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11,100 | Brian Milch, and Stuart J. Russell: First-Order Probabilistic Languages: Into the Unknown, Inductive Logic Programming, volume 4455 of Lecture Notes in Computer Science, page 10–24. Springer, 2006 Rodrigo de Salvo Braz, Eyal Amir, and Dan Roth: A Survey of First-Order Probabilistic Models, Innovations in Bayesian Netwo... | Statistical relational learning | 0.818281 |
11,101 | Depending on the properties on which the emphasis is given, there are several non equivalent such formalizations, that are all called surface, sometimes with some qualifier, such as algebraic surface, smooth surface or fractal surface. The concept of surface and its mathematical abstraction are both widely used in phys... | Curved surface | 0.818279 |
11,102 | A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is the portion with which other materials first interact. The surface of an obje... | Curved surface | 0.818279 |
11,103 | Other surfaces considered in physics are wavefronts. One of these, discovered by Fresnel, is called wave surface by mathematicians. The surface of the reflector of a telescope is a paraboloid of revolution. Other occurrences: Soap bubbles, which are physical examples of minimal surfaces Equipotential surface in, e.g., ... | Curved surface | 0.818279 |
11,104 | Many surfaces considered in physics and chemistry (physical sciences in general) are interfaces. For example, a surface may be the idealized limit between two fluids, liquid and gas (the surface of the sea in air) or the idealized boundary of a solid (the surface of a ball). In fluid dynamics, the shape of a free surfa... | Curved surface | 0.818279 |
11,105 | Input Gene Regulatory Networks can be drawn by hand. Networks can be built using lists of interactions entered via dialog boxes. Lists of interactions can be input using comma-separated-value (CSV) files. Networks can be built using SIF files as input. BioTapestry can accept network definitions via the Gaggle framework... | BioTapestry | 0.818279 |
11,106 | BioTapestry was created at the Institute of Systems Biology in Seattle, in collaboration with the Davidson Lab at the California Institute of Technology. The project was initiated to support the ongoing development of the model of the GRN regulating the development of the endomesoderm in the sea urchin Strongylocentrot... | BioTapestry | 0.818279 |
11,107 | For chiral objects it is the same as the full symmetry group. Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of Noether's theorem, the rotational symmetry of a physical system is equivalent to the angular momentum conservation law. | Rotation symmetry | 0.818273 |
11,108 | Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are p... | Rotation symmetry | 0.818273 |
11,109 | The transfer principle however doesn't mean that R and *R have identical behavior. For instance, in *R there exists an element ω such that 1 < ω , 1 + 1 < ω , 1 + 1 + 1 < ω , 1 + 1 + 1 + 1 < ω , … {\displaystyle 1<\omega ,\quad 1+1<\omega ,\quad 1+1+1<\omega ,\quad 1+1+1+1<\omega ,\ldots } but there is no such number i... | Transfer principles | 0.81827 |
11,110 | In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first-order language of fields that is true for the complex numbers is also true... | Transfer principles | 0.81827 |
11,111 | Modular notebooks may connect to a variety of computational back ends, called "kernels". Notebook interfaces are widely used for statistics, data science, machine learning, and computer algebra.At the notebook core is the idea of literate programming tools which "let you arrange the parts of a program in any order and ... | Notebook interface | 0.818267 |
11,112 | In physics, interaction energy is the contribution to the total energy that is caused by an interaction between the objects being considered. The interaction energy usually depends on the relative position of the objects. For example, Q 1 Q 2 / ( 4 π ε 0 Δ r ) {\displaystyle Q_{1}Q_{2}/(4\pi \varepsilon _{0}\Delta r)} ... | Interaction energy | 0.818266 |
11,113 | Entry-level microbiology jobs generally require at least a bachelor's degree in microbiology or a related field. These degree programs frequently include courses in chemistry, physics, statistics, biochemistry, and genetics, followed by more specialized courses in sub-fields of interest. Many of these courses have labo... | Microbiologist | 0.818264 |
11,114 | Microbiologists will continue to be needed to advance basic science knowledge and to contribute to development of pharmaceuticals and biotechnology products. However, job prospects vary widely by job and location.In the United States, the Bureau of Labor Statistics predicts that employment of microbiologists will grow ... | Microbiologist | 0.818264 |
