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18,000 | The beginnings of the field of floral biology is generally traced to Christian Konrad Sprengel's Entdeckte Geheimniss der Natur im Bau in der Befruchtung der Blumen (The Secret of Nature in the Form and Fertilization of Flowers Discovered) (1793). Sprengel may however have been influenced by the earlier work of Joseph ... | Floral biology | 0.81253 |
18,001 | Differentials are also important in algebraic geometry, and there are several important notions. Abelian differentials usually mean differential one-forms on an algebraic curve or Riemann surface. Quadratic differentials (which behave like "squares" of abelian differentials) are also important in the theory of Riemann ... | Differential (mathematics) | 0.812529 |
18,002 | In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions.The term is used in various branches of mathematics such as calculus, differential geometry, algebraic geometry and algebr... | Differential (mathematics) | 0.812529 |
18,003 | In nuclear physics and chemistry, the Q value for a reaction is the amount of energy absorbed or released during the nuclear reaction. The value relates to the enthalpy of a chemical reaction or the energy of radioactive decay products. It can be determined from the masses of reactants and products. | Q value (nuclear science) | 0.812528 |
18,004 | Chemical Q values are measurement in calorimetry. Exothermic chemical reactions tend to be more spontaneous and can emit light or heat, resulting in runaway feedback(i.e. explosions). Q values are also featured in particle physics. | Q value (nuclear science) | 0.812528 |
18,005 | Consequently, the Chilton-Colburn analogy of St•Pr2/3 increasing with increasing cf is qualitatively valid whenever the Reynolds’ analogy is valid. Further, the validity of the Reynolds’ analogy is linked to the applicability of Prigogine's Theorem of Minimum Entropy Production. Thus, Reynolds' analogy is valid for flo... | Reynolds analogy | 0.812525 |
18,006 | Mary Ann Liebert, Inc. is a privately held independent publishing company founded by its president, Mary Ann Liebert, in 1980. The company publishes peer-reviewed academic journals, books, and trade magazines in the areas of biotechnology, biomedical sciences, medical research, and life sciences; clinical medicine, sur... | Genetic Engineering & Biotechnology News | 0.812522 |
18,007 | Each of the application areas described above employ a range of computer vision tasks; more or less well-defined measurement problems or processing problems, which can be solved using a variety of methods. Some examples of typical computer vision tasks are presented below. Computer vision tasks include methods for acqu... | Computer Vision | 0.812521 |
18,008 | Outline of computer vision List of emerging technologies Outline of artificial intelligence | Computer Vision | 0.812521 |
18,009 | This decade also marked the first time statistical learning techniques were used in practice to recognize faces in images (see Eigenface). Toward the end of the 1990s, a significant change came about with the increased interaction between the fields of computer graphics and computer vision. This included image-based re... | Computer Vision | 0.812521 |
18,010 | Research in projective 3-D reconstructions led to better understanding of camera calibration. With the advent of optimization methods for camera calibration, it was realized that a lot of the ideas were already explored in bundle adjustment theory from the field of photogrammetry. This led to methods for sparse 3-D rec... | Computer Vision | 0.812521 |
18,011 | These include the concept of scale-space, the inference of shape from various cues such as shading, texture and focus, and contour models known as snakes. Researchers also realized that many of these mathematical concepts could be treated within the same optimization framework as regularization and Markov random fields... | Computer Vision | 0.812521 |
18,012 | In the late 1960s, computer vision began at universities that were pioneering artificial intelligence. It was meant to mimic the human visual system, as a stepping stone to endowing robots with intelligent behavior. In 1966, it was believed that this could be achieved through a summer project, by attaching a camera to ... | Computer Vision | 0.812521 |
18,013 | For example, they are not good at classifying objects into fine-grained classes, such as the particular breed of dog or species of bird, whereas convolutional neural networks handle this with ease.Several specialized tasks based on recognition exist, such as: Content-based image retrieval – finding all images in a larg... | Computer Vision | 0.812521 |
18,014 | An illustration of their capabilities is given by the ImageNet Large Scale Visual Recognition Challenge; this is a benchmark in object classification and detection, with millions of images and 1000 object classes used in the competition. Performance of convolutional neural networks on the ImageNet tests is now close to... | Computer Vision | 0.812521 |
