id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
18,700 | CSAIL Alliances is the industry connection arm of MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL). CSAIL Alliances offers companies programs to connect with the research, faculty, students, and startups of CSAIL by providing organizations with opportunities to learn about the research, engage with... | MIT Computer Science and Artificial Intelligence Laboratory | 0.811981 |
18,701 | (Including members and alumni of CSAIL's predecessor laboratories) MacArthur Fellows Tim Berners-Lee, Erik Demaine, Dina Katabi, Daniela L. Rus, Regina Barzilay, Peter Shor, Richard Stallman, and Joshua Tenenbaum Turing Award recipients Leonard M. Adleman, Fernando J. Corbató, Shafi Goldwasser, Butler W. Lampson, John ... | MIT Computer Science and Artificial Intelligence Laboratory | 0.811981 |
18,702 | On the fortieth anniversary of Project MAC's establishment, July 1, 2003, LCS was merged with the AI Lab to form the MIT Computer Science and Artificial Intelligence Laboratory, or CSAIL. This merger created the largest laboratory (over 600 personnel) on the MIT campus and was regarded as a reuniting of the diversified... | MIT Computer Science and Artificial Intelligence Laboratory | 0.811981 |
18,703 | To this end, Corbató brought the first computer time-sharing system, Compatible Time-Sharing System (CTSS), with him from the MIT Computation Center, using the DARPA funding to purchase an IBM 7094 for research use. One of the early focuses of Project MAC would be the development of a successor to CTSS, Multics, which ... | MIT Computer Science and Artificial Intelligence Laboratory | 0.811981 |
18,704 | Project MAC would become famous for groundbreaking research in operating systems, artificial intelligence, and the theory of computation. Its contemporaries included Project Genie at Berkeley, the Stanford Artificial Intelligence Laboratory, and (somewhat later) University of Southern California's (USC's) Information S... | MIT Computer Science and Artificial Intelligence Laboratory | 0.811981 |
18,705 | Those researchers who did not join the smaller AI Lab formed the Laboratory for Computer Science and continued their research into operating systems, programming languages, distributed systems, and the theory of computation. Two professors, Hal Abelson and Gerald Jay Sussman, chose to remain neutral — their group was r... | MIT Computer Science and Artificial Intelligence Laboratory | 0.811981 |
18,706 | In the late 1960s, Minsky's artificial intelligence group was seeking more space, and was unable to get satisfaction from project director Licklider. Minsky found that although Project MAC as a single entity could not get the additional space he wanted, he could split off to form his own laboratory and then be entitled... | MIT Computer Science and Artificial Intelligence Laboratory | 0.811981 |
18,707 | A bioinformatics workflow management system is a specialized form of workflow management system designed specifically to compose and execute a series of computational or data manipulation steps, or a workflow, that relate to bioinformatics. There are currently many different workflow systems. Some have been developed m... | Bioinformatics workflow management systems | 0.811978 |
18,708 | Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and Prigogine 1998, page 323). The stable stationary state has a local maximum of entropy and is locally the most reproducible state of the system. There are theorems about the irreversible dissipation of fluctuations... | Disequilibrium (thermodynamics) | 0.811978 |
18,709 | After initial addition of monomer to the initiator system, the viscosity increased (due to increased polymer chain growth), but eventually cease after depletion of monomer concentration. However, he found that addition of more monomer caused an increase in viscosity, indicating growth of the polymer chain, and thus con... | Living polymerization | 0.811978 |
18,710 | Ansa-cyclopentadienyl-amido (CpA) initiatorsCpA initiators have one cyclopentadienyl substituent and one or more nitrogen substituents coordinated to the metal center (generally a Zr or Ti) (Odian). The dimethyl(pentamethylcyclopentyl)zirconium acetamidinate in figure___ has been used for a stereospecific living polyme... | Living polymerization | 0.811978 |
18,711 | α-olefins can be polymerized through an anionic coordination polymerization in which the metal center of the catalyst is considered the counter cation for the anionic end of the alkyl chain (through a M-R coordination). Ziegler-Natta initiators were developed in the mid-1950s and are heterogeneous initiators used in th... | Living polymerization | 0.811978 |
