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13,400 | SIMD use for 4D vectors can be conveniently wrapped in a vector maths library (commonly implemented in C or C++) commonly used in video game development, along with 4×4 matrix support. These are distinct from more general linear algebra libraries in other domains focussing on matrices of arbitrary size. Such libraries ... | 4D vector | 0.816241 |
13,401 | In non-standard calculus the limit of a function is defined by: if and only if for all x ∈ R ∗ , {\displaystyle x\in \mathbb {R} ^{*},} f ∗ ( x ) − L {\displaystyle f^{*}(x)-L} is infinitesimal whenever x − a is infinitesimal. Here R ∗ {\displaystyle \mathbb {R} ^{*}} are the hyperreal numbers and f* is the natural ext... | Epsilon, delta | 0.816238 |
13,402 | Misha Mahowald Erik Winfree – Professor, California Institute of Technology. Laurent Itti – Professor, University of Southern California. Tobias Delbruck – Professor of Physics and Electrical Engineering, ETH Zurich. Shih-Chii Liu – Professor at the University of Zürich and co-leader of the Sensors Group at the Institu... | Computation and Neural Systems | 0.816238 |
13,403 | David J. C. MacKay – Professor of Natural Philosophy in the Department of Physics at the University of Cambridge and chief scientific adviser to the UK Department of Energy and Climate Change (DECC). Bruno Olshausen – Director, Redwood Center for Theoretical Neuroscience, University of California, Berkeley. Kwabena Boa... | Computation and Neural Systems | 0.816238 |
13,404 | CNS faculty founded and co-founded a number of conferences and workshops: Snowbird Meeting on Neural Networks for Computing, in 1984. Neural Information Processing Systems (NIPS) in 1987. Methods in Computational Neuroscience at the Marine Biological Laboratory in Woods Hole, in 1988. The Telluride Summer School in Neu... | Computation and Neural Systems | 0.816238 |
13,405 | The creation of this multidisciplinary program stems largely from progress on several previously unrelated fronts: the analysis of complex neural systems at both the single-cell and the network levels using a variety of techniques (in particular, patch clamp recordings, intracellular and extra-cellular single and multi... | Computation and Neural Systems | 0.816238 |
13,406 | program to give birth to a scholarly community studying questions arising at the interface between neurobiology and electrical engineering, computer science and physics. It was called Computation and Neural Systems (CNS). The unifying theme of the program was the relationship between the physical structure of a computa... | Computation and Neural Systems | 0.816238 |
13,407 | Richard Feynman joined them and three separate courses resulted: Hopfield's on neural networks, Mead's on neuromorphic analog circuits, and Feynman's course on the physics of computation. At this point, Mead and Hopfield realized that a new field was emerging with neural scientists and the people doing the computer mod... | Computation and Neural Systems | 0.816238 |
13,408 | In the early 1980s, having laid out the foundations of VLSI, Carver Mead became interested in exploring the similarities between computation done in the brain and the type of computations that could be carried out in analog silicon electronic circuits. Mead joined with John Hopfield, who was studying the theoretical fo... | Computation and Neural Systems | 0.816238 |
13,409 | A form of double-blind study called a "double-dummy" design allows additional insurance against bias. In this kind of study, all patients are given both placebo and active doses in alternating periods. Placebo-controlled: The use of a placebo (fake treatment) allows the researchers to isolate the effect of the study tr... | Clinical study | 0.816237 |
13,410 | In theoretical physics and other fields, differential equations posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to differential calculus. Even to formulate such equations requires a fresh no... | Tensor field | 0.816236 |
13,411 | The concept of a tensor field can be generalized by considering objects that transform differently. An object that transforms as an ordinary tensor field under coordinate transformations, except that it is also multiplied by the determinant of the Jacobian of the inverse coordinate transformation to the wth power, is c... | Tensor field | 0.816236 |
13,412 | Nascent delta functions often arise as convolution semigroups. This amounts to the further constraint that the convolution of ηε with ηδ must satisfy for all ε, δ > 0. Convolution semigroups in L1 that form a nascent delta function are always an approximation to the identity in the above sense, however the semigroup co... | Unit impulse | 0.816231 |
