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9,400 | In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.For example, the function f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} has a s... | Mathematical singularities | 0.820258 |
9,401 | Water is an excellent solvent and has two other useful properties: the fact that ice floats enables aquatic organisms to survive beneath it in winter; and its molecules have electrically negative and positive ends, which enables it to form a wider range of compounds than other solvents can. Other good solvents, such as... | Terrestrial life | 0.820251 |
9,402 | Sexual recombination gives good traits the opportunities to become linked with other good traits, and mathematical models suggest this may be more than enough to offset the disadvantages of sexual reproduction. Other combinations of hypotheses that are inadequate on their own are also being examined.The adaptive functi... | Terrestrial life | 0.820251 |
9,403 | The human brain accounts for around 80 billion of these cells.The study of cells and how they work has led to many other studies in related areas of biology, including: discovery of DNA, cancer systems biology, aging and developmental biology. Cell biology is the study of cells, which were discovered by Robert Hooke in... | Cell components | 0.82025 |
9,404 | Here, it makes sense to define 1⁄0 to be infinity; and, in fact, the results are profound and applicable to many problems in engineering and physics. Some prominent mathematicians argued for such a definition long before either number system was developed. Negative zero is another redundant feature of many ways of writ... | 0.999... | 0.820234 |
9,405 | When asked about 0.999..., novices often believe there should be a "final 9", believing 1 − 0.999... to be a positive number which they write as "0.000...1". Whether or not that makes sense, the intuitive goal is clear: adding a 1 to the final 9 in 0.999... would carry all the 9s into 0s and leave a 1 in the ones place... | 0.999... | 0.820234 |
9,406 | So b0.b1b2b3... is defined to be the unique number contained within all the intervals , , and so on. 0.999... is then the unique real number that lies in all of the intervals , , , and for every finite string of 9s. Since 1 is an element of each of these intervals, 0.999... = 1.The Nested Intervals Theorem is usually ... | 0.999... | 0.820234 |
9,407 | Continuing this process yields an infinite sequence of nested intervals, labeled by an infinite sequence of digits b0, b1, b2, b3, ..., and one writes In this formalism, the identities 1 = 0.999... and 1 = 1.000... reflect, respectively, the fact that 1 lies in both and , so one can choose either subinterval when find... | 0.999... | 0.820234 |
9,408 | The result that 0.999... = 1 generalizes readily in two ways. First, every nonzero number with a finite decimal notation (equivalently, endless trailing 0s) has a counterpart with trailing 9s. For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions... | 0.999... | 0.820234 |
9,409 | A corresponding proof of the theorem explicitly computes that sequence; it can be found in any proof-based introduction to calculus or analysis.A sequence (x0, x1, x2, ...) has a limit x if the distance |x − xn| becomes arbitrarily small as n increases. The statement that 0.999... = 1 can itself be interpreted and prov... | 0.999... | 0.820234 |
9,410 | A common development of decimal expansions is to define them as sums of infinite series. In general: For 0.999... one can apply the convergence theorem concerning geometric series: Since 0.999... is such a sum with a = 9 and common ratio r = 1⁄10, the theorem makes short work of the question: This proof appears as earl... | 0.999... | 0.820234 |
9,411 | Under these assumptions, the above argument shows that an order preserving map from the collection of strings to an interval of the real numbers cannot be a bijection: either some numbers do not correspond to any string, or some of them correspond to more than one string. Marko Petkovšek has proven that for any positio... | 0.999... | 0.820234 |
9,412 | Nor are more sophisticated methods foolproof: students who are fully capable of applying rigorous definitions may still fall back on intuitive images when they are surprised by a result in advanced mathematics, including 0.999.... For example, one real analysis student was able to prove that 0.333... = 1⁄3 using a supr... | 0.999... | 0.820234 |
9,413 | In particular "0.333...;...000..." and "0.999...;...000..." do not correspond to any number. The standard definition of the number 0.999... is the limit of the sequence 0.9, 0.99, 0.999, ... A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class of this sequence in the ultr... | 0.999... | 0.820234 |
