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16,100 | In physics, thermal contact conductance is the study of heat conduction between solid or liquid bodies in thermal contact. The thermal contact conductance coefficient, h c {\displaystyle h_{c}} , is a property indicating the thermal conductivity, or ability to conduct heat, between two bodies in contact. The inverse of... | Thermal contact resistance | 0.813957 |
16,101 | In materials science and molecular biology, thermostability is the ability of a substance to resist irreversible change in its chemical or physical structure, often by resisting decomposition or polymerization, at a high relative temperature. Thermostable materials may be used industrially as fire retardants. A thermos... | Thermostability | 0.813956 |
16,102 | Algebraically, negation (NOT) is replaced with 1 − x, conjunction (AND) is replaced with multiplication ( x y {\displaystyle xy} ), and disjunction (OR) is defined via De Morgan's law. Interpreting these values as logical truth values yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic ... | Contact algebra | 0.813954 |
16,103 | The candidates for membership in a set work just like the wires in a digital computer: each candidate is either a member or a nonmember, just as each wire is either high or low. Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine to make the algebra of two value... | Contact algebra | 0.813954 |
16,104 | Other areas where two values is a good choice are the law and mathematics. In everyday relaxed conversation, nuanced or complex answers such as "maybe" or "only on the weekend" are acceptable. In more focused situations such as a court of law or theorem-based mathematics however it is deemed advantageous to frame quest... | Contact algebra | 0.813954 |
16,105 | The pi-pi charge-transfer interactions of bipyridinium with dioxyarenes or diaminoarenes have been used extensively for the construction of mechanically interlocked systems and in crystal engineering. The use of crown ether binding with metal or ammonium cations is ubiquitous in supramolecular chemistry. The formation ... | Supermolecular chemistry | 0.813952 |
16,106 | The existence of intermolecular forces was first postulated by Johannes Diderik van der Waals in 1873. However, Nobel laureate Hermann Emil Fischer developed supramolecular chemistry's philosophical roots. In 1894, Fischer suggested that enzyme–substrate interactions take the form of a "lock and key", the fundamental p... | Supermolecular chemistry | 0.813952 |
16,107 | The development of selective "host–guest" complexes in particular, in which a host molecule recognizes and selectively binds a certain guest, was cited as an important contribution. In the 1990s, supramolecular chemistry became even more sophisticated, with researchers such as James Fraser Stoddart developing molecular... | Supermolecular chemistry | 0.813952 |
16,108 | The breakthrough came in the 1960s with the synthesis of the crown ethers by Charles J. Pedersen. Following this work, other researchers such as Donald J. Cram, Jean-Marie Lehn and Fritz Vögtle became active in synthesizing shape- and ion-selective receptors, and throughout the 1980s research in the area gathered a rap... | Supermolecular chemistry | 0.813952 |
16,109 | This pre-organization also serves purposes such as minimizing side reactions, lowering the activation energy of the reaction, and producing desired stereochemistry. After the reaction has taken place, the template may remain in place, be forcibly removed, or may be "automatically" decomplexed on account of the differen... | Supermolecular chemistry | 0.813952 |
16,110 | Molecular recognition and self-assembly may be used with reactive species in order to pre-organize a system for a chemical reaction (to form one or more covalent bonds). It may be considered a special case of supramolecular catalysis. Non-covalent bonds between the reactants and a "template" hold the reactive sites of ... | Supermolecular chemistry | 0.813952 |
16,111 | It is sometimes called integer division, and denoted by "//".Dividing integers in a computer program requires special care. Some programming languages treat integer division as in case 5 above, so the answer is an integer. Other languages, such as MATLAB and every computer algebra system return a rational number as the... | Division (mathematics) | 0.813947 |
16,112 | An automorphism of a differentiable manifold M is a diffeomorphism from M to itself. The automorphism group is sometimes denoted Diff(M). In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorph... | Trivial automorphism | 0.813945 |
