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Topic: Basis (mathematics)

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Technical Reports 2002 We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig's approach to the canonical basis and the string parametrization of the canonical basis, are given on a large region of linearity by linear functions arising from these monomial bases for a quantized enveloping algebra. In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author arising from representations of quivers. The seemingly distinct areas of group theory, formal language theory, and complexity theory interact in an important way when one considers decision problems in groups, such as the question of whether a word in the generators of the group represents the identity or not. www.mcs.le.ac.uk /research/publications/TechReports02.html   (5276 words)

Hamel Hamel was appointed to the chair of mechanics at the Rheinisch-Westfälische Hochschule in Aachen on 1 October 1912, then in 1919 he moved to the Technical University of Charlottenburg in Berlin where he was appointed as professor of mathematics and mechanics. Hamel was clearly associated with the views of National Socialism and in 1933 spoke of a spiritual bond between mathematics and the "Third Reich". Hamel worked in function theory, mechanics and the foundations of mathematics. www.gap-system.org /~history/Mathematicians/Hamel.html   (926 words)

PlanetMath: linear algebra Linear algebra is the branch of mathematics devoted to the theory of linear structure. From the geometric point of view, “linear” is synonymous with “straight”, and consequently linear algebra can be regarded as the branch of mathematics dealing with lines and planes, as well as with transformations of space that preserve “straightness”, e.g. Linearity is a very basic notion, and consequently linear algebra has applications in numerous areas of mathematics, science, and engineering. planetmath.org /encyclopedia/LinearAlgebra.html   (482 words)

Bengt Fornberg Observations on the behavior of radial basis functions near boundaries (BF, T.A. Driscoll, G. Wright and R. Charles), Computers and Mathematics with Applications 43(2002), 473-490. A new class of oscillatory radial basis functions (BF, E. Larsson and G. Wright), to appear in Computers and Mathematics with Applications. A numerical study of some radial basis function based solution methods for elliptic PDEs (E. Larsson and BF), Computers and Mathematics with Applications, 46 (2003), 891-902. amath.colorado.edu /faculty/fornberg   (825 words)

CITIDEL: Viewing 'On the power of polynomial discriminators and radial basis function networks' M.J.D.Powell, The Theory of Radial Basis Function Approximation in 1990. Adaptive radial basis function nonlinearities and the problem of generalization. Technical report #DAMTP 1990/NAll, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, December 1990. www.citidel.org /?op=getobj&identifier=oai:ACMDL:articles.168324   (676 words)

Functional Data Analysis - The Basics of Basis Functions - Page1 A system developed in early nineteenth century France is the Fourier basis, consisting of 1 and a series of pairs of sines and cosines of increasing frequency; that is, 1, sin(ωt), cos(ωt), sin(2ωt), cos(2ωt), sin(3ωt), cos(3ωt),.... There are plenty of other basis systems around, and it is worth mentioning wavelets at this point, without ordering any description of this system in these introductory notes. Figure 1 shows the first five basis functions in the series with a period of one unit. www.psych.mcgill.ca /misc/fda/ex-basis-a1.html   (676 words)

Articles - Harmonic Harmonics may be used as the basis of just intonation systems or considered as the basis of all just intonation systems. In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. Harmonics may be used to check at a unison the tuning of strings which are not tuned to the unison. www.storegolf.com /articles/Harmonic   (676 words)

Romance Of Mathematics mathematics is the basis of any reality, conceivable or inconceivable, as it is the basis for physical reality... Interestingly, while they reject the beliefs about mathematics outlined above, they are adamant that the embodied mind theory is neither relativist (where the romance of mathematics is "absolutist"), nor postmodern (where the romance of mathematics is profoundly modernist). Human knowledge of mathematics is neither abstract nor transcendent, and may well be dependent on the innate ability of subitizing and developed by means of metaphorical thinking. clublet.com /why?page=RomanceOfMathematics   (676 words)

Mathematics Archives - Topics in Mathematics - Linear Algebra Linear Transformations in 2-Dimensions, Products of Linear Transformations in 2-Dimensions, Linear Transformations in 3-Dimensions, Products of Linear Transformations in 3-Dimensions, Eigenvalues and the Characteristic Polynomial, Effect of a Linear Transformation on its Eigenvectors, Change of Basis Course materials, lecture notes, linear functions, linear algebra review, orthonormal vectors and QR factorization, least-squares methods, regularized least-squares and minimum norm methods, autonomous linear dynamical systems, eigenvectors and diagonalization, Jordan canonical form, aircraft dynamics, symmetric matrices, quadratic forms, matrix norm, and SVD, quantum mechanics, controllability and state transfer Worksheets, Introduction to Matrices, Definitions, Matrix arithmetic, Identity matrices, Inverse matrices, Finding eigenvalues and eigenvectors, Using eigenvalues and eigenvectors to solve differential equations and discrete systems. archives.math.utk.edu /topics/linearAlgebra.html   (676 words)