9,414 | Some proofs that 0.999... = 1 rely on the Archimedean property of the real numbers: that there are no nonzero infinitesimals. Specifically, the difference 1 − 0.999... must be smaller than any positive rational number, so it must be an infinitesimal; but since the reals do not contain nonzero infinitesimals, the differ... | 0.999... | 0.820234 |
9,415 | Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of one or more dig... | 0.999... | 0.820234 |
9,416 | If it can be proved that if a decimal of the form 0.b1b2b3... is a positive integer, then it must be 0.999..., which is then the source of the 9s in the theorem. Investigations in this direction can motivate such concepts as greatest common divisors, modular arithmetic, Fermat primes, order of group elements, and quadr... | 0.999... | 0.820234 |
9,417 | 1⁄73 = 0.01369863 and 0136 + 9863 = 9999.E. Midy proved a general result about such fractions, now called Midy's theorem, in 1836. The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does. | 0.999... | 0.820234 |
9,418 | One application of 0.999... as a representation of 1 occurs in elementary number theory. In 1802, H. Goodwin published an observation on the appearance of 9s in the repeating-decimal representations of fractions whose denominators are certain prime numbers. Examples include: 1⁄7 = 0.142857 and 142 + 857 = 999. | 0.999... | 0.820234 |
9,419 | Many algebraic arguments have been provided, which suggest that 1 = 0.999 … {\displaystyle 1=0.999\ldots } They are not mathematical proofs since they are typically based on the fact that the rules for adding and multiplying finite decimals extend to infinite decimals. This is true, but the proof is essentially the sam... | 0.999... | 0.820234 |
9,420 | A discussion in 2006 took place between UCSB physicists at KITP and science journalist George Johnson regarding the controversy caused by the books of Smolin (The Trouble with Physics) and Woit (Not Even Wrong). The meeting was titled "The String Wars" to reflect the impression the media has given people regarding the ... | The Trouble with Physics | 0.820234 |
9,421 | Hossenfelder believes that Smolin's book attempts to restore the relation physics once had with philosophy, quoting him as follows: Philosophy used to be part of the natural sciences – for a long time. For long centuries during which our understanding of the world we live in has progressed tremendously. There is no dou... | The Trouble with Physics | 0.820234 |
9,422 | Sabine Hossenfelder, in a review written a year later and titled "The Trouble With Physics: Aftermath" alludes to the book's polarising effect on the scientific community. She explores the author's views as a contrast in generations, while supporting his right to them. | The Trouble with Physics | 0.820234 |
9,423 | The book generated much controversy and debate about the merits of string theory, and was criticised by some prominent physicists including Sean Carroll and string theorists Joseph Polchinski and Luboš Motl.Polchinski's review states, "In the end, these books fail to capture much of the spirit and logic of string theo... | The Trouble with Physics | 0.820234 |
9,424 | In the book, Smolin claims that string theory makes no new testable predictions; that it has no coherent mathematical formulation; and that it has not been mathematically proved finite. Some experts in the theoretical physics community disagree with these statements.Smolin states that to propose a string theory landsca... | The Trouble with Physics | 0.820234 |
9,425 | Smolin suggests both that there appear to be serious deficiencies in string theory and that string theory has an unhealthy near-monopoly on fundamental physics in the United States, and that a diversity of approaches is needed. He argues that more attention should instead be paid to background independent theories of q... | The Trouble with Physics | 0.820234 |
9,426 | The Trouble with Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next is a 2006 book by the theoretical physicist Lee Smolin about the problems with string theory. The book strongly criticizes string theory and its prominence in contemporary theoretical physics, on the grounds that string theo... | The Trouble with Physics | 0.820234 |
9,427 | In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation. | Stiffness matrix | 0.820224 |
9,428 | Soil biology plays a vital role in determining many soil characteristics. The decomposition of organic matter by soil organisms has an immense influence on soil fertility, plant growth, soil structure, and carbon storage. As a relatively new science, much remains unknown about soil biology and its effect on soil ecosys... | Soil life | 0.82022 |