16,113 | In particular, if two nodes are joined by an edge, so are their images under the permutation. In geometry, an automorphism may be called a motion of the space. Specialized terminology is also used: In metric geometry an automorphism is a self-isometry. | Trivial automorphism | 0.813945 |
16,114 | This group is isomorphic to SO(3), the group of rotations in 3-dimensional space. The automorphism group of the octonions (O) is the exceptional Lie group G2. In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges. | Trivial automorphism | 0.813945 |
16,115 | Field automorphisms are important to the theory of field extensions, in particular Galois extensions. In the case of a Galois extension L/K the subgroup of all automorphisms of L fixing K pointwise is called the Galois group of the extension. The automorphism group of the quaternions (H) as a ring are the inner automor... | Trivial automorphism | 0.813945 |
16,116 | In linear algebra, an endomorphism of a vector space V is a linear operator V → V. An automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is the same as the general linear group, GL(V). (The algebraic structure of all endomorphisms of V is itself ... | Trivial automorphism | 0.813945 |
16,117 | A common form for closed-core coils is a toroidal core coil, in which the core has the shape of a torus or doughnut, with either a circular or rectangular cross section. This geometry has minimum leakage flux and radiates minimum electromagnetic interference (EMI). A coil with a core which is a straight bar or other no... | Magnetic coil | 0.813944 |
16,118 | A coil with a core which forms a closed loop, possibly with some narrow air gaps, is called a closed-core coil. By providing a closed path for the magnetic field lines, this geometry minimizes the magnetic reluctance and produces the strongest magnetic field. It is often used in transformers. | Magnetic coil | 0.813944 |
16,119 | Each layer is bonded strongly in plane, but weakly in interlayers. Therefore, TMDCs can be easily exfoliated into atomically thin layers through various methods. TMDCs show layer-dependent optical and electrical properties. When exfoliated into monolayers, the band gaps of several TMDCs change from indirect to direct, ... | Two-dimensional semiconductor | 0.813943 |
16,120 | Some applications include electronic devices, photonic and energy harvesting devices, and flexible and transparent substrates. Other applications include on quantum computing qubit devices solar cells, and flexible electronics. | Two-dimensional semiconductor | 0.813943 |
16,121 | Kuhn's view of demarcation is most clearly expressed in his comparison of astronomy with astrology. Since antiquity, astronomy has been a puzzle-solving activity and therefore a science. If an astronomer's prediction failed, then this was a puzzle that he could hope to solve for instance with more measurements or with ... | Problem of demarcation | 0.813939 |
16,122 | Concern for informed human nutrition resulted in the following note in 1942: If our boys and girls are to be exposed to the superficial and frequently ill-informed statements about science and medicine made over the radio and in the daily press, it is desirable, if not necessary, that some corrective in the form of acc... | Problem of demarcation | 0.813939 |
16,123 | In order to be verifiable, the statement must be verifiable in the observable world, or facts that can be induced from "derived experience". This is referred to as the "verifiability" criterion.This distinction between science, which in the opinion of the Vienna Circle possessed empirically verifiable statements, and w... | Problem of demarcation | 0.813939 |
16,124 | Logical positivism, formulated during the 1920s, is the idea that only statements about matters of fact or logical relations between concepts are meaningful. All other statements lack sense and are labelled "metaphysics" (see the verifiability theory of meaning also known as verificationism). According to A. J. Ayer, m... | Problem of demarcation | 0.813939 |
16,125 | Phyloscan is a web service for DNA sequence analysis that is free and open to all users (without login requirement). For locating matches to a user-specified sequence motif for a regulatory binding site, Phyloscan provides a statistically sensitive scan of user-supplied mixed aligned and unaligned DNA sequence data. Ph... | Phyloscan | 0.813937 |
16,126 | Many enzymatic reactions are oxidation–reduction reactions, in which one compound is oxidized and another compound is reduced. The ability of an organism to carry out oxidation–reduction reactions depends on the oxidation–reduction state of the environment, or its reduction potential ( E h {\displaystyle E_{h}} ). Stri... | Oxidation-reduction potential | 0.813937 |