18,712 | The dormant species (Pn-X) can be activated to regenerate the active propagating species (P*) spontaneously, thermally, using a catalyst and optically. The second strategy is based on a degenerative transfer (DT) of the propagating radical between transfer agent that acts as a dormant species (i.e. Reversible addition−... | Living polymerization | 0.811978 |
18,713 | Starting in the 1970s several new methods were discovered which allowed the development of living polymerization using free radical chemistry. These techniques involved catalytic chain transfer polymerization, iniferter mediated polymerization, stable free radical mediated polymerization (SFRP), atom transfer radical p... | Living polymerization | 0.811978 |
18,714 | In polymer chemistry, living polymerization is a form of chain growth polymerization where the ability of a growing polymer chain to terminate has been removed. This can be accomplished in a variety of ways. Chain termination and chain transfer reactions are absent and the rate of chain initiation is also much larger t... | Living polymerization | 0.811978 |
18,715 | Teaching about the functioning of electrical circuits is often on primary and secondary school curricula. Students are expected to understand the rudiments of circuit diagrams and their functioning. Use of diagrammatic representations of circuit diagrams can aid understanding of principles of electricity. Principles of... | Circuit schematics | 0.811977 |
18,716 | The concept of imbuing products with similar life-like qualities has been an area of increasing research within academia and industry alike. However, researchers have used a variety of different terms to describe this research, for example transformational products,. shape changing, kinetic, or in a more general sense,... | Four-dimensional product | 0.811976 |
18,717 | A physics game is a type of logical puzzle video game wherein the player must use the game's physics and environment to complete each puzzle. Physics games use consistent physics to make games more challenging. The genre is popular in online flash games and mobile games. Educators have used these games to demonstrate p... | Puzzle computer game | 0.811975 |
18,718 | Different charges are associated with different eigenspaces of the Casimir invariants of the universal enveloping algebra for those symmetries. This is the case for both the Lorentz symmetry of the underlying spacetime manifold, as well as the symmetries of any fibers in the fiber bundle posed above the spacetime manif... | Charge conjugation symmetry | 0.811971 |
18,719 | This prescription for a quantum field naturally generalizes to any situation where one can enumerate the continuous symmetries of the system, and define duals in a coherent, consistent fashion. The pairing ties together opposite charges in the fully abstract sense. In physics, a charge is associated with a generator of... | Charge conjugation symmetry | 0.811971 |
18,720 | In quantum field theory, charge conjugation can be understood as the exchange of particles with anti-particles. To understand this statement, one must have a minimal understanding of what quantum field theory is. In (vastly) simplified terms, it is a technique for performing calculations to obtain solutions for a syste... | Charge conjugation symmetry | 0.811971 |
18,721 | In the modern setting, information geometry applies to a much wider context, including non-exponential families, nonparametric statistics, and even abstract statistical manifolds not induced from a known statistical model. The results combine techniques from information theory, affine differential geometry, convex anal... | Information Geometry | 0.811971 |
18,722 | In this case, the manifold naturally inherits two flat affine connections, as well as a canonical Bregman divergence. Historically, much of the work was devoted to studying the associated geometry of these examples. | Information Geometry | 0.811971 |
18,723 | Historically, information geometry can be traced back to the work of C. R. Rao, who was the first to treat the Fisher matrix as a Riemannian metric. The modern theory is largely due to Shun'ichi Amari, whose work has been greatly influential on the development of the field.Classically, information geometry considered a... | Information Geometry | 0.811971 |
18,724 | Information geometry is an interdisciplinary field that applies the techniques of differential geometry to study probability theory and statistics. It studies statistical manifolds, which are Riemannian manifolds whose points correspond to probability distributions. | Information Geometry | 0.811971 |
18,725 | An academic study states that the cookbook method of teaching introductory statistics leaves no time for history, philosophy or controversy. Hypothesis testing has been taught as received unified method. Surveys showed that graduates of the class were filled with philosophical misconceptions (on all aspects of statisti... | Test of significance | 0.811963 |
