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Topic: Irreducible (mathematics)

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	ipedia.com: Irreducible complexity Article (Site not responding. Last check: 2007-11-07)
		Irreducible complexity is not recognised by the wider scientific community who point to lack of evidence and problems with its philosophy.
		The argument from irreducible complexity is a descendant of the teleological argument for God (the argument from design or argument from complexity).
		Proponents of irreducible complexity argue that the transition from a reptile lung (bellows lung) to a bird lung (circulatory lung) is unlikely since intermediate stages would be a detriment to the organism.
	www.ipedia.com /irreducible_complexity.html (1842 words)

	Mathematics Other Homework Help
		Irreducible Polynomials - Show that there are exactly (p^2-p)/2 monic irreducible polynomials of degree 2 over Z_p, where p is any prime.
		Using the definition of irreducibility, Theorem: A polynomial of degree 2 or 3 is irr...
		Matrix irreducible representations - Let G be the group of matrices 1 x y 0 1 z 0 0 1 where x, y, z are elements of the finite field F_5.
	www.brainmass.com /homeworkhelp/math/other/37924 (204 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-07)
		Frequently, an irreducible representation in a topological vector space is called a topologically-irreducible representation.
		There are a number of concepts close to that of an irreducible representation, among them an operator-irreducible representation, and a completely-irreducible representation (one for which the family of operators forms a completely-irreducible set, cf.
		A completely-irreducible representation is (topologically) irreducible and operator irreducible; the converse assertions are not true, in general.
	eom.springer.de /i/i052630.htm (132 words)

	International Mathematics Olympiad
		The Bay Area Mathematical Olympiad (BAMO) is a contest for high school students sponsored jointly by the Mathematical Sciences Research Institute (MSRI), the American Institute of Mathematics (AIM), the University of California at Berkeley (UCB), and the University of San Francisco (USF).
		The Colorado Mathematical Olympiad (CMO) is the largest essay-type mathematical competition in the United States, with 600 to 1,000 participants competing annually for fine prizes.
		The Olimpíada Matemática Argentina (OMA) is the Argentinian Mathematics Olympiad.
	olympiads.win.tue.nl /imo/index.html (2153 words)

	Irreducible polynomial - Wikipedia, the free encyclopedia
		It is helpful to compare irreducible polynomials to prime numbers: prime numbers (together with the corresponding negative numbers of equal modulus) are the irreducible integers.
		Hence, all irreducible polynomials are of degree 1.
		The existence of irreducible polynomials of degree greater than one (without zeros in the original field) historically motivated the extension of that original number field so that even these polynomials can be reduced into linear factors: from rational numbers to real numbers and further to complex numbers.
	en.wikipedia.org /wiki/Irreducible_polynomial (773 words)

	Rose-Hulman Undergraduate Mathematics Conference ::: Invited Speakers, Their Bios and Topic Abstracts
		Dr. Holder began a career in mathematics in 1998 after completing Ph.D. studies at the University of Colorado at Denver under the tutelage of Dr. Harvey Greenberg.
		His primary interest was in applied mathematics and specifically in optimization, a field that intersects numerous scientific disciplines.
		The interaction between the mathematical field of optimization and the medical field of oncology has provided some fertile and fascinating research over the last decade.
	www.rose-hulman.edu /mathconf/invited.php (980 words)

	Senior Thesis Pages: Melissa Banister (Site not responding. Last check: 2007-11-07)
		We present the results of an investigation into the representation theory of the alternating and dihedral groups and explore how their irreducible representations can be distinguished with the use of class sums.
		The four class sums corresponding to the 2-cycles, 3-cycles, 4-cycles, and 5-cycles are sufficient to distinguish all of the irreducible representations of the symmetric group of order 41.
		A natural question to ask is whether the irreducible representations of other finite groups can be distinguished with a small number of class sums relative to the order of the group.
	www.math.hmc.edu /seniorthesis/archives/2004/mbanister (245 words)

	irreducible - Search Results - MSN Encarta
		- impossible to factor into lesser polynomials: in mathematics, used to describe a polynomial that cannot be factored into two polynomials of a lesser degree
		Associationism, in psychology, the theory that the mind learns by combining simple, irreducible elements through association.
		Dualism, in philosophy, the theory that the universe is explicable only as a whole composed of two distinct and mutually irreducible elements.
	encarta.msn.com /irreducible.html (102 words)