11,115 | Some microbiologists work in the field of patent law, either with national patent offices or private law practices. Her duties include research and navigation of intellectual property regulations. Clinical microbiologists tend to work in government or hospital laboratories where their duties include analyzing clinical ... | Microbiologist | 0.818264 |
11,116 | A microbiologist (from Greek μῑκρος) is a scientist who studies microscopic life forms and processes. This includes study of the growth, interactions and characteristics of microscopic organisms such as bacteria, algae, fungi, and some types of parasites and their vectors. Most microbiologists work in offices and/or re... | Microbiologist | 0.818264 |
11,117 | David N. Yetter proposed a weaker structural rule in place of the exchange rule of linear logic, yielding cyclic linear logic. Sequents of cyclic linear logic form a ring, and so are invariant under rotation, where multipremise rules glue their rings together at the formulae described in the rules. The calculus support... | Noncommutative logic | 0.818258 |
11,118 | By extension, the term noncommutative logic is also used by a number of authors to refer to a family of substructural logics in which the exchange rule is inadmissible. The remainder of this article is devoted to a presentation of this acceptance of the term. The oldest noncommutative logic is the Lambek calculus, whic... | Noncommutative logic | 0.818258 |
11,119 | Alessio Guglielmi proposed a variation of Retoré's calculus, BV, in which the two noncommutative operations are collapsed onto a single, self-dual, operator, and proposed a novel proof calculus, the calculus of structures to accommodate the calculus. The principal novelty of the calculus of structures was its pervasive... | Noncommutative logic | 0.818258 |
11,120 | Structads are an approach to the semantics of logic that are based upon generalising the notion of sequent along the lines of Joyal's combinatorial species, allowing the treatment of more drastically nonstandard logics than those described above, where, for example, the ',' of the sequent calculus is not associative. | Noncommutative logic | 0.818258 |
11,121 | Pomset logic was proposed by Christian Retoré in a semantic formalism with two dual sequential operators existing together with the usual tensor product and par operators of linear logic, the first logic proposed to have both commutative and noncommutative operators. A sequent calculus for the logic was given, but it l... | Noncommutative logic | 0.818258 |
11,122 | Noncommutative logic is an extension of linear logic that combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties (a family of cyclic orders that may be viewed as a species of structur... | Noncommutative logic | 0.818258 |
11,123 | Joachim Lambek proposed the first noncommutative logic in his 1958 paper Mathematics of Sentence Structure to model the combinatory possibilities of the syntax of natural languages. His calculus has thus become one of the fundamental formalisms of computational linguistics. | Noncommutative logic | 0.818258 |
11,124 | Civil structural engineering includes all structural engineering related to the built environment. It includes: The structural engineer is the lead designer on these structures, and often the sole designer. In the design of structures such as these, structural safety is of paramount importance (in the UK, designs for d... | Engineering structure | 0.818253 |
11,125 | Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of human-made structures. Structural engineers also must understand and calculate the stability, strength, rigidity and earthquake-susceptibility of ... | Engineering structure | 0.818253 |
11,126 | Therefore, the structural strength of the pyramid stems from the material properties of the stones from which it was built rather than the pyramid's geometry. Throughout ancient and medieval history most architectural design and construction were carried out by artisans, such as stonemasons and carpenters, rising to th... | Engineering structure | 0.818253 |
11,127 | Results from the theory of schemata suggest that in general the smaller the alphabet, the better the performance, but it was initially surprising to researchers that good results were obtained from using real-valued chromosomes. This was explained as the set of real values in a finite population of chromosomes as formi... | Speciation (genetic algorithm) | 0.818246 |
11,128 | Although reproduction methods that are based on the use of two parents are more "biology inspired", some research suggests that more than two "parents" generate higher quality chromosomes. These processes ultimately result in the next generation population of chromosomes that is different from the initial generation. G... | Speciation (genetic algorithm) | 0.818246 |
