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In this paper, we introduce the notion of commutator of two elements in a specific NeutroGroup. Then we define the notion of a NeutroNilpotentGroup and we study some of their properties. Moreover, we show that the intersection of two NeutroNilpotentGroups is a NeutroNilpotentGroup. Also, we show that the quotient of a NeutroNilpotentGroup is a NeutroNilpotentGroup. Specially, using NeutroHomomorphism we prove the NeutroNilpotentcy is closed with respect to homomorphic image.
Journals
by Florentin Smarandache
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In all classical algebraic structures, the Laws of Compositions on a given set are well-defined. But this is a restrictive case, because there are many more situations in science and in any domain of knowledge when a law of composition defined on a set may be only partially-defined (or partially true) and partially-undefined (or partially false), that we call NeutroDefined, or totally undefined (totally false) that we call AntiDefined.
Journals
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This paper introduces the novel concept of Neutro-BCK-algebra. In Neutro-BCK-algebra, the outcome of any given two elements under an underlying operation (neutro-sophication procedure) has three cases, such as: appurtenance, non-appurtenance, or indeterminate. While for an axiom: equal, non-equal, or indeterminate. This study investigates the Neutro-BCK-algebra and shows that Neutro-BCK-algebra are different from BCKalgebra.
Journals
by F. Smarandache
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In any science, a classical Theorem, defined on a given space, is a statement that is 100% true (i.e. true for all elements of the space). To prove that a classical theorem is false, it is sufficient to get a single counter-example where the statement is false. Therefore, the classical sciences do not leave room for partial truth of a theorem (or a statement). But, in our world and in our everyday life, we have many more examples of statements that are only partially true, than statements that...
Journals
by M.A. Ibrahim; A.A.A. Agboola
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NeutroSophication and AntiSophication are processes through which NeutroAlgebraic and AntiAlgebraic structures can be generated from any classical structures.
Journals
by Madeleine Al-Tahan; F. Smarandache; Bijan Davvaz
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Starting with a partial order on a NeutroAlgebra, we get a NeutroStructure. The latter if it satisfies the conditions of NeutroOrder, it becomes a NeutroOrderedAlgebra. In this paper, we apply our new defined notion to semigroups by studying NeutroOrderedSemigroups. More precisely, we define some related terms like NeutrosOrderedSemigroup, NeutroOrderedIdeal, NeutroOrderedFilter, NeutroOrderedHomomorphism, etc., illustrate them via some examples, and study some of their properties.
Journals
by A.A.A. Agboola
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NeutroRings are alternatives to the classical rings and they are of different types. NeutroRings in some cases exhibit different algebraic properties, and in some cases they exhibit algebraic properties similar to the classical rings. The objective of this paper is to revisit the concept of NeutroRings and study finite and infinite NeutroRings of type-NR.
Journals
by F. Smarandache; A. Rezaei; S. Mirvakilii
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As generalizations and alternatives of classical algebraic structures Florentin Smarandache has introduced in 2019 the NeutroAlgebraic structures (or NeutroAlgebras) and AntiAlgebraic structures (or AntiAlgebras). Unlike the classical algebraic structures, where all operations are well-defined and all axioms are totally true, in NeutroAlgebras and AntiAlgebras the operations may be partially well-defined and the axioms partially true or respectively totally outerdefined and the axioms totally...
Journals
by Madeleine Al-Tahan; Bijan Davvaz; Florentin Smarandache
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Neutrosophy, the study of neutralities, is a new branch of Philosophy that has applications in many different fields of science. Inspired by the idea of Neutrosophy, Smarandache introduced NeutroAlgebraicStructures (or NeutroAlgebras) by allowing the partiality and indeterminacy to be included in the structuresí operations and/or axioms. The aim of this paper is to combine the concept of Neutrosophy with hyperstructures theory. In this regard, we introduce NeutroSemihypergroups as well as...
Journals
by A.A.A. Agboola; M.A. Ibrahim; E.O. Adeleke
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The objective of this paper is to examine NeutroAlgebras and AntiAlgebras viz-a-viz the classical number systems.
Journals
by Florentin Smarandache
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In this paper we present the development from paradoxism to neutrosophy, which gave birth to neutrosophic set and logic and especially to NeutroAlgebraic Structures (or NeutroAlgebras) and AntiAlgebraic Structures (or AntiAlgebras) that are generalizations and alternatives of the classical algebraic structures.