18,015 | Detection – the image data are scanned for a specific objects along with their locations. Examples include the detection of an obstacle in the car's field of view and possible abnormal cells or tissues in medical images or the detection of a vehicle in an automatic road toll system. Detection based on relatively simple... | Computer Vision | 0.812521 |
18,016 | The classical problem in computer vision, image processing, and machine vision is that of determining whether or not the image data contains some specific object, feature, or activity. Different varieties of recognition problem are described in the literature. Object recognition (also called object classification) – on... | Computer Vision | 0.812521 |
18,017 | The feature space of the kernel has an infinite number of dimensions; for σ = 1 {\displaystyle \sigma =1} , its expansion using the multinomial theorem is: exp ( − 1 2 ‖ x − x ′ ‖ 2 ) = exp ( 2 2 x ⊤ x ′ − 1 2 ‖ x ‖ 2 − 1 2 ‖ x ′ ‖ 2 ) = exp ( x ⊤ x ′ ) exp ( − 1 2 ‖ x ‖ 2 ) exp ( − 1 2 ‖ x ′ ‖ 2 ) = ∑ j = 0 ... | Radial basis function kernel | 0.812512 |
18,018 | In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular, it is commonly used in support vector machine classification.The RBF kernel on two samples x ∈ R k {\displaystyle \mathbf {x} \in \mathbb {R} ^{k}} and x', re... | Radial basis function kernel | 0.812512 |
18,019 | Theorem: E = e ‖ x − y ‖ 2 2 σ 2 {\displaystyle \mathbb {E} =e^{\frac {\|x-y\|^{2}}{2\sigma ^{2}}}} . Proof: It suffices to prove the case of D = 1 {\displaystyle D=1} . Use the trigonometric identity cos ( a − b ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) {\displaystyle \cos(a-b)=\cos(a)\cos(b)+\sin(a)\si... | Radial basis function kernel | 0.812512 |
18,020 | The most commonly used compounds as a matrix are variations of glycerol, such as glycerol, deuteroglycerol, thioglycerol, and aminoglycerol. If the sample cannot dissolve in the chosen matrix, such as glycerol, a cosolvent or additive can be mixed with the matrix to facilitate the dissolving of the solid analyte. For e... | Sample preparation in mass spectrometry | 0.812511 |
18,021 | Sample preparation for mass spectrometry is used for the optimization of a sample for analysis in a mass spectrometer (MS). Each ionization method has certain factors that must be considered for that method to be successful, such as volume, concentration, sample phase, and composition of the analyte solution. Quite pos... | Sample preparation in mass spectrometry | 0.812511 |
18,022 | The Association grew out of the major international conference series Computability in Europe (CiE); the first CiE conference was held in Amsterdam in June 2005. CiE is an interdisciplinary annual conference series promoting the development of computability-related science, ranging over mathematics, computer science, a... | Computability in Europe | 0.812508 |
18,023 | The Association Computability in Europe was founded in Athens, Greece in 2008. Its founding president (2008 to 2015) was Professor S. Barry Cooper; its current president is Elvira Mayordomo and its current secretary general is Giuseppe Primiero. The Association is promoting the development, particularly in Europe, of c... | Computability in Europe | 0.812508 |
18,024 | The Knoop hardness test is a microhardness test – a test for mechanical hardness used particularly for very brittle materials or thin sheets, where only a small indentation may be made for testing purposes. A pyramidal diamond point is pressed into the polished surface of the test material with a known (often 100g) loa... | Knoop hardness | 0.812492 |
18,025 | Kernels are familiar in many categories from abstract algebra, such as the category of groups or the category of (left) modules over a fixed ring (including vector spaces over a fixed field). To be explicit, if f: X → Y is a homomorphism in one of these categories, and K is its kernel in the usual algebraic sense, then... | Kernel (category theory) | 0.812489 |
18,026 | Universal algebra defines a notion of kernel for homomorphisms between two algebraic structures of the same kind. This concept of kernel measures how far the given homomorphism is from being injective. There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the sit... | Kernel (category theory) | 0.812489 |
18,027 | In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism f: X → Y is the "most general" morphism k: K → X that yields z... | Kernel (category theory) | 0.812489 |
18,028 | In physics and chemistry, it is common to measure energy on the atomic scale in the non-SI, but convenient, units electronvolts (eV). 1 eV is equivalent to the kinetic energy acquired by an electron in passing through a potential difference of 1 volt in a vacuum. It is common to use the SI magnitude prefixes (eg milli-... | Units of energy | 0.812475 |
18,029 | In theoretical computer science, a Markov algorithm is a string rewriting system that uses grammar-like rules to operate on strings of symbols. Markov algorithms have been shown to be Turing-complete, which means that they are suitable as a general model of computation and can represent any mathematical expression from... | Markov algorithm | 0.812475 |
18,030 | There are several conserved residues, include Glutamate (E) 91 and Lysine (K) 72, that mediate the positioning of alpha- and beta-phosphate groups. The hydroxyl group of the peptide substrate's Serine/Threonine attacks the gamma phosphate group at the phosphorus via an SN2 nucleophilic reaction, which results in the tr... | CAMP-dependent protein kinase | 0.812474 |
