologies despite the lack of a granular frontal cortex in nonprimates. The projection
Prefrontal_cortex
Protein family
cytolethal distending toxins in subsequent studies. The discovery of the homology of cdtB to mammalian DNase I and the current AB model for the toxin were
Cytolethal_distending_toxin
For example, recA recombinase, that catalyses the key functions of DNA homology search and strand exchange in the bacterial sexual process of transformation
Evolution of sexual reproduction
Evolution_of_sexual_reproduction
Lung cancer of glandular origin or characteristics
to other forms of lung cancer, and patients most commonly complain of persistent cough and shortness of breath. Adenocarcinoma is more common in patients
Adenocarcinoma_of_the_lung
1093/aob/mcag082. PMC 13274997. PMID 41981894. Tremblay, S.; Mercadal, J. (2026). "Homology of the dark cells of Paleozoic liverworts with the specialized oil body
2026_in_paleobotany
Gonadotropin secreted by the adenohypophysis
This beta subunit contains an amino acid sequence that exhibits large homologies with that of the beta subunit of hCG and both stimulate the same receptor
Luteinizing_hormone
Family of regulator genes
discovered in this family, due to homology with the viral gene v-myc. In cancer, c-myc is often constitutively (persistently) expressed. This leads to the
Myc
Sendai virus, virus of rodents
antigenic determinants. Thus, porcine parainfluenza 1, which has high sequence homology with SeV and also belongs to the same genus Respirovirus as SeV, probably
Murine_respirovirus
Type of virus that infects the domain of unicellular, prokaryotic organisms or Archaea
portion of the genes encoded by archaeal viruses have no known function or homology to any other genes. Compared to bacterial and eukaryotic viruses, few archaeal
Archaeal_virus
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PERSISTENT HOMOLOGY
PERSISTENT HOMOLOGY
Surname or Lastname
German and Dutch
German and Dutch : from Middle High German bloch, Middle Dutch blok ‘block of wood’, ‘stocks’. The surname probably originated as a nickname for a large, lumpish man, or perhaps as a nickname for a persistent lawbreaker who found himself often in the stocks.English : possibly a metonymic occupational name for someone who blocks, as in shoemaking and bookbinding, from Middle English blok ‘block’.Jewish (Ashkenazic) : Americanized spelling of Bloch (see Vlach).Adriaen Coertsz Block was a Dutch-born merchant-explorer who traded along the CT coast and Long Island shortly after Hudson’s voyage to the region in 1609. Block Island, between the north fork of Long Island and RI, which he used as a base of operations, is named after him.
Surname or Lastname
English
English : from an Old English personal name or byname Tutta, preserved in place names such as Tutnall (Worcestershire) and Tuttington (Norfolk), and apparently persisting into the Middle Ages. Its origin and meaning are unclear.German (also Tütt) : from a short form of a Germanic personal name formed with Diet- (see Dietrich), or from a short form of Dudo, originally a name from nursery talk.
Surname or Lastname
Jewish (Ashkenazic)
Jewish (Ashkenazic) : ornamental name from modern German Gold, Yiddish gold ‘gold’. In North America it is often a reduced form of one of the many compound ornamental names of which Gold is the first element.English and German : from Old English, Old High German gold ‘gold’, applied as a metonymic occupational name for someone who worked in gold, i.e. a refiner, jeweler, or gilder, or as a nickname for someone who either had many gold possessions or bright yellow hair.English : from an Old English personal name Golda (or the feminine Golde), which persisted into the Middle Ages as a personal name. The name was in part a byname from gold ‘gold’, and in part a short form of the various compound names with this first element.
Male
African
obstinate, persistent.
Surname or Lastname
English (Devon)
English (Devon) : from Middle English hauek ‘hawk’, applied as a metonymic occupational name for a hawker (see Hawker), a name denoting a tenant who held land in return for providing hawks for his lord, or a nickname for someone supposedly resembling a hawk. There was an Old English personal name (originally a byname) H(e)afoc ‘hawk’, which persisted into the early Middle English period as a personal name and may therefore also be a source.English (Devon) : topographic name for someone who lived in an isolated nook, from Middle English halke (derived from Old English halh + the diminutive suffix -oc), or a habitational name from some minor place named with this word, such as Halke in Sheldwich, Kent.
Girl/Female
Muslim
Assiduous, Persistent
Boy/Male
Hindu, Indian, Kannada, Marathi, Sanskrit, Telugu
Never Ending; Persistence; Continuity; Perpetuity; Eternity; Uninterrupted Duration; Diligence; Conscientiousness; Truthful; Straightforward; Honest
Boy/Male
Arabic, Muslim
To Attack Violently and Persistently
Girl/Female
Indian
Assiduous, Persistent
Girl/Female
Arabic, Muslim
Assiduous; Persistent; Devoted
Surname or Lastname
English
English : from northern Middle English gad ‘goad’, ‘spike’, ‘sting’ (Old Norse gaddr), hence a metonymic occupational name for a cattle driver or, more likely, a nickname for a persistent and irritating person. The Old Norse word is attested as a byname (see Gadsby).
Female
Bulgarian
, persistent, resolute.
PERSISTENT HOMOLOGY
PERSISTENT HOMOLOGY
PERSISTENT HOMOLOGY
PERSISTENT HOMOLOGY
PERSISTENT HOMOLOGY
PERSISTENT HOMOLOGY
PERSISTENT HOMOLOGY
n.
Alt. of Persistency
v. t.
Figuratively, to draw out or obtain, as secrets or money, by persistent questioning or plying; to question or ply persistently in order to elicit something, as information, money, etc.
v. t.
To harass with importunity; to pursue with persistent solicitations; to annoy.
a.
Characterized by perseverance; persistent.
a.
Sullenly obstinate; obstinately determined or persistent; as, dogged resolution; dogged work.
a.
Remaining beyond the period when parts of the same kind sometimes fall off or are absorbed; permanent; as, persistent teeth or gills; a persistent calyx; -- opposed to deciduous, and caducous.
a.
See Persistent.
a.
Performed with constant diligence or attention; unremitting; persistent; as, assiduous labor.
a.
Persistent.
n.
The quality or state of being persistent; staying or continuing quality; hence, in an unfavorable sense, doggedness; obstinacy.
n.
The persistence of motion.
n.
Visual persistence, or persistence of the visual impression; auditory persistence, etc.
v. t.
To hold or maintain persistently.
n.
The act of soliciting; earnest request; persistent asking; importunity.
a.
Persistently opposed.
adv.
In a persistent manner.
a.
Inclined to persist; tenacious of purpose; persistent.
n.
Indefatigable quality; unweariedness; persistency.
a.
Inclined to persist; having staying qualities; tenacious of position or purpose.
a.
Insisting; persistent; persevering.
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