Journals
by Florentin Smarandache; Akbar Rezaei; Hee Sik Kim
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In this paper, as an extension of CI-algebras, we discuss the new notions of Neutro-CI-algebras and Anti-CI-algebras. First, some examples are given to show that these definitions are different. We prove that any proper CI-algebra is a Neutro-BE-algebra or Anti-BE-algebra. Also, we show that any NeutroSelf-distributive and AntiCommutative CIalgebras are not BE-algebras.
Journals
by A. Rezaei; F. Smarandache; S. Mirvakili
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In this paper, we extend the notion of semi-hypergroups (resp. hypergroups) to neutro-semihypergroups (resp. neutrohypergroups). We investigate the property of anti-semihypergroups (resp. anti-hypergroups). We also give a new alternative of neutro-hyperoperations (resp. anti-hyperoperations), neutro-hyperoperation-sophications (resp. anti-hypersophications). Moreover, we show that these new concepts are different from classical concepts by several examples.
Journals
by A.A.A. Agboola
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In this paper, we are going to study a class of NeutroGroups of type-NG[1,2,4]. In this class of NeutroGroups, the closure law, the axiom of associativity and existence of inverse are taking to be either partially true or partially false for some elements; while the existence of identity element and axiom of commutativity are taking to be totally true for all the elements.
Journals
by A.A.A. Agboola
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The objective of this paper is to introduce the concept of NeutroRings by considering three NeutroAxioms (NeutroAbelianGroup (additive), NeutroSemigroup (multiplicative) and NeutroDistributivity (multiplication over addition)). Several interesting results and examples on NeutroRings, NeutroSubgrings, NeutroIdeals, NeutroQuotientRings and NeutroRingHomomorphisms are presented. It is shown that the 1st isomorphism theorem of the classical rings holds in the class of NeutroRings.
Journals
by Diego Silva JimÈnez; Juan Alexis Valenzuela Mayorga; Mara Esther Roja Ubilla
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In any science, a classical Theorem, defined on a given space, is a statement that is 100% true (i.e. true for all elements of the space). To prove that a classical theorem is false, it is sufficient to get a single counter-example where the statement is false. Therefore, the classical sciences do not leave room for partial truth of a theorem (or a statement). But, in our world and in our everyday life, we have many more examples of statements that are only partially true, than statements that...
Journals
by Akbar Rezaei; Florentin Smarandache
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In this paper, the concepts of Neutro-algebra and Anti-algebra are introduced, and some related properties and four theoremsare investigated. We show that the classes of Neutro-algebra and Anti-algebras are alternatives of the class of algebras.
Journals
by A.A.A. Agboola
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The notion of AntiGroups is formally presented in this paper. A particular class of AntiGroups of type-AG is studied with several examples and basic properties presented. In AntiGroups of type-AG, the existence of an inverse is taking to be totally false for all the elements while the closure law, the existence of identity element, the axioms of associativity and commutativity are taking to be either partially true, partially indeterminate or partially false for some elements
Journals
by Florentin Smarandache
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We recall and improve our 2019 concepts of n-Power Set of a Set, n-SuperHyperGraph, Plithogenic n-SuperHyperGraph, and n-ary HyperAlgebra, n-ary NeutroHyperAlgebra, n-ary AntiHyperAlgebra respectively, and we present several properties and examples connected with the real world.
Journals
by A.A.A. Agboola
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The objective of this paper is to formally present the concept of NeutroGroups by considering three NeutroAxioms (NeutroAssociativity, existence of NeutroNeutral element and existence of NeutroInverse element). Several interesting results and examples of NeutroGroups, NeutroSubgroups, NeutroCyclicGroups, NeutroQuotientGroups and NeutroGroupHomomorphisms are presented.
Journals
by Florentin Smarandache
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In all classical algebraic structures, the Axioms (Associativity, Commutativity, etc.) defined on a set are totally true, but it is again a restrictive case, because similarly there are numerous situations in science and in any domain of knowledge when an Axiom defined on a set may be only partially-true (and partially-false), that we call NeutroAxiom, or totally false that we call AntiAxiom. Therefore, we open for the first time in 2019 new fields of research called NeutroStructures and...