18,726 | Statisticians learn how to create good statistical test procedures (like z, Student's t, F and chi-squared). Statistical hypothesis testing is considered a mature area within statistics, but a limited amount of development continues. | Test of significance | 0.811963 |
18,727 | Such fields as literature and divinity now include findings based on statistical analysis (see the Bible Analyzer). An introductory statistics class teaches hypothesis testing as a cookbook process. Hypothesis testing is also taught at the postgraduate level. | Test of significance | 0.811963 |
18,728 | Statistics is increasingly being taught in schools with hypothesis testing being one of the elements taught. Many conclusions reported in the popular press (political opinion polls to medical studies) are based on statistics. Some writers have stated that statistical analysis of this kind allows for thinking clearly ab... | Test of significance | 0.811963 |
18,729 | One property worth noting is that the normalized states | ψ ⟩ {\displaystyle |\psi \rangle } are characterized by and for orthonormal basis this translates to Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the | k i ⟩ {\displaystyle |{k_{i}}\rangle } are eigenstate... | Mixed state (physics) | 0.811961 |
18,730 | Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on ... | Mixed state (physics) | 0.811961 |
18,731 | A statistical mixture of states is a statistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using r... | Mixed state (physics) | 0.811961 |
18,732 | The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring wi... | Mixed state (physics) | 0.811961 |
18,733 | While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with Dirac picture. : 65 | Mixed state (physics) | 0.811961 |
18,734 | Gaussian model is one of the simplest methods to create one-class classifiers. Due to Central Limit Theorem (CLT), these methods work best when large number of samples are present, and they are perturbed by small independent error values. The probability distribution for a d-dimensional object is given by: p N ( x ; μ ... | One-class classification | 0.811957 |
18,735 | In machine learning, one-class classification (OCC), also known as unary classification or class-modelling, tries to identify objects of a specific class amongst all objects, by primarily learning from a training set containing only the objects of that class, although there exist variants of one-class classifiers where... | One-class classification | 0.811957 |
18,736 | A similar problem is PU learning, in which a binary classifier is learned in a semi-supervised way from only positive and unlabeled sample points.In PU learning, two sets of examples are assumed to be available for training: the positive set P {\displaystyle P} and a mixed set U {\displaystyle U} , which is assumed to ... | One-class classification | 0.811957 |
18,737 | . . , n {\displaystyle {\text{subject to, }}||\Phi (x_{i})-c||^{2}\leq r^{2}+\zeta _{i}\;\;\forall i=1,2,...,n} From Karush-Kuhn-Tucker (KKT) optimality conditions, we get c = ∑ i = 1 n α i Φ ( x i ) , {\displaystyle c=\sum _{i=1}^{n}\alpha _{i}\Phi (x_{i}),} where the α i {\displaystyle \alpha _{i}} 's are the solutio... | One-class classification | 0.811957 |
18,738 | SVM based one-class classification (OCC) relies on identifying the smallest hypersphere (with radius r, and center c) consisting of all the data points. This method is called Support Vector Data Description (SVDD). Formally, the problem can be defined in the following constrained optimization form, However, the above f... | One-class classification | 0.811957 |
18,739 | In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars. In the 20th century, understanding... | N-Body Problem | 0.811942 |
18,740 | The last general constant of the motion is given by the conservation of energy H. Hence, every n-body problem has ten integrals of motion. Because T and U are homogeneous functions of degree 2 and −1, respectively, the equations of motion have a scaling invariance: if qi(t) is a solution, then so is λ−2/3qi(λt) for any... | N-Body Problem | 0.811942 |
18,741 | Summing over all masses yields the n-body equations of motion:where U is the self-potential energy Defining the momentum to be pi = mi dqi/dt, Hamilton's equations of motion for the n-body problem become where the Hamiltonian function is and T is the kinetic energy Hamilton's equations show that the n-body problem is a... | N-Body Problem | 0.811942 |