16,127 | Phase distinction is used in conjunction with static checking. By using a calculus based system, phase distinction removes the need to enforce linear logic between different types and terms of programming. | Phase distinction | 0.813937 |
16,128 | Linkage and association analysis are primary tools for gene discovery, localization and functional analysis. While conceptual underpinning of these approaches have been long known, advances in recent decades in molecular genetics, development in efficient algorithms, and computing power have enabled the large scale app... | Family-based QTL mapping | 0.813936 |
16,129 | The technology is used to process unstructured content. Common applications include self-driving cars, self-checkouts at grocery stores, smart home assistants, and appliances. Businesses can apply data and machine learning to build predictive analytics that react to consumer behavior changes, or to implement RPA to imp... | Intelligent automation | 0.813935 |
16,130 | Intelligent automation, or alternately intelligent process automation, is a software term that refers to a combination of artificial intelligence (AI) and robotic process automation (RPA). Companies use intelligent automation to cut costs by using artificial-intelligence-powered robotic software to replace workers who ... | Intelligent automation | 0.813935 |
16,131 | When computing accuracy in multiclass classification, accuracy is simply the fraction of correct classifications: This is usually expressed as a percentage. For example, if a classifier makes ten predictions and nine of them are correct, the accuracy is 90%. Accuracy is also called top-1 accuracy to distinguish it from... | Accuracy and precision | 0.813935 |
16,132 | In that case, the term standard error is properly applied: the precision of the average is equal to the known standard deviation of the process divided by the square root of the number of measurements averaged. Further, the central limit theorem shows that the probability distribution of the averaged measurements will ... | Accuracy and precision | 0.813935 |
16,133 | In the fields of science and engineering, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity's true value. The precision of a measurement system, related to reproducibility and repeatability, is the degree to which repeated measurements under unchanged conditi... | Accuracy and precision | 0.813935 |
16,134 | In psychometrics and psychophysics, the term accuracy is interchangeably used with validity and constant error. Precision is a synonym for reliability and variable error. The validity of a measurement instrument or psychological test is established through experiment or correlation with behavior. Reliability is establi... | Accuracy and precision | 0.813935 |
16,135 | The SKI combinator calculus is a combinatory logic system and a computational system. It can be thought of as a computer programming language, though it is not convenient for writing software. Instead, it is important in the mathematical theory of algorithms because it is an extremely simple Turing complete language. I... | SKI calculus | 0.813932 |
16,136 | A number of iodine radioisotopes are also used in medical applications. Iodine is found on Earth mainly as the highly water-soluble iodide I−, which concentrates it in oceans and brine pools. Like the other halogens, free iodine occurs mainly as a diatomic molecule I2, and then only momentarily after being oxidized fro... | Period 5 element | 0.81393 |
16,137 | Bradwardine expressed this by a series of specific examples, but although the logarithm had not yet been conceived, we can express his conclusion anachronistically by writing: V = log (F/R). Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantif... | Roman mathematics | 0.813919 |
16,138 | Many Greek and Arabic texts on mathematics were translated into Latin from the 12th century onward, leading to further development of mathematics in Medieval Europe. From ancient times through the Middle Ages, periods of mathematical discovery were often followed by centuries of stagnation. Beginning in Renaissance Ita... | Roman mathematics | 0.813919 |
16,139 | Although they made virtually no contributions to theoretical mathematics, the ancient Romans used applied mathematics in surveying, structural engineering, mechanical engineering, bookkeeping, creation of lunar and solar calendars, and even arts and crafts. Chinese mathematics made early contributions, including a plac... | Roman mathematics | 0.813919 |
16,140 | All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the Pythagoreans, wh... | Roman mathematics | 0.813919 |