Journals
by Florentin Smarandache
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In this paper we recall, improve, and extend several definitions, properties and applications of our previous 2019 research referred to NeutroAlgebras and AntiAlgebras, also called NeutroAlgebraic Structures and respectively AntiAlgebraic Structures
Neutrosophic Statistics
by Soumyadip Dhar; Malay K. Kundu
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Neutrosophic Statistics
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In this talk, we firstly review some basic concepts related to neutrosophy. Also, we discuss NeutroAlgebra. Next, we present some of our results related to our new defined concept NeutroOrderedAlgebra and compare it to the well known concept of Ordered Algebra. Finally, we leave with some questions that open new research options in this field.
Neutrosophic Statistics
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Neutrosophic Statistics
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by Daniel H. Rothman
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The chaotic phenomena of the Lorenz equations may be exhibited by even simpler systems.
Topics: Maths, Dynamics and Relativity, Dynamical Systems, Mathematics
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by Tomasz S. Mrowka
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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
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by Chiang C. Mei
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Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
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by Chiang C. Mei
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Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
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by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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Introduction: Before we try to solve higher order equations with discontinuous or impulsive input we need to think carefully about what happens to the solution at the point of discontinuity.
Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs), Second...
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by Dmitry Panchenko
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Covariance of X and Y is de�ned as: ... Positive when both high or low in deviation
Topics: Maths, Statistics and Probability, Mathematics
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by Scott Aaronson
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Topics: Maths, Mathematics
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by Albert R. Meyer
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We need to know one more property of planar graphs in order to prove that planar graphs are 5-colorable.
Topics: Maths, Mathematics
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by Chiang C. Mei
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Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Fluid Dynamics,...
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by David Tong
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So far, we�ve only discussed strings propagating in �at spacetime. In this section we will consider strings propagating in di_erent backgrounds. This is equivalent to having di_erent CFTs on the world sheet of the string.
Topics: Physics, Particle Physics and Fields, General Theory of Fields and Particles, Strings and Branes,...
Neutrosophic Statistics
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PubMed Central
by Schmidt, Julius J; Lorenzen, Johan; Chatzikyrkou, Christos; Lichtinghagen, Ralf; Kielstein, Jan T
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This article is from BMC Pharmacology & Toxicology , volume 15 . Abstract Background: Lithium intoxication has potentially fatal neurologic and cardiac side effects. Extracorporeal removal can therefore be lifesaving. The dialysance of lithium is high as it is a small molecule. Comparable to its neighbor in the periodic table, sodium, its intracellular accumulation hampers its removal by renal replacement therapy, despite its favorable size. For this reason the combination of short...
Source: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4172392
PubMed Central
by Sato, Hiroe; Kobayashi, Daisuke; Abe, Asami; Ito, Satoshi; Ishikawa, Hajime; Nakazono, Kiyoshi; Murasawa, Akira; Kuroda, Takeshi; Nakano, Masaaki; Narita, Ichiei
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This article is from BMC Research Notes , volume 7 . Abstract Background: Multiple sclerosis is a relatively rare disease, and complications of multiple sclerosis and rheumatoid arthritis are much rarer. Since anti-tumor necrosis factor therapy increases exacerbations of multiple sclerosis, complications of demyelinating diseases contraindicate anti-tumor necrosis factor therapy. There have been few reports of anti-interleukin-6 receptor therapy for patients with rheumatoid arthritis...
Source: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4171561
PubMed Central
by Verdegem, Dries; Moens, Stijn; Stapor, Peter; Carmeliet, Peter
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This article is from Cancer & Metabolism , volume 2 . Abstract The stromal vasculature in tumors is a vital conduit of nutrients and oxygen for cancer cells. To date, the vast majority of studies have focused on unraveling the genetic basis of vessel sprouting (also termed angiogenesis). In contrast to the widely studied changes in cancer cell metabolism, insight in the metabolic regulation of angiogenesis is only just emerging. These studies show that metabolic pathways in endothelial...
Source: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4171726
Neutrosophic Statistics
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by Chiang C. Mei
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In this chapter we examine high-speed �ows of a viscous �uid. As a prelude, the limit ofinviscid �ows is brei�y discussed.
Topics: Physics, Classical Mechanics, Electromagnetism and Electromagnetic Radiation, Fluid Mechanics�,...
