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25,700 | Agricultural biotechnology has been used to improve the nutritional content of a variety of crops in an effort to meet the needs of an increasing population. Genetic engineering can produce crops with a higher concentration of vitamins. For example, golden rice contains three genes that allow plants to produce compound... | Agricultural biotechnology | 0.807499 |
25,701 | Often, crops are afflicted by disease spread through insects (like aphids). Spreading disease among crop plants is incredibly difficult to control and was previously only managed by completely removing the affected crop. The field of agricultural biotechnology offers a solution through genetically engineering virus res... | Agricultural biotechnology | 0.807499 |
25,702 | The base-a logarithm of e is 1, if a equals e. So symbolically, d d x log e x = 1 x . {\displaystyle {\frac {d}{dx}}\log _{e}x={\frac {1}{x}}.} The logarithm with this special base is called the natural logarithm, and is denoted as ln; it behaves well under differentiation since there is no undetermined limit to carr... | Euler’s number | 0.807497 |
25,703 | Consequently, the exponential function with base e is particularly suited to doing calculus. Choosing e (as opposed to some other number as the base of the exponential function) makes calculations involving the derivatives much simpler. Another motivation comes from considering the derivative of the base-a logarithm (i... | Euler’s number | 0.807497 |
25,704 | The principal motivation for introducing the number e, particularly in calculus, is to perform differential and integral calculus with exponential functions and logarithms. A general exponential function y = ax has a derivative, given by a limit: d d x a x = lim h → 0 a x + h − a x h = lim h → 0 a x a h − a x h = a x ⋅... | Euler’s number | 0.807497 |
25,705 | Steiner's problem asks to find the global maximum for the function This maximum occurs precisely at x = e. (One can check that the derivative of ln f(x) is zero only for this value of x.) Similarly, x = 1/e is where the global minimum occurs for the function The infinite tetration x x x ⋅ ⋅ ⋅ {\displaystyle x^{x^{x^{\c... | Euler’s number | 0.807497 |
25,706 | Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any non-zero polynomial equation with rational coefficients. It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the... | Euler’s number | 0.807497 |
25,707 | Terahertz radiation – also known as submillimeter radiation, terahertz waves, tremendously high frequency (THF), T-rays, T-waves, T-light, T-lux or THz – consists of electromagnetic waves within the ITU-designated band of frequencies from 0.3 to 3 terahertz (THz), although the upper boundary is somewhat arbitrary and i... | Terahertz band | 0.807496 |
25,708 | While the work function of a semiconductor can be changed by doping, the electron affinity ideally does not change with doping and so it is closer to being a material constant. However, like work function the electron affinity does depend on the surface termination (crystal face, surface chemistry, etc.) and is strictl... | Electron gain enthalpy | 0.807495 |
25,709 | In the field of solid state physics, the electron affinity is defined differently than in chemistry and atomic physics. For a semiconductor-vacuum interface (that is, the surface of a semiconductor), electron affinity, typically denoted by EEA or χ, is defined as the energy obtained by moving an electron from the vacuu... | Electron gain enthalpy | 0.807495 |
25,710 | It has been surmised that a drop kick has a slightly longer range than the standard place kick, but since these kicks are so rare, that is not known for sure. During the early NFL era, this was generally true, and drop kicks were the norm for longer field goals; in fact, the first unofficial NFL record kick of 55 yards... | Field goal range | 0.807494 |
25,711 | Biodiversity informatics is the application of informatics techniques to biodiversity information, such as taxonomy, biogeography or ecology. It is defined as the application of Information technology technologies to management, algorithmic exploration, analysis and interpretation of primary data regarding life, partic... | Biodiversity Informatics | 0.807493 |
