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Topic: Infinite product

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	Euler product - Wikipedia, the free encyclopedia
		In mathematics, an Euler product is an infinite product expansion, indexed by prime numbers p, of a Dirichlet series.
		The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Euler.
		This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the infinite series is not zero in such a half-plane.
	en.wikipedia.org /wiki/Euler_product (287 words)

	Cartesian product - Wikipedia, the free encyclopedia
		In the infinite case this is a family.
		The assertion that the Cartesian product of arbitrary non-empty collection of non-empty sets is non-empty is equivalent to the axiom of choice.
		Categorically, the cartesian product is the product in the Category of sets.
	en.wikipedia.org /wiki/Cartesian_Product (550 words)

	Infinite product - Wikipedia, the free encyclopedia
		The product is said to converge when the limit exists and is not zero.
		The best known examples of infinite products are probably some of the formulae for π, such as the following two products, respectively by Viète and John Wallis (Wallis product):
		One important result concerning infinite products is that every entire function f(z) (i.e., every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions each with at most a single zero.
	en.wikipedia.org /wiki/Infinite_product (434 words)

	PlanetMath: convergence condition of infinite product
		When the infinite product converges, we say that the value of the infinite product is equal to
		"convergence condition of infinite product" is owned by pahio.
		This is version 9 of convergence condition of infinite product, born on 2004-09-21, modified 2005-03-27.
	planetmath.org /encyclopedia/ConvergenceConditionOfInfiniteProduct.html (101 words)

	Infinite product: Facts and details from Encyclopedia Topic (Site not responding. Last check: 2007-11-06)
		The value zero is treated specially in order to get results analogous to those for infinite sums[For more facts and a topic of this subject, click this link].
		The best known examples of infinite products are probably some of the formulae for π[For more facts and a topic of this subject, click this link], EHandler: no quick summary.
		An asymptote to a curve is a straight line that the curve approaches in such a manner that it becomes as close as one might wish to the line by going far enough...
	www.absoluteastronomy.com /encyclopedia/i/in/infinite_product.htm (1093 words)

	Cartesian product - Wikipedia, the free encyclopedia
		The Cartesian product is named after René Descartes whose formulation of analytic geometry gave rise to this concept.
		The Cartesian square (or binary Cartesian product) of a set X is the Cartesian product X × X.
		However, it is possible to define the Cartesian product over an arbitrary (possibly infinite) collection of sets.
	en.wikipedia.org /wiki/Cartesian_product (550 words)

	SERIES (a Latin word f... - Online Information article about SERIES (a Latin word f...
		In a singly infinite series we have a general term u,., where n is an integer positive in the case of an ordinary series, and positive or negative in the case of a back-and-forwards series.
		When the product of two series is arranged as a doubly infinite series, summing for the rectangular boundary x= aT, y =/3T we obtain the product of the sums of the series.
		One of the simplest cases of a function ex-pressed as an infinite product is that of sin z/z, which is the value of the absolutely convergent infinite product.
	encyclopedia.jrank.org /SCY_SHA/SERIES_a_Latin_word_from_serere.html (5488 words)

	CD Baby: INFINITE MONOPOLY: Product '01A'
		Product 01A is the result of a six-month collaborative effort between composer/songwriter/producer Barak Moffitt and producer/engineer Jim Schultz.
		Of course Infinite Monopoly is not Pink or Led and I'm not 18.
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	cdbaby.com /cd/infinitemonopoly?cdbaby=787c8abf08b7473ca4493602921f9637 (546 words)

	Infinite Products (Site not responding. Last check: 2007-11-06)
		Like an infinite series, the infinite product of s is the limit of the sequence of partial products.
		Now partial products corespond to partial sums, and since log() is bicontinuous, the product converges iff the series converges.
		Just as the sum of a linear combination of several series gives the linear combination of their limits, infinite products may be raised to various exponents and multiplied together.
	www.mathreference.com /lc-prod,intro.html (294 words)

	Infinite Products and a Tangent Fan
		we have the trivial infinite product 1/2 1/4 1/8 a = a a a....
		The "identity" of this simple circular locus with the infinite product on the right is non-trivial, because the right-hand side is evidently discontinuous at every real value of z.
		Also, the right side is infinite if any of the arguments 2^k z is of the form n PI + PI/2, which implies it is infinite for z = (n PI + PI/2)/2^k for any integers n and k.
	www.mathpages.com /home/kmath088.htm (1531 words)