13,413 | In January 2017, D-Wave Systems announced that its qbsolv open-source quantum computing software had been successfully used to find a faster solution to the MSA problem. | Multiple alignment | 0.816231 |
13,414 | Standard optimization techniques in computer science — both of which were inspired by, but do not directly reproduce, physical processes — have also been used in an attempt to more efficiently produce quality MSAs. One such technique, genetic algorithms, has been used for MSA production in an attempt to broadly simulat... | Multiple alignment | 0.816231 |
13,415 | Multiple sequence alignment lectures — from the Max Planck Institute for Molecular Genetics Lecture Notes and practical exercises on multiple sequence alignments at the EMBL Molecular Bioinformatics Lecture Notes Molecular Evolution and Bioinformatics Lecture Notes | Multiple alignment | 0.816231 |
13,416 | The use of spectrophotometers spans various scientific fields, such as physics, materials science, chemistry, biochemistry, Chemical Engineering, and molecular biology. They are widely used in many industries including semiconductors, laser and optical manufacturing, printing and forensic examination, as well as in lab... | Spectrophotometer | 0.816231 |
13,417 | The names are due to the geometry of the light source, observer and interior of the measurement chamber. Scientists use this instrument to measure the amount of compounds in a sample. If the compound is more concentrated more light will be absorbed by the sample; within small ranges, the Beer–Lambert law holds and the ... | Spectrophotometer | 0.816231 |
13,418 | Machine differential diagnosis is the use of computer software to partly or fully make a differential diagnosis. It may be regarded as an application of artificial intelligence. Alternatively, it may be seen as "Augmented Intelligence" if it meets the FDA criteria, namely that (1) it reveals the underlying data, (2) re... | Differential diagnosis | 0.816226 |
13,419 | Abraham, Ralph; Shaw, C.D. (1992). Dynamics--the geometry of behavior. Redwood City, Calif: Addison-Wesley, Advanced Book Program. | Structural complexity (applied mathematics) | 0.816224 |
13,420 | Structural complexity is a science of applied mathematics, that aims at relating fundamental physical or biological aspects of a complex system with the mathematical description of the morphological complexity that the system exhibits, by establishing rigorous relations between mathematical and physical properties of s... | Structural complexity (applied mathematics) | 0.816224 |
13,421 | Protonolysis is the cleavage of a chemical bond by acids. Many examples are found in organometallic chemistry since the reaction requires polar Mδ+-Rδ- bonds, where δ+ and δ- signify partial positive and negative charges associated with the bonding atoms. When compounds containing these bonds are treated with acid (HX)... | Protonolysis | 0.816224 |
13,422 | In statistics, the score test assesses constraints on statistical parameters based on the gradient of the likelihood function—known as the score—evaluated at the hypothesized parameter value under the null hypothesis. Intuitively, if the restricted estimator is near the maximum of the likelihood function, the score sho... | Score test | 0.816222 |
13,423 | However, rarely, multiple thresholds may be chosen and the stump therefore contains three or more leaves. Decision stumps are often used as components (called "weak learners" or "base learners") in machine learning ensemble techniques such as bagging and boosting. For example, a Viola–Jones face detection algorithm emp... | Decision stump | 0.81622 |
13,424 | A decision stump is a machine learning model consisting of a one-level decision tree. That is, it is a decision tree with one internal node (the root) which is immediately connected to the terminal nodes (its leaves). A decision stump makes a prediction based on the value of just a single input feature. Sometimes they ... | Decision stump | 0.81622 |
13,425 | An example of formal vs. informal usage may be obtained from the right-hand photo, in which the metal bar is "conducting heat" from its hot end to its cold end, but if the metal bar is considered a thermodynamic system, then the energy flowing within the metal bar is called internal energy, not heat. The hot metal bar ... | Heat | 0.816219 |
13,426 | Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry. There are also substantial connections with complexity theory, mathematical logic, the study of Lie groups and their discrete subgroups, dynamical system... | Geometric group theory | 0.816192 |
13,427 | Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmet... | Geometric group theory | 0.816192 |