9,429 | Soil biology is the study of microbial and faunal activity and ecology in soil. Soil life, soil biota, soil fauna, or edaphon is a collective term that encompasses all organisms that spend a significant portion of their life cycle within a soil profile, or at the soil-litter interface. These organisms include earthworm... | Soil life | 0.82022 |
9,430 | A reversed-field pinch (RFP) is a device used to produce and contain near-thermonuclear plasmas. It is a toroidal pinch which uses a unique magnetic field configuration as a scheme to magnetically confine a plasma, primarily to study magnetic confinement fusion. Its magnetic geometry is somewhat different from that of ... | Reversed field pinch | 0.820217 |
9,431 | The Reversed Field Pinch is also interesting from a physics standpoint. RFP dynamics are highly turbulent. RFPs also exhibit a strong plasma dynamo, similar to many astrophysical bodies. Basic plasma science is another important aspect of Reversed Field Pinch research. | Reversed field pinch | 0.820217 |
9,432 | A rational curve, also called a unicursal curve, is any curve which is birationally equivalent to a line, which we may take to be a projective line; accordingly, we may identify the function field of the curve with the field of rational functions in one indeterminate F(x). If F is algebraically closed, this is equivale... | Sextic plane curve | 0.820214 |
9,433 | In this section, we consider a plane algebraic curve defined by a bivariate polynomial p(x, y) and its projective completion, defined by the homogenization P ( x , y , z ) = h p ( x , y , z ) {\displaystyle P(x,y,z)={}^{h}p(x,y,z)} of p. | Sextic plane curve | 0.820214 |
9,434 | In this sense, the one-to-one correspondence between irreducible algebraic curves over F (up to birational equivalence) and algebraic function fields in one variable over F holds in general. Two curves can be birationally equivalent (i.e. have isomorphic function fields) without being isomorphic as curves. The situatio... | Sextic plane curve | 0.820214 |
9,435 | If the field F is not algebraically closed, the point of view of function fields is a little more general than that of considering the locus of points, since we include, for instance, "curves" with no points on them. For example, if the base field F is the field R of real numbers, then x2 + y2 = −1 defines an algebraic... | Sextic plane curve | 0.820214 |
9,436 | The study of algebraic curves can be reduced to the study of irreducible algebraic curves: those curves that cannot be written as the union of two smaller curves. Up to birational equivalence, the irreducible curves over a field F are categorically equivalent to algebraic function fields in one variable over F. Such an... | Sextic plane curve | 0.820214 |
9,437 | More generally, if p is a prime and n, m are positive integers with n dividing m, then = m/n. The field extension C(T)/C, where C(T) is the field of rational functions over C, has infinite degree (indeed it is a purely transcendental extension). This can be seen by observing that the elements 1, T, T2, etc., are linea... | Finite field extension | 0.820214 |
9,438 | Physical activity has a significant influence on the hippocampus, since this is the part of the brain that is responsible for encoding information into memory. With physical activity having such an impact on the hippocampus this can regulate other parts of the body as well like weight, memory, daily function, and many ... | Cued recall | 0.820212 |
9,439 | One lone pair is used as a sigma X donor, the other two lone pairs are available as L-type pi donors. If both lone pairs are used in pi bonds then the M−N−R geometry is linear. However, if one or both these lone pairs is nonbonding then the M−N−R bond is bent and the extent of the bend speaks to how much pi bonding the... | Ligand exchange | 0.820201 |
9,440 | In von Neumann–Bernays–Gödel set theory, a distinction is made between sets and classes. A class C is a set if and only if it belongs to some class E. In this theory, there is a theorem schema that reads that is, provided that the quantifiers in the predicate P are restricted to sets. This theorem schema is itself a re... | Unrestricted comprehension | 0.820193 |
9,441 | The Virtual breakdown mechanism is a concept in the field of electrochemistry. In electrochemical reactions, when the cathode and the anode are close enough to each other (i.e., so-called "nanogap electrochemical cells"), the double layer of the regions from the two electrodes is overlapped, forming a large electric fi... | Virtual breakdown mechanism | 0.820192 |
9,442 | While DNA encodes all of the functional elements of life, the information encoded must be converted into functional form. Following the central dogma of molecular biology, messenger RNA encodes genetic information for producing proteins, which, alongside functional RNA carry out the majority of cellular processes requi... | Time-resolved RNA sequencing | 0.820186 |