18,031 | Reject the null hypothesis, in favor of the alternative hypothesis, if and only if the p-value is less than (or equal to) the significance level (the selected probability) threshold (α), for example 0.05 or 0.01.The former process was advantageous in the past when only tables of test statistics at common probability th... | Statistical hypothesis test | 0.812473 |
18,032 | In the statistics literature, statistical hypothesis testing plays a fundamental role. There are two mathematically equivalent processes that can be used.The usual line of reasoning is as follows: There is an initial research hypothesis of which the truth is unknown. The first step is to state the relevant null and alt... | Statistical hypothesis test | 0.812473 |
18,033 | The group is currently led by 9 faculty members, and has a dozen postdoctoral researchers and students at any given time.Notable physicists who were students or postdoctoral researchers in the SLAC Theory Group include Nima Arkani-Hamed, Thomas Appelquist, Mirjam Cvetic, Michael Dine, John Ellis, Rouven Essig, Edward F... | SLAC Theory Group | 0.812469 |
18,034 | The group has a diverse research program, specializing in areas of quantum field theory, beyond the standard model physics, dark matter, neutrinos, and collider phenomenology. | SLAC Theory Group | 0.812469 |
18,035 | Current and former faculty members in the SLAC Theory Group include: James Bjorken, discoverer of Bjorken Scaling (light-cone scaling) and Bjorken Sum Rule, 2004 Dirac Medal recipient Stanley Brodsky, 2007 Sakurai Prize recipient for applications of perturbative quantum field theory to the analysis of hard exclusive st... | SLAC Theory Group | 0.812469 |
18,036 | The SLAC Theory Group is the hub of theoretical particle physics research at the SLAC National Accelerator Laboratory at Stanford University. It is a subdivision of the Elementary Particle Physics (EPP) Division at SLAC. | SLAC Theory Group | 0.812469 |
18,037 | In polymer chemistry, there are several mechanisms by which a polymerization reaction can terminate depending on the mechanism and circumstances of the reaction. A method of termination that applies to all polymer reactions is the depletion of monomer. In chain growth polymerization, two growing chains can collide head... | Termination reaction | 0.812468 |
18,038 | Science of Logic (SL; German: Wissenschaft der Logik, WdL), first published between 1812 and 1816, is the work in which Georg Wilhelm Friedrich Hegel outlined his vision of logic. Hegel's logic is a system of dialectics, i.e., a dialectical metaphysics: it is a development of the principle that thought and being consti... | Science of Logic | 0.812462 |
18,039 | From the standpoint of this Ideal One, both Repulsion and Attraction now presuppose each other, and, taken one step further, each presupposes itself as mediated by the other. The One is only a One with reference to another One―Repulsion; but this "other" One is in itself identical to, is in fact, the original One―Attra... | Science of Logic | 0.812462 |
18,040 | This gesture is made explicit in the Quantitative Ratio, where two Quanta are brought into relationship with one another in such a way that neither one in itself is self-determined, but in relating to each other, they Qualitatively determine something beyond themselves, e.g., a line or a curve.Hegel here engages in a l... | Science of Logic | 0.812462 |
18,041 | The Void can be said to be the Quality of the One.At this stage, the Logic has incorporated the ancient atomism of Leucippus and Democritus. Hegel actually held the ancient philosophical notion of atomism in higher esteem than the scientific one of modern physics because the former understood the void not just as the e... | Science of Logic | 0.812462 |
18,042 | Given a set of experimental data that looks to be clustered about a line, a linear ansatz could be made to find the parameters of the line by a least squares curve fit. Variational approximation methods use ansätze and then fit the parameters. Another example could be the mass, energy, and entropy balance equations tha... | Ansatz | 0.812462 |
18,043 | An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the boundary conditions (in fact, ... | Ansatz | 0.812462 |
18,044 | In physics and mathematics, an ansatz (; German: , meaning: "initial placement of a tool at a work piece", plural ansätze ; German: ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the solution by its results. | Ansatz | 0.812462 |
18,045 | The third law was developed by chemist Walther Nernst during the years 1906 to 1912 and is therefore often referred to as the Nernst heat theorem, or sometimes the Nernst-Simon heat theorem to include the contribution of Nernst's doctoral student Francis Simon. The third law of thermodynamics states that the entropy of... | Third law of thermodynamics | 0.812459 |
18,046 | The conflict is resolved as follows: At a certain temperature the quantum nature of matter starts to dominate the behavior. Fermi particles follow Fermi–Dirac statistics and Bose particles follow Bose–Einstein statistics. In both cases the heat capacity at low temperatures is no longer temperature independent, even for... | Third law of thermodynamics | 0.812459 |