16,141 | He also wrote influential work on Euclid's parallel postulate. In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots, which was a special case of the methods given many centuries later by Ruffini and Horner. Other achievements of M... | Roman mathematics | 0.813919 |
16,142 | He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.In the late 11th century, Omar Khayyam wrote Discussions of the Difficulties in Euclid, a book about what he perceived as flaws in Euclid's Elements, especially t... | Roman mathematics | 0.813919 |
16,143 | Further developments in algebra were made by Al-Karaji in his treatise al-Fakhri, where he extends the methodology to incorporate integer powers and integer roots of unknown quantities. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the b... | Roman mathematics | 0.813919 |
16,144 | He also developed techniques used to solve three non-linear simultaneous equations with three unknown variables. One unique feature of his works was trying to find all the possible solutions to some of his problems, including one where he found 2676 solutions. His works formed an important foundation for the developmen... | Roman mathematics | 0.813919 |
16,145 | His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." He also studied an equation for its own s... | Roman mathematics | 0.813919 |
16,146 | He gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots, and he was the first to teach algebra in an elementary form and for its own sake. He also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the ot... | Roman mathematics | 0.813919 |
16,147 | The Islamic Empire established across the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics. Although most Islamic texts on mathematics were written in Arabic, most of them were not written by Arabs, since much like the status of... | Roman mathematics | 0.813919 |
16,148 | Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem). Other 19th-century mathematicians used this in their proofs that straight edge and compass alone are not sufficient to t... | Roman mathematics | 0.813919 |
16,149 | Augustin-Louis Cauchy, Bernhard Riemann, and Karl Weierstrass reformulated the calculus in a more rigorous fashion. Also, for the first time, the limits of mathematics were explored. | Roman mathematics | 0.813919 |
16,150 | Hermann Grassmann in Germany gave a first version of vector spaces, William Rowan Hamilton in Ireland developed noncommutative algebra. The British mathematician George Boole devised an algebra that soon evolved into what is now called Boolean algebra, in which the only numbers were 0 and 1. Boolean algebra is the star... | Roman mathematics | 0.813919 |
16,151 | Elliptic geometry was developed later in the 19th century by the German mathematician Bernhard Riemann; here no parallel can be found and the angles in a triangle add up to more than 180°. Riemann also developed Riemannian geometry, which unifies and vastly generalizes the three types of geometry, and he defined the co... | Roman mathematics | 0.813919 |
16,152 | This century saw the development of the two forms of non-Euclidean geometry, where the parallel postulate of Euclidean geometry no longer holds. The Russian mathematician Nikolai Ivanovich Lobachevsky and his rival, the Hungarian mathematician János Bolyai, independently defined and studied hyperbolic geometry, where u... | Roman mathematics | 0.813919 |
16,153 | Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory p... | Roman mathematics | 0.813919 |
16,154 | The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian sta... | Roman mathematics | 0.813919 |
16,155 | In the 16th century, Jyesthadeva consolidated many of the Kerala School's developments and theorems in the Yukti-bhāṣā. It has been argued that the advances of the Kerala school, which laid the foundations of the calculus, were transmitted to Europe in the 16th century via Jesuit missionaries and traders who were activ... | Roman mathematics | 0.813919 |
16,156 | To what extent he anticipated the invention of calculus is a controversial subject among historians of mathematics. In the 14th century, Narayana Pandita completed his Ganita Kaumudi.Also in the 14th century, Madhava of Sangamagrama, the founder of the Kerala School of Mathematics, found the Madhava–Leibniz series and ... | Roman mathematics | 0.813919 |
16,157 | Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, Halayudha's commentary on Pingala's work contains a study of the Fibonacci sequence and Pascal's triangle, and describes the formation of a matrix.In the 12th century, Bhāskara II, who lived in southern India, wrote extens... | Roman mathematics | 0.813919 |