18,742 | Knowing three orbital positions of a planet's orbit – positions obtained by Sir Isaac Newton from astronomer John Flamsteed – Newton was able to produce an equation by straightforward analytical geometry, to predict a planet's motion; i.e., to give its orbital properties: position, orbital diameter, period and orbital ... | N-Body Problem | 0.811942 |
18,743 | In 1963, Vladimir Arnold proved using KAM theory a kind of stability of the planetary problem: there exists a set of positive measure of quasiperiodic orbits in the case of the planetary problem restricted to the plane. In the KAM theory, chaotic planetary orbits would be bounded by quasiperiodic KAM tori. Arnold's res... | N-Body Problem | 0.811942 |
18,744 | For every solution of the problem, not only applying an isometry or a time shift but also a reversal of time (unlike in the case of friction) gives a solution as well.In the physical literature about the n-body problem (n ≥ 3), sometimes reference is made to "the impossibility of solving the n-body problem" (via employ... | N-Body Problem | 0.811942 |
18,745 | For a small number of bodies, an n-body problem can be solved using direct methods, also called particle–particle methods. These methods numerically integrate the differential equations of motion. Numerical integration for this problem can be a challenge for several reasons. | N-Body Problem | 0.811942 |
18,746 | Newton stated (in section 11 of the Principia) that the main reason, however, for failing to predict the forces for elliptical orbits was that his math model was for a body confined to a situation that hardly existed in the real world, namely, the motions of bodies attracted toward an unmoving center. Some present phys... | N-Body Problem | 0.811942 |
18,747 | A central configuration q1(0), …, qN(0) is an initial configuration such that if the particles were all released with zero velocity, they would all collapse toward the center of mass C. Such a motion is called homothetic. Central configurations may also give rise to homographic motions in which all masses moves along K... | N-Body Problem | 0.811942 |
18,748 | One way of solving the classical n-body problem is "the n-body problem by Taylor series". We start by defining the system of differential equations: As xi(t0) and dxi(t0)/dt are given as initial conditions, every d2xi(t)/dt2 is known. Differentiating d2xi(t)/dt2 results in d3xi(t)/dt3 which at t0 which is also known, a... | N-Body Problem | 0.811942 |
18,749 | These are often used with periodic boundary conditions on the region simulated and Ewald summation techniques are used to speed up computations.In statistics and machine learning, some models have loss functions of a form similar to that of the gravitational potential: a sum of kernel functions over all pairs of object... | N-Body Problem | 0.811942 |
18,750 | Most work done on the n-body problem has been on the gravitational problem. But there exist other systems for which n-body mathematics and simulation techniques have proven useful. In large scale electrostatics problems, such as the simulation of proteins and cellular assemblies in structural biology, the Coulomb poten... | N-Body Problem | 0.811942 |
18,751 | In quantum physics and quantum mechanics, the MOSFET is the basis for two-dimensional electron gas (2DEG) and the quantum Hall effect. The MOSFET enables physicists to study electron behavior in a two-dimensional gas, called a two-dimensional electron gas. In a MOSFET, conduction electrons travel in a thin surface laye... | MOSFET applications | 0.81194 |
18,752 | 3DLT was a Covington, Kentucky based 3D printing marketplace for 3D printing as-a-service. Users of 3DLT designed and uploaded 3D printable files and 3DLT worked to print and sell those products. The company was formed in 2012 and ceased operations at the end of 2015. | 3DLT | 0.811939 |
18,753 | leaving the company. Shortly after, 3DLT partnered Authentise to stream licensed 3D content.In mid-2014, 3DLT and PieceMaker announced plans to collaborate and share content, technology and in-store marketing and production. In particular, the company was promoting 3-D printing kiosks made by Piecemaker that used its d... | 3DLT | 0.811939 |
18,754 | During his 37-year tenure at KU, Jenks developed the Cartography program into one of three programs renowned for their graduate education in the field; the others being the University of Wisconsin and the University of Washington. Much of his time was spent developing and promoting improved cartographic training techni... | Jenks natural breaks optimization | 0.811938 |
18,755 | The Jenks optimization method, also called the Jenks natural breaks classification method, is a data clustering method designed to determine the best arrangement of values into different classes. This is done by seeking to minimize each class's average deviation from the class mean, while maximizing each class's deviat... | Jenks natural breaks optimization | 0.811938 |