	How Leibniz Might Have Anticipated Euler
		4 \1 + 1/3/ \1 - 1/5/ \1 + 1/7/ \1 + 1/11/ \1 - 1/13/ where the product is taken over all odd primes p, and the sign in the denominator is + or - depending on whether p is congruent to -1 or +1 modulo 4.
		It's easy to form the conjecture based on numerical evidence that this series (and the infinite product) converges on pi/2, i.e., exactly twice the former series (and product).
		Returning to our hypothetical Leibniz and his reversal of the signs in the infinite product that formally corresponds to his famous series for pi/4, notice what happens if we multiply his two results together.
	www.mathpages.com /home/kmath477.htm (1007 words)

	Roman Redziejowski Publications
		The notion of associative infinite product is applied to processes.
		The notion of associative infinite product is applied to traces, resulting in an alternative approach to introducing infinite traces.
		Four different versions of the product are explored, two of them identical to known definitions of infinite trace.
	home.swipnet.se /redz/roman/pubs.htm (587 words)

	Euler
		Which means that the infinite series by which f (x) was originally defined has been equated to the infinite product on the right.
		Euler then went on by multiplying out the infinite product on the right side of the preceding equation and then collecting all terms having the same power of x.
		After Euler had multiplied out the infinite product to get two infinite sums equal to each other, he went on to equate the like powers of x.
	members.tripod.com /egyptonline/euler.htm (329 words)

	Email Server Product Review
		Infinite Interchange email server, from Infinite Technologies, is a complete remote-access electronic mail software solution that can bring order to your company's email systems.
		Infinite Interchange lets remote users access your company's email server over the Internet or a LAN using various methods.
		Because Infinite Interchange email server is a middleware layer between your users and their electronic mailboxes, it requires a large amount of time and administration knowledge to properly set up.
	www.ssimail.com /PR19.htm (679 words)

	Infinite Interchange To Bring Wireless E-Mail to Corporations (Site not responding. Last check: 2007-11-06)
		Infinite Technologies is the first company to offer WAP support in a commercial e-mail product.
		Infinite InterChange will allow users to access the same mailbox from their wireless phone as they do via the wired Internet.
		Infinite InterChange is a server-based application that integrates seamlessly with any IMAP4 (Internet Messaging Access Protocol Version 4), POP (Post Office Protocol Version 3), SMTP (Simple Mail Transport Protocol), Microsoft Mail, Microsoft Exchange, Lotus cc:Mail, Lotus Notes, or Connect2 based e-mail system, or a mixed environment containing multiple protocols.
	www.mobic.com /oldnews/9909/infinite_interchange_to_bring_wi.htm (473 words)

	Amazon.com: The Infinite (Leisure Horror): Books: Douglas Clegg (Site not responding. Last check: 2007-11-06)
		A kind of riff on Shirley Jackson's classic The Haunting of Hill House, The Infinite tells the story of three psychically-gifted individuals invited by the leader of a psychic research foundation to spend a few days in a haunted mansion in the Hudson River Valley of New York state.
		THE INFINITE is the third cycle in a series revolving around an allegedly haunted house called Harrow (and I'm sure there will be more episodes to follow).
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	www.amazon.com /exec/obidos/tg/detail/-/0843950552?v=glance (3350 words)

	Pi - Wikipedia, the free encyclopedia
		Although this precision is more than sufficient for use in engineering and science, the exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number (and indeed, a transcendental number).
		The most pressing open question about π is whether it is a normal number -- whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every base, not just base 10.
		Jonathan Sondow, "A faster product for pi and a new integral for ln pi/2," Amer.
	en.wikipedia.org /wiki/Pi (4116 words)

	Infinite Papilloma/Acanthoma: Infinite Product Model for Unbounded Surface Tumor Growth. 11/20/2005.
		The infinite product, P, is the limit of the n-products as n approaches infinity.
		So, demonstrating that the infinite product equals zero for a particular terminal volume means that the papilla summation equals the area/volume under the curve.
		We are assuming that a malignant cancer-cell population has a genetic mandate to proliferate, unconstrained by the usual limiting processes, such as p53; or equivalently, a genetic mandate to not-to-die (apoptosis), unconstrained by the usual limiting processes, such as bcl-2.
	www.medparse.com /infnpapl.htm (4093 words)

	Infinite Products
		Infinite product expressions for analytic functions of two variables.
		An analogue of a theorem of Hahn for infinite products.
		Representation of an analytic function of two variables by means of a double infinite product.
	math.fullerton.edu /mathews/c2003/InfiniteProductBib/Links/InfiniteProductBib_lnk_3.html (575 words)

