Definition of topology in English:

topology

nounPlural topologies təˈpɒlədʒitəˈpɑlədʒi

1Mathematics
mass noun The study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures.
〔数〕拓扑学
Example sentencesExamples
- He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups.
- He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well.
- For example, algebraic geometry, the field I am most familiar with, combines algebra, geometry, topology, and analysis.
- He started research in algebra, geometry and topology as a student but did not consider his results sufficiently important to merit publication.
- Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.
- 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.
- Euclidean geometry studies Euclidean-space-structure, topology studies topological structures, and so on.
- He established a geometry and topology based on group theory without the concept of a limit.
- This book is a classic in the study of the geometric topology of 3-manifolds.
- For my Ph.D. 1 was required to study analysis, algebra, and algebraic topology.
- His aim was to bring together point-set topology and algebraic topology with his 1932 paper.
- His research interests include topology, algebraic geometry, and Lie theory.
- The mathematical ballistics of 1918 was neither as refined as axiomatic geometry nor as theoretical as algebraic topology.
- All mathematicians in geometry, complex analysis, low-dimensional topology, and geometric group theory will want to have it on their bookshelves.
- The Poincare conjecture belongs to the field of topology, which studies properties that are preserved when a shape is stretched or twisted without tearing.
- Geometry, topology, and algebraic geometry and group theory, almost anything you want, seems to be thrown into the mixture.
- His main work was in set theory, general topology, and measure theory.
- Ktheory is a mathematical theory that studies topology using matrices, using operators that don't commute with each another.
- They relate Boolean algebras to general topology and to the theory of rings and ideals, and include what is called Stone-tech compactification today.
- 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.
1. 1.1count noun A 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.
  拓扑
  Example sentencesExamples
  For each of the predetermined phenetic topologies, the least number of steps required to separate the members of a demographic group completely was determined.
  Tree topologies gather together members of the same families and fit the taxonomic classification of these dipteran species.
  The methodology is case studies which, although mainly historical in format, do periodically furnish topologies and descriptive statistics.
  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.
  An identical topology was obtained with the parsimony method.
  The MP tree showed a topology almost identical to that of the NJ tree (data not shown).
  However, on increasing the number of sequences, this uniform prior on the topologies leads to a superexponential explosion of the parameter space.
  It therefore has a different topology from the torus of fig. 4, which was be deformed into a mug with only one handle.
  The genealogy estimated for the coding region of 48A alone showed an identical topology.
  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.
  If our three-dimensional universe possesses one of these identified topologies, then this may in some way be dictated by laws of Nature.
  With a number of methods a topology in which the grasses clade is basal within the angiosperm group was found with high bootstrap support.
  The resultant tree topologies were used to construct two subsets of data within and among Pan I allelic lineages for analyses with likelihood methods.
  Other algorithms such as Jukes-Cantor or Tamura-Nei produced identical tree topologies.
  An identical topology is obtained when each of the three regions is considered separately
  Inferred relationships are based on maximum parsimony, with this tree being one of eight identified topologies with the smallest number of steps.
  A single most parsimonious cladogram was recovered for the ‘archaic’ subset with a topology identical to that seen in all of the MMPC.
  Distances were used in MEGA to construct a neighbor-joining tree with a topology identical to that of Lanyon's maximum parsimony tree.
  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.
  It is important to realize that different genetic operators have different ‘neighborhood’ space topologies.
2The way in which constituent parts are interrelated or arranged.
结构，构造
the topology of a computer network
计算机网络结构。
Example sentencesExamples
- In this work, the topology of protein domain networks generated with data from the ProDom, Pfam, and Prosite domain databases was studied.
- 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.
- Firstly, if you happen to run an extensive subterranean mass transit system, you should familiarise yourself with the topology of the network.
- The genealogy can be viewed as consisting of two components: the topology of the tree structure and the times between coalescent events.
- The actual fiber path plays a big part in how the topology of the network will look.
- I've been doing a lot of stuff at work which is about designing social networks and their topologies, and peterme seems similarly preoccupied.
- On a molecular level, stress relaxation occurs by a change in the topology of the connections within the skeletal network.
- We explored forcing our data to conform to the topologies of previous among-group arrangements.
- Thus, the strength of selective constraints on gene sequences is affected by the topology of the gene coexpression network.
- 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.
- 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.
- 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.
- A ring network is a topology of computer networks where each user is connected
- We also have no indication that the topology of the phylogenetic trees is biased by the alignments or the tree-building algorithms.
- This means that the topology of the metabolic network is only partially able to explain the observed data.
- After you've installed the Mazu hardware, you need to teach the system about the topology of your internal network.
- The stoichiometry matrix: Structural modelling in this context is so called because the focus is on the network topology of the system.
- Other than an unresolved reticulation at the most ancestral node, the topology of this network is straightforward and there are three noteworthy results.
- What is the power of a computer which can change the very notion of the topology of the underlying circuit connectivity?
- As chol was added, the topology of the domains was changed (formation of lo domains), and eventually phase separation disappeared at high chol fractions.