9,443 | As with strengths, this method shares many weaknesses with affinity purification methods. Notably, 4-sU uptake and increased sample handling. Since Timelapse-seq relies upon synthetic chemistry methods to convert nucleotides, incomplete reactions result in an underestimation of the abundance of newly synthesized RNA an... | Time-resolved RNA sequencing | 0.820186 |
9,444 | Since NET-seq extracts transcripts that have not completed transcription, it is possible to obtain single-nucleotide resolution on the most recently synthesized nucleotide of transcripts. This is valuable in the study of phenomena such as transcriptional kinetics. Furthermore, it allows for the study of unstable transc... | Time-resolved RNA sequencing | 0.820186 |
9,445 | However, starting with version 3.0 (including its preview version, Keras Core), Keras will become multi-backend again, supporting TensorFlow, JAX, and PyTorch. Designed to enable fast experimentation with deep neural networks, it focuses on being user-friendly, modular, and extensible. It was developed as part of the r... | Keras | 0.820186 |
9,446 | Keras is an open-source library that provides a Python interface for artificial neural networks. Keras acts as an interface for the TensorFlow library. Up until version 2.3, Keras supported multiple backends, including TensorFlow, Microsoft Cognitive Toolkit, Theano, and PlaidML. As of version 2.4, only TensorFlow is s... | Keras | 0.820186 |
9,447 | Keras contains numerous implementations of commonly used neural-network building blocks such as layers, objectives, activation functions, optimizers, and a host of tools for working with image and text data to simplify programming in deep neural network area. The code is hosted on GitHub, and community support forums i... | Keras | 0.820186 |
9,448 | That single center of attraction along with a roughly equal single repeller appear to be the most significant contributors to the CMB dipole. The authors of the article published in Nature Astronomy in January 2017 argue that the distance velocity measurements of the Dipole Repeller are incompatible with an explanation... | Dipole repeller | 0.820182 |
9,449 | GBB1, one of GBB's control shells implements BB1's style of control while adding efficiency improvements. Other well-known of early academic blackboard systems are the Hearsay II speech recognition system and Douglas Hofstadter's Copycat and Numbo projects. Some more recent examples of deployed real-world applications ... | Blackboard system | 0.820179 |
9,450 | Blackboard-like systems have been constructed within modern Bayesian machine learning settings, using agents to add and remove Bayesian network nodes. In these 'Bayesian Blackboard' systems, the heuristics can acquire more rigorous probabilistic meanings as proposal and acceptances in Metropolis Hastings sampling thoug... | Blackboard system | 0.820179 |
9,451 | A blackboard system is an artificial intelligence approach based on the blackboard architectural model, where a common knowledge base, the "blackboard", is iteratively updated by a diverse group of specialist knowledge sources, starting with a problem specification and ending with a solution. Each knowledge source upda... | Blackboard system | 0.820179 |
9,452 | In 1885, Michelson began a collaboration with Edward Morley, spending considerable time and money to confirm with higher accuracy Fizeau's 1851 experiment on Fresnel's drag coefficient, to improve on Michelson's 1881 experiment, and to establish the wavelength of light as a standard of length. At this time Michelson wa... | Michelson–Morley experiment | 0.82017 |
9,453 | Designing experiments to investigate these properties was a high priority of 19th-century physics. : 411ff Earth orbits around the Sun at a speed of around 30 km/s (18.64 mi/s), or 108,000 km/h (67,000 mph). The Earth is in motion, so two main possibilities were considered: (1) The aether is stationary and only partial... | Michelson–Morley experiment | 0.82017 |
9,454 | Physics theories of the 19th century assumed that just as surface water waves must have a supporting substance, i.e., a "medium", to move across (in this case water), and audible sound requires a medium to transmit its wave motions (such as air or water), so light must also require a medium, the "luminiferous aether", ... | Michelson–Morley experiment | 0.82017 |
9,455 | A gravity turn or zero-lift turn is a maneuver used in launching a spacecraft into, or descending from, an orbit around a celestial body such as a planet or a moon. It is a trajectory optimization that uses gravity to steer the vehicle onto its desired trajectory. It offers two main advantages over a trajectory control... | Gravity turn | 0.820164 |