25,712 | One major goal for biodiversity informatics is the creation of a complete master list of currently recognised species of the world. This goal has been achieved to a large extent by the Catalogue of Life project which lists >2 million species in its 2022 Annual Checklist. A similar effort for fossil taxa, the Paleobiolo... | Biodiversity Informatics | 0.807493 |
25,713 | For finite dimensional manifolds M {\displaystyle M} and N {\displaystyle N} with M {\displaystyle M} compact, the space Emb ( M , N ) {\displaystyle \operatorname {Emb} (M,N)} of all smooth embeddings of M {\displaystyle M} into N {\displaystyle N} , is open in C ∞ ( M , N ) {\displaystyle C^{\infty }(M,N)} , so it ... | Convenient vector space | 0.807492 |
25,714 | Convenient calculus of smooth mappings appeared for the first time in , . Convenient calculus (having properties 6 and 7) exists also for: Real analytic mappings (Kriegl, Michor, 1990; see also , chapter II). Holomorphic mappings (Kriegl, Nel, 1985; see also , chapter II). | Convenient vector space | 0.807492 |
25,715 | An overview of applications using geometry of shape spaces and diffeomorphism groups can be found in . | Convenient vector space | 0.807492 |
25,716 | Here it is a theorem. This property is the source of the name convenient, which was borrowed from (Steenrod 1967). 7. | Convenient vector space | 0.807492 |
25,717 | C ∞ ( U , C ∞ ( V , G ) ) ≅ C ∞ ( U × V , G ) , f ↦ g , f ( u ) ( v ) = g ( u , v ) . {\displaystyle C^{\infty }(U,C^{\infty }(V,G))\cong C^{\infty }(U\times V,G),\qquad f\mapsto g,\qquad f(u)(v)=g(u,v).} This is the main assumption of variational calculus. | Convenient vector space | 0.807492 |
25,718 | 5. The space C ∞ ( U , F ) {\displaystyle C^{\infty }(U,F)} of all smooth mappings U → F {\displaystyle U\to F} is again a convenient vector space where the structure is given by the following injection, where C ∞ ( R , R ) {\displaystyle C^{\infty }(\mathbb {R} ,\mathbb {R} )} carries the topology of compact convergen... | Convenient vector space | 0.807492 |
25,719 | If f: E ⊇ U → F {\displaystyle f:E\supseteq U\to F} is smooth then the derivative d f: U × E → F {\displaystyle df:U\times E\to F} is smooth, and also d f: U → L ( E , F ) {\displaystyle df:U\to L(E,F)} is smooth where L ( E , F ) {\displaystyle L(E,F)} denotes the space of all bounded linear mappings with the topology... | Convenient vector space | 0.807492 |
25,720 | 1. For maps on Fréchet spaces this notion of smoothness coincides with all other reasonable definitions. On R 2 {\displaystyle \mathbb {R} ^{2}} this is a non-trivial theorem, proved by Boman, 1967. See also , 3.4. | Convenient vector space | 0.807492 |
25,721 | {\displaystyle \partial _{t}\operatorname {Evol} (X)(t,x)=X(t,\operatorname {Evol} (X)(t,x)).} Given a smooth curve in the Lie algebra, X ( s , t , x ) ∈ C ∞ ( R 2 , X ( M ) ) {\displaystyle X(s,t,x)\in C^{\infty }(\mathbb {R} ^{2},{\mathfrak {X}}(M))} , then the solution of the ordinary differential equation depends s... | Convenient vector space | 0.807492 |
25,722 | Proof: The diffeomorphism group Diff ( M ) {\displaystyle \operatorname {Diff} (M)} is a smooth manifold since it is an open subset in C ∞ ( M , M ) {\displaystyle C^{\infty }(M,M)} . Composition is smooth by restriction. Inversion is smooth: If t → f ( t , ) {\displaystyle t\to f(t,\ )} is a smooth curve in Diff (... | Convenient vector space | 0.807492 |
25,723 | Theorem. For each compact manifold M {\displaystyle M} , the diffeomorphism group Diff ( M ) {\displaystyle \operatorname {Diff} (M)} is a regular Lie group. Its Lie algebra is the space X ( M ) {\displaystyle {\mathfrak {X}}(M)} of all smooth vector fields on M {\displaystyle M} , with the negative of the usual brac... | Convenient vector space | 0.807492 |
25,724 | The following mapping is required to be smooth: evol G r: C ∞ ( R , g ) → G . {\displaystyle \operatorname {evol} _{G}^{r}:C^{\infty }(\mathbb {R} ,{\mathfrak {g}})\to G.} If X {\displaystyle X} is a constant curve in the Lie algebra, then evol G r ( X ) = exp G ( X ) {\displaystyle \operatorname {evol} _{G}^{r}(X)... | Convenient vector space | 0.807492 |
25,725 | {\displaystyle \mu :G\times G\to G,\quad \mu (x,y)=x.y=\mu _{x}(y)=\mu ^{y}(x),\qquad \nu :G\to G,\nu (x)=x^{-1}.} The notion of a regular Lie group is originally due to Omori et al. for Fréchet Lie groups, was weakened and made more transparent by J. Milnor, and was then carried over to convenient Lie groups; see , 38... | Convenient vector space | 0.807492 |