13,428 | Bass–Serre theory, introduced in the 1977 book of Serre, derives structural algebraic information about groups by studying group actions on simplicial trees. External precursors of geometric group theory include the study of lattices in Lie groups, especially Mostow's rigidity theorem, the study of Kleinian groups, and... | Geometric group theory | 0.816192 |
13,429 | Other precursors of geometric group theory include small cancellation theory and Bass–Serre theory. Small cancellation theory was introduced by Martin Grindlinger in the 1960s and further developed by Roger Lyndon and Paul Schupp. It studies van Kampen diagrams, corresponding to finite group presentations, via combinat... | Geometric group theory | 0.816192 |
13,430 | Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, while an earl... | Geometric group theory | 0.816192 |
13,431 | Interactions with finite group theory, particularly progress in the study of subgroup growth. Studying subgroups and lattices in linear groups, such as S L ( n , R ) {\displaystyle SL(n,\mathbb {R} )} , and of other Lie groups, via geometric methods (e.g. buildings), algebro-geometric tools (e.g. algebraic groups and r... | Geometric group theory | 0.816192 |
13,432 | A particularly important development here is the work of Gromov who used probabilistic methods to prove the existence of a finitely generated group that is not uniformly embeddable into a Hilbert space. Other notable developments include introduction and study of the notion of generic-case complexity for group-theoreti... | Geometric group theory | 0.816192 |
13,433 | The study of group actions on CAT(0) spaces and CAT(0) cubical complexes, motivated by ideas from Alexandrov geometry. Interactions with low-dimensional topology and hyperbolic geometry, particularly the study of 3-manifold groups (see, e.g.,), mapping class groups of surfaces, braid groups and Kleinian groups. Introdu... | Geometric group theory | 0.816192 |
13,434 | This notion has been developed by many authors for both finitely presented and finitely generated groups. Connections with geometric analysis, the study of C*-algebras associated with discrete groups and of the theory of free probability. This theme is represented, in particular, by considerable progress on the Novikov... | Geometric group theory | 0.816192 |
13,435 | Interactions with computer science, complexity theory and the theory of formal languages. This theme is exemplified by the development of the theory of automatic groups, a notion that imposes certain geometric and language theoretic conditions on the multiplication operation in a finitely generated group. The study of ... | Geometric group theory | 0.816192 |
13,436 | Interactions with mathematical logic and the study of the first-order theory of free groups. Particularly important progress occurred on the famous Tarski conjectures, due to the work of Sela as well as of Olga Kharlampovich and Alexei Myasnikov. The study of limit groups and introduction of the language and machinery ... | Geometric group theory | 0.816192 |
13,437 | Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example: Gromov's polynomial growth theorem; Stallings' ends theorem; Mostow rigidity theorem. Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or met... | Geometric group theory | 0.816192 |
13,438 | Notable themes and developments in geometric group theory in 1990s and 2000s include: Gromov's program to study quasi-isometric properties of groups.A particularly influential broad theme in the area is Gromov's program of classifying finitely generated groups according to their large scale geometry. Formally, this mea... | Geometric group theory | 0.816192 |
13,439 | In general, neuroimaging studies have found that reading involves three overlapping neural systems: networks active in visual, orthography-phonology (angular gyrus), and semantic functions (anterior temporal lobe with Broca's and Wernicke's areas). However, these neural networks are not discrete, meaning these areas ha... | Reading comprehension | 0.816192 |
13,440 | A phase portrait represents the directional behavior of a system of ordinary differential equations (ODEs). The phase portrait can indicate the stability of the system. The phase portrait behavior of a system of ODEs can be determined by the eigenvalues or the trace and determinant (trace = λ1 + λ2, determinant = λ1 x ... | Phase portrait | 0.816192 |
13,441 | In mesoscopic physics, ballistic conduction (ballistic transport) is the unimpeded flow (or transport) of charge carriers (usually electrons), or energy-carrying particles, over relatively long distances in a material. In general, the resistivity of a material exists because an electron, while moving inside a medium, i... | Ballistic conductivity | 0.816189 |
13,442 | In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant s... | Symmetry group | 0.816183 |