11,129 | They use self-adaptation to adjust parameters, and can include other variation operations such as combining information from multiple parents. Estimation of Distribution Algorithm (EDA) substitutes traditional reproduction operators by model-guided operators. Such models are learned from the population by employing mac... | Speciation (genetic algorithm) | 0.818246 |
11,130 | Recent approaches use more abstract variables for deciding pc and pm. Examples are dominance & co-dominance principles and LIGA (levelized interpolative genetic algorithm), which combines a flexible GA with modified A* search to tackle search space anisotropicity.It can be quite effective to combine GA with other optim... | Speciation (genetic algorithm) | 0.818246 |
11,131 | There are more examples of AGA variants: Successive zooming method is an early example of improving convergence. In CAGA (clustering-based adaptive genetic algorithm), through the use of clustering analysis to judge the optimization states of the population, the adjustment of pc and pm depends on these optimization sta... | Speciation (genetic algorithm) | 0.818246 |
11,132 | Swarm intelligence is a sub-field of evolutionary computing. Ant colony optimization (ACO) uses many ants (or agents) equipped with a pheromone model to traverse the solution space and find locally productive areas. Although considered an Estimation of distribution algorithm, Particle swarm optimization (PSO) is a comp... | Speciation (genetic algorithm) | 0.818246 |
11,133 | "Gravitomagnetic Barnett Effect". Indian Journal of Physics B. 75: 459–461. | Gravitoelectromagnetism | 0.818244 |
11,134 | Annals of Physics. 215 (1): 1–50. arXiv:gr-qc/0106043. | Gravitoelectromagnetism | 0.818244 |
11,135 | Forward (1963). "Guidelines to Antigravity". American Journal of Physics. | Gravitoelectromagnetism | 0.818244 |
11,136 | This allows a test of the genetic overlap between different phenotypes: for instance hair color and eye color. Environment and genetics may also interact, and heritability analyses can test for and examine these interactions (GxE models). A prerequisite for heritability analyses is that there is some population variati... | Heritability | 0.818235 |
11,137 | Individuals with the same genotype can also exhibit different phenotypes through a mechanism called phenotypic plasticity, which makes heritability difficult to measure in some cases. Recent insights in molecular biology have identified changes in transcriptional activity of individual genes associated with environment... | Heritability | 0.818235 |
11,138 | Heritability increases when genetics are contributing more variation or because non-genetic factors are contributing less variation; what matters is the relative contribution. Heritability is specific to a particular population in a particular environment. High heritability of a trait, consequently, does not necessaril... | Heritability | 0.818235 |
11,139 | Heritability measures the fraction of phenotype variability that can be attributed to genetic variation. This is not the same as saying that this fraction of an individual phenotype is caused by genetics. For example, it is incorrect to say that since the heritability of personality traits is about 0.6, that means that... | Heritability | 0.818235 |
11,140 | Such experimental control is generally not possible when gathering human data, relying on naturally occurring relationships and environments. In classical quantitative genetics, there were two schools of thought regarding estimation of heritability. One school of thought was developed by Sewall Wright at The University... | Heritability | 0.818235 |
11,141 | When genome-wide genotype data and phenotypes from large population samples are available, one can estimate the relationships between individuals based on their genotypes and use a linear mixed model to estimate the variance explained by the genetic markers. This gives a genomic heritability estimate based on the varia... | Heritability | 0.818235 |
11,142 | Heritability is estimated by comparing individual phenotypic variation among related individuals in a population, by examining the association between individual phenotype and genotype data, or even by modeling summary-level data from genome-wide association studies (GWAS). Heritability is an important concept in quant... | Heritability | 0.818235 |
11,143 | Heritability is a statistic used in the fields of breeding and genetics that estimates the degree of variation in a phenotypic trait in a population that is due to genetic variation between individuals in that population. The concept of heritability can be expressed in the form of the following question: "What is the p... | Heritability | 0.818235 |