25,726 | Let G {\displaystyle G} be a connected smooth Lie group modeled on convenient vector spaces, with Lie algebra g = T e G {\displaystyle {\mathfrak {g}}=T_{e}G} . Multiplication and inversion are denoted by: μ: G × G → G , μ ( x , y ) = x . y = μ x ( y ) = μ y ( x ) , ν: G → G , ν ( x ) = x − 1 . | Convenient vector space | 0.807492 |
25,727 | This leads to a Cartesian closed category of smooth mappings between c ∞ {\displaystyle c^{\infty }} -open subsets of convenient vector spaces (see property 6 below). The corresponding calculus of smooth mappings is called convenient calculus. It is weaker than any other reasonable notion of differentiability, it is ea... | Convenient vector space | 0.807492 |
25,728 | In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition. Traditional differential calculus is effective in the analysis of finite-dimensional vector spaces and for Banach spaces. Beyond Banach spaces, difficulties begin to arise; in particular, composition... | Convenient vector space | 0.807492 |
25,729 | See Rational difference equation and Matrix difference equation for example of uses of "difference equation" instead of "recurrence relation" Difference equations resemble to differential equations, and this resemblance is often used to mimic methods for solving differentiable equations to apply to solving difference e... | Recursive relation | 0.807491 |
25,730 | In the 11th century, Avicenna (Ibn Sīnā) discussed Philoponus' theory in The Book of Healing, in Physics IV.14 he says: When we independently verify the issue (of projectile motion), we find the most correct doctrine is the doctrine of those who think that the moved object acquires an inclination from the mover Ibn Sīn... | Theory of impetus | 0.80749 |
25,731 | Mathematics has been used in biology as early as the 13th century, when Fibonacci used the famous Fibonacci series to describe a growing population of rabbits. In the 18th century, Daniel Bernoulli applied mathematics to describe the effect of smallpox on the human population. Thomas Malthus' 1789 essay on the growth o... | Theoretical biologist | 0.80749 |
25,732 | Computational neuroscience (also known as theoretical neuroscience or mathematical neuroscience) is the theoretical study of the nervous system. | Theoretical biologist | 0.80749 |
25,733 | An elaboration of systems biology to understand the more complex life processes was developed since 1970 in connection with molecular set theory, relational biology and algebraic biology. | Theoretical biologist | 0.80749 |
25,734 | Acta Biotheoretica Bulletin of Mathematical Biology Journal of Theoretical Biology BioSystems Mathematical Biosciences Theoretical Population Biology Journal of Mathematical Biology Artificial Life Theory in Biosciences Biological Theory Theoretical Ecology Philosophy, Theory, and Practice in Biology | Theoretical biologist | 0.80749 |
25,735 | Abstract relational biology (ARB) is concerned with the study of general, relational models of complex biological systems, usually abstracting out specific morphological, or anatomical, structures. Some of the simplest models in ARB are the Metabolic-Replication, or (M,R)--systems introduced by Robert Rosen in 1957–195... | Theoretical biologist | 0.80749 |
25,736 | The book is aimed at a "general mathematical audience" including undergraduate mathematics students with an introductory-level background in real analysis. It is intended both to excite mathematicians, physicists, and computer scientists about the foundational issues in their fields, and to provide an accessible introd... | Reverse Mathematics: Proofs from the Inside Out | 0.807488 |
25,737 | The book begins with a historical overview of the long struggles with the parallel postulate in Euclidean geometry, and of the foundational crisis of the late 19th and early 20th centuries, Then, after reviewing background material in real analysis and computability theory, the book concentrates on the reverse mathemat... | Reverse Mathematics: Proofs from the Inside Out | 0.807488 |