Functional Data Analysis - The Basics of Basis Functions - Page1 A system developed in early nineteenth century France is the Fourier basis, consisting of 1 and a series of pairs of sines and cosines of increasing frequency; that is, 1, sin(ωt), cos(ωt), sin(2ωt), cos(2ωt), sin(3ωt), cos(3ωt),.... Figure 1 shows the first five basis functions in the series with a period of one unit. By "stacking" in mathematics we mean adding things, possibly after multiplying each of them by its own constant. www.psych.mcgill.ca /misc/fda/ex-basis-a1.html   (558 words)

Geoff's Website—Science of the Soul Article Either way, however—and even we continue to narrowly regard mathematics as merely a human conceptual construct—the ideas that concepts cannot express paradox, or that paradox conversely arises from the "dualistic nature of all concepts," are incorrect, and so cannot be the basis of any valid understanding as to what duality or transcendence are. Conversely, as we shall see, the most interesting "paradoxical" descriptions of transcendence have their basis, not in the "dualistic nature of all concepts," but rather in accurate mystical reportings of the subtle cosmic structures underlying human consciousness. If one follows the mathematics of that through, it turns out that higher levels of consciousness will initially expand, as literal spheres of consciousness, farther into space, then contracting intermittently to a point more frequently, in the interval following the impulse, than will lower states. www.angelfire.com /trek/geoffreyfalk/books/sosarticle.html   (558 words)

Mathematics Graduate Financial Support Since the number of graduate students eligible for support exceeds the number of available positions, support is allocated competitively on a year to year basis. Financial support is available on a competitive basis in the form of Campus Fellowships, Teaching Assistantships and Associateships, Readerships, Graduate Student Research positions, Non-Resident Tuition Fellowships, and University Financial Aid (loans and work-study for domestic students only, based on need). Students apply every quarter in the Mathematics Department office in advance of the quarter for which they wish to read. www.math.ucsb.edu /grad/financial.php   (1061 words)

Josiah Royce: The Problem of Truth in Light of Recent Discussion The new logic to which I refer is especially a study of the logic of mathematics. But the methods that they used proved to be inapplicable to many of the problems of modern mathematics. At the basis of every mathematical theory,--as, for instance, at the basis of pure geometry, or pure number theory,--one finds a set of fundamental concepts, the so-called "indefinables" of the theory in question, and a set of fundamental "propositions," the so-called "axioms" of this theory. spartan.ac.brocku.ca /~lward/Royce/Royce_1908.html   (10569 words)

The Kinematics of Mixing - Cambridge University Press In spite of its universality, mixing is poorly understood and generally speaking, mixing problems are attacked on a case-by-case basis. The author’s aim is to provide a conceptually clear basis from which to launch analysis and to facilitate an understanding of the numerous mixing problems encountered in nature and technology. Series: Cambridge Texts in Applied Mathematics (No. 3) www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521368782   (279 words)

Mathematics You should be able to use mathematics at an ABE Advanced level with competence: In addition, you should be able to use foundational algebraic skills to solve problems and use your knowledge of algebra as a basis for further study in pre-calculus, the sciences, and other fields. The goal of Advanced Business/Technical Mathematics is to provide you with practical applications useful in future vocational training, careers, or personal life. In addition you should be able to use foundational algebraic skills to solve problems and have the ability to apply your knowledge of algebra as a basis for further study in algebra, statistics, business, and other programs. www.bccampus.ca /Page141.aspx   (279 words)

Geometry & Topology Publications The purpose of all Geometry and Topology Publications is the advancement of mathematics. Editors evaluate submissions strictly on the basis of scientific merit, without regard to authors' nationality, country of residence, institutional affiliation, sex, ethnic origin and political views. GTP is based in the Mathematics Department of the University of Warwick at Coventry, UK. www.maths.warwick.ac.uk /gt/gtp.html   (146 words)

UC Berkeley Mathematics It is known from algebraic methods that the canonical basis is compatible with this filtration. We will demonstrate that using our new geometric realization of Demazure modules, one can show that the semicanonical basis is also compatible with the Demazure filtration, thus adding to its list of nice properties. Quiver varieties can be used to provide a geometric realization of representations of simply laced Kac-Moody algebras. math.berkeley.edu /calendar-event633.html   (170 words)

God Constrained By Mathematics If instead, mathematics is the basis of any reality, conceivable or inconceivable, as it is the basis for physical reality, then the notion of a god unconstrained by mathematics is gibberish. Mathematics is partly empirical, it discovers facts about this universe, but these facts do not, in general have any connection with each other. So the structures and relations seen in the mathematics we study, most of which are ultimately derived from observation, are unrepresentative of mathematics (about almost all of which we know nothing as yet). clublet.com /why?GodConstrainedByMathematics   (353 words)