18,047 | The entropy of a perfect crystal lattice as defined by Nernst's theorem is zero provided that its ground state is unique, because ln(1) = 0. If the system is composed of one-billion atoms that are all alike and lie within the matrix of a perfect crystal, the number of combinations of one billion identical things taken ... | Third law of thermodynamics | 0.812459 |
18,048 | In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant. The shape of different probability distributions can be compared using st... | Standardized moment | 0.812458 |
18,049 | Another scale invariant, dimensionless measure for characteristics of a distribution is the coefficient of variation, σ μ {\displaystyle {\frac {\sigma }{\mu }}} . However, this is not a standardized moment, firstly because it is a reciprocal, and secondly because μ {\displaystyle \mu } is the first moment about zero (... | Standardized moment | 0.812458 |
18,050 | The term elementary generally denotes a method that does not use complex analysis. For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg. The term is somewhat ambiguous: for example, proofs based on complex Tauberian th... | Number theoretic | 0.812454 |
18,051 | In the early ninth century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind, which may or may not be Brahmagupta's Brāhmasphuṭasiddhānta). Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820–912). Part of the ... | Number theoretic | 0.812454 |
18,052 | First, regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. Second, in some situations regression analysis can be used to infer causal relationships between the independent and dependent variables. Importantly, regressions by themse... | Regression prediction | 0.812448 |
18,053 | In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable (often called the 'outcome' or 'response' variable, or a 'label' in machine learning parlance) and one or more independent variables (often called 'predictors', 'covariates', 'exp... | Regression prediction | 0.812448 |
18,054 | The abelian group Z 2 {\displaystyle \mathbb {Z} ^{2}} maps the complex plane into the fundamental parallelogram. That is, every point z ∈ C {\displaystyle z\in \mathbb {C} } can be written as z = p + m ω 1 + n ω 2 {\displaystyle z=p+m\omega _{1}+n\omega _{2}} for integers m , n {\displaystyle m,n} with a point p {\dis... | Lattice basis | 0.812443 |
18,055 | A rubber sheet, for example, behaves more like glass at typical bullet speeds. That is, it fractures, and does not stretch or vibrate. The field of applications of impact theory ranges from the optimization of material processing, impact testing, dynamics of granular media to medical applications related to the biomech... | Mechanical impact | 0.812443 |
18,056 | In propositional calculus a literal is simply a propositional variable or its negation. In predicate calculus a literal is an atomic formula or its negation, where an atomic formula is a predicate symbol applied to some terms, P ( t 1 , … , t n ) {\displaystyle P(t_{1},\ldots ,t_{n})} with the terms recursively defined... | Boolean literal | 0.812441 |
18,057 | Quantum mechanics shows that light, along with all other forms of electromagnetic radiation, comes in discrete units, called photons, and predicts its spectral energies (corresponding to pure colors), and the intensities of its light beams. A single photon is a quantum, or smallest observable particle, of the electroma... | Introduction to quantum mechanics | 0.812433 |
18,058 | The physical measurements, equations, and predictions pertinent to quantum mechanics are all consistent and hold a very high level of confirmation. However, the question of what these abstract models say about the underlying nature of the real world has received competing answers. These interpretations are widely varyi... | Introduction to quantum mechanics | 0.812433 |
18,059 | The uncertainty principle is not only a statement about the accuracy of our measuring equipment but, more deeply, is about the conceptual nature of the measured quantities—the assumption that the car had simultaneously defined position and speed does not work in quantum mechanics. On a scale of cars and people, these u... | Introduction to quantum mechanics | 0.812433 |
18,060 | In 1927, Heisenberg proved that this last assumption is not correct. Quantum mechanics shows that certain pairs of physical properties, for example, position and speed, cannot be simultaneously measured, nor defined in operational terms, to arbitrary precision: the more precisely one property is measured, or defined in... | Introduction to quantum mechanics | 0.812433 |
18,061 | This applies even though the particles may be so far apart that it is impossible for the result of the first measurement to have been transmitted to the second particle before the second measurement takes place. A further example is superfluidity. Liquid helium in a container, cooled to a temperature near absolute zero... | Introduction to quantum mechanics | 0.812433 |