Derivatives

topological

adjectivetɒpəˈlɒdʒɪk(ə)lˌtɑpəˈlɑdʒək(ə)l

His idea was to use a topological - functional approach to the study of differential equations.
Example sentencesExamples
- There is a third way in which topological concepts entered mathematics, namely via functional analysis.
- The object of the course was to study the modular group and some of its subgroups, with help of algebraic rather than analytic or topological methods.
- This work continued through the early 1930s then in the late 1930s he studied ordered topological vector spaces.
- Most of the features for surfaces appearing in this book are closely related to topological geometry.
- One of the main problems has been and remains the classification of these algebras as intrinsic algebraic and topological objects.
- Euclidean geometry studies Euclidean-space-structure, topology studies topological structures, and so on.
- The author has succeeded in writing a book understandable to readers with very little knowledge of functional analysis and topological vector spaces.
- In the first addition most of the topological concepts are dealt with in an supplement to Volume 3.
- He constructed the classifying space of a topological group and gave a geometric realisation of a semi-simplicial complex.
- They have very rich topological and geometrical structure; they may possess peculiar properties that do not occur in ordinary L p spaces.
- He gave a topological characterisation of the plane which simplified considerably the Jordan curve theorem.
- Three important papers on plane topology proved the topological invariance of the dimension of the square.
- In this work he coined the now familiar term topological algebra but the thesis is memorable in other ways too.
- He seems to be just as good at bringing mathematics to life, especially rather elaborate topological constructions.
- He discovered characterisations of topological mappings of the Cartesian plane and a number of fixed point theorems.
- Morse theory is important in the field of global analysis which is the study of ordinary and partial differential equations from a global or topological point of view.
- However he continued to work on topological ideas, in particular embedding complexes in Euclidean space.
- He worked, in particular, on the theory of topological and metric mappings on 2-dimensional space.
- Before 1920 he had shown that the homology of a simplicial complex is a topological invariant.