11,144 | Heritability estimates' prominent critics, such as Steven Rose, Jay Joseph, and Richard Bentall, focus largely on heritability estimates in behavioral sciences and social sciences. Bentall has claimed that such heritability scores are typically calculated counterintuitively to derive numerically high scores, that herit... | Heritability | 0.818235 |
11,145 | The Oxford English Dictionary records the use of the word injection as a noun by S. Mac Lane in Bulletin of the American Mathematical Society (1950), and injective as an adjective by Eilenberg and Steenrod in Foundations of Algebraic Topology (1952). However, it was not until the French Bourbaki group coined the inject... | Bijection, injection and surjection | 0.818233 |
11,146 | Pinning force is a force acting on a pinned object from a pinning center. In solid state physics, this most often refers to the vortex pinning, the pinning of the magnetic vortices (magnetic flux quanta, Abrikosov vortices) by different kinds of the defects in a type II superconductor. Important quantities are the indi... | Pinning force | 0.818227 |
11,147 | The mathēmatikoi acknowledged the religious underpinning of Pythagoreanism and engaged in mathēma (translated as "learning" or "studying") as part of their practice. While their scientific pursuits were largely mathematical, they also promoted other fields of scientific study in which Pythagoras had engaged during his ... | Pythagorean mathematics | 0.818226 |
11,148 | Ortho-Planar Mechanisms are an earlier concept similar to LEMs. More well known LEMs include pop-up books, flat-folding origami mechanisms, origami stents, and deployable mechanisms. The research in LEMs also overlaps with deployable structures, origami, kirigami, compliant mechanisms, microelectromechanical systems, p... | Lamina emergent mechanisms | 0.818226 |
11,149 | Pythagoras was already well known in ancient times for the mathematical achievement of the Pythagorean theorem. Pythagoras had been credited with discovering that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. In ancient times Pythagoras was also noted... | Pythagorean mathematics | 0.818226 |
11,150 | (In biochemistry, a residue refers to a specific monomer within the polymeric chain of a polysaccharide, protein or nucleic acid.) The integral membrane proteins tend to have outer rings of exposed hydrophobic amino acids that anchor them in the lipid bilayer. Some peripheral membrane proteins have a patch of hydrophob... | Hydrophobic residues | 0.818223 |
11,151 | Such a proof eluded mathematicians for centuries, however, and as such his statement became famous as Fermat's Last Theorem. It was not until 1995 that it was proven by the British mathematician Andrew Wiles. In 1657, Fermat attempted to solve the Diophantine equation 61x2 + 1 = y2 (solved by Brahmagupta over 1000 year... | Diophantine problem | 0.81822 |
11,152 | Solving a homogeneous Diophantine equation is generally a very difficult problem, even in the simplest non-trivial case of three indeterminates (in the case of two indeterminates the problem is equivalent with testing if a rational number is the dth power of another rational number). A witness of the difficulty of the ... | Diophantine problem | 0.81822 |
11,153 | A homogeneous Diophantine equation is a Diophantine equation that is defined by a homogeneous polynomial. A typical such equation is the equation of Fermat's Last Theorem x d + y d − z d = 0. {\displaystyle x^{d}+y^{d}-z^{d}=0.} As a homogeneous polynomial in n indeterminates defines a hypersurface in the projective sp... | Diophantine problem | 0.81822 |
11,154 | By the 1930s his table was appearing in handbooks and encyclopedias of chemistry. It was also distributed for many years by the Sargent-Welch Scientific Company.The advantages of the medium form are that it correlates the positions of the elements with their electronic structures, accommodates the vertical, horizontal ... | Alternative periodic tables | 0.818216 |
11,155 | Medium tables have around 18 columns. The popularity of this form is thought to be a result of it having a good balance of features in terms of ease of construction and size, and its depiction of atomic order and periodic trends.Deming's version of a medium table, which appeared in the first edition of his 1923 textboo... | Alternative periodic tables | 0.818216 |
11,156 | Selected area (electron) diffraction (abbreviated as SAD or SAED) is a crystallographic experimental technique typically performed using a transmission electron microscope (TEM). It is a specific case of electron diffraction used primarily in material science and solid state physics as one of the most common experiment... | Selected area diffraction | 0.818213 |
