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1Mathematics The study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.
‘He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well.’
‘All mathematicians in geometry, complex analysis, low-dimensional topology, and geometric group theory will want to have it on their bookshelves.’
‘Ktheory is a mathematical theory that studies topology using matrices, using operators that don't commute with each another.’
‘The key divergence from traditional modelling methods was to think of the form as a topological entity, with topology being the study of geometry unaffected by change in scale or shape.’
‘His research interests include topology, algebraic geometry, and Lie theory.’
‘The largest number of paper-cutting/tearing opportunities related to Ending the sum of the interior angles in a triangle along with spatial visualization and topology.’
‘Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.’
‘He established a geometry and topology based on group theory without the concept of a limit.’
‘Euclidean geometry studies Euclidean-space-structure, topology studies topological structures, and so on.’
‘They relate Boolean algebras to general topology and to the theory of rings and ideals, and include what is called Stone-tech compactification today.’
‘He started research in algebra, geometry and topology as a student but did not consider his results sufficiently important to merit publication.’
‘The Poincare conjecture belongs to the field of topology, which studies properties that are preserved when a shape is stretched or twisted without tearing.’
‘For my Ph.D. 1 was required to study analysis, algebra, and algebraic topology.’
‘The mathematical ballistics of 1918 was neither as refined as axiomatic geometry nor as theoretical as algebraic topology.’
‘His main work was in set theory, general topology, and measure theory.’
‘For example, algebraic geometry, the field I am most familiar with, combines algebra, geometry, topology, and analysis.’
‘Geometry, topology, and algebraic geometry and group theory, almost anything you want, seems to be thrown into the mixture.’
‘He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups.’
‘This book is a classic in the study of the geometric topology of 3-manifolds.’
‘His aim was to bring together point-set topology and algebraic topology with his 1932 paper.’
1.1A family of open subsets of an abstract space such that the union and the intersection of any two of them are members of the family, and which includes the space itself and the empty set.
‘It is important to realize that different genetic operators have different ‘neighborhood’ space topologies.’
‘An identical topology is obtained when each of the three regions is considered separately’
‘A single most parsimonious cladogram was recovered for the ‘archaic’ subset with a topology identical to that seen in all of the MMPC.’
‘With a number of methods a topology in which the grasses clade is basal within the angiosperm group was found with high bootstrap support.’
‘Membrane topology of the FNT family members is consistent with six putative transmembrane helical spanning domains, as established for the FOCA protein of E.coli.’
‘Other algorithms such as Jukes-Cantor or Tamura-Nei produced identical tree topologies.’
‘The genealogy estimated for the coding region of 48A alone showed an identical topology.’
‘If our three-dimensional universe possesses one of these identified topologies, then this may in some way be dictated by laws of Nature.’
‘Distances were used in MEGA to construct a neighbor-joining tree with a topology identical to that of Lanyon's maximum parsimony tree.’
‘It therefore has a different topology from the torus of fig. 4, which was be deformed into a mug with only one handle.’
‘In this analysis, such is not the case; the Adams consensus produces a topology that is identical to one of the 24 most parsimonious trees.’
‘Inferred relationships are based on maximum parsimony, with this tree being one of eight identified topologies with the smallest number of steps.’
‘The methodology is case studies which, although mainly historical in format, do periodically furnish topologies and descriptive statistics.’
‘An identical topology was obtained with the parsimony method.’
‘The upshot of all this is that only the possession of a Moebius-like topology by a space genuinely eliminates the possibility of there being pairs of objects in that space which are incongruent counterparts.’
‘The resultant tree topologies were used to construct two subsets of data within and among Pan I allelic lineages for analyses with likelihood methods.’
‘For each of the predetermined phenetic topologies, the least number of steps required to separate the members of a demographic group completely was determined.’
‘However, on increasing the number of sequences, this uniform prior on the topologies leads to a superexponential explosion of the parameter space.’
‘Tree topologies gather together members of the same families and fit the taxonomic classification of these dipteran species.’
‘The MP tree showed a topology almost identical to that of the NJ tree (data not shown).’
2The way in which constituent parts are interrelated or arranged.
‘the topology of a computer network’
‘This means that the topology of the metabolic network is only partially able to explain the observed data.’
‘On a molecular level, stress relaxation occurs by a change in the topology of the connections within the skeletal network.’
‘The null space of the stoichiometric matrix that describes the topology of metabolic networks and that contains the steady-state flux distributions has been analyzed in detail.’
‘Although this relationship may vary between species, for a particular species it is constant, indicating that the polarity of the pollen grain owes its origin to the topology of microspore tetrad formation.’
‘We also have no indication that the topology of the phylogenetic trees is biased by the alignments or the tree-building algorithms.’
‘We explored forcing our data to conform to the topologies of previous among-group arrangements.’
‘This result is not merely a sampled version of Fig 7A; rather, it incorporates the constraints imposed by the edge distributions on the topology of the network.’
‘Firstly, if you happen to run an extensive subterranean mass transit system, you should familiarise yourself with the topology of the network.’
‘I've been doing a lot of stuff at work which is about designing social networks and their topologies, and peterme seems similarly preoccupied.’
‘A ring network is a topology of computer networks where each user is connected’
‘The actual fiber path plays a big part in how the topology of the network will look.’
‘In this work, the topology of protein domain networks generated with data from the ProDom, Pfam, and Prosite domain databases was studied.’
‘The genealogy can be viewed as consisting of two components: the topology of the tree structure and the times between coalescent events.’
‘After you've installed the Mazu hardware, you need to teach the system about the topology of your internal network.’
‘As chol was added, the topology of the domains was changed (formation of lo domains), and eventually phase separation disappeared at high chol fractions.’
‘Thus, the strength of selective constraints on gene sequences is affected by the topology of the gene coexpression network.’
‘The stoichiometry matrix: Structural modelling in this context is so called because the focus is on the network topology of the system.’
‘A schematic of region I of the germarium showing the topology of the germline stem cell region and the expression pattern of genes relevant to this study.’
‘What is the power of a computer which can change the very notion of the topology of the underlying circuit connectivity?’
‘Other than an unresolved reticulation at the most ancestral node, the topology of this network is straightforward and there are three noteworthy results.’
Origin
Late 19th century: via German from Greek topos ‘place’ + -logy.