topologically

adverbtɒpəˈlɒdʒɪk(ə)li

For the case of a finite-dimensional real linear space these are infinite systems whose adjoint cone is topologically closed.
Example sentencesExamples
- According to embedding theory, the Poincare map is topologically equivalent to a Poincare surface of section that intersects the system's trajectory in its characteristic state-space.
- A Mobius Strip made of paper: if you consider the thickness of the paper, a ring that makes a mobius strip is topologically the same as solid tori.
- In particular, he conjectured that any smooth 3-surface that has no edges, no corkscrew-like twists, and no doughnut-like holes must be topologically equivalent to a 3-sphere.
- For example, tethering of nucleosomes to the surface may at least transiently strengthen their ability to create a local topologically closed domain, leading to enhanced torsional stress within that region when hSwi-Snf is active.
- On the other hand, information is preserved in topologically trivial metrics.
- He was able to show that any 2-dimensional surface having the same fundamental group as the 2-dimensional surface of a sphere is topologically equivalent to a sphere.
- It implies that contact may obtain between A and B only if either A or B is topologically closed while the other is topologically open in the relevant contact area; but the appeal to indeterminacy allows one to leave the matter unsettled.
- In effect, any object that mathematically behaves like a 3-sphere is topologically a 3-sphere, no matter how distorted or twisted its shape may be.
- This is because every organism is a closed surface, topologically speaking, and it has been known for many years what kinds of closed surfaces can and cannot be continuously mapped to each other.
- These two-dimensional examples give us an idea of how to build the more difficult three-dimensional, topologically compact spaces.
- For a space that is topologically equivalent to the real line and for metrics that are consistent with that topology, each coordinate system determines a metric and each metric determines a coordinate system.
- I've heard you refer to the inverse as being topologically finite.
- In addition, the measurement results can be displayed dynamically, topologically, statistically, or in a tabular format on a monitor.
- Apart from the importance in understanding the mechanisms involved in vivo, the interplay between conformation and interactions in topologically constrained polymers is of interest in its own right from a biophysical point of view.
- Unless matter is to be created or destroyed, we seem to have to say that the break is half-open, with one new matter-surface being topologically closed and the other being topologically open.
- We then propose a classification of bacterial phenotypes based on the topology of the metabolic phase diagram, and enumerate the possible topologically distinct phenotypes that can be achieved through mutations of the PTS.
- Highly significant correlations are found between the most topologically relevant sites in our analysis and the single point mutations known to be associated with the arousal of the genetic forms of prion disease.
- We show that this small and topologically simple protein is remarkably mechanically resistant, reinforcing the view that extension of hydrogen-bonded parallel -strands is key to defining mechanical strength.
- Thus, the situation in much of the remainder of the dermal skeleton, where individual elements are indistinguishable morphologically or topologically, ensures that discussion of patterns such as fusion or loss are entirely vacuous.

topologist

noun

To a topologist, the fact that the fish species Scarus sp. can be continuously transformed into the species Pomacanthus is entirely unremarkable.
Example sentencesExamples
- It was enough to make a topologist weep with joy.
- He left topologists a treasure-trove of theorems and techniques and left the University of Texas Mathematics Department with a goal and thirteen years of good progress toward it.
- For now we note that his General Topology became a bible for aspiring topologists and analysts from the time it first appeared in 1955.
- Now, when modern topologists study knots, they assume the knots are constructed out of perfectly flexible, perfectly stretchable, infinitely thin string.
- Of special interest to topologists are manifolds, which means ‘having multiple features or forms’.
- The Poincaré Conjecture remained unsolved during the entire twentieth century and defeated the efforts of many of the best topologists and geometers of the time.
- If the book has a thesis, it is that a functional analyst is an analyst, first and foremost, and not a degenerate species of a topologist.
- Certainly his work in topology would make him one of the leading topologists of all time but he may be known principally for other work.
- When he produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration.
- In fact, essentially every invariant studied by topologists can be made into a representable or corepresentable functor.
- A simple way to describe topology is as a ‘rubber sheet geometry’ - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed.
- It was an exciting time for the topologists in Moscow for he lectured on the topology of continua and often his latest results were presented in the course shortly after he had proved them.
- In this spirit, topologists regard any two objects that can be deformed into each other without tearing to be one and the same thing - have a look at Plus article In space, do all roads lead home?
- June 13, 2003 the bewildered topologist couldn't understand why it wasn't called ‘Lord of the doughnuts’.
- For a long time, the Alexander polynomial was one of the few tools topologists had for telling knots apart.
- His invariant had been missed completely by topologists, in spite of intense activity in closely related areas during the preceding 60 years, and it was a complete surprise.
- Your comments about the cube and venus de milo are excellent - a born topologist!
- When topologists started to explore the world of geometric forms in higher dimensions, they found that neither the intuition nor the vocabulary of ordinary geometry was sufficient to describe and classify the new forms they discovered.
- Topology is a branch of math that is sometimes called ‘rubber-sheet geometry’ because topologists study objects that retain their spatial properties even when they are twisted into odd shapes.

Origin

Late 19th century: via German from Greek topos 'place' + -logy.