25,738 | All of the light that is striking the concentrator across the full circular area is reflected to the same spot, as long as the concentrator is pointed at the sun. The wide hemisphere faces the sun. The concentrators are mounted on poles in rigid rigs (ten feet above the ground); the rigs allow a computer system to move... | Cool Earth Solar | 0.807488 |
25,739 | There is rising demand for solar energy in the United States based on government clean energy mandates. Many states require energy supply companies to produce a specific portion of their electricity from renewable sources. As of July 2010, 29 states plus the District of Columbia have state-mandated renewable portfolio ... | Cool Earth Solar | 0.807487 |
25,740 | The aerobic enzyme has a structure homologous to the complex alpha/beta topology found in the gene product of human sulfatase-modifying factor 1 (SUMF1). Aerobic FGE converts a cysteine residue in the highly conserved consensus sequence CXPXR to fGly. To do so, FGE “activates” its target by utilizing mononuclear copper... | Formylglycine-generating enzyme | 0.807487 |
25,741 | In molecular biology, "formylglycine-generating enzyme" (sometimes annotated as formylglycine-generating sulfatase enzyme) is the name of the FGE protein domain, whether or not the protein is catalytically active. Both prokaryotic and eukaryotic homologs of FGE possess highly conserved active sites — including the cata... | Formylglycine-generating enzyme | 0.807487 |
25,742 | An n-ellipse is the set of points all having the same sum of distances to n foci (the n = 2 case being the conventional ellipse). The concept of a focus can be generalized to arbitrary algebraic curves. Let C be a curve of class m and let I and J denote the circular points at infinity. Draw the m tangents to C through ... | Focus (geometry) | 0.807482 |
25,743 | Let P1, P2, …, Pm be given as foci of a curve C of class m. Let P be the product of the tangential equations of these points and Q the product of the tangential equations of the circular points at infinity. Then all the lines which are common tangents to both P = 0 and Q = 0 are tangent to C. So, by the AF+BG theorem, ... | Focus (geometry) | 0.807482 |
25,744 | Nearline storage (a portmanteau of "near" and "online storage") is a term used in computer science to describe an intermediate type of data storage that represents a compromise between online storage (supporting frequent, very rapid access to data) and offline storage/archiving (used for backups or long-term storage, w... | Nearline storage | 0.807481 |
25,745 | For some centuries they confined themselves to using common and sexagesimal fractions... This half-heartedness has never been completely overcome, and sexagesimal fractions still form the basis of our trigonometry, astronomy and measurement of time. ¶ ... Mathematicians sought to avoid fractions by taking the radius R ... | Positional numeral system | 0.807481 |
25,746 | This was a driving motivation behind rule-based expert systems. IF-THEN rules provide a subset of FOL but a very useful one that is also very intuitive. The history of most of the early AI knowledge representation formalisms; from databases to semantic nets to theorem provers and production systems can be viewed as var... | Knowledge model | 0.807476 |
25,747 | Knowledge-representation is a field of artificial intelligence that focuses on designing computer representations that capture information about the world that can be used for solving complex problems. The justification for knowledge representation is that conventional procedural code is not the best formalism to use t... | Knowledge model | 0.807476 |
25,748 | To solve this problem, surface treatment is used as a preparation step before adhesive bonding. It cleans the surface from the organic contaminants, removes a weak boundary layer, chemically bonds to the substrate a strong layer with high surface energy and chemical affinity to the adhesive, and modifies the surface to... | Plasma functionalization | 0.807474 |
25,749 | Importantly, the plasma activation can be performed at atmospheric pressure using air or typical industrial gases including hydrogen, nitrogen and oxygen. Thus, the surface functionalization is achieved without expensive vacuum equipment or wet chemistry, which positively affects its costs, safety and environmental imp... | Plasma functionalization | 0.807474 |