11,157 | SAD analysis is widely used in material research for its relative simplicity and high information value. Once the sample is prepared and examined in a modern transmission electron microscope, the device allows for a routine diffraction acquisition in a matter of seconds. If the images are interpreted correctly, they ca... | Selected area diffraction | 0.818213 |
11,158 | Inclinometers are also used in civil engineering, for example, to measure the inclination of land to be built upon. Some inclinometers provide an electronic interface based on CAN (Controller Area Network). In addition, those inclinometers may support the standardized CANopen profile (CiA 410). In this case, these incl... | Inclinometer | 0.818211 |
11,159 | The tilt angle range is the range of desired linear output. Common implementations of tilt sensors and inclinometers are accelerometer, Liquid Capacitive, electrolytic, gas bubble in liquid, and pendulum. Tilt sensor technology has also been implemented in video games. | Inclinometer | 0.818211 |
11,160 | Nintendo used tilt sensor technology in five games for its Game Boy series of hand-held game systems. The tilt sensor allows players to control aspects of the game by twisting the game system. Games that use this feature: Yoshi's Universal Gravitation (Game Boy Advance) WarioWare: Twisted! (Game Boy Advance)(not releas... | Inclinometer | 0.818211 |
11,161 | Measuring the orientation of planes and lineations in rocks, in combination with a compass, in structural geology. Measuring range of motion in the joints of the body Measuring the inclination angle of the pelvis. Numerous neck and back measurements require the simultaneous use of two inclinometers. | Inclinometer | 0.818211 |
11,162 | Adjusting a solar panel to the optimal angle to maximize its output. Measuring the slope angle of a tape or chain during distance measurement. Measuring the height of a building, tree, or other feature using a vertical angle and a distance (determined by taping or pacing), using trigonometry. | Inclinometer | 0.818211 |
11,163 | Such inclinometers are, for instance, used for monitoring volcanoes, or for measuring the depth and rate of landslide movement. Measuring movements in walls or the ground in civil engineering projects. Determining the dip of beds or strata, or the slope of an embankment or cutting; a kind of plumb level. | Inclinometer | 0.818211 |
11,164 | Alerting an equipment operator that it may tip over. Measuring angles of elevation, slope, or incline, e.g. of an embankment. Measuring slight differences in slopes, particularly for geophysics. | Inclinometer | 0.818211 |
11,165 | In mathematics and computer science, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x).For examp... | Floor function | 0.81821 |
11,166 | Orbital uncertainty is related to several parameters used in the orbit determination process including the number of observations (measurements), the time spanned by those observations (observation arc), the quality of the observations (e.g. radar vs. optical), and the geometry of the observations. Of these parameters,... | Uncertainty parameter | 0.81821 |
11,167 | While this need not be true in general, a result of Philip Hall and others is that every infinite locally finite group contains an infinite abelian group. The proof of this fact in infinite group theory relies upon the Feit–Thompson theorem on the solubility of finite groups of odd order (Robinson 1996, p. 432). | Locally finite group | 0.818209 |
11,168 | The class of locally finite groups is closed under subgroups, quotients, and extensions (Robinson 1996, p. 429). Locally finite groups satisfy a weaker form of Sylow's theorems. If a locally finite group has a finite p-subgroup contained in no other p-subgroups, then all maximal p-subgroups are finite and conjugate. | Locally finite group | 0.818209 |
11,169 | There is a unique circle passing through the top and bottom of the painting and tangent to the eye-level line. By elementary geometry, if the viewer's position were to move along the circle, the angle subtended by the painting would remain constant. All positions on the eye-level line except the point of tangency are o... | Regiomontanus' angle maximization problem | 0.818201 |
11,170 | In the present day, this problem is widely known because it appears as an exercise in many first-year calculus textbooks (for example that of Stewart ). Let a = the height of the bottom of the painting above eye level; b = the height of the top of the painting above eye level; x = the viewer's distance from the wall; α... | Regiomontanus' angle maximization problem | 0.818201 |
11,171 | In mathematics, the Regiomontanus's angle maximization problem, is a famous optimization problem posed by the 15th-century German mathematician Johannes Müller (also known as Regiomontanus). The problem is as follows: A painting hangs from a wall. Given the heights of the top and bottom of the painting above the viewer... | Regiomontanus' angle maximization problem | 0.818201 |