Definition of topology in US English:

topology

nountəˈpɑlədʒitəˈpäləjē

1Mathematics
The study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.
〔数〕拓扑学
Example sentencesExamples
- 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.
- The Poincare conjecture belongs to the field of topology, which studies properties that are preserved when a shape is stretched or twisted without tearing.
- Ktheory is a mathematical theory that studies topology using matrices, using operators that don't commute with each another.
- He established a geometry and topology based on group theory without the concept of a limit.
- His main work was in set theory, general topology, and measure theory.
- Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.
- All mathematicians in geometry, complex analysis, low-dimensional topology, and geometric group theory will want to have it on their bookshelves.
- The mathematical ballistics of 1918 was neither as refined as axiomatic geometry nor as theoretical as algebraic topology.
- For my Ph.D. 1 was required to study analysis, algebra, and algebraic topology.
- For example, algebraic geometry, the field I am most familiar with, combines algebra, geometry, topology, and analysis.
- They relate Boolean algebras to general topology and to the theory of rings and ideals, and include what is called Stone-tech compactification today.
- 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.
- He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups.
- Geometry, topology, and algebraic geometry and group theory, almost anything you want, seems to be thrown into the mixture.
- He started research in algebra, geometry and topology as a student but did not consider his results sufficiently important to merit publication.
- Euclidean geometry studies Euclidean-space-structure, topology studies topological structures, and so on.
- He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well.
- His research interests include topology, algebraic geometry, and Lie theory.
- 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. 1.1 A 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.
  拓扑
  Example sentencesExamples
  A single most parsimonious cladogram was recovered for the ‘archaic’ subset with a topology identical to that seen in all of the MMPC.
  Other algorithms such as Jukes-Cantor or Tamura-Nei produced identical tree topologies.
  Inferred relationships are based on maximum parsimony, with this tree being one of eight identified topologies with the smallest number of steps.
  Distances were used in MEGA to construct a neighbor-joining tree with a topology identical to that of Lanyon's maximum parsimony tree.
  With a number of methods a topology in which the grasses clade is basal within the angiosperm group was found with high bootstrap support.
  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.
  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.
  The resultant tree topologies were used to construct two subsets of data within and among Pan I allelic lineages for analyses with likelihood methods.
  Tree topologies gather together members of the same families and fit the taxonomic classification of these dipteran species.
  It therefore has a different topology from the torus of fig. 4, which was be deformed into a mug with only one handle.
  The MP tree showed a topology almost identical to that of the NJ tree (data not shown).
  The methodology is case studies which, although mainly historical in format, do periodically furnish topologies and descriptive statistics.
  It is important to realize that different genetic operators have different ‘neighborhood’ space topologies.
  If our three-dimensional universe possesses one of these identified topologies, then this may in some way be dictated by laws of Nature.
  The genealogy estimated for the coding region of 48A alone showed an identical topology.
  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.
  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.
  An identical topology was obtained with the parsimony method.
  An identical topology is obtained when each of the three regions is considered separately
2The way in which constituent parts are interrelated or arranged.
结构，构造
the topology of a computer network
计算机网络结构。
Example sentencesExamples
- We explored forcing our data to conform to the topologies of previous among-group arrangements.
- 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.
- Other than an unresolved reticulation at the most ancestral node, the topology of this network is straightforward and there are three noteworthy results.
- The stoichiometry matrix: Structural modelling in this context is so called because the focus is on the network topology of the system.
- The actual fiber path plays a big part in how the topology of the network will look.
- 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.
- The genealogy can be viewed as consisting of two components: the topology of the tree structure and the times between coalescent events.
- 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.
- Firstly, if you happen to run an extensive subterranean mass transit system, you should familiarise yourself with the topology of the network.
- In this work, the topology of protein domain networks generated with data from the ProDom, Pfam, and Prosite domain databases was studied.
- On a molecular level, stress relaxation occurs by a change in the topology of the connections within the skeletal network.
- This means that the topology of the metabolic network is only partially able to explain the observed data.
- What is the power of a computer which can change the very notion of the topology of the underlying circuit connectivity?
- We also have no indication that the topology of the phylogenetic trees is biased by the alignments or the tree-building algorithms.
- 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.
- 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.
- 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
- After you've installed the Mazu hardware, you need to teach the system about the topology of your internal network.