16,158 | Around 500 AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.It is in the Aryabhatiya that the decimal place-value system first appears. Several c... | Roman mathematics | 0.813919 |
16,159 | Pingala's work also contains the basic ideas of Fibonacci numbers (called mātrāmeru).The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas, astronomical treatises from the 4th and 5th centuries AD (Gupta period) showing strong Hellenistic influence. They are significant in tha... | Roman mathematics | 0.813919 |
16,160 | His notation was similar to modern mathematical notation, and used metarules, transformations, and recursion. Pingala (roughly 3rd–1st centuries BC) in his treatise of prosody uses a device corresponding to a binary numeral system. His discussion of the combinatorics of meters corresponds to an elementary version of th... | Roman mathematics | 0.813919 |
16,161 | The earliest civilization on the Indian subcontinent is the Indus Valley civilization (mature second phase: 2600 to 1900 BC) that flourished in the Indus river basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.The oldest extant mathematical r... | Roman mathematics | 0.813919 |
16,162 | Although ethnic Greek mathematicians continued under the rule of the late Roman Republic and subsequent Roman Empire, there were no noteworthy native Latin mathematicians in comparison. Ancient Romans such as Cicero (106–43 BC), an influential Roman statesman who studied mathematics in Greece, believed that Roman surve... | Roman mathematics | 0.813919 |
16,163 | Emmy Noether has been described by many as the most important woman in the history of mathematics. She studied the theories of rings, fields, and algebras. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: by the end of the century there were hundreds of specialized ... | Roman mathematics | 0.813919 |
16,164 | Hence mathematics cannot be reduced to mathematical logic, and David Hilbert's dream of making all of mathematics complete and consistent needed to be reformulated. One of the more colorful figures in 20th-century mathematics was Srinivasa Aiyangar Ramanujan (1887–1920), an Indian autodidact who conjectured or proved o... | Roman mathematics | 0.813919 |
16,165 | In 1931, Kurt Gödel found that this was not the case for the natural numbers plus both addition and multiplication; this system, known as Peano arithmetic, was in fact incompletable. (Peano arithmetic is adequate for a good deal of number theory, including the notion of prime number.) A consequence of Gödel's two incom... | Roman mathematics | 0.813919 |
16,166 | Some of the most important methods and algorithms of the 20th century are: the simplex algorithm, the fast Fourier transform, error-correcting codes, the Kalman filter from control theory and the RSA algorithm of public-key cryptography. At the same time, deep insights were made about the limitations to mathematics. In... | Roman mathematics | 0.813919 |
16,167 | The development and continual improvement of computers, at first mechanical analog machines and then digital electronic machines, allowed industry to deal with larger and larger amounts of data to facilitate mass production and distribution and communication, and new areas of mathematics were developed to deal with thi... | Roman mathematics | 0.813919 |
16,168 | Lie theory with its Lie groups and Lie algebras became one of the major areas of study. Non-standard analysis, introduced by Abraham Robinson, rehabilitated the infinitesimal approach to calculus, which had fallen into disrepute in favour of the theory of limits, by extending the field of real numbers to the Hyperreal ... | Roman mathematics | 0.813919 |
16,169 | Knot theory greatly expanded. Quantum mechanics led to the development of functional analysis. Other new areas include Laurent Schwartz's distribution theory, fixed point theory, singularity theory and René Thom's catastrophe theory, model theory, and Mandelbrot's fractals. | Roman mathematics | 0.813919 |
16,170 | Large advances were made in the qualitative study of dynamical systems that Poincaré had begun in the 1890s. Measure theory was developed in the late 19th and early 20th centuries. Applications of measures include the Lebesgue integral, Kolmogorov's axiomatisation of probability theory, and ergodic theory. | Roman mathematics | 0.813919 |
16,171 | Entirely new areas of mathematics such as mathematical logic, topology, and John von Neumann's game theory changed the kinds of questions that could be answered by mathematical methods. All kinds of structures were abstracted using axioms and given names like metric spaces, topological spaces etc. As mathematicians do,... | Roman mathematics | 0.813919 |