18,756 | Other methods of data classification include Head/tail Breaks, Natural Breaks (without Jenks Optimization), Equal Interval, Quantile, and Standard Deviation. | Jenks natural breaks optimization | 0.811938 |
18,757 | Jenks was a cartographer by profession. His work with statistics grew out of a desire to make choropleth maps more visually accurate for the viewer. In his paper, The Data Model Concept in Statistical Mapping, he claims that by visualizing data in a three dimensional model cartographers could devise a “systematic and r... | Jenks natural breaks optimization | 0.811938 |
18,758 | There are two well-known uniqueness theorems for Leavitt path algebras: the graded uniqueness theorem and the Cuntz-Krieger uniqueness theorem. These are analogous, respectively, to the gauge-invariant uniqueness theorem and Cuntz-Krieger uniqueness theorem for graph C*-algebras. Formal statements of the uniqueness the... | Leavitt path algebra | 0.811938 |
18,759 | As with graph C*-algebras, graph-theoretic properties of E {\displaystyle E} correspond to algebraic properties of L K ( E ) {\displaystyle L_{K}(E)} . Interestingly, it is often the case that the graph properties of E {\displaystyle E} that are equivalent to an algebraic property of L K ( E ) {\displaystyle L_{K}(E)} ... | Leavitt path algebra | 0.811938 |
18,760 | Theorem: Fix a field K {\displaystyle K} , and let E {\displaystyle E} be a row-finite graph. Then the following hold: The function H ↦ I H {\displaystyle H\mapsto I_{H}} is a lattice isomorphism from the lattice of saturated hereditary subsets of E {\displaystyle E} onto the lattice of graded ideals of L K ( E ) {\dis... | Leavitt path algebra | 0.811938 |
18,761 | The graded ideals are partially ordered by inclusion and form a lattice with meet I 1 ∧ I 2 := I 1 ∩ I 2 {\displaystyle I_{1}\wedge I_{2}:=I_{1}\cap I_{2}} and joint I 1 ∨ I 2 {\displaystyle I_{1}\vee I_{2}} defined to be the ideal generated by I 1 ∪ I 2 {\displaystyle I_{1}\cup I_{2}} . For any saturated hereditary su... | Leavitt path algebra | 0.811938 |
18,762 | We use the term ideal to mean "two-sided ideal" in our Leavitt path algebras. The ideal structure of L K ( E ) {\displaystyle L_{K}(E)} can be determined from E {\displaystyle E} . A subset of vertices H ⊆ E 0 {\displaystyle H\subseteq E^{0}} is called hereditary if for all e ∈ E 1 {\displaystyle e\in E^{1}} , s ( e ) ... | Leavitt path algebra | 0.811938 |
18,763 | Let E {\displaystyle E} be a graph, and let L K ( E ) {\displaystyle L_{K}(E)} be the associated Leavitt path algebra. If A {\displaystyle A} is a graded K {\displaystyle K} -algebra and ϕ: L K ( E ) → A {\displaystyle \phi :L_{K}(E)\to A} is a graded algebra homomorphism with ϕ ( p v ) ≠ 0 {\displaystyle \phi (p_{v})\... | Leavitt path algebra | 0.811938 |
18,764 | A cycle e 1 e 2 … e n {\displaystyle e_{1}e_{2}\ldots e_{n}} is called a simple cycle if s ( e i ) ≠ s ( e 1 ) {\displaystyle s(e_{i})\neq s(e_{1})} for all 2 ≤ i ≤ n {\displaystyle 2\leq i\leq n} . The following are two important graph conditions that arise in the study of Leavitt path algebras. | Leavitt path algebra | 0.811938 |
18,765 | The theory of Leavitt path algebras uses terminology for graphs similar to that of C*-algebraists, which differs slightly from that used by graph theorists. The term graph is typically taken to mean a directed graph E = ( E 0 , E 1 , r , s ) {\displaystyle E=(E^{0},E^{1},r,s)} consisting of a countable set of vertices ... | Leavitt path algebra | 0.811938 |
18,766 | For a path α := e 1 … e n {\displaystyle \alpha :=e_{1}\ldots e_{n}} we let | α | := n {\displaystyle |\alpha |:=n} denote the length of α {\displaystyle \alpha } . For each integer n ∈ Z {\displaystyle n\in \mathbb {Z} } we define L K ( E ) n := span K { s α s β ∗: | α | − | β | = n } {\displaystyle L_{K}(E)_{n}:=\o... | Leavitt path algebra | 0.811938 |
18,767 | Leavitt path algebras were simultaneously introduced in 2005 by Gene Abrams and Gonzalo Aranda Pino as well as by Pere Ara, María Moreno, and Enrique Pardo, with neither of the two groups aware of the other's work. Leavitt path algebras have been investigated by dozens of mathematicians since their introduction, and in... | Leavitt path algebra | 0.811938 |
18,768 | Leavitt path algebras has been computed for many graphs, and the following table shows some particular graphs and their Leavitt path algebras. We use the convention that a double arrow drawn from one vertex to another and labeled ∞ {\displaystyle \infty } indicates that there are a countably infinite number of edges fr... | Leavitt path algebra | 0.811938 |
18,769 | In mathematics, a Leavitt path algebra is a universal algebra constructed from a directed graph. Leavitt path algebras generalize Leavitt algebras and may be considered as algebraic analogues of graph C*-algebras. | Leavitt path algebra | 0.811938 |