Origin

Late 19th century: via German from Greek topos ‘place’ + -logy.

词汇	topology
释义	Definition of topology in English: topology nounPlural topologies təˈpɒlədʒitəˈpɑlədʒi 1Mathematics mass noun The study of geometrical properties and spatial relations unaffected by the continuous change of shape or size of figures. 〔数〕拓扑学 Example sentencesExamples He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups. He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well. For example, algebraic geometry, the field I am most familiar with, combines algebra, geometry, topology, and analysis. He started research in algebra, geometry and topology as a student but did not consider his results sufficiently important to merit publication. Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry. 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. Euclidean geometry studies Euclidean-space-structure, topology studies topological structures, and so on. He established a geometry and topology based on group theory without the concept of a limit. This book is a classic in the study of the geometric topology of 3-manifolds. For my Ph.D. 1 was required to study analysis, algebra, and algebraic topology. His aim was to bring together point-set topology and algebraic topology with his 1932 paper. His research interests include topology, algebraic geometry, and Lie theory. The mathematical ballistics of 1918 was neither as refined as axiomatic geometry nor as theoretical as algebraic topology. All mathematicians in geometry, complex analysis, low-dimensional topology, and geometric group theory will want to have it on their bookshelves. The Poincare conjecture belongs to the field of topology, which studies properties that are preserved when a shape is stretched or twisted without tearing. Geometry, topology, and algebraic geometry and group theory, almost anything you want, seems to be thrown into the mixture. His main work was in set theory, general topology, and measure theory. Ktheory is a mathematical theory that studies topology using matrices, using operators that don't commute with each another. They relate Boolean algebras to general topology and to the theory of rings and ideals, and include what is called Stone-tech compactification today. 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. 1.1count noun A 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. 拓扑 Example sentencesExamples For each of the predetermined phenetic topologies, the least number of steps required to separate the members of a demographic group completely was determined. Tree topologies gather together members of the same families and fit the taxonomic classification of these dipteran species. The methodology is case studies which, although mainly historical in format, do periodically furnish topologies and descriptive statistics. 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. An identical topology was obtained with the parsimony method. The MP tree showed a topology almost identical to that of the NJ tree (data not shown). However, on increasing the number of sequences, this uniform prior on the topologies leads to a superexponential explosion of the parameter space. It therefore has a different topology from the torus of fig. 4, which was be deformed into a mug with only one handle. The genealogy estimated for the coding region of 48A alone showed an identical topology. 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. If our three-dimensional universe possesses one of these identified topologies, then this may in some way be dictated by laws of Nature. With a number of methods a topology in which the grasses clade is basal within the angiosperm group was found with high bootstrap support. The resultant tree topologies were used to construct two subsets of data within and among Pan I allelic lineages for analyses with likelihood methods. Other algorithms such as Jukes-Cantor or Tamura-Nei produced identical tree topologies. An identical topology is obtained when each of the three regions is considered separately Inferred relationships are based on maximum parsimony, with this tree being one of eight identified topologies with the smallest number of steps. A single most parsimonious cladogram was recovered for the ‘archaic’ subset with a topology identical to that seen in all of the MMPC. Distances were used in MEGA to construct a neighbor-joining tree with a topology identical to that of Lanyon's maximum parsimony tree. 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. It is important to realize that different genetic operators have different ‘neighborhood’ space topologies. 2The way in which constituent parts are interrelated or arranged. 结构，构造 the topology of a computer network 计算机网络结构。 Example sentencesExamples In this work, the topology of protein domain networks generated with data from the ProDom, Pfam, and Prosite domain databases was studied. 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. Firstly, if you happen to run an extensive subterranean mass transit system, you should familiarise yourself with the topology of the network. The genealogy can be viewed as consisting of two components: the topology of the tree structure and the times between coalescent events. The actual fiber path plays a big part in how the topology of the network will look. I've been doing a lot of stuff at work which is about designing social networks and their topologies, and peterme seems similarly preoccupied. On a molecular level, stress relaxation occurs by a change in the topology of the connections within the skeletal network. We explored forcing our data to conform to the topologies of previous among-group arrangements. Thus, the strength of selective constraints on gene sequences is affected by the topology of the gene coexpression network. 