25,750 | In molecular biology the SeqA protein is found in bacteria and archaea. The function of this protein is highly important in DNA replication. The protein negatively regulates the initiation of DNA replication at the origin of replication, in Escherichia coli, OriC. | SeqA protein | 0.807472 |
25,751 | In 2001, the US National Association of State Departments of Agriculture (NASDA) defined biosecurity as "the sum of risk management practices in defense against biological threats", and its main goal as "protect against the risk posed by disease and organisms".The USDA Animal and Plant Health Inspection Service (APHIS)... | Biosecurity in the United States | 0.807471 |
25,752 | In the US, biosecurity is governed by the Bureau of Western Hemisphere Affairs, which is within the Department of State. The Bureau promotes global health security as part of its role in the biodefense network, "because infectious disease threats, whether naturally occurring, deliberate, or accidental, have the potenti... | Biosecurity in the United States | 0.807471 |
25,753 | In theoretical physics, a mixed anomaly is an example of an anomaly: it is an effect of quantum mechanics — usually a one-loop diagram — that implies that the classically valid general covariance and gauge symmetry of a theory of general relativity combined with gauge fields and fermionic fields cannot be preserved sim... | Mixed anomaly | 0.807469 |
25,754 | The IEEE Annals of the History of Computing is a quarterly peer-reviewed academic journal published by the IEEE Computer Society. It covers the history of computing, computer science, and computer hardware. It was founded in 1979 by the AFIPS, in particular by Saul Rosen, who was an editor until his death in 1991.The j... | IEEE Annals of the History of Computing | 0.807468 |
25,755 | "Common sense" ideas tend to relate to events within human experience, and thus commensurate with these scales. There is thus no commonsense intuition of, for example, interstellar distances or speeds approaching the speed of light. Weights and measures tend to reflect human scale, and many older systems of measurement... | Human scale | 0.807465 |
25,756 | In the context of machine learning, the exploration-exploitation tradeoff is often encountered in reinforcement learning, a type of machine learning that involves training agents to make decisions based on feedback from the environment. The agent must decide whether to exploit the current best-known policy or explore n... | Exploration-exploitation dilemma | 0.807464 |
25,757 | Quantum algebra is one of the top-level mathematics categories used by the arXiv. It is the study of noncommutative analogues and generalizations of commutative algebras, especially those arising in Lie theory.Subjects include: Quantum groups Skein theories Operadic algebra Diagrammatic algebra Quantum field theory Rac... | Quantum algebra | 0.807464 |
25,758 | Randomness extractors are used widely in cryptographic applications, whereby a cryptographic hash function is applied to a high-entropy, but non-uniform source, such as disk drive timing information or keyboard delays, to yield a uniformly random result. Randomness extractors have played a part in recent developments i... | Randomness extractor | 0.807462 |
25,759 | The value of k {\displaystyle k} is calculated by using the definition of the extractor, where we know: ( n , k ) = ( n , δ n ) ⇒ k = δ n {\displaystyle (n,k)=(n,\delta n)\Rightarrow k=\delta n} and by using the value of m {\displaystyle m} we have: m = δ 2 n = n 2 γ {\displaystyle m=\delta ^{2}n=n^{2\gamma }} Using th... | Randomness extractor | 0.807462 |
25,760 | The lemma is proved by examining the distance from uniform of the output, which in a 2 − m ϵ ( n ) {\displaystyle 2^{-m}\epsilon (n)} -extractor obviously is at most 2 − m ϵ ( n ) {\displaystyle 2^{-m}\epsilon (n)} , which satisfies the condition of the APRF. The lemma leads to the following theorem, stating that there... | Randomness extractor | 0.807462 |
25,761 | For this purpose Almost-Perfect Resilient Functions (APRF) are used. The definition of an APRF is as follows: Definition (k-APRF): A k = k ( n ) {\displaystyle k=k(n)} APRF is a function f {\displaystyle f} where, for any setting of n − k {\displaystyle n-k} bits of the input r {\displaystyle r} to any fixed values, th... | Randomness extractor | 0.807462 |