13,443 | In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Each type of mathematical structure has invertible mappings which preserve the structure. Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or inv... | Symmetry group | 0.816183 |
13,444 | Radiopharmacology is radiochemistry applied to medicine and thus the pharmacology of radiopharmaceuticals (medicinal radiocompounds, that is, pharmaceutical drugs that are radioactive). Radiopharmaceuticals are used in the field of nuclear medicine as radioactive tracers in medical imaging and in therapy for many disea... | Radioactive pharmaceuticals | 0.816182 |
13,445 | The journal is indexed in Google Scholar, Baidu, Elsevier (Ei Compendex), Clarivate Analytics (Web of Science), ProQuest, Civil engineering database, TRDI, OCLC (WorldCat), IET/INSPEC, Crossref, Scopus, and EBSCOHost. | Journal of Engineering Mechanics | 0.816182 |
13,446 | The Journal of Engineering Mechanics is a peer-reviewed scientific journal published by the American Society of Civil Engineers, and covers activity and development in the field of applied mechanics as it relates to civil engineering. Published papers typically describe the development and implementation of new analyti... | Journal of Engineering Mechanics | 0.816182 |
13,447 | Mathematical physics is an enormous and diverse subject. Some indications of future research directions are given in "New Trends in Mathematical Physics: Selected Contributions of the XVth International Congress on Mathematical Physics". | Future of mathematics | 0.81618 |
13,448 | In the year 2000, Mathematical logic was discussed in "The Prospects For Mathematical Logic In The Twenty-First Century", including set theory, mathematical logic in computer science, and proof theory. | Future of mathematics | 0.81618 |
13,449 | In 2000, Lloyd N. Trefethen wrote "Predictions for scientific computing 50 years from now", which concluded with the theme that "Human beings will be removed from the loop" and writing in 2008 in The Princeton Companion to Mathematics predicted that by 2050 most numerical programs will be 99% intelligent wrapper and on... | Future of mathematics | 0.81618 |
13,450 | In 1998, Mikhail Gromov in "Possible Trends in Mathematics in the Coming Decades", says that traditional probability theory applies where global structure such as the Gauss Law emerges when there is a lack of structure between individual data points, but that one of today's problems is to develop methods for analyzing ... | Future of mathematics | 0.81618 |
13,451 | In "Rough structure and classification", Timothy Gowers writes about three stages: 1) at the moment computers are just slaves doing boring calculations, 2) soon databases of mathematical concepts and proof methods will lead to an intermediate stage where computers are very helpful with theorem proving but unthreatening... | Future of mathematics | 0.81618 |
13,452 | Doron Zeilberger considers a time when computers become so powerful that the predominant questions in mathematics change from proving things to determining how much it would cost: "As wider classes of identities, and perhaps even other kinds of classes of theorems, become routinely provable, we might witness many resul... | Future of mathematics | 0.81618 |
13,453 | Mathematical biology is one of the fastest expanding areas of mathematics at the beginning of the 21st century. "Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better" is an essay by Joel E. Cohen. | Future of mathematics | 0.81618 |
13,454 | In the field of biology, the biotechnology revolution in the 1980s grew from the development of reagents that could be used to identify and manipulate the chemical matter in and on cells. These reagents included antibodies (polyclonal and monoclonal), oligomers, all sorts of model organisms and immortalised cell lines,... | Chemical reagents | 0.816171 |
13,455 | An open source simulation of C. elegans at the cellular level is being pursued by the OpenWorm community. So far the physics engine Gepetto has been built and models of the neural connectome and a muscle cell have been created in the NeuroML format. | Modelling biological systems | 0.816168 |
13,456 | The Blue Brain Project is an attempt to create a synthetic brain by reverse-engineering the mammalian brain down to the molecular level. The aim of this project, founded in May 2005 by the Brain and Mind Institute of the École Polytechnique in Lausanne, Switzerland, is to study the brain's architectural and functional ... | Modelling biological systems | 0.816168 |
13,457 | In spherical geometry, a monogon can be constructed as a vertex on a great circle (equator). This forms a dihedron, {1,2}, with two hemispherical monogonal faces which share one 360° edge and one vertex. Its dual, a hosohedron, {2,1} has two antipodal vertices at the poles, one 360° lune face, and one edge (meridian) b... | Monogonal dihedron | 0.816156 |