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by Kiran S. Kedlaya
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In this unit, we derive von Mangoldt�s formula estimating _(x) _ x in terms of the critical zeroes of the Riemann zeta function. This �nishes the derivation of a form of the prime number theorem with error bounds. It also serves as another good example of how to use contour integration to derive bounds on number-theoretic quantities; we will return to this strategy in the context of the work of Goldston-Pintz-Y�ld�r�m.
Topics: Maths, Algebra, Number Theory, Mathematics
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by Kiran S. Kedlaya
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Topics: Maths, Number Theory, Mathematics
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by K. Venkatram
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Topics: Maths, Linear Algebra and Geometry, Topology and Metric Spaces, Geometry, Differential Geometry,...
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by Emma Carberry
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Topics: Maths, Linear Algebra and Geometry, Geometry, Differential Geometry, Inverse Function Theorem,...
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by Dmitry Panchenko
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Topics: Maths, Analysis and Calculus, Statistics and Probability, Measure Theory, Probability,...
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by Andrew Snowden
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In this paper, we �rst develop and prove a special case of the Hurewicz theorem. We then give a few results from the theory of the higher homotopy groups. Finally, we state the full form of the Hurewicz theorem (without proof). We discuss some applications throughout the paper.
Topics: Maths, Algebra, Topology and Metric Spaces, Mathematics
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by Hung Cheng
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In this lecture and the next two lectures, we�ll briefly review partial differential equations (PDEs). As PDEs are much more difficult to solve than ODEs, we shall start with the simplest of PDEs, those of the first order.
Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
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by Katrin Wehrheim
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Topics: Maths, Analysis and Calculus, Topology and Metric Spaces, Analysis, Measure Theory, Cauchy...
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by Hung Cheng
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Solutions expanded around an irregular singular point are distinctive in one aspect: they are usually in the form of an exponential function times a Frobenius series. Due to the factor of the exponential function, a solution near an irregular singular point behaves very differently from that near a regular singular point. It may blow up exponentially, or vanish exponentially, or oscillate wildly.
Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...
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by Abhinav Kumar
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First, what is number theory? At the most basic level, it�s the study of the properties of the integers Z = {. . . , _2, _1, 0, 1, 2, . . . } or the natu-ral numbers N = {0, 1, 2, . . . }
Topics: Maths, Algebra, Number Theory, Mathematics
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by Vera Mikyoung Hur
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We give a comprehensive development of the theory of linear differential equations with constant coef�cients. We use the operator calculus to deduce the existence and uniqueness. We presents techniques for �nding a complete solution of the inhomogeneous equation from solu-tions of the homogeneous equation. We also give qualitative results on asymptotic stability.
Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...
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by Vera Mikyoung Hur
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Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
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by Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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Our goal in this session is to derive formulas for solving both homo_geneous and inhomogeneous �rst order linear ODE�s. For the inhomoge_neous equations we will use what are called integrating factors.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
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by Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
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by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
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by Pavel Etingof
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In this section, we begin a systematic development of representation theory of �nite groups.
Topics: Maths, Algebra, Representation Theory, Mathematics
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by Dmitry Panchenko
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Topics: Maths, Statistics and Probability, Mathematics
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by Richard Melrose
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I �rst discussed the de�nition of preHilbert and Hilbert spaces and proved Cauchy�s inequality and the parallelogram law.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
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by Albert R. Meyer
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So far we focused on probabilities of events �that you win the Monty Hall game; that you have a rare medical condition, given that you tested positive; . . . . Now we focus on quantitative questions: How many contestants must play the Monty Hall game until one of them �nally wins? . . . How long will this condition last? How much will I lose playing 6.042 games all day? Random variables are the mathematical tool for addressing such questions.
Topics: Maths, Mathematics
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by Dmitry Panchenko
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Topics: Maths, Statistics and Probability, Mathematics
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by Gilbert Strang
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One theme of this book is the relation of equations to minimum principles. To minimize P is to solve P' = 0. There may be more to it, but that is the main point. This section is also the opening to control theory � the modern form of the calculus of variations. Its constraints are di_erential equations, and Pontryagin�s maximum principle yields solutions.
Topics: Maths, Mathematics
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by Matthew Herman
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We can now define an abstract root system in a Eucledian space.
Topics: Maths, Linear Algebra and Geometry, Linear Algebra, Mathematics
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by Jared Speck
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In this section, we will study (scalar-valued) functions _ on R^(1+n). They are sometimes called (scalar-valued) �elds on R^(1+n)
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
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by Albert R. Meyer
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Recursive data types play a central role in programming. From a mathematical point of view, recursive data types are what induction is about. Recursive data types are speci�ed by recursive de�nitions that say how to build something from its parts.