	Homomorphic Images Of An Infinite Product Of Zero-Dimensional Rings - Gilmer, Heinzer (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Let R = Q a2A Ra be an infinite product of zero-dimensional chained rings.
		2 The prime spectrum of an infinite product of copies of Z (context) - Levy, Loustaunau et al.
		2 The prime spectrum of an infinite product of zero-dimensiona..
	citeseer.ist.psu.edu /372029.html (434 words)

	Press Release
		The Infinite Enterprise WAP Server, a more powerful version of the popular WAPLite Corporate WAP Server product, was selected after extensive testing in a live service environment over the past three months.
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		Infinite products include the award-winning WAPLite WAP Server, the Infinite Enterprise WAP Server, and Infinite InterChange, the most powerful WAP e-mail solution available for corporate, ISP and mobile operator environments.
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	Latest INFINITE NET Release Begins Shipping Today (Site not responding. Last check: 2007-11-06)
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		INFINITE NET is a suite of products that creates a graphical front end for IBM AS/400 and Mainframe screens.
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	www.prnewswire.com /cgi-bin/stories.pl?ACCT=104&STORY=/www/story/02-17-2004/0002111420&EDATE= (300 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		I guess this is not the case for a direct product of > infinitely many summands.
		The Artinian condition is not preserved by infinite products.
		Even the infinitely sum of semisimple ring is not semisimple because of possessing infinitely many nonisomorphic minimal ideals.
	www.math.niu.edu /~rusin/known-math/00_incoming/semisimple (493 words)

	Vodavi Telecommunication Systems
		With more than one million phone systems installed, Vodavi is a leader in providing product solutions that meet the quality, flexibility, budgetary and reliability requirements of any business or call center.
		The infinite digital platform provides for applications such as digital voice mail, fax mail, ISDN and wireless to a system from as small as 3 lines by 8 stations to 384 ports.
		The infinite DVXPlus family provides affordable, easy-to-use systems that start as small as 3 lines and 8 telephones and expand to 376 universal ports.
	www.grimesgrafix.com /cartertelnet/vodavi.html (465 words)

	ORCON: Infinite Broadband
		Infinite can also be bundled with other Orcon products (such as hosting or domain name registration).
		The Infinite product is totally flat rate - there are no ‘overage’ or ‘excess usage’ fees to pay in any way, just one fixed amount every month.
		Each infinite customer is assigned a static IP address — this alone costs another $20 per month from some providers.
	www.orcon.net.nz /products/broadband/67132/Infinite (562 words)

	Math Forum - Ask Dr. Math
		Date: 12/01/97 at 02:13:53 From: Thang Nguyen Subject: Infinite product Hi, my name is Thang and I am having a hard time with this problem and wondering if you can help me. My problem is how to find the infinite product of this: 3^1/3 * 9^1/9 * 27^1/27 *.....
		Then, your infinite product becomes 3^(1/3) * 3^(2/9) * 3^(3/27) *..., which can now be written as 3^[1/3 + 2/9 + 3/27 +...
		This can be done by writing it as an infinite sum of infinite sums, like this 1/3 + 1/9 + 1/27 + 1/81 +...
	mathforum.org /library/drmath/view/53614.html (261 words)

	Infinite Product
		The above product is from the infinite product page at mathworld, their products were presumably derived using the infinite product for the Euler Gamma Function.
		Is there a 'model' infinite product which is used to evaluate other infinite products (like what shmoe appears to have alluded to).
		If you look at the product form of sine, it's essentially factoring sine as though it were an "infinite polynomial", it's a product over its zeros (paired up to simplify convergence).
	www.physicsforums.com /showthread.php?threadid=106300 (822 words)

	Infinite Products And Integration. 8/4/2005.
		A brisk treatment of infinite products, which lead to Lebesgue integration, is presented at an elementary (undergraduate) level in mathematics.
		This familiar infinite product is equal to the reciprocal of the Riemann-zeta-function:
		In the form (3.5), the mixed product (3.1) is subject to the filling lemma and to all the discussions and examples of Chapter 1: Rectangle filling and infinite products, and the inequalities (3.6) furnish useful information as to the numerical value of P.
	www.infiniteproduct.info /struitgr.htm (7472 words)

	PC Pro: Product Reviews: Infinite Sudoku Focus
		Infinite Sudoku allows you to create billions of puzzles at different levels of difficulty, starting from easy and working their way up to devilish.
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		One of the features that fascinated us immediately was its claim to let you e-mail puzzles to friends and family.
	www.pcpro.co.uk /reviews/77751/infinite-sudoku-focus.html (381 words)

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