11,172 | A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. : 19–22 For example, there are scalar functions of two variables with points in their domain which give different limits when approached along different paths. E.g., the functio... | Multivariate calculus | 0.818195 |
11,173 | It is used in regression analysis to derive formulas for estimating relationships among various sets of empirical data. Multivariable calculus is used in many fields of natural and social science and engineering to model and study high-dimensional systems that exhibit deterministic behavior. In economics, for example, ... | Multivariate calculus | 0.818195 |
11,174 | Techniques of multivariable calculus are used to study many objects of interest in the material world. In particular, Multivariable calculus can be applied to analyze deterministic systems that have multiple degrees of freedom. Functions with independent variables corresponding to each of the degrees of freedom are oft... | Multivariate calculus | 0.818195 |
11,175 | Quantitative genetics focuses on genetic variance due to genetic interactions. Any two locus interactions at a particular gene frequency can be decomposed into eight independent genetic effects using a weighted regression. In this regression, the observed two locus genetic effects are treated as dependent variables and... | Gene interaction | 0.818194 |
11,176 | In statistics, the backfitting algorithm is a simple iterative procedure used to fit a generalized additive model. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the Gauss–Seidel method, an algorithm used for... | Backfitting algorithm | 0.818192 |
11,177 | Maximizing the likelihood function: The next step is to find the set of parameter values that maximizes the likelihood function. This can be achieved using optimization techniques such as numerical optimization algorithms. | Statistical inference | 0.818191 |
11,178 | Likelihood-based inference is a paradigm used to estimate the parameters of a statistical model based on observed data. Likelihoodism approaches statistics by using the likelihood function, denoted as L ( θ | x ) {\displaystyle L(\theta |x)} , quantifies the probability of observing the given data x {\displaystyle x} ,... | Statistical inference | 0.818191 |
11,179 | The second, and more important, reason is to allow a system containing both mechanical and electrical parts to be analysed as a unified whole. This is of great benefit in the fields of mechatronics and robotics. In such cases the mechanical domain is most often converted to the electrical domain because network analysi... | Elastance | 0.81819 |
11,180 | The first is to allow electrical phenomena to be explained in terms of the more familiar mechanical systems. For instance, an electrical inductor-capacitor-resistor circuit has differential equations of the same form as a mechanical mass-spring-damper system. In such cases the electrical domain is converted to the mech... | Elastance | 0.81819 |
11,181 | In a June 2008 article on personal genomics, a doctor from the Southern Illinois University School of Medicine said: The availability of online tools such as SNPedia means we are now in the position where the patient often knows more about their risk implications than their doctor In January 2011, technology journalis... | SNPedia | 0.818187 |
11,182 | SNPedia (pronounced "snipedia") is a wiki-based bioinformatics web site that serves as a database of single nucleotide polymorphisms (SNPs). Each article on a SNP provides a short description, links to scientific articles and personal genomics web sites, as well as microarray information about that SNP. Thus SNPedia ma... | SNPedia | 0.818187 |
11,183 | An associated computer program called Promethease, also developed by the SNPedia team, allows users to compare personal genetics results against the SNPedia database, generating a report with information about a person's attributes, such as propensity to diseases, based on the presence of specific SNPs within their gen... | SNPedia | 0.818187 |
11,184 | These subjective judgments vary not only from person to person, but also with time and culture: for example, as a proof is obtained, simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be stated simply, but its proof may involve surprising and sub... | Mathematical theorem | 0.818181 |
11,185 | In some cases, one might even be able to substantiate a theorem by using a picture as its proof. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". | Mathematical theorem | 0.818181 |