18,062 | The desire to resolve inconsistencies between observed phenomena and classical theory led to a revolution in physics, a shift in the original scientific paradigm: the development of quantum mechanics. Many aspects of quantum mechanics are counterintuitive and can seem paradoxical because they describe behavior quite di... | Introduction to quantum mechanics | 0.812433 |
18,063 | Quantum mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the moon. Classical physics is still use... | Introduction to quantum mechanics | 0.812433 |
18,064 | Years later, renormalization largely solved this problem. Initially viewed as a provisional, suspect procedure by some of its originators, renormalization eventually was embraced as an important and self-consistent tool in QED and other fields of physics. | Introduction to quantum mechanics | 0.812433 |
18,065 | Maxwell's unification of electricity, magnetism, and light in the 1880s led to experiments on the interaction of light and matter. Some of these experiments had aspects which could not be explained. Quantum mechanics emerged in the early part of the 20th century from efforts to explain these results. | Introduction to quantum mechanics | 0.812433 |
18,066 | In nuclear physics, atomic physics, and nuclear chemistry, the nuclear shell model is a model of the atomic nucleus which uses the Pauli exclusion principle to describe the structure of the nucleus in terms of energy levels. The first shell model was proposed by Dmitri Ivanenko (together with E. Gapon) in 1932. The mod... | Nuclear shell model | 0.812432 |
18,067 | This one is in the first (i.e. lowest energy) state of the 4th shell, which is a d-shell (ℓ = 2), and since p = (−1)ℓ, this gives the nucleus an overall parity of +1. This 4th d-shell has a j = 5⁄2, thus the nucleus of 178O is expected to have positive parity and total angular momentum 5⁄2, which indeed it has. The rul... | Nuclear shell model | 0.812432 |
18,068 | The Journal of Aerospace Engineering is a peer-reviewed scientific journal published by the American Society of Civil Engineers and combines civil engineering with aerospace technology (but also incorporates other elements of civil engineering) to develop structures for space and extreme conditions. Topics of interest ... | Journal of Aerospace Engineering | 0.812431 |
18,069 | Biotechcellence is a national level technical symposium. It was established by the co-operative efforts of Department of Biotechnology (DBT) and Association of Bio-technologists of Anna University. The symposium aims to highlight advancements in biotechnology that have taken place over the years in the medical, industr... | Biotechcellence | 0.81243 |
18,070 | The Centre for Biotechnology (CBT) was established in 1987 at the Anna University- supported by the Department of Biotechnology, Delhi and the University Grants Commission, Delhi, with the following objectives: To provide educational and training facilities in different areas of Biotechnology To carry out fundamental r... | Biotechcellence | 0.81243 |
18,071 | Let X be a topological space and let x and y be points in X. We say that x and y are separated if each lies in a neighbourhood that does not contain the other point. X is called a T1 space if any two distinct points in X are separated. X is called an R0 space if any two topologically distinguishable points in X are sep... | T1 separation axiom | 0.812429 |
18,072 | While topological spaces can be extremely varied and exotic, many areas of topology focus on the more familiar class of spaces known as manifolds. A manifold is a topological space that resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighborhood that is homeomor... | Topology | 0.812429 |
18,073 | Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if: Both the empty set and X are elements of τ. Any union of elements of τ is an element of τ. Any intersection of finitely many elements of τ is an element of τ.If τ is a topology on X, then the pair (X, τ) is called a top... | Topology | 0.812429 |
18,074 | The term topology also refers to a specific mathematical idea central to the area of mathematics called topology. Informally, a topology describes how elements of a set relate spatially to each other. The same set can have different topologies. For instance, the real line, the complex plane, and the Cantor set can be t... | Topology | 0.812429 |
18,075 | The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional... | Topology | 0.812429 |
18,076 | This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is immediately convincing to most people, even though they might not re... | Topology | 0.812429 |
18,077 | The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects (from a topological point of view) and b... | Topology | 0.812429 |
18,078 | Currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski.Modern topology depends strongly on the ideas of set theory, developed by Georg Cantor in the later part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered ... | Topology | 0.812429 |
18,079 | Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print. The English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geom... | Topology | 0.812429 |