Electromagnetic conic sections curl in terms of a Cartesian basis, an introductory mathematics However, if the basis vectors were both curl-free First of all, the rectangular basis { î, www.physics.oregonstate.edu /bridge/papers/ConicSections.html   (353 words)

Citations: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties - Mnev (ResearchIndex) ....2 First is the observation of Gelfand Goresky MacPherson Serganova that the matroid stratification is precisely the coarsest common refinement of all of the Schubert cell decompositions given by the standard basis and permutations of the standard basis. In O.Y. Viro, ed., Topology and Geometry - Rohlin Seminar, LN Mathematics, 1346, pp. Viro, editor), Lecture Notes in Mathematics, volume 1346, pages 527--543. citeseer.ist.psu.edu /context/217609/0   (3610 words)

harmonic Harmonics may be used as the basis of just intonation systems or considered as the basis of all just intonation systems. In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integral multiple of the fundamental frequency. Harmonics may be used to check at a unison the tuning of strings which are not tuned to the unison. www.fact-library.com /harmonic.html   (544 words)

Maths All this will be under-pinned by detailed planning on a termly and weekly basis. Numeracy homework will be used to provide extension, support, reinforcement, revision and enjoyment on a regular basis from year 2 onwards When planning the mathematics curriculum teachers will follow the National Curroculum for mathematics as interpreted in the National Numeracy Straegy. www.greatworth.northants.sch.uk /Curriculum/Maths.html   (544 words)

Geometry.Net - Theorems And Conjectures Books: Algebra Theorem A student who uses this text will have seen a lot of really interesting mathematics by the end and will have a sense of how the mathematics can be used. This title compiles and synthesizes a wealth of definitions, theorems, and techniques classically used in asymptotic theory of estimators. As a graduate student in statistics, I have had plenty of opportunities to browse through the pages of this book in search of that theorem, technique, or tool that would be applicable to my own work. www.geometry.net /theorems_and_conjectures_bk/algebra_theorem.html   (544 words)

A Linear Algebra View of the Wavelet Transform A wavelet basis function refers to the number of coefficients in the scaling and wavelet function. The wavelet literature in some cases seems to be written for those with a graduate level background in mathematics. As with any field of specialty, mathematics assumes a shared basic set of knowledge, which includes areas like linear algebra. www.bearcave.com /misl/misl_tech/wavelets/matrix   (1660 words)

My World of Linear Algebra In my own work I've encountered many flavors of mathematics (group theory, ring theory, commutative algebra, applied mathematics and numerical analysis, etc) and a common thread throughout my mathematical experiences has been linear algebra. Maple Notebooks Tutorial notebooks in Maple, some of which are the basis for linear algebra projects. Linear Algebra Octave Files Program files for a Matlab-like program called Octave which I have found very useful in linear algebra. www.math.unl.edu /~tshores1/mylinalg.html   (816 words)

Amazing Science The Indian numeral system and its place value, decimal system of enumeration came to the attention of the Arabs in the seventh or eighth century, and served as the basis for the well known advancement in Arab mathematics, represented by figures such as al-Khwarizmi. G G Joseph, in The crest of the peacock argues that the algebraic nature of Indian mathematics arises as a consequence of the structure of the Sanskrit language. An indirect consequence of Panini's efforts to increase the linguistic facility of Sanskrit soon became apparent in the character of scientific and mathematical literature. www.hinduism.co.za /amazing.htm   (816 words)

Mathematics - Simple English Wikipedia Mathematical prediction is the basis of hard science, which almost always uses equations to predict events in physics. Mathematics (math or maths for short) is the study of quantity (how many things there are), structure (how things are organized), space (where things are) and change. The philosophy of mathematics explores these issues, and others regarding how mathematics fits in with philosophy, ethics and real life. simple.wikipedia.org /wiki/Mathematics   (807 words)

Basis - Wikipedia, the free encyclopedia In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. In economics or accounting, a basis is the cost or value of an asset as adjusted for tax purposes. This is a disambiguation page, a list of pages that otherwise might share the same title. en.wikipedia.org /wiki/Basis   (807 words)

Hilbert's basis theorem Hilbert produced an innovative proof by contradiction using mathematical induction ; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. In mathematics, Hilbert's basis theorem, first proved by David Hilbert in 1888, states that, if ''k is a field, then every ideal in the ring of multivariate polynomials k''[''x A slightly more general statement of Hilbert's basis theorem is: if R is a left (respectively right) Noetherian ring, then the polynomial ring R''[''X''] is also left (respectively right) Noetherian. www.serebella.com /encyclopedia/article-Hilbert's_basis_theorem.html   (807 words)

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