16,172 | The resulting several dozen volumes has had a controversial influence on mathematical education. Differential geometry came into its own when Albert Einstein used it in general relativity. | Roman mathematics | 0.813919 |
16,173 | An example is the classification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages. A group of French mathematicians, including Jean Dieudonné and André Weil, publishing under the... | Roman mathematics | 0.813919 |
16,174 | Notable historical conjectures were finally proven. In 1976, Wolfgang Haken and Kenneth Appel proved the four color theorem, controversial at the time for the use of a computer to do so. Andrew Wiles, building on the work of others, proved Fermat's Last Theorem in 1995. | Roman mathematics | 0.813919 |
16,175 | On the other hand, the limitation of three dimensions in geometry was surpassed in the 19th century through considerations of parameter space and hypercomplex numbers. Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and th... | Roman mathematics | 0.813919 |
16,176 | There are many observable trends in mathematics, the most notable being that the subject is growing ever larger, computers are ever more important and powerful, the application of mathematics to bioinformatics is rapidly expanding, and the volume of data being produced by science and industry, facilitated by computers,... | Roman mathematics | 0.813919 |
16,177 | This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system.Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular numbers, and their reciprocal pairs. The tablets al... | Roman mathematics | 0.813919 |
16,178 | Korean and Japanese mathematics were heavily influenced by the algebraic works produced during China's Song dynasty, whereas Vietnamese mathematics was heavily indebted to popular works of China's Ming dynasty (1368–1644). For instance, although Vietnamese mathematical treatises were written in either Chinese or the na... | Roman mathematics | 0.813919 |
16,179 | Though more of a matter of computational stamina than theoretical insight, in the 5th century AD Zu Chongzhi computed the value of π to seven decimal places (between 3.1415926 and 3.1415927), which remained the most accurate value of π for almost the next 1000 years. He also established a method which would later be ca... | Roman mathematics | 0.813919 |
16,180 | The most important of these is The Nine Chapters on the Mathematical Art, the full title of which appeared by AD 179, but existed in part under other titles beforehand. It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda... | Roman mathematics | 0.813919 |
16,181 | The Mo Jing described various aspects of many fields associated with physical science, and provided a small number of geometrical theorems as well. It also defined the concepts of circumference, diameter, radius, and volume. | Roman mathematics | 0.813919 |
16,182 | The date of the invention of the suan pan is not certain, but the earliest written mention dates from AD 190, in Xu Yue's Supplementary Notes on the Art of Figures. The oldest existent work on geometry in China comes from the philosophical Mohist canon c. 330 BC, compiled by the followers of Mozi (470–390 BC). | Roman mathematics | 0.813919 |
16,183 | Leibniz has been called the "founder of computer science".Science and mathematics had become an international endeavor, which would soon spread over the entire world.In addition to the application of mathematics to the studies of the heavens, applied mathematics began to expand into new areas, with the correspondence o... | Roman mathematics | 0.813919 |
16,184 | Building on earlier work by many predecessors, Isaac Newton discovered the laws of physics that explain Kepler's Laws, and brought together the concepts now known as calculus. Independently, Gottfried Wilhelm Leibniz, developed calculus and much of the calculus notation still in use today. He also refined the binary nu... | Roman mathematics | 0.813919 |
16,185 | Kepler's calculations were made simpler by the contemporaneous invention of logarithms by John Napier and Jost Bürgi. Kepler succeeded in formulating mathematical laws of planetary motion. The analytic geometry developed by René Descartes (1596–1650) allowed those orbits to be plotted on a graph, in Cartesian coordinat... | Roman mathematics | 0.813919 |
16,186 | Medieval European interest in mathematics was driven by concerns quite different from those of modern mathematicians. One driving element was the belief that mathematics provided the key to understanding the created order of nature, frequently justified by Plato's Timaeus and the biblical passage (in the Book of Wisdom... | Roman mathematics | 0.813919 |