13,458 | Hestenes (1998) found that while "nearly 80% of the could state Newton's Third Law at the beginning of the course, FCI data showed that less than 15% of them fully understood it at the end". These results have been replicated in a number of studies involving students at a range of institutions (see sources section bel... | Force Concept Inventory | 0.816155 |
13,459 | The Force Concept Inventory is a test measuring mastery of concepts commonly taught in a first semester of physics developed by Hestenes, Halloun, Wells, and Swackhamer (1985). It was the first such "concept inventory" and several others have been developed since for a variety of topics. The FCI was designed to assess ... | Force Concept Inventory | 0.816155 |
13,460 | The acquisition of labeled data for a learning problem often requires a skilled human agent (e.g. to transcribe an audio segment) or a physical experiment (e.g. determining the 3D structure of a protein or determining whether there is oil at a particular location). The cost associated with the labeling process thus may... | Semisupervised learning | 0.816154 |
13,461 | Geometry can be defined as the study of shapes their size, angles, dimensions and proportions | Mathematical visualization | 0.816149 |
13,462 | Proofs without words have existed since antiquity, as in the Pythagorean theorem proof found in the Zhoubi Suanjing Chinese text which dates from 1046 BC to 256 BC. The Clebsch diagonal surface demonstrates the 27 lines on a cubic surface.Sphere eversion – that a sphere can be turned inside out in 3 dimension if allowe... | Mathematical visualization | 0.816149 |
13,463 | Mathematical visualization is used throughout mathematics, particularly in the fields of geometry and analysis. Notable examples include plane curves, space curves, polyhedra, ordinary differential equations, partial differential equations (particularly numerical solutions, as in fluid dynamics or minimal surfaces such... | Mathematical visualization | 0.816149 |
13,464 | Mathematical phenomena can be understood and explored via visualization. Classically this consisted of two-dimensional drawings or building three-dimensional models (particularly plaster models in the 19th and early 20th century), while today it most frequently consists of using computers to make static two or three di... | Mathematical visualization | 0.816149 |
13,465 | The flow around a lifting airfoil is a fluid mechanics phenomenon that can be understood on essentially two levels: There are mathematical theories, which are based on established laws of physics and represent the flow accurately, but which require solving partial differential equations. And there are physical explanat... | Dynamic lift | 0.816145 |
13,466 | Mathematical theories of lift are based on continuum fluid mechanics, assuming that air flows as a continuous fluid. Lift is generated in accordance with the fundamental principles of physics, the most relevant being the following three principles: Conservation of momentum, which is a consequence of Newton's laws of mo... | Dynamic lift | 0.816145 |
13,467 | Stark spectroscopy (sometimes known as electroabsorption/emission spectroscopy) is a form of spectroscopy based on the Stark effect. In brief, this technique makes use of the Stark effect (or electrochromism) either to reveal information about the physiochemical or physical properties of a sample using a well-character... | Stark spectroscopy | 0.816142 |
13,468 | Three sequences, UAG, UGA, and UAA, known as stop codons, do not code for an amino acid but instead signal the release of the nascent polypeptide from the ribosome. In the standard code, the sequence AUG—read as methionine—can serve as a start codon and, along with sequences such as an initiation factor, initiates tran... | DNA and RNA codon tables | 0.81614 |
13,469 | The original, modern synthesis view of population genetics assumes that mutations provide ample raw material, and focuses only on the change in frequency of alleles within populations. The main processes influencing allele frequencies are natural selection, genetic drift, gene flow and recurrent mutation. Fisher and Wr... | Population genetic | 0.816133 |
13,470 | No population genetics perspective have ever given genetic drift a central role by itself, but some have made genetic drift important in combination with another non-selective force. The shifting balance theory of Sewall Wright held that the combination of population structure and genetic drift was important. Motoo Kim... | Population genetic | 0.816133 |
13,471 | The effect of genetic drift is larger for alleles present in few copies than when an allele is present in many copies. The population genetics of genetic drift are described using either branching processes or a diffusion equation describing changes in allele frequency. These approaches are usually applied to the Wrigh... | Population genetic | 0.816133 |