9,456 | In the context of atomic physics, nondimensionalization using the defining constants of the Hartree atomic system can be a convenient shortcut, since it can be thought of as eliminating these constants wherever they occur. Nondimesionalization involves a substitution of variables that results in equations in which thes... | Atomic units system | 0.82016 |
9,457 | The Hartree atomic units are a system of natural units of measurement which is especially convenient for calculations in atomic physics and related scientific fields, such as computational chemistry and atomic spectroscopy. They are named after the physicist Douglas Hartree. Atomic units are often abbreviated "a.u." or... | Atomic units system | 0.82016 |
9,458 | In the late 2000's, scale-free and small-world networks began shaping the emergence of systems biology, network biology, and network medicine. In 2014, graph theoretical methods were used by Frank Emmert-Streib to analyze biological networks. In the 1980s, researchers started viewing DNA or genomes as the dynamic stora... | Biological network | 0.820159 |
9,459 | As early as 1736 Leonhard Euler analyzed a real-world issue known as the Seven Bridges of Königsberg, which established the foundation of graph theory. From the 1930's-1950's the study of random graphs were developed. During the mid 1990's, it was discovered that many different types of "real" networks have structural ... | Biological network | 0.820159 |
9,460 | All organisms are connected through feeding interactions. If a species eats or is eaten by another species, they are connected in an intricate food web of predator and prey interactions. The stability of these interactions has been a long-standing question in ecology. That is to say if certain individuals are removed, ... | Biological network | 0.820159 |
9,461 | This greatly improves the speed of merging nodes. Another optimization is in the refinement phase in-which the algorithm randomly chooses for a node from a set of communities to merge with. This allows for greater depth in choosing communities as Louvain solely focuses on maximizing the modularity that was chosen. The ... | Biological network | 0.820159 |
9,462 | Studying the community structure of a network by subdividing groups of nodes into like-regions can be an integral tool for bioinformatics when exploring data as a network. A food web of The Secaucus High School Marsh exemplifies the benefits of grouping as the relationships between nodes are far easier to analyze with ... | Biological network | 0.820159 |
9,463 | Protein-protein interaction networks (PINs) represent the physical relationship among proteins present in a cell, where proteins are nodes, and their interactions are undirected edges. Due to their undirected nature, it is difficult to identify all the proteins involved in an interaction. Protein–protein interactions (... | Biological network | 0.820159 |
9,464 | Changes in an individual's social network environment can also influence characteristics such as 'personality': for example, social spiders that huddle with bolder neighbors tend to increase also in boldness. This is a very small set of broad examples of how researchers can use network analysis to study animal behavior... | Biological network | 0.820159 |
9,465 | Network analysis provides the ability to quantify associations between individuals, which makes it possible to infer details about the network as a whole at the species and/or population level. One of the most attractive features of the network paradigm would be that it provides a single conceptual framework in which t... | Biological network | 0.820159 |
9,466 | LINPACK 100 is very similar to the original benchmark published in 1979 along with the LINPACK users' manual. The solution is obtained by Gaussian elimination with partial pivoting, with 2/3n³ + 2n² floating-point operations where n is 100, the order of the dense matrix A that defines the problem. Its small size and th... | LINPACK benchmark | 0.820158 |
9,467 | We say that a variety V of loops satisfies the Moufang theorem if for every loop Q in V the following implication holds: for every x, y, z in Q, if x(yz) = (xy)z then the subloop generated by x, y, z is a group. Is every variety that satisfies Moufang theorem contained in the variety of Moufang loops? Proposed by: Andr... | List of problems in loop theory and quasigroup theory | 0.820154 |
9,468 | Conjecture: Let M be a finite Moufang loop of exponent n with m generators. Then there exists a function f(n,m) such that |M| < f(n,m). Proposed: by Alexander Grishkov at Loops '11, Třešť 2011 Comments: In the case when n is a prime different from 3 the conjecture was proved by Grishkov. If p = 3 and M is commutative, ... | List of problems in loop theory and quasigroup theory | 0.820154 |