	UBC Mathematics Department - Colloquium (Site not responding. Last check: 2007-11-07)
		In particular, the algorithms lay out the decomposition of the solution set into irreducible components, and give upper bounds for the multiplicities of the irreducible components, with the upper bound of an irreducible component equal to one if and only if the irreducible component is reduced.
		The basic data in a numerical primary decomposition are generic points, that certify the existence of irreducible components of the solution set their dimensions, and their degrees.
		The algorithms, which make essential use of generic projection and interpolation, can also be used to produce, for each irreducible component of the reduced solution set, a finite number of polynomials that vanish precisely on the irreducible component.
	www.math.ubc.ca /Dept/Events/colloquia/sommese.html (231 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-07)
		An explicit construction of all irreducible rational representations can be obtained in the following way.
		To obtain irreducible rational representations, one often applies the above-mentioned operations to given rational representations.
		Moreover, all the irreducible rational representations can be constructed using the infinitesimally-irreducible ones.
	eom.springer.de /R/r077630.htm (501 words)

	Temple REU (Site not responding. Last check: 2007-11-07)
		The fundamental mathematical prerequisite is elementary linear algebra.
		Of particular interest are the irreducible representations, which correspond to points in ``quantum'' or ``noncommutative'' geometric spaces.
		Irreducible representations -- of certain special types of systems -- play a prominent role in particle physics.
	www.math.temple.edu /~reu (1110 words)

	Irreducible (mathematics) - Wikipedia, the free encyclopedia
		In commutative algebra, a commutative ring R is irreducible if its prime spectrum, that is, the topological space Spec R, is an irreducible topological space.
		A directed graph is irreducible if, given any two vertices, there exists a path from the first vertex to the second.
		The notions of irreducibility in algebra and manifold theory are related.
	en.wikipedia.org /wiki/Irreducible_(mathematics) (386 words)

	Brainstorms: IC vs SC: Never Should the Twain Meet
		One of the advocates of "Irreducible Complexity" in particle physics won the Nobel Prize, although I believe he is not an ID advocate, he is considered a renegade.
		From the philosophical and personal side, because of irreducibility, there is justification to believe many of the concepts in our hearts such as intelligence, consciousness, life, health, and joy may be not only be irreducible, unprovable truths, but also the most important truths.
		Hofstadter provides a mathematical basis from Typographical Number Theory as to how to arrive at a minimum number of parts, but he does not actually come right out and say at what number of parts is needed for "critical mass".
	www.iscid.org /boards/ubb-get_topic-f-6-t-000452.html (4665 words)

	The Limits of Reason
		It reveals that certain mathematical facts are true for no reason, a discovery that flies in the face of the principle of sufficient reason.
		Mathematics therefore has infinite complexity, whereas any individual theory of everything would have only finite complexity and could not capture all the richness of the full world of mathematical truth.
		Mathematics, in contrast, is somehow independent of the universe.
	cs.umaine.edu /~chaitin/sciamer3.html (5081 words)

	Canadian Mathematics Competition Mathematics Magazine
		The problems come from various areas of mathematics, such as are included in math curricula at secondary schools.
		The solution of these problems, however, requires exceptional mathematical ability and excellent mathematical knowledge on the part of the contestants.
		IMO (International Mathematical Olympiad) was hosted by Japan in Tokyo during July 7- 20, 2003.
	www.mathematicsmagazine.com /Info/InternationalMathematicsOlympiad.htm (268 words)

	5.2.3 Lie Groups and Representation Theory
		In a very broad sense the two underlying problems in representation theory are to decompose ``large'' representations into their irreducible components and to understand in great detail the irreducible ones.
		The former is often called noncommutative harmonic analysis, as it is the analog for a noncommutative group G of Fourier analysis for the group R or the circle group, where the irreducible representations are one-dimensional.
		The latter involves classification and construction of irreducible representations as well as determining certain of their properties such as unitarity.
	www.math.okstate.edu /grad/long-hbk/5_2_3Lie_Groups_Representat.html (592 words)