13,472 | In this way, natural selection converts differences in the fitness of individuals with different phenotypes into changes in allele frequency in a population over successive generations.Before the advent of population genetics, many biologists doubted that small differences in fitness were sufficient to make a large dif... | Population genetic | 0.816133 |
13,473 | Natural selection, which includes sexual selection, is the fact that some traits make it more likely for an organism to survive and reproduce. Population genetics describes natural selection by defining fitness as a propensity or probability of survival and reproduction in a particular environment. The fitness is norma... | Population genetic | 0.816133 |
13,474 | Given this perspective, and borrowing from Kepler, Newton attributed the term "inertia" to mean "the innate force possessed by an object which resists changes in motion"; thus, Newton defined "inertia" to mean the cause of the phenomenon, rather than the phenomenon itself. However, Newton's original ideas of "innate re... | Principle of inertia | 0.816132 |
13,475 | An aspect of this property is the tendency of objects to keep moving in a straight line at a constant speed when no forces act upon them. Inertia is one of the primary manifestations of mass, which is a quantitative property of physical systems.In his monumental Philosophiæ Naturalis Principia Mathematica, Isaac Newton... | Principle of inertia | 0.816132 |
13,476 | The measure-theoretically inclined may prefer to substitute events { X ∈ A } {\displaystyle \{X\in A\}} for events { X ≤ x } {\displaystyle \{X\leq x\}} in the above definition, where A {\displaystyle A} is any Borel set. That definition is exactly equivalent to the one above when the values of the random variables are... | Independent random variables | 0.816127 |
13,477 | In robotics, robot kinematics applies geometry to the study of the movement of multi-degree of freedom kinematic chains that form the structure of robotic systems. The emphasis on geometry means that the links of the robot are modeled as rigid bodies and its joints are assumed to provide pure rotation or translation. R... | Robot kinematics | 0.816121 |
13,478 | A representation of an algebra A is an algebra homomorphism ρ: A → End(V) from A to the endomorphism algebra of some vector space (or module) V. The property of ρ being an algebra homomorphism means that ρ preserves the multiplicative operation (that is, ρ(xy) = ρ(x)ρ(y) for all x and y in A), and that ρ sends the unit... | Linear associative algebra | 0.816119 |
13,479 | Given an associative algebra A over a commutative ring R, the enveloping algebra A e {\displaystyle A^{e}} of A is the algebra A ⊗ R A o p {\displaystyle A\otimes _{R}A^{op}} or A o p ⊗ R A {\displaystyle A^{op}\otimes _{R}A} , depending on authors.Note that a bimodule over A is exactly a left module over A e {\display... | Linear associative algebra | 0.816119 |
13,480 | Let A be a finite-dimensional algebra over a field k. Then A is an Artinian ring. | Linear associative algebra | 0.816119 |
13,481 | We will also assume that all rings are unital, and all ring homomorphisms are unital. Many authors consider the more general concept of an associative algebra over a commutative ring R, instead of a field: An R-algebra is an R-module with an associative R-bilinear binary operation, which also contains a multiplicative ... | Linear associative algebra | 0.816119 |
13,482 | In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field K. The addition and multiplication operations together give A the structure of a ring; the addition and scalar mult... | Linear associative algebra | 0.816119 |
13,483 | See for example Fig. 4; there, the arrow from "real linear topological" to "real linear" is two-headed, since every real linear space admits some (at least one) topology compatible with its linear structure. Such topology is non-unique in general, but unique when the real linear space is finite-dimensional. For these s... | Space (mathematics) | 0.816113 |
13,484 | The transition from "Euclidean" to "topological" is forgetful. Topology distinguishes continuous from discontinuous, but does not distinguish rectilinear from curvilinear. Intuition tells us that the Euclidean structure cannot be restored from the topology. | Space (mathematics) | 0.816113 |
13,485 | Relations of this kind are treated in more detail in the Section "Types of spaces". It is not always clear whether a given mathematical object should be considered as a geometric "space", or an algebraic "structure". A general definition of "structure", proposed by Bourbaki, embraces all common types of spaces, provide... | Space (mathematics) | 0.816113 |