9,469 | Conjecture: Any finite commutative Moufang loop of period 3 can be embedded into a commutative alternative algebra. Proposed: by Alexander Grishkov at Loops '03, Prague 2003 | List of problems in loop theory and quasigroup theory | 0.820154 |
9,470 | Classify nonassociative Moufang loops of order 64. Proposed: at Milehigh conference on quasigroups, loops, and nonassociative systems, Denver 2005 Solved by: Gábor P. Nagy and Petr Vojtěchovský Solution: There are 4262 nonassociative Moufang loops of order 64. They were found by the method of group modifications in (Vo... | List of problems in loop theory and quasigroup theory | 0.820154 |
9,471 | Furthermore, the class of cores of groups is not closed under subalgebras. Here is an outline of the proof: Cores of abelian groups are medial, by (Romanowska and Smith, 1985), (Rozskowska-Lech, 1999). The smallest nonabelian group S 3 {\displaystyle S_{3}} has core containing a submagma G {\displaystyle G} of order 4 ... | List of problems in loop theory and quasigroup theory | 0.820154 |
9,472 | Is the class of cores of Bol loops a quasivariety? Proposed: by Jonathan D. H. Smith and Alena Vanžurová at Loops '03, Prague 2003 Solved by: Alena Vanžurová, 2004. Solution: No, the class of cores of Bol loops is not closed under subalgebras. | List of problems in loop theory and quasigroup theory | 0.820154 |
9,473 | In mathematics, especially abstract algebra, loop theory and quasigroup theory are active research areas with many open problems. As in other areas of mathematics, such problems are often made public at professional conferences and meetings. Many of the problems posed here first appeared in the Loops (Prague) conferenc... | List of problems in loop theory and quasigroup theory | 0.820154 |
9,474 | In computer science, hardness of approximation is a field that studies the algorithmic complexity of finding near-optimal solutions to optimization problems. | Hardness of approximation | 0.820154 |
9,475 | For an example of an NP-hard optimization problem that is hard to approximate, see set cover. | Hardness of approximation | 0.820154 |
9,476 | Since the early 1970s it was known that many optimization problems could not be solved in polynomial time unless P = NP, but in many of these problems the optimal solution could be efficiently approximated to a certain degree. In the 1970s, Teofilo F. Gonzalez and Sartaj Sahni began the study of hardness of approximati... | Hardness of approximation | 0.820154 |
9,477 | In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. | Arithmetic scheme | 0.820123 |
9,478 | At present, single entity electrochemistry is not sensitive enough to quantify the turnover of a single enzyme. == References == | Single-entity electrochemistry | 0.820113 |
9,479 | Single-Molecule electrochemistry is an electrochemical technique used to study the faradaic response of redox molecules in electrochemical environments. The ability to study singular molecules gives rise to the potential of developing ultra-sensitive sensors which are necessary in SEE. From the work of Bard and Fan, th... | Single-entity electrochemistry | 0.820113 |
9,480 | The ability to monitor the movement of one electron or ion from one unit to another is valuable, as many vital reactions and mechanisms undergo this process. Electrochemistry is well suited for this measurement due to its incredible sensitivity. Single-Entity Electrochemistry can be used to investigate nanoparticles, w... | Single-entity electrochemistry | 0.820113 |
9,481 | Single-Entity Electrochemistry (SEE) refers to the electroanalysis of an individual unit of interest. A unique feature of SEE is that it unifies multiple different branches of electrochemistry. Single-Entity Electrochemistry pushes the bounds of the field as it can measure entities on a scale of 100 microns to angstrom... | Single-entity electrochemistry | 0.820113 |
9,482 | Plasmonic nanoparticles can be individually analyzed through optoelectrochemical imaging (in which electrochemical processes are measured by optical means). When electrochemistry is performed on a nanoparticle, the refractive index of its environment will change resulting in a shift of the localized surface plasmon res... | Single-entity electrochemistry | 0.820113 |
9,483 | In the following we will calculate the internal energy of the box at absolute temperature T. According to statistical mechanics, the equilibrium probability distribution over the energy levels of a particular mode is given by:where we use the reciprocal temperatureThe denominator Z(β), is the partition function of a si... | Planck's Law | 0.82011 |
9,484 | This has at times been called Planck's "second theory".It was not till 1919 that Planck in the third edition of his monograph more or less accepted his 'third theory', that both emission and absorption of light were quantal.The colourful term "ultraviolet catastrophe" was given by Paul Ehrenfest in 1911 to the paradoxi... | Planck's Law | 0.82011 |