	PlanetMath: irreducible polynomial (Site not responding. Last check: 2007-11-07)
		are irreducible (although they split in linear factors in the fields
		The above definition of irreducible polynomial is special case of the more general setting where
		This is version 13 of irreducible polynomial, born on 2004-06-10, modified 2006-10-02.
	www.planetmath.org /encyclopedia/IrreduciblePolynomial2.html (242 words)

	Dr. P. G. Dixon, School of Mathematics & Statistics, University of Sheffield (Site not responding. Last check: 2007-11-07)
		As an undergraduate, I was chiefly interested in mathematical logic, but as a postgraduate I moved into functional analysis, specialising in general Banach algebra theory (MR classes 46H, 46J).
		The great attraction of this field as a research area is that it involves both analysis and algebra in proportions which vary from problem to problem.
		In 1998-9, I gave a postgraduate course on set theory; a PDF file of the notes may be downloaded by clicking here.
	www.peter-dixon.staff.shef.ac.uk (393 words)

	SL(2,C) and sl(2,C)
		So basically,i don't need an irreducible representation of the algebra sl(2,C) (and that chapter from Fulton and Harris (note [1],post #10),but i need the irreducible finite dimensional representations of the group SL(2,C),because i'm interested in finding the finite dimensional irreducible representations of Lor(1,3).
		The Lor(1,3) group is semisimple,therefore all his reducible finite dimensional representations are completely reducible,so it suffices to use only the irreducible ones.I'm not interested in those infinite dimensional ones.
		Oh, and we recover the reps of G be exponetiating reps of g where g is G's lie alg.
	www.physicsforums.com /showthread.php?t=72727 (2137 words)

	HJM, Vol. 26, No. 3, 2000
		It is known that if G is a finite non-abelian group in which every irreducible character vanishes on at most one conjugacy class, then G is a Frobenius group with a Frobenius complement of order 2 and Frobenius kernel of odd order.
		This function is used to perturb the metric, and an application of the ergodic theorem shows that there exists a bundle-like metric for which the basic projection of the mean curvature is basic-harmonic.
		Suzanne Lenhart and Min Liang, Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300 (lenhart@math.utk.edu), (mliang@math.utk.edu).
	www.math.uh.edu /~hjm/Vol26-3.html (1234 words)

	HJM, Vol. 32, No. 2, 2006
		The domain R is idpf (irreducible divisors of powers finite) if for every non-zero element a in R, the set D(a) is finite.
		The domain is idf (irreducible divisors finite) if the set of non-associate irreducible divisors of a is finite for every non-zero element a.
		Pengtong Li, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China (pengtonglee@vip.sina.com), Jipu Ma, Department of Mathematics, Nanjing University, Nanjing 210093, China, and Jing Wu, Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA (ucfjing@yahoo.com).
	www.math.uh.edu /~hjm/Vol32-2.html (2425 words)

	Amazon.com: Fields and Rings (Chicago Lectures in Mathematics): Books: Irving Kaplansky (Site not responding. Last check: 2007-11-07)
		Irving Kaplansky is Director Emeritus of the Mathematical Sciences Research Institute and George Herbert Mead Distinguished Service Professor Emeritus in the Department of Mathematics at the University of Chicago.
		The subject matter used to be thought of as purely mathematical, but due to the influence of the field of cryptography, it now has many applications.
		He shows first that for the Galois group of a separable irreducible cubic over a field K is either the alternating group A(3) or the symmetric group S(3).
	www.amazon.com /Fields-Rings-Chicago-Lectures-Mathematics/dp/0226424510 (1490 words)