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. 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. 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. A ring network is a topology of computer networks where each user is connected We also have no indication that the topology of the phylogenetic trees is biased by the alignments or the tree-building algorithms. This means that the topology of the metabolic network is only partially able to explain the observed data. After you've installed the Mazu hardware, you need to teach the system about the topology of your internal network. The stoichiometry matrix: Structural modelling in this context is so called because the focus is on the network topology of the system. Other than an unresolved reticulation at the most ancestral node, the topology of this network is straightforward and there are three noteworthy results. What is the power of a computer which can change the very notion of the topology of the underlying circuit connectivity? As chol was added, the topology of the domains was changed (formation of lo domains), and eventually phase separation disappeared at high chol fractions. Derivatives topological adjectivetɒpəˈlɒdʒɪk(ə)lˌtɑpəˈlɑdʒək(ə)l His idea was to use a topological - functional approach to the study of differential equations. Example sentencesExamples There is a third way in which topological concepts entered mathematics, namely via functional analysis. The object of the course was to study the modular group and some of its subgroups, with help of algebraic rather than analytic or topological methods. This work continued through the early 1930s then in the late 1930s he studied ordered topological vector spaces. Most of the features for surfaces appearing in this book are closely related to topological geometry. One of the main problems has been and remains the classification of these algebras as intrinsic algebraic and topological objects. Euclidean geometry studies Euclidean-space-structure, topology studies topological structures, and so on. The author has succeeded in writing a book understandable to readers with very little knowledge of functional analysis and topological vector spaces. In the first addition most of the topological concepts are dealt with in an supplement to Volume 3. He constructed the classifying space of a topological group and gave a geometric realisation of a semi-simplicial complex. They have very rich topological and geometrical structure; they may possess peculiar properties that do not occur in ordinary L p spaces. He gave a topological characterisation of the plane which simplified considerably the Jordan curve theorem. Three important papers on plane topology proved the topological invariance of the dimension of the square. In this work he coined the now familiar term topological algebra but the thesis is memorable in other ways too. He seems to be just as good at bringing mathematics to life, especially rather elaborate topological constructions. He discovered characterisations of topological mappings of the Cartesian plane and a number of fixed point theorems. Morse theory is important in the field of global analysis which is the study of ordinary and partial differential equations from a global or topological point of view. However he continued to work on topological ideas, in particular embedding complexes in Euclidean space. He worked, in particular, on the theory of topological and metric mappings on 2-dimensional space. Before 1920 he had shown that the homology of a simplicial complex is a topological invariant. topologically adverbtɒpəˈlɒdʒɪk(ə)li For the case of a finite-dimensional real linear space these are infinite systems whose adjoint cone is topologically closed. Example sentencesExamples According to embedding theory, the Poincare map is topologically equivalent to a Poincare surface of section that intersects the system's trajectory in its characteristic state-space. A Mobius Strip made of paper: if you consider the thickness of the paper, a ring that makes a mobius strip is topologically the same as solid tori. In particular, he conjectured that any smooth 3-surface that has no edges, no corkscrew-like twists, and no doughnut-like holes must be topologically equivalent to a 3-sphere. For example, tethering of nucleosomes to the surface may at least transiently strengthen their ability to create a local topologically closed domain, leading to enhanced torsional stress within that region when hSwi-Snf is active. On the other hand, information is preserved in topologically trivial metrics. He was able to show that any 2-dimensional surface having the same fundamental group as the 2-dimensional surface of a sphere is topologically equivalent to a sphere. It implies that contact may obtain between A and B only if either A or B is topologically closed while the other is topologically open in the relevant contact area; but the appeal to indeterminacy allows one to leave the matter unsettled. In effect, any object that mathematically behaves like a 3-sphere is topologically a 3-sphere, no matter how distorted or twisted its shape may be. This is because every organism is a closed surface, topologically speaking, and it has been known for many years what kinds of closed surfaces can and cannot be continuously mapped to each other. These two-dimensional examples