Topics: Maths, Mathematics
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by Albert R. Meyer
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We will start the chapter with a motivating example involving annuities. Figuring out the value of the annuity will involve a large and nasty-looking sum. We will then describe several methods for �nding closed forms for all sorts of sums, including the annuity sums. In some cases, a closed form for a sum may not exist and so we will provide a general method for �nding good upper and lower bounds on the sum (which are closed form, of course).
Topics: Maths, Mathematics
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by Jeff Viaclovsky
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Convergence almost everywhere. Lebesgue�s dominated conver_gence theorem (LDCT) in the case of almost everywhere.
Topics: Maths, Analysis and Calculus, Analysis, Integration, Mathematics
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by Daniel Kleitman;Peter Shor
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We now explain the algorithm that Lempel and Ziv gave in a 1978 paper, and generally called LZ78. This is opposed to LZ77, an earlier algorithm which di_ered signi�cantly in the implementation details but is based on the same general idea. This idea is that if some text is not random, a substring that you see once is more likely to appear again than substrings you haven�t seen.
Topics: Maths, Algebra, Coding and Cryptography, Codes, Mathematics
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by Daniel Kleitman;Peter Shor
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Unfortunately, the OCW notes on Kuratowski�s theorem seem to have several things substantially wrong with the proof, and the notes from Prof. Kleitman�s website are too vague to be able to deduce the proof from them. I�m just going to type in the OCW notes, changing things to make the proofs correct.
Topics: Maths, Mathematics
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by Daniel H. Rothman
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Summary Finally, we summarize the characteristics of dissipative systems: � Energy not conserved. � Irreversible. � Contraction of areas (volumes) in phase space.
Topics: Maths, Dynamics and Relativity, Dynamical Systems, Mathematics
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by Daniel H. Rothman
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Topics: Maths, Dynamics and Relativity, Dynamical Systems, Mathematics
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by Kiran S. Kedlaya
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We�re going to use the language of category theory freely. Fortunately, it�s easy to learn because it corresponds naturally to the way you (hopefully) already think about mathemat_ical objects. (I could give a reference, but in fact you should be �ne just looking these things up in Wikipedia.)
Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics
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by Chiang C. Mei
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Topics: Physics, Classical Mechanics, Fluid Mechanics�, General Theory in Fluid Dynamics, Fluid Dynamics,...
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by Denis Auroux
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Topics: Maths, Linear Algebra and Geometry, Geometry of Manifolds, Mathematics
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by Tomasz S. Mrowka
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Topics: Maths, Linear Algebra and Geometry, Differential Geometry, Geometry of Manifolds, Smooth Manifold,...
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by Chiang C. Mei
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Topics: Physics, Classical Mechanics, Fluid Mechanics, General Theory in Fluid Dynamics, Fluid Dynamics,...
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by Jerry Griffiths
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It is �nally appropriate to review a series of results which are related to the analysis of exact solutions describing the collision of plane waves in a �at background, which is the main subject of this book. The results described in this chapter di_er from those given in previous chapters in a variety of ways. A number of interesting results have been obtained using di_erent approaches, by considering the background in region I tobe non-�at, or by considering alternative gravitational...
Topics: Physics, Special Relativity, General Relativity and Gravitation, Classical General Relativity,...
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by Peter Dourmashkin;Kate Scholberg
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In this demo we measured the value of Young�s modulus for a copper wire. The setup is shown in the �gure (not to scale!).
Topics: Physics, Classical Mechanics, Continuum Mechanics of Solids�, Stress, Physics
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by Joel Kamnitzer
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Topic: Maths
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by Victor Flynn
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These notes are modi�ed from previous versions (due to Neil Dummigan,Alan Lauder and Roger Heath-Brown) and have been recently revised by me. They draw mainly upon �A Classical Introduction to Modern Number Theory�, by Ireland and Rosen, and �Algebraic Number Theory�, by Stewart and Tall. While I take full responsibility for their current contents, considerable thanks are clearly due to Neil, Alan and Roger.
Topics: Maths, Mathematics
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by Dmitry Panchenko
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In this section we will make several connections between the Kantorovich-Rubinstein theorem and other classical objects. Let us start with the following classical inequality.