	Nasty feelings in the OOL community toward Hubert Yockey? \| Uncommon Descent
		In mathematics “irreducible” refers to something uncomputable, like a fact that has no reason, as something that must be taken as an axiom itself (if it the universe was truly random, every phenomena would be “irreducible”).
		A Godel formula is such an irreducible statement – and I hold that the origin of life is such an “irreducible phenomenon”.
		Irreducible complexity refers to something different, but it could be that the flagellum is an “irreducible” phenomenon in the sense that it does not follow from natural law or some genetic algorithm, but by design inference.
	www.uncommondescent.com /index.php/archives/918 (2300 words)

	Template of MITACS Project Website
		O. Ahmadi and G. Vega, "On the parity of the number of irreducible factors of self-reciprocal polynomials over finite fields", Finite Fields and Their Applications, to appear.
		Y. Lee, R. Scheidler and C. Yarrish, "Computation of the Fundamental Units and the Regulator of a Cyclic Cubic Function Field", Experimental Mathematics 12 (2003), 211-225.
		Stein and E. Teske, Explicit bounds and heuristics on class numbers in hyperelliptic function fields, Mathematics of Computation, 71 (2002), 837-861.
	www.cacr.math.uwaterloo.ca /mitacs/Publication.htm (2424 words)

	irreducible - OneLook Dictionary Search
		irreducible, irreducible, irreducible, irreducible, irreducible, irreducible : PlanetMath Encyclopedia [home, info]
		Phrases that include irreducible: irreducible polynomial, irreducible case, irreducible complexity, irreducible component, irreducible matrix, more...
		Words similar to irreducible: irreducibility, irreducibly, basic, more...
	www.onelook.com /?w=irreducible (226 words)

	Publications
		Irreducible tempered representations, Lie Theories and Their Applications, Proceedings of the 1977 Annual Seminar of the Canadian Mathematical Congress, Queen's Papers in Pure and Applied Mathematics, No. 48, Queen's University, Kingston, Ontario, 1978, pp.
		Survey of irreducible unitary representations of semisimple Lie groups, GROUP21: Physical Applications and Mathematical Aspects of Geometry, Groups, and Algebras, Proceedings of the XXI International Colloquium, vol.
		Access to files of corrections, the table of contents, the preface, and abstracts of chapters new or substantially changed from the first edition.
	www.math.sunysb.edu /~aknapp/publications.html (2263 words)

	Crystal page (Site not responding. Last check: 2007-11-07)
		Basically this amounts to implementing a theorem of Kashiwara : the irreducible constituents of such a tensor product are in a natural 1-1 correspondence with the connected components of the crystal graph of the tensor product.
		compute the restriction of an irreducible representation obtained in #1 to a simple subalgebra obtained by omitting one of the simple roots.
		An irreducible representation is denoted by its highest dominant weight vector, written as a linear combination of the ei's.
	web.usna.navy.mil /~wdj/crystal.htm (1093 words)

	Andrew Sommese - Department of Mathematics - University of Notre Dame
		A recent success has been our carrying out the irreducible decomposition of a complex algebraic variety numerically.
		One consequence of this decomposition is the first numerical algorithm to compute the precise set of isolated solutions of a polynomial system.
		As applications of our new algorithms, we have been investigating applied questions from the design of mechanisms and developing algorithms for theoretical questions such as how to factor polynomials in several variables over the complex numbers.
	www.nd.edu /~mathwww/faculty/sommese.shtml (234 words)

	G J Chaitin Home Page
		On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility
		Randomness in arithmetic and the decline & fall of reductionism in pure mathematics
		Responses to "Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics'', by A. Jaffe and F. Quinn
	www.umcs.maine.edu /~chaitin/latest.html (195 words)

	References
		F. ``Constructive problems for irreducible polynomials over finite fields,'' in Information Theory and Applications (A. Gulliver and N. Secord, eds), LNCS 793, Springer-Verlag, 1994, 1-23.
		V. ``Mathematical Models in Classical Cryptology,'' to appear in Modeling in Mathematical Sciences, a volume dedicated to Clayton Aucoin, CRC Press.
		V. ``Mathematical Models in Public-key Cryptology,'' to appear in Modeling in Mathematical Sciences, a volume dedicated to Clayton Aucoin, CRC Press.
	www.math.clemson.edu /faculty/Gao/res_sum/node1.html (553 words)

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