18,080 | On 14 November 1750, Euler wrote to a friend that he had realized the importance of the edges of a polyhedron. This led to his polyhedron formula, V − E + F = 2 (where V, E, and F respectively indicate the number of vertices, edges, and faces of the polyhedron). Some authorities regard this analysis as the first theore... | Topology | 0.812429 |
18,081 | Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one o... | Topology | 0.812429 |
18,082 | This area of research is commonly known as spacetime topology. In condensed matter a relevant application to topological physics comes from the possibility to obtain one-way current, which is a current protected from backscattering. It was first discovered in electronics with the famous quantum Hall effect, and then ge... | Topology | 0.812429 |
18,083 | Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to topological field theory. The topological classification of Calabi–Yau manifolds has important implications in string theory, as different manifolds can sustain different kinds of strings.In cosmology, topology can be used to descri... | Topology | 0.812429 |
18,084 | Topology is of further significance in Contact mechanics where the dependence of stiffness and friction on the dimensionality of surface structures is the subject of interest with applications in multi-body physics. A topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that ... | Topology | 0.812429 |
18,085 | Topology is relevant to physics in areas such as condensed matter physics, quantum field theory and physical cosmology. The topological dependence of mechanical properties in solids is of interest in disciplines of mechanical engineering and materials science. Electrical and mechanical properties depend on the arrangem... | Topology | 0.812429 |
18,086 | These enzymes cut, twist, and reconnect the DNA, causing knotting with observable effects such as slower electrophoresis. Topology is also used in evolutionary biology to represent the relationship between phenotype and genotype. Phenotypic forms that appear quite different can be separated by only a few mutations depe... | Topology | 0.812429 |
18,087 | Topology has been used to study various biological systems including molecules and nanostructure (e.g., membraneous objects). In particular, circuit topology and knot theory have been extensively applied to classify and compare the topology of folded proteins and nucleic acids. Circuit topology classifies folded molecu... | Topology | 0.812429 |
18,088 | Topological data analysis uses techniques from algebraic topology to determine the large scale structure of a set (for instance, determining if a cloud of points is spherical or toroidal). The main method used by topological data analysis is to: Replace a set of data points with a family of simplicial complexes, indexe... | Topology | 0.812429 |
18,089 | Geometric topology is a branch of topology that primarily focuses on low-dimensional manifolds (that is, spaces of dimensions 2, 3, and 4) and their interaction with geometry, but it also includes some higher-dimensional topology. Some examples of topics in geometric topology are orientability, handle decompositions, l... | Topology | 0.812429 |
18,090 | In a metric space, an open set is a union of open disks, where an open disk of radius r centered at x is the set of all points whose distance to x is less than r. Many common spaces are topological spaces whose topology can be defined by a metric. This is the case of the real line, the complex plane, real and complex v... | Topology | 0.812429 |
18,091 | Changing a topology consists of changing the collection of open sets. This changes which functions are continuous and which subsets are compact or connected. Metric spaces are an important class of topological spaces where the distance between any two points is defined by a function called a metric. | Topology | 0.812429 |
18,092 | The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. | Topology | 0.812429 |
18,093 | General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. T... | Topology | 0.812429 |
18,094 | Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds. More specifically, differential topology considers the properties and structures that require o... | Topology | 0.812429 |
18,095 | Algebraic topology is a branch of mathematics that uses tools from algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. The most important of these invariants are homotopy groups... | Topology | 0.812429 |
18,096 | A function or map from one topological space to another is called continuous if the inverse image of any open set is open. If the function maps the real numbers to the real numbers (both spaces with the standard topology), then this definition of continuous is equivalent to the definition of continuous in calculus. If ... | Topology | 0.812429 |
18,097 | Occasionally, one needs to use the tools of topology but a "set of points" is not available. In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are structures defined on arbitrary categories that allow the definition of sheaves on those ... | Topology | 0.812429 |
18,098 | The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 1... | Topology | 0.812429 |
18,099 | In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or p... | Topology | 0.812429 |
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