25,762 | The following paragraphs define and establish an important relationship between two kinds of ERF--k-ERF and k-APRF--which are useful in Exposure-Resilient cryptography. Definition (k-ERF): An adaptive k-ERF is a function f {\displaystyle f} where, for a random input r {\displaystyle r} , when a computationally unbounde... | Randomness extractor | 0.807462 |
25,763 | For example, if the source is known but the seed is not known (or vice versa). This property of extractors is particularly useful in what is commonly called Exposure-Resilient cryptography in which the desired extractor is used as an Exposure-Resilient Function (ERF). Exposure-Resilient cryptography takes into account ... | Randomness extractor | 0.807462 |
25,764 | One of the most important aspects of cryptography is random key generation. It is often necessary to generate secret and random keys from sources that are semi-secret or which may be compromised to some degree. By taking a single, short (and secret) random key as a source, an extractor can be used to generate a longer ... | Randomness extractor | 0.807462 |
25,765 | Kazhdan's theorem says the following: | Arithmetic variety | 0.807461 |
25,766 | In mathematics, an arithmetic variety is the quotient space of a Hermitian symmetric space by an arithmetic subgroup of the associated algebraic Lie group. | Arithmetic variety | 0.807461 |
25,767 | == V Curve == In power engineering & electrical engineering, V curve is the graph showing the relation of armature current as a function of field current in synchronous machines keeping the load constant. The purpose of the curve is to show the variation in the magnitude of the armature current as the excitation voltag... | V curve | 0.80746 |
25,768 | Marks' Standard Handbook for Mechanical Engineers is a comprehensive handbook for the field of mechanical engineering. Originally based on the even older German Hütte, it was first published in 1916 by Lionel Simeon Marks. In 2017, its 12th edition, published by McGraw-Hill, marked the 100th anniversary of the work. Th... | Marks' Standard Handbook for Mechanical Engineers | 0.807456 |
25,769 | Readily-accessible debug interfaces are not available in the product's final form factor. This hampers the identification of bugs and performance optimization in the end product. Since the debug logic is still present in the end product, an alternative access path is needed. An effective way is to equip a mobile termin... | MIPI Debug Architecture | 0.807454 |
25,770 | The MIPI System Trace Protocol (MIPI STP) specifies a generic protocol that allows the merging of trace streams originated from anywhere in the SoC to a trace stream of 4-bit frames. It was intentionally designed to merge system trace information. The MIPI System Trace Protocol uses a channel/master topology that allow... | MIPI Debug Architecture | 0.807453 |
25,771 | Several layers of superveniency are possible. Chemical properties can be used for building chemical classifications. They can also be useful to identify an unknown substance or to separate or purify it from other substances. Materials science will normally consider the chemical properties of a substance to guide its ap... | Chemical properties | 0.807453 |
25,772 | Chemical properties can be contrasted with physical properties, which can be discerned without changing the substance's structure. However, for many properties within the scope of physical chemistry, and other disciplines at the boundary between chemistry and physics, the distinction may be a matter of researcher's per... | Chemical properties | 0.807453 |
25,773 | If m(B) is finite, the invariant n(B) is the number of atoms of Bm(B) if this number is finite, or ∞ if this number is infinite. The invariant l(B) is 0 if Bm(B) is atomic or if m(B) is ∞, and 1 otherwise.Then two Boolean algebras are elementarily equivalent if and only if their invariants l, m, and n are the same. In ... | First-order theories | 0.807453 |
25,774 | the ideal I(B) consists of elements that are the sum of an atomic and an atomless element (an element with no atoms below it). The quotient algebras Bi of B are defined inductively by B0=B, Bk+1 = Bk/I(Bk). The invariant m(B) is the smallest integer such that Bm+1 is trivial, or ∞ if no such integer exists. | First-order theories | 0.807453 |