18,770 | The Turing Talk, previously known as the Turing Lecture, is an annual award lecture delivered by a noted speaker on the subject of Computer Science. Sponsored and co-hosted by the Institution of Engineering and Technology (IET) and the British Computer Society, the talk has been delivered at different locations in the ... | Turing Talk | 0.811937 |
18,771 | Previous speakers have included: 2022: Julie McCann, a day in the life of a smart city 2021: Cecilia Mascolo, Sounding out wearable and audio data for health diagnostics 2020: Mark Girolami, Digital Twins: The Next Phase of the AI Revolution 2019: Krishna Gummadi Engineering a fair future: Why we need to train unbiased... | Turing Talk | 0.811937 |
18,772 | Logical criticisms of this text led to a highly revised second edition. In reaction to the launching of the Russian satellite Sputnik there was a call in the United States to revise the school mathematics curriculum. From this effort there arose the New Math program of the 1960s. With this as a background, many individ... | Foundations of geometry | 0.811937 |
18,773 | Whether or not it is wise to teach Euclidean geometry from an axiomatic viewpoint at the high school level has been a matter of debate. There have been many attempts to do so and not all of them have been successful. In 1904, George Bruce Halsted published a high school geometry text based on Hilbert's axiom set. | Foundations of geometry | 0.811937 |
18,774 | Moore and R.L. Moore independently proved that this axiom is redundant, and the former published this result in an article appearing in the Transactions of the American Mathematical Society in 1902. Hilbert moved the axiom to Theorem 5 and renumbered the axioms accordingly (old axiom II-5 (Pasch's axiom) now became II-... | Foundations of geometry | 0.811937 |
18,775 | By the 7th edition of the Grundlagen, this axiom had been replaced by the axiom of line completeness given above and the old axiom V.2 became Theorem 32. Also to be found in the 1899 monograph (and appearing in the Townsend translation) is: II.4. Any four points A, B, C, D of a line can always be labeled so that B shal... | Foundations of geometry | 0.811937 |
18,776 | If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I.1 to I.3 can be proved trivially by using superposition.To address these issues in Euclid's work, later authors have either attempted to fill in the holes in Euclid's present... | Foundations of geometry | 0.811937 |
18,777 | The standards of mathematical rigor have changed since Euclid wrote the Elements. Modern attitudes towards, and viewpoints of, an axiomatic system can make it appear that Euclid was in some way sloppy or careless in his approach to the subject, but this is an ahistorical illusion. It is only after the foundations were ... | Foundations of geometry | 0.811937 |
18,778 | In Euclid's Elements, the first 28 propositions and Proposition I.31 avoid using the parallel postulate, and therefore are valid theorems in absolute geometry. Proposition I.31 proves the existence of parallel lines (by construction). Also, the Saccheri–Legendre theorem, which states that the sum of the angles in a tri... | Foundations of geometry | 0.811937 |
18,779 | In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be experimentally verified with a scale and protractor. In a radical departure from the synthetic approach of Hilbert, Birkhoff was the fir... | Foundations of geometry | 0.811937 |
18,780 | Some of his models were very complex and other mathematicians tried to simplify them. For instance, Hilbert's model for showing the independence of Desargues theorem from certain axioms ultimately led Ray Moulton to discover the non-Desarguesian Moulton plane. These investigations by Hilbert virtually inaugurated the m... | Foundations of geometry | 0.811937 |
18,781 | Going beyond the establishment of a satisfactory set of axioms, Hilbert also proved the consistency of his system relative to the theory of real numbers by constructing a model of his axiom system from the real numbers. He proved the independence of some of his axioms by constructing models of geometries which satisfy ... | Foundations of geometry | 0.811937 |
18,782 | Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can b... | Foundations of geometry | 0.811937 |
18,783 | The following partial list indicates some of these theorems that are of historical interest. Parallel straight lines are equidistant. | Foundations of geometry | 0.811937 |
18,784 | Equivalence here means that in the presence of the other axioms of the geometry each of these theorems can be assumed to be true and the parallel postulate can be proved from this altered set of axioms. This is not the same as logical equivalence. In different sets of axioms for Euclidean geometry, any of these can rep... | Foundations of geometry | 0.811937 |