16,187 | The analytical solution of the equation, the wave function, is therefore proportional to a product of two infinite power series. The numerical evaluation of the series can be readily performed on a computer. The analytical solutions for the electronic energy eigenvalues are also a generalization of the Lambert W functi... | Hydrogen molecule ion | 0.813918 |
16,188 | The equation can be solved in a relatively straightforward way due to the lack of electron–electron repulsion (electron correlation). The wave equation (a partial differential equation) separates into two coupled ordinary differential equations when using prolate spheroidal coordinates instead of cartesian coordinates. | Hydrogen molecule ion | 0.813918 |
16,189 | The accuracy has steadily improved over more than half a century, eventually resulting in a theoretical framework allowing ultra-high-accuracy predictions for the energies of the rotational and vibrational levels in the electronic ground state, which are mostly metastable. In parallel, the experimental approach to the ... | Hydrogen molecule ion | 0.813918 |
16,190 | The first such solution was derived by Ø. Burrau in 1927, just one year after the wave theory of quantum mechanics was published. The theoretical interest arises because an accurate mathematical solution, taking into account the motion of all constituents, is feasible. | Hydrogen molecule ion | 0.813918 |
16,191 | It is commonly formed in molecular clouds in space by the action of cosmic rays. The dihydrogen cation is of great historical, theoretical, and experimental interest. Historically it is of interest because, having only one electron, the equations of quantum mechanics that describe its structure can be solved approximat... | Hydrogen molecule ion | 0.813918 |
16,192 | In theoretical computer science, a nondeterministic Turing machine (NTM) is a theoretical model of computation whose governing rules specify more than one possible action when in some given situations. That is, an NTM's next state is not completely determined by its action and the current symbol it sees, unlike a deter... | Non-deterministic Turing machine | 0.813915 |
16,193 | Packed columns utilizing filter media for chemical exchange are the most common devices used in the chemical industry for reactant contact optimization. Packed columns are used in a range of industries to allow intimate contact between two immiscible/partly immiscible fluids, which can be liquid/gas or liquid/liquid. T... | Random column packing | 0.813914 |
16,194 | This also parallels the popular (and engineering) usage of stress, but note that some literature defines stress as the response to stressors, especially where usage in biology influences neuroscience. It has been long recognized that psychological stress plays a significant role in understanding how psychopathology dev... | Diathesis stress model | 0.813912 |
16,195 | A graph that can be assigned a k-edge-coloring is said to be k-edge-colorable. The smallest number of colors needed in a (proper) edge coloring of a graph G is the chromatic index, or edge chromatic number, χ′(G). The chromatic index is also sometimes written using the notation χ1(G); in this notation, the subscript on... | K-edge colorable | 0.813907 |
16,196 | A cycle graph may have its edges colored with two colors if the length of the cycle is even: simply alternate the two colors around the cycle. However, if the length is odd, three colors are needed. A complete graph Kn with n vertices is edge-colorable with n − 1 colors when n is an even number; this is a special case ... | K-edge colorable | 0.813907 |
16,197 | David Chalmers has argued that simulated beings might wonder whether their mental lives are governed by the physics of their environment, when in fact these mental lives are simulated separately (and are thus, in fact, not governed by the simulated physics). Chalmers claims that they might eventually find that their th... | Simulation argument | 0.813905 |
16,198 | Economist Robin Hanson argues that a self-interested occupant of a high-fidelity simulation should strive to be entertaining and praiseworthy in order to avoid being turned off or being shunted into a non-conscious low-fidelity part of the simulation. Hanson additionally speculates that someone who is aware that he mig... | Simulation argument | 0.813905 |
16,199 | If many layers of reality exist, each layer contributes some noise = mistakes impossible to be explained exclusively by the physics or the program of the particular examined world. In the infinite series of simulations hypothesis, the first layer is physical and noise builds up to infinity. Weird deviations in the phys... | Simulation argument | 0.813905 |
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