13,486 | Topological notions such as continuity have natural definitions in every Euclidean space. However, topology does not distinguish straight lines from curved lines, and the relation between Euclidean and topological spaces is thus "forgetful". | Space (mathematics) | 0.816113 |
13,487 | A solid arrow denotes a prevalent, so-called "canonical" transition that suggests itself naturally and is widely used, often implicitly, by default. For example, speaking about a continuous function on a Euclidean space, one need not specify its topology explicitly. In fact, alternative topologies exist and are used so... | Space (mathematics) | 0.816113 |
13,488 | The arrow from "topological" to "measurable" is dashed for the reason explained there: "In order to turn a topological space into a measurable space one endows it with a σ-algebra. The σ-algebra of Borel sets is the most popular, but not the only choice." | Space (mathematics) | 0.816113 |
13,489 | First, a 3-dim Euclidean space is a special (not general) case of a Euclidean space. Second, a topology of a Euclidean space is a special case of topology (for instance, it must be non-compact, and connected, etc). We denote surjective transitions by a two-headed arrow, "↠" rather than "→". | Space (mathematics) | 0.816113 |
13,490 | One may expect that the structures called "spaces" are perceived more geometrically than other mathematical objects, but this is not always true. According to the famous inaugural lecture given by Bernhard Riemann in 1854, every mathematical object parametrized by n real numbers may be treated as a point of the n-dimen... | Space (mathematics) | 0.816113 |
13,491 | This shift in foundations requires a new set of axioms, and if these axioms are adopted, the classical axioms of geometry become theorems. A space now consists of selected mathematical objects (for instance, functions on another space, or subspaces of another space, or just elements of a set) treated as points, and sel... | Space (mathematics) | 0.816113 |
13,492 | For instance, in Richard Dedekind's 1872 essay Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), he asserts that points on a line ought to have the properties of Dedekind cuts, and that therefore a line was the same thing as the set of real numbers. Dedekind is careful to note that this is an assum... | Space (mathematics) | 0.816113 |
13,493 | : 136 However, the heritage of classical geometry was not lost. According to Bourbaki,: 138 "passed over in its role as an autonomous and living science, classical geometry is thus transfigured into a universal language of contemporary mathematics". Simultaneously, numbers began to displace geometry as the foundation o... | Space (mathematics) | 0.816113 |
13,494 | These axiom systems describe the space via primitive notions (such as "point", "between", "congruent") constrained by a number of axioms. Analytic geometry made great progress and succeeded in replacing theorems of classical geometry with computations via invariants of transformation groups. : 134, 5 Since that time, n... | Space (mathematics) | 0.816113 |
13,495 | The word "geometry" (from Ancient Greek: geo- "earth", -metron "measurement") initially meant a practical way of processing lengths, regions and volumes in the space in which we live, but was then extended widely (as well as the notion of space in question here). According to Bourbaki,: 131 the period between 1795 (Géo... | Space (mathematics) | 0.816113 |
13,496 | Generally, finitely many principal base sets and finitely many auxiliary base sets are stipulated by Bourbaki. Many mathematical structures of geometric flavor treated in the "Non-commutative geometry", "Schemes" and "Topoi" subsections above do not stipulate a base set of points. For example, "pointless topology" (in ... | Space (mathematics) | 0.816113 |
13,497 | For example, two-dimensional projective geometry may be formalized via two base sets, the set of points and the set of lines. Moreover, a striking feature of projective planes is the symmetry of the roles played by points and lines. A less geometric example: a graph may be formalized via two base sets, the set of verti... | Space (mathematics) | 0.816113 |
13,498 | Every space treated in Section "Types of spaces" above, except for "Non-commutative geometry", "Schemes" and "Topoi" subsections, is a set (the "principal base set" of the structure, according to Bourbaki) endowed with some additional structure; elements of the base set are usually called "points" of this space. In con... | Space (mathematics) | 0.816113 |
13,499 | field of complex numbers, is the same as the (geometric?) complex plane. It is first of all "a place we do analysis" (rather than algebra or geometry). | Space (mathematics) | 0.816113 |
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