18,785 | The success of the Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are supposedly his. Euclid's systematic development of his subject, from a small set of axioms to deep results, and... | Foundations of geometry | 0.811937 |
18,786 | Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):"Let the following be postulated": "To draw a straight line from any point to any point." "To produce a finite straight line continuously in a... | Foundations of geometry | 0.811937 |
18,787 | Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base. | Foundations of geometry | 0.811937 |
18,788 | The Pythagorean theorem is proved.Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as prime numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved. | Foundations of geometry | 0.811937 |
18,789 | It is assumed that its superiority over earlier treatments was recognized, with the consequence that there was little interest in preserving the earlier ones, and they are now nearly all lost. Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, e.g., If a triangle has two equal angles... | Foundations of geometry | 0.811937 |
18,790 | Being first set in type in Venice in 1482, it is one of the very earliest mathematical works to be printed after the invention of the printing press and was estimated by Carl Benjamin Boyer to be second only to the Bible in the number of editions published, with the number reaching well over one thousand. For centuries... | Foundations of geometry | 0.811937 |
18,791 | Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books ... | Foundations of geometry | 0.811937 |
18,792 | But the influence of Hilbert's work went far beyond this, for, backed by the author's great mathematical authority, it firmly implanted the postulational method, not only in the field of geometry, but also in essentially every other branch of mathematics. The stimulus to the development of the foundations of mathematic... | Foundations of geometry | 0.811937 |
18,793 | The influence of the book was immediate. According to Eves (1963, pp. 384–5): By developing a postulate set for Euclidean geometry that does not depart too greatly in spirit from Euclid's own, and by employing a minimum of symbolism, Hilbert succeeded in convincing mathematicians to a far greater extent than had Pasch ... | Foundations of geometry | 0.811937 |
18,794 | At the University of Göttingen, during the 1898–1899 winter term, the eminent German mathematician David Hilbert (1862–1943) presented a course of lectures on the foundations of geometry. At the request of Felix Klein, Professor Hilbert was asked to write up the lecture notes for this course in time for the summer 1899... | Foundations of geometry | 0.811937 |
18,795 | That is, the defect of a triangle is negative. All the lines perpendicular to a given line meet at a common point in elliptic geometry, called the pole of the line. In hyperbolic geometry these lines are mutually non-intersecting, while in Euclidean geometry they are mutually parallel.Other results, such as the exterio... | Foundations of geometry | 0.811937 |
18,796 | Some of the propositions which exhibit this property are: The fourth angle of a Lambert quadrilateral is an obtuse angle in elliptic geometry. The summit angles of a Saccheri quadrilateral are obtuse in elliptic geometry. The sum of the measures of the angles of any triangle is greater than 180° if the geometry is elli... | Foundations of geometry | 0.811937 |
18,797 | Together with the new axiom on the nonexistence of parallel lines we have a consistent system of axioms giving a new geometry. The geometry that results is called (plane) Elliptic geometry. Even though elliptic geometry is not an extension of absolute geometry (as Euclidean and hyperbolic geometry are), there is a cert... | Foundations of geometry | 0.811937 |
18,798 | A line segment in this new geometry is determined by three collinear points A, B and C and consists of those three points and all the points not separated from B by A and C. There are further consequences. Since two points do not determine a line segment uniquely, three noncollinear points do not determine a unique tri... | Foundations of geometry | 0.811937 |
18,799 | For any five distinct collinear points A, B, C, D and E, if (A,B|D,E), then either (A,B|C,D) or (A,B|C,E). Perspectivities preserve separation.Since the Hilbert notion of "betweeness" has been removed, terms which were defined using that concept need to be redefined. Thus, a line segment AB defined as the points A and ... | Foundations of geometry | 0.811937 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.