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	Multiset - Wikipedia, the free encyclopedia
		In mathematics, a multiset (sometimes also called a bag) differs from a set in that each member has a multiplicity, which is a natural number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset.
		A submultiset (B, n) of a multiset (A, m) is a subset B ⊆ A and a function n : B → N such that n(a) ≤ m(a).
		The cumulant-generating function of the multiset of sums of elements of two multisets of numbers is the sum of the two cumulant-generating functions.
	en.wikipedia.org /wiki/Multiset (1427 words)

	PlanetMath: multiset
		A multiset is a set for which repeated elements are considered.
		This is version 7 of multiset, born on 2002-02-18, modified 2005-03-18.
		Object id is 2090, canonical name is Multiset.
	planetmath.org /encyclopedia/Multiset.html (95 words)

	Amenable number - Wikipedia, the free encyclopedia
		An amenable number is an integer for which there exists a multiset of as many integers as the original number that can be either added up or multiplied together to give the original number.
		All composite numbers would be amenable if the multiset was allowed to be of any length, because, even if other solutions are available, one can always make an inelegant solution by taking the prime factorization (expressed with repeated factors rather than exponents) and add as many 1s as necessary to add up to n.
		Because of the multiplicative identity, multiplying this set of integers will yield n no matter how many 1s there are in the set.
	en.wikipedia.org /wiki/Amenable_number (292 words)

	multiset - Dev Shed
		multiset C programming forum discussing all C derivatives, including C#, C++, Object-C, and even plain old vanilla C. These languages are low level languages, and used on projects such as device drivers, compilers, and even whole computer operating systems.
		If you were not aware of it, the multiset is implemented as a red-fl tree.
		That's because the second parameter of a multiset<> is a function object.
	forums.devshed.com /c-programming-42/multiset-268919.html (1149 words)

	Multiset
		Formally, a multiset consists of a set S and a function m(x) that maps each element x of S to the number of times it occurs in the multiset.
		The Multiset interface extends the Collection interface, adding methods for getting the number of distinct elements and the number of times a given element occurs in the multiset.
		An entry is a pair (e, n) where e is an element of the multiset and n is the number of times e occurs in the multiset.
	www.cs.berkeley.edu /~milch/blog/apidocs/common/Multiset.html (273 words)

	Pike: src/multiset.h Source File
		The node 00268 * offset within the multiset data block is used then, which together 00269 * with the multiset struct can access the node.
		This has the effect that if nodes are added and removed 00284 * in a multiset that is being traversed in some direction, then no 00285 * newly added nodes in the vicinity of the current one are missed.
		It 00286 * also has the effect that the node returned by multiset_next for a 00287 * deleted node might be before the one returned by multiset_prev.
	pike.ida.liu.se /generated/manual/c_api/html/multiset_8h-source.html (1421 words)

	Problem Set (Site not responding. Last check: 2007-10-13)
		A multiset (or bag) is a set that may contain more than one occurrence of a particular element.
		multisets are useful for a variety of purposes; for example, keeping track of histograms during experimental measurements.
		We expect your implementation to be reasonably efficient, especially for multisets with many occurrences of the same element; the run-time efficiency will carry some weight but will be a secondary consideration.
	www.cs.cornell.edu /courses/cs312/2005sp/hw/ps3/ps3.htm (1914 words)

	Info on Multisets
		A multiset is like a set, except that repeated elements are allowed.
		When referring to permutations of a multiset we assume that the elements of the multiset are 0,1,..., t, that the number of occurrences of i is n
		If S is a multiset, then such k-subsets have several applications, notably in statistics, where they arise in permutation tests with repeated data values and as 2 x n contingency tables.
	www.theory.csc.uvic.ca /~cos/inf/mult/Multiset.html (248 words)

	KCGL1 Help CXXLSTD Containers multiset (Site not responding. Last check: 2007-10-13)
		Constructs an empty multiset which will use the optional relation comp to order keys, if it is supplied, and the allocator alloc for all storage management.
		iterator erase(iterator first, iterator last); Providing the iterators first and last point to the same multiset and last is reachable from first, all elements in the range (first, last) will be deleted from the multiset.
		For example, if your compiler does not support member function templates, you can construct a multiset in the following two ways: int intarray[10]; multiset first_multiset(intarray, intarray +10); multiset second_multiset(first_multiset.begin(), first_multiset.end()); but not this way: multiset long_multiset(first_multiset.begin(),first_multiset.end()); since the long_multiset and first_multiset are not the same type.
	cctr.umkc.edu /HELP/CXXLSTD/CONTAINERS/MULTISET (1116 words)

	Problem Set 3 (Site not responding. Last check: 2007-10-13)
		Background: A multiset (or bag) is a set that may contain more than one occurrence of a particular element.
		Note that multiset do not keep track of any ordering of their elements, so the multisets {1,2,2,3} and {2,3,1,2} are equal.
		Multisets are useful for a variety of purposes; for example, keeping track of a collection of experimental measurements.
	www.cs.cornell.edu /courses/cs312/2003sp/hw/ps3/ps3.html (1435 words)

	WebmasterFree - MultiSet
		MultiSet is a solution to the problem of automatic program installation.
		MultiSet has been created to free you from routine work such as program installations.
		MultiSet will: Run the installation program; Click the necessary buttons such as Next, I Agree..., Finish, etc. Specify a destination folder for the application; Specify the necessary values during the installation; Enter the registration information (the key, user name, etc.) if necessary; Reproduce all your actions but much faster!
	www.webmasterfree.com /software/SystemUtilities/AutomationTools/multiset.html (225 words)

	multiset (Site not responding. Last check: 2007-10-13)
		Multiset is a sorted associative container that can store multiple copies of the same key.
		The elements in a multiset are ordered using the function object specified.
		The default function object used to order elements of a multiset is less<Key>.
	babbage.cs.qc.edu /STL_Docs/multiset.htm (40 words)