Topics: Maths, Analysis and Calculus, Statistics and Probability, Mathematics
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by Dmitry Panchenko
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Topics: Maths, Analysis and Calculus, Statistics and Probability, Measure Theory, Probability, ?-Algebras,...
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by Rodolfo R. Rosales
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When nonlinearities are "small" there are various ways one can exploit this fact - and the fact that the linearized problem can be solved exactly - to produce useful approximations to the solutions. We illustrate two of these techniques here, with examples from phase plane analysis: The Poincare-Lindstedt method and the (more exible) Two Timing method. This second method is a particular case of the Multiple Scales approximation technique, which is useful whenever the solution of a...
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Mathematics
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by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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De�nition: 1. The gain is de�ned to be the the ratio of the amplitude of the output sinusoid to the amplitude of the input sinusoid. 2. The phase lag is de�ned to be the angle by which the output sinusoid is shifted relative to the input sinusoid.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
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by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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In this section we will compute the unit impulse response as the limit of the responses to these box functions.
Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...
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by Scott Aaronson
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Of course the real question is: can quantum computers actually do something more e_ciently than classical computers? In this lecture, we�ll see why the modern consensus is that they can.
Topics: Maths, Mathematics
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by Jeff Viaclovsky
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In this course, we will mainly be concerned with the following problems: 1) Harmonic functions. 2) Heat equation. 3) Poisson Equation
Topics: Maths, Analysis and Calculus, Differential Equations (ODEs & PDEs), Mathematics
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by Arthur Mattuck;Haynes Miller;Jeremy Orloff;John Lewis
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Topics: Maths, Differential Equations (ODEs & PDEs), Ordinary Differential Equations (ODEs),...
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by John Lewis;Arthur Mattuck;Haynes Miller;Jeremy Orloff
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We say an eigenvalue _1 of A is repeated if it is a multiple root of the characteristic equation of A; in our case, as this is a quadratic equation, the only possible case is when _1 is a double real root.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
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by Vera Mikyoung Hur
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We discuss various techniques for solving inhomogeneous linear differential equations.
Topics: Maths, Differential Equations (ODEs & PDEs), Mathematics
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by Lawrence Baggett
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Abstract: We motivate our third definition of an integral over a curve by returning to physics. This definition is very much a real variable one, so that we think of the plane as R^2 instead of C. A connection between this real variable definition and the complex variable definition of a contour integral will emerge later.
Topics: Maths, Mathematics
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by Richard Melrose
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So, recall the de�nitions of a Lebesgue integrable function on the line (forming the linear space L^1(R)) and of a set of measure zero E _ R. The �rst thing we want to show is that the putative norm on L^1 does make sense.
Topics: Maths, Differential Equations (ODEs & PDEs), Partial Differential Equations (PDEs), Functional...
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by Jeff Viaclovsky
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Lebesgue measurable sets (with �nite outer measure). Let A _ Rn and __(A) < � (A has �nite outer measure). Then we write that ... and de�ne measure of A to be ...
Topics: Maths, Analysis and Calculus, Analysis, Integration, Mathematics
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by Tom Leighton
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This class will discuss several research problems that are related to the Internet. Each lecture will discuss How a particular Internet component or technology works today: Issues and problems with the current technology, Potential new lines of research, Formulation of concrete problems and solutions.
Topics: Maths, Mathematics
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by Michel X. Goemans
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Last lecture we covered matroid intersection, and de�ned matroid union. In this lecture we review the de�nitions of matroid intersection, and then show that the matroid intersection polytope is TDI. This is Chapter 41 in Schrijver�s book. Next we review matroid union, and show that unlike matroid intersection, the union of two matroids is again a matroid. This material is largely contained in Chapter 42 in Schrijver�s book. We leave testing independence in the union matroid for the next...
Topics: Maths, Optimization and Control, Statistics and Probability, Optimization, Probability,...
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by Rahul Hariharan
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Topics: Maths, Mathematics
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by Jonathan Kelner
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Today�s lecture covers three main parts: � Courant-Fischer formula and Rayleigh quotients � The connection of _2 to graph cutting � Cheeger�s Inequality
Topics: Maths, Linear Algebra and Geometry, Graph Theory, Optimization and Control, Vectors and Matrices,...
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by Abhinav Kumar
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Topics: Maths, Linear Algebra and Geometry, Algebraic Geometry, Mathematics 