25,775 | We write x ≤ y as an abbreviation for x∧y = x, and atom(x) as an abbreviation for ¬x = 0 ∧ ∀y y ≤ x → y = 0 ∨ y = x, read as "x is an atom", in other words a non-zero element with nothing between it and 0. Here are some first-order properties of Boolean algebras: Atomic: ∀x x = 0 ∨ ∃y y ≤ x ∧ atom(y) Atomless: ∀x ¬atom... | First-order theories | 0.807453 |
25,776 | Unfortunately, this convention clashes badly with the next convention: In algebra, the usual convention is that the language has two constants, 0 and 1, and two binary functions · and +. The function · has the same meaning as ∧, but a+b means a∨b∧¬(a∧b). The reason for this is that the axioms for a Boolean algebra are ... | First-order theories | 0.807453 |
25,777 | There are several different signatures and conventions used for Boolean algebras: The signature has two constants, 0 and 1, and two binary functions ∧ and ∨ ("and" and "or"), and one unary function ¬ ("not"). This can be confusing as the functions use the same symbols as the propositional functions of first-order logic... | First-order theories | 0.807453 |
25,778 | (x\vee z))=(x\vee y)\wedge (x\vee z)} (modular lattices)Heyting algebras can be defined as lattices with certain extra first-order properties. Completeness is not a first order property of lattices. | First-order theories | 0.807453 |
25,779 | Lattices can be considered either as special sorts of partially ordered sets, with a signature consisting of one binary relation symbol ≤, or as algebraic structures with a signature consisting of two binary operations ∧ and ∨. The two approaches can be related by defining a ≤ b to mean a∧b = a. For two binary operatio... | First-order theories | 0.807453 |
25,780 | Related techniques include product breakdown, systems analysis, systems engineering, value engineering, value analysis and functional analysis. Product breakdown: Recursively divide the product into components and subcomponents. | Product analysis | 0.807452 |
25,781 | "Skolem rejected the result as meaningless; Tarski very reasonably responded that Skolem's formalist viewpoint ought to reckon the downward Löwenheim-Skolem theorem meaningless just like the upward." – Hodges (1993). "Legend has it that Thoralf Skolem, up until the end of his life, was scandalized by the association ... | Löwenheim–Skolem theorem | 0.807451 |
25,782 | It is somewhat ironic that Skolem's name is connected with the upward direction of the theorem as well as with the downward direction: "I follow custom in calling Corollary 6.1.4 the upward Löwenheim-Skolem theorem. But in fact Skolem didn't even believe it, because he didn't believe in the existence of uncountable set... | Löwenheim–Skolem theorem | 0.807451 |
25,783 | He cited a note by Skolem, according to which the theorem had been proved by Alfred Tarski in a seminar in 1928. Therefore, the general theorem is sometimes known as the Löwenheim–Skolem–Tarski theorem. But Tarski did not remember his proof, and it remains a mystery how he could do it without the compactness theorem. | Löwenheim–Skolem theorem | 0.807451 |
25,784 | In a revisionist account, Badesa (2004) considers that Löwenheim's proof was complete. Skolem (1920) gave a (correct) proof using formulas in what would later be called Skolem normal form and relying on the axiom of choice: Every countable theory which is satisfiable in a model M, is satisfiable in a countable substruc... | Löwenheim–Skolem theorem | 0.807451 |
25,785 | This account is based mainly on Dawson (1993). To understand the early history of model theory one must distinguish between syntactical consistency (no contradiction can be derived using the deduction rules for first-order logic) and satisfiability (there is a model). Somewhat surprisingly, even before the completeness... | Löwenheim–Skolem theorem | 0.807451 |
25,786 | The Löwenheim–Skolem theorem shows that these axiomatizations cannot be first-order. For example, in the theory of the real numbers, the completeness of a linear order used to characterize R as a complete ordered field, is a non-first-order property. Another consequence that was considered particularly troubling is the... | Löwenheim–Skolem theorem | 0.807451 |