11,186 | Although theorems can be written in a completely symbolic form (e.g., as propositions in propositional calculus), they are often expressed informally in a natural language such as English for better readability. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arg... | Mathematical theorem | 0.818181 |
11,187 | Many mathematical theorems are conditional statements, whose proofs deduce conclusions from conditions known as hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses. Namely, that the conclusion is true in c... | Mathematical theorem | 0.818181 |
11,188 | Initially, many mathematicians did not accept this form of proof, but it has become more widely accepted. The mathematician Doron Zeilberger has even gone so far as to claim that these are possibly the only nontrivial results that mathematicians have ever proved. Many mathematical theorems can be reduced to more straig... | Mathematical theorem | 0.818181 |
11,189 | Other theorems have a known proof that cannot easily be written down. The most prominent examples are the four color theorem and the Kepler conjecture. Both of these theorems are only known to be true by reducing them to a computational search that is then verified by a computer program. | Mathematical theorem | 0.818181 |
11,190 | Some, on the other hand, may be called "deep", because their proofs may be long and difficult, involve areas of mathematics superficially distinct from the statement of the theorem itself, or show surprising connections between disparate areas of mathematics. A theorem might be simple to state and yet be deep. An excel... | Mathematical theorem | 0.818181 |
11,191 | These hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of mathematics known as proof theory studies formal languages, axioms and the structure of proofs. Some theorems are "trivial", in the sense that they follow from definitions, axioms, and other theorems in obvious ... | Mathematical theorem | 0.818181 |
11,192 | In order for a theorem to be proved, it must be in principle expressible as a precise, formal statement. However, theorems are usually expressed in natural language rather than in a completely symbolic form—with the presumption that a formal statement can be derived from the informal one. It is common in mathematics to... | Mathematical theorem | 0.818181 |
11,193 | Logically, many theorems are of the form of an indicative conditional: If A, then B. Such a theorem does not assert B — only that B is a necessary consequence of A. In this case, A is called the hypothesis of the theorem ("hypothesis" here means something very different from a conjecture), and B the conclusion of the t... | Mathematical theorem | 0.818181 |
11,194 | It has been estimated that over a quarter of a million theorems are proved every year.The well-known aphorism, "A mathematician is a device for turning coffee into theorems", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős (and Rényi may have been thinking of Erdős), who... | Mathematical theorem | 0.818181 |
11,195 | What makes formal theorems useful and interesting is that they may be interpreted as true propositions and their derivations may be interpreted as a proof of their truth. A theorem whose interpretation is a true statement about a formal system (as opposed to within a formal system) is called a metatheorem. Some importa... | Mathematical theorem | 0.818181 |
11,196 | The definition of theorems as sentences of a formal language is useful within proof theory, which is a branch of mathematics that studies the structure of formal proofs and the structure of provable formulas. It is also important in model theory, which is concerned with the relationship between formal theories and stru... | Mathematical theorem | 0.818181 |
11,197 | The closure of the empty set under the relation of logical consequence yields the set that contains just those sentences that are the theorems of the deductive system. In the broad sense in which the term is used within logic, a theorem does not have to be true, since the theory that contains it may be unsound relative... | Mathematical theorem | 0.818181 |
11,198 | Some accounts define a theory to be closed under the semantic consequence relation ( ⊨ {\displaystyle \models } ), while others define it to be closed under the syntactic consequence, or derivability relation ( ⊢ {\displaystyle \vdash } ). For a theory to be closed under a derivability relation, it must be associated w... | Mathematical theorem | 0.818181 |
11,199 | In mathematical logic, a formal theory is a set of sentences within a formal language. A sentence is a well-formed formula with no free variables. A sentence that is a member of a theory is one of its theorems, and the theory is the set of its theorems. Usually a theory is understood to be closed under the relation of ... | Mathematical theorem | 0.818181 |
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