9,485 | Heisenberg's explanation of the Planck oscillators, as non-linear effects apparent as Fourier modes of transient processes of emission or absorption of radiation, showed why Planck's oscillators, viewed as enduring physical objects such as might be envisaged by classical physics, did not give an adequate explanation of... | Planck's Law | 0.82011 |
9,486 | The frequency of a quantum of radiation was that of a definite coupling between internal atomic meta-stable oscillatory quantum states. At that time, Heisenberg knew nothing of matrix algebra, but Max Born read the manuscript of Heisenberg's paper and recognized the matrix character of Heisenberg's theory. Then Born an... | Planck's Law | 0.82011 |
9,487 | The above-mentioned linearity of Planck's mechanical assumptions, not allowing for energetic interactions between frequency components, was superseded in 1925 by Heisenberg's original quantum mechanics. In his paper submitted on 29 July 1925, Heisenberg's theory accounted for Bohr's above-mentioned formula of 1913. It ... | Planck's Law | 0.82011 |
9,488 | Lewis in 1926, who mistakenly believed that photons were conserved, contrary to Bose–Einstein statistics; nevertheless the word 'photon' was adopted to express the Einstein postulate of the packet nature of light propagation. In an electromagnetic field isolated in a vacuum in a vessel with perfectly reflective walls, ... | Planck's Law | 0.82011 |
9,489 | In 1910, criticizing a manuscript sent to him by Planck, knowing that Planck was a steady supporter of Einstein's theory of special relativity, Einstein wrote to Planck: "To me it seems absurd to have energy continuously distributed in space without assuming an aether. "According to Thomas Kuhn, it was not till 1908 th... | Planck's Law | 0.82011 |
9,490 | Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift.Non-Euclidean geometry is an example of a scientific revolution in the history of science, in which mathematicians and scientists changed... | Non-Euclidean geometries | 0.82011 |
9,491 | The philosopher Immanuel Kant's treatment of human knowledge had a special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterab... | Non-Euclidean geometries | 0.82011 |
9,492 | Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the mathematical model of space. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute... | Non-Euclidean geometries | 0.82011 |
9,493 | The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry. Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry parabolic, a term that generally fell out of use). His influence has led to the current... | Non-Euclidean geometries | 0.82011 |
9,494 | It was Gauss who coined the term "non-Euclidean geometry". He was referring to his own work, which today we call hyperbolic geometry or Lobachevskian geometry. Several modern authors still use the generic term non-Euclidean geometry to mean hyperbolic geometry.Arthur Cayley noted that distance between points inside a c... | Non-Euclidean geometries | 0.82011 |
9,495 | There are many discrete entities that are easily detected but difficult to count. Any sort of amplification reaction or catalysis reaction obliterates easy quantification but allows presence to be detected very sensitively. Common examples include microorganism growth, enzyme action, or catalytic chemistry. The MPN met... | Most probable number | 0.820099 |
9,496 | In microbiology, the cultures are incubated and assessed by eye, bypassing tedious colony counting or expensive and tedious microscopic counts. Presumptive, confirmative and completed tests are a part of MPN.In molecular biology, a common application involves DNA templates diluted into polymerase chain reactions (PCR).... | Most probable number | 0.820099 |
9,497 | There are a variety of career paths within the field of medical genetics, and naturally the training required for each area differs considerably. The information included in this section applies to the typical pathways in the United States and there may be differences in other countries. US practitioners in clinical, c... | Clinical genetics | 0.820095 |
9,498 | Basic research geneticists usually undertake research in universities, biotechnology firms and research institutes. | Clinical genetics | 0.820095 |
9,499 | Population genetics was created. Medical genetics was a late developer, emerging largely after the close of World War II (1945) when the eugenics movement had fallen into disrepute. The Nazi misuse of eugenics sounded its death knell. Shorn of eugenics, a scientific approach could be used and was applied to human and m... | Clinical genetics | 0.820095 |
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