25,787 | Let N denote the natural numbers and R the reals. It follows from the theorem that the theory of (N, +, ×, 0, 1) (the theory of true first-order arithmetic) has uncountable models, and that the theory of (R, +, ×, 0, 1) (the theory of real closed fields) has a countable model. There are, of course, axiomatizations char... | Löwenheim–Skolem theorem | 0.807451 |
25,788 | Later, in 1931, the hope was shattered completely by Gödel's incompleteness theorem. Many consequences of the Löwenheim–Skolem theorem seemed counterintuitive to logicians in the early 20th century, as the distinction between first-order and non-first-order properties was not yet understood. One such consequence is the... | Löwenheim–Skolem theorem | 0.807451 |
25,789 | The statement given in the introduction follows immediately by taking M to be an infinite model of the theory. The proof of the upward part of the theorem also shows that a theory with arbitrarily large finite models must have an infinite model; sometimes this is considered to be part of the theorem.A theory is called ... | Löwenheim–Skolem theorem | 0.807451 |
25,790 | In its general form, the Löwenheim–Skolem Theorem states that for every signature σ, every infinite σ-structure M and every infinite cardinal number κ ≥ |σ|, there is a σ-structure N such that |N| = κ and such that if κ < |M| then N is an elementary substructure of M; if κ > |M| then N is an elementary extension of M.T... | Löwenheim–Skolem theorem | 0.807451 |
25,791 | First, one extends the signature by adding a new constant symbol for every element of M. The complete theory of M for the extended signature σ' is called the elementary diagram of M. In the next step one adds κ many new constant symbols to the signature and adds to the elementary diagram of M the sentences c ≠ c' for a... | Löwenheim–Skolem theorem | 0.807451 |
25,792 | Badesa, Calixto (2004), The Birth of Model Theory: Löwenheim's Theorem in the Frame of the Theory of Relatives, Princeton, NJ: Princeton University Press, ISBN 978-0-691-05853-5; A more concise account appears in chapter 9 of Leila Haaparanta, ed. (2009), The Development of Modern Logic, Oxford University Press, ISBN 9... | Löwenheim–Skolem theorem | 0.807451 |
25,793 | Although the (classical) Löwenheim–Skolem theorem is tied very closely to first-order logic, variants hold for other logics. For example, every consistent theory in second-order logic has a model smaller than the first supercompact cardinal (assuming one exists). The minimum size at which a (downward) Löwenheim–Skolem–... | Löwenheim–Skolem theorem | 0.807451 |
25,794 | ), Cambridge, Massachusetts: Harvard University Press, pp. 290–301, ISBN 0-674-32449-8 (online copy, p. 290, at Google Books) Skolem, Thoralf (1929), "Über einige Grundlagenfragen der Mathematik", Skrifter Utgitt av Det Norske Videnskaps-Akademi I Oslo, I. Matematisk-naturvidenskabelig Klasse, 7: 1–49 Veblen, Oswald (1... | Löwenheim–Skolem theorem | 0.807451 |
25,795 | Löwenheim, Leopold (1915), "Über Möglichkeiten im Relativkalkül" (PDF), Mathematische Annalen, 76 (4): 447–470, doi:10.1007/BF01458217, ISSN 0025-5831, S2CID 116581304 Löwenheim, Leopold (1977), "On possibilities in the calculus of relatives", From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.... | Löwenheim–Skolem theorem | 0.807451 |
25,796 | As a consequence, first-order theories are unable to control the cardinality of their infinite models. The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic. In general, the Löwenheim–Skolem th... | Löwenheim–Skolem theorem | 0.807451 |
25,797 | In mathematical logic, the Löwenheim–Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Löwenheim and Thoralf Skolem. The precise formulation is given below. It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has... | Löwenheim–Skolem theorem | 0.807451 |
25,798 | In organic chemistry, an alkyl group is an alkane missing one hydrogen. The term alkyl is intentionally unspecific to include many possible substitutions. An acyclic alkyl has the general formula of −CnH2n+1. | Tertiary alkyl | 0.807451 |
25,799 | Although the Reuleaux triangle is not smooth, curves of constant width can always be approximated arbitrarily closely by smooth curves of the same constant width. Cylinders with constant-width cross-section can be used as rollers to support a level surface. Another application of curves of constant width is for coinage... | Curve of constant width | 0.80745 |
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