give us an idea of how to build the more difficult three-dimensional, topologically compact spaces. For a space that is topologically equivalent to the real line and for metrics that are consistent with that topology, each coordinate system determines a metric and each metric determines a coordinate system. I've heard you refer to the inverse as being topologically finite. In addition, the measurement results can be displayed dynamically, topologically, statistically, or in a tabular format on a monitor. Apart from the importance in understanding the mechanisms involved in vivo, the interplay between conformation and interactions in topologically constrained polymers is of interest in its own right from a biophysical point of view. Unless matter is to be created or destroyed, we seem to have to say that the break is half-open, with one new matter-surface being topologically closed and the other being topologically open. We then propose a classification of bacterial phenotypes based on the topology of the metabolic phase diagram, and enumerate the possible topologically distinct phenotypes that can be achieved through mutations of the PTS. Highly significant correlations are found between the most topologically relevant sites in our analysis and the single point mutations known to be associated with the arousal of the genetic forms of prion disease. We show that this small and topologically simple protein is remarkably mechanically resistant, reinforcing the view that extension of hydrogen-bonded parallel -strands is key to defining mechanical strength. Thus, the situation in much of the remainder of the dermal skeleton, where individual elements are indistinguishable morphologically or topologically, ensures that discussion of patterns such as fusion or loss are entirely vacuous. topologist noun To a topologist, the fact that the fish species Scarus sp. can be continuously transformed into the species Pomacanthus is entirely unremarkable. Example sentencesExamples It was enough to make a topologist weep with joy. He left topologists a treasure-trove of theorems and techniques and left the University of Texas Mathematics Department with a goal and thirteen years of good progress toward it. For now we note that his General Topology became a bible for aspiring topologists and analysts from the time it first appeared in 1955. Now, when modern topologists study knots, they assume the knots are constructed out of perfectly flexible, perfectly stretchable, infinitely thin string. Of special interest to topologists are manifolds, which means ‘having multiple features or forms’. The Poincaré Conjecture remained unsolved during the entire twentieth century and defeated the efforts of many of the best topologists and geometers of the time. If the book has a thesis, it is that a functional analyst is an analyst, first and foremost, and not a degenerate species of a topologist. Certainly his work in topology would make him one of the leading topologists of all time but he may be known principally for other work. When he produced his first few results on 4-manifolds, the ideas were so new and foreign to geometers and topologists that they merely gazed in bewildered admiration. In fact, essentially every invariant studied by topologists can be made into a representable or corepresentable functor. A simple way to describe topology is as a ‘rubber sheet geometry’ - topologists study those properties of shapes that remain the same when the shapes are stretched or compressed. It was an exciting time for the topologists in Moscow for he lectured on the topology of continua and often his latest results were presented in the course shortly after he had proved them. In this spirit, topologists regard any two objects that can be deformed into each other without tearing to be one and the same thing - have a look at Plus article In space, do all roads lead home? June 13, 2003 the bewildered topologist couldn't understand why it wasn't called ‘Lord of the doughnuts’. For a long time, the Alexander polynomial was one of the few tools topologists had for telling knots apart. His invariant had been missed completely by topologists, in spite of intense activity in closely related areas during the preceding 60 years, and it was a complete surprise. Your comments about the cube and venus de milo are excellent - a born topologist! When topologists started to explore the world of geometric forms in higher dimensions, they found that neither the intuition nor the vocabulary of ordinary geometry was sufficient to describe and classify the new forms they discovered. Topology is a branch of math that is sometimes called ‘rubber-sheet geometry’ because topologists study objects that retain their spatial properties even when they are twisted into odd shapes. Origin Late 19th century: via German from Greek topos 'place' + -logy. Definition of topology in US English: topology nountəˈpɑlədʒitəˈpäləjē 1Mathematics The study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures. 〔数〕拓扑学 Example sentencesExamples 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. The Poincare conjecture belongs to the field of topology, which studies properties that are preserved when a shape is stretched or twisted without tearing. Ktheory is a mathematical theory that studies topology using matrices, using operators that don't commute with each another. He established a geometry and topology based on group theory without the concept of a limit. His main work was in set theory, general topology, and measure theory. Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry. All mathematicians in geometry, complex analysis, low-dimensional topology, and geometric group theory will want to have it on their bookshelves. The mathematical ballistics of 1918 was neither as refined as axiomatic geometry nor as theoretical as algebraic topology. For my Ph.D. 1 was required to study analysis, algebra, and algebraic topology. For example, algebraic geometry, the field I am most familiar with, combines algebra, geometry, topology, and analysis. They relate Boolean algebras to general topology and to the theory of rings and ideals, and include what is called Stone-tech compactification today. 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. He worked in a wide variety of mathematical areas including general topology, topological vector spaces, algebraic geometry, invariant theory and the classical groups. Geometry, topology, and algebraic geometry and group theory, almost anything you want, seems to be thrown into the mixture. He started research in algebra, geometry and topology as a student but did not consider his results sufficiently important to merit publication. Euclidean geometry studies Euclidean-space-structure, topology studies topological structures, and so on. He was always full of mathematical ideas, not only on game theory, but in geometry and topology as well. His research interests include topology, algebraic geometry, and Lie theory. 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. 1.1 A 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. 拓扑 Example sentencesExamples A single most parsimonious cladogram was recovered for the ‘archaic’ subset with a topology identical to that seen in all of the MMPC. Other algorithms such as Jukes-Cantor or Tamura-Nei produced identical tree topologies. Inferred relationships are based on maximum parsimony, with this tree being one of eight identified topologies with the smallest number of steps. Distances were used in MEGA to construct a neighbor-joining tree with a topology identical to that of Lanyon's maximum parsimony tree. With a number of methods a topology in which the grasses clade is basal within the angiosperm group was found with high bootstrap support. 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. 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. The resultant tree topologies were used to construct two subsets of data within and among Pan I allelic lineages for analyses with likelihood methods. Tree topologies gather together members of the same families and fit the taxonomic classification of these dipteran species. It therefore has a different topology from the torus of fig. 4, which was be deformed into a mug with only one handle. The MP tree showed a topology almost identical to that of the NJ tree (data not shown). The methodology is case studies which, although mainly historical in format, do periodically furnish topologies and descriptive statistics. It is important to realize that different genetic operators have different ‘neighborhood’ space topologies. If our three-dimensional universe possesses one of these identified topologies, then this may in some way be dictated by laws of Nature. The genealogy estimated for the coding region of 48A alone showed an identical topology. 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. 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. An identical topology was obtained with the parsimony method. An identical topology is obtained when each of the three regions is considered separately 2The way in which constituent parts are interrelated or arranged. 结构，构造 the topology of a computer network 计算机网络结构。 Example sentencesExamples We explored forcing our data to conform to the topologies of previous among-group arrangements. 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. Other than an unresolved reticulation at the most ancestral node, the topology of this network is straightforward and there are three noteworthy results. The stoichiometry matrix: Structural modelling in this context is so called because the focus is on the network topology of the system. The actual fiber path plays a big part in how the topology of the network will look. 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. The genealogy can be viewed as consisting of two components: the topology of the tree structure and the times between coalescent events. 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. Firstly, if you happen to run an extensive subterranean mass transit system, you should familiarise yourself with the topology of the network. In this work, the topology of protein domain networks generated with data from the ProDom, Pfam, and Prosite domain databases was studied. On a molecular level, stress relaxation occurs by a change in the topology of the connections within the skeletal network. This means that the topology of the metabolic network is only partially able to explain the observed data. What is the power of a computer which can change the very notion of the topology of the underlying circuit connectivity? We also have no indication that the topology of the phylogenetic trees is biased by the alignments or the tree-building algorithms. 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. 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. 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 After you've installed the Mazu hardware, you need to teach the system about the topology of your internal network. Origin Late 19th century: via German from Greek topos ‘place’ + -logy.
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