	Lighting Control & Design - Link-To MultiSet - Overview
		MultiSet may be put in "Timer Mode" automatically after normal business hours.
		Functionally expand MultiSet network size to include as many as 12,000 GR 2400 devices.
		Any GR 2400 device may control any MultiSet device for scene control, on, off, raise and lower functions.
	www.lightingcontrols.com /products/new/closeup_multiset.asp (121 words)

	Almeza Research LLC Releases Multiset !
		Almeza Research LLC announces the release of Multiset, a new automatic program installer giving you freedom from such routine work as program installation.
		The Multiset's main task is to record and playback the user's actions precisely as it was done by the user before.
		A trial 30-day version of the program with a nag screen and a few locked features is available as a free download at http://www.almeza.com/download/multiset.exe.
	www.cdrinfo.com /forum/tm.asp?m=123221&mpage=1𞅕 (588 words)

	Amazon.com: Multiset Processing : Mathematical, Computer Science, and Molecular Computing Points of View (Lecture Notes ... (Site not responding. Last check: 2007-10-13)
		The multiset, as a set with multiplicities associated with its elements in the form of natural numbers, is a notation which has appeared again and again in various areas of mathematics and computer science.
		As a data structure, multisets stand in-between strings/lists, where a linear ordering of symbols/items is present, and sets, where no ordering and no multiplicity is considered.
		All in all, the book assesses the state of the art of the notion of multisets, the mathematical background, and the computer science and molecular computing relevance.
	www.amazon.com /exec/obidos/tg/detail/-/3540430636?v=glance (646 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		MULTISET REDUCTION Overview of Multiset Extension C. Norris Ip Feb 1995 Content: 1) Introduction 2) Specification of symmetry in Murphi 2.x a) new datatype: multiset 3) Verification using multiset reduction 4) Underlying mechanism 1) Introduction: Symmetry can be used to reduce the amount of time and memory used in verification.
		Multiset is used to handle such aspect of the system.
		Basically a multiset is very similar to an array indexed by a scalarset, except that the "multiset index" cannot be stored in any state variable and most of the local variables.
	www.cs.ubc.ca /~ajh/courses/cpsc513/Murphi/doc/User.Manual (5861 words)

	Download MultiSet 2.5 Free Trial - MultiSet is an automatic program installer - Softpedia
		MultiSet frees you from the routine work of application installation.
		A user is to install the application only once with MultiSet run in the recording mode.
		MultiSet will record all your actions and generate the automatic installation package after the recording process is completed.
	www.softpedia.com /get/System/System-Miscellaneous/MultiSet.shtml (558 words)

	multiset1
		This symbol is used to denote the n-ary intersection of multisets.
		It is used to denote that the first multiset is a proper subset of the second, that is a subset of the second multiset but not actually equal to it.
		A proper subset of a multiset is a subset of the multiset but not actually equal to it.
	www.win.tue.nl /~amc/oz/om/cds/multiset1.xml (1396 words)

	Multiset - (Site not responding. Last check: 2007-10-13)
		Within set theory, a multiset can be formally defined as a pair (A, m) where A is some set and m : A → N is a function from A to the set N of (positive) natural numbers.
		A submultiset (B, n) of a multiset (A, m) is a subset B &sube; A and a function n : B → N such that n(a) ≤ m(a).
		The special case y=x : The infinite multiset of finite multisets of elements from the multiset x
	psychcentral.com /psypsych/Multiset (1518 words)

	CPS 100 : Spring 2000 : Fun With Linked Lists
		MSCounter in mscounter.h: This object prints the number of unique/different words in a multiset, i.e., the number of non-header nodes in the linked list implementations.
		The union of two multisets contains every element in either multiset, i.e., every element in one multiset or in the other including duplicates.
		The intersection of two multisets is the elements that are in common to both.
	www.cs.duke.edu /courses/cps100/spring00/assign/wordtrack (1718 words)

	Multiset Operators
		Multiset operators combine the results of two nested tables into a single nested table.
		The examples related to multiset operators require that two nested tables be created and loaded with data as follows:
		Please refer to "Multiset Operators " to create this table and nested table columns.
	www.stanford.edu /dept/itss/docs/oracle/10g/server.101/b10759/operators006.htm (588 words)

	Automatic Software Installer - MultiSet - Automatic Software installer (Site not responding. Last check: 2007-10-13)
		I have tried nearly every macro based script recorder available, but none could do what I needed the most: install an application not based on time, but rather the state of the buttons in the install window.
		Multiset's recording and playback are exceptional, accurate, and most important, not based on time, so the scripts will work on any computer regardless of speed.
		All you have to do is to install the application only once on your PC and MultiSet will record all your actions.
	www.almeza.com (370 words)

	WinSite: MultiSet
		MultiSet is an automatic program installer with a simple and convenient interface.
		MultiSet frees you from the routine work of program installation.
		The MultiSet program will install all the necessary programs for you automatically, handle registration data, and more.
	www.winsite.com /bin/Info?26000000038593 (47 words)

	[Coq-Club] Multiset order. (Site not responding. Last check: 2007-10-13)
		There is a nice proof of well-foundedness of multiset order due to Wilfried Buchholz presented by Tobias Nipkow (http://www4.informatik.tu-muenchen.de/~nipkow/misc/multiset.ps).
		However it uses induction on the size of multiset which is impossible to perform with this definition (the problem is somehow more serious because this definition allows infinite multisets in general which is undesirable in this case of course).
		Is there any chance to stick to library definition of multisets, like, let's say, by refining it somehow (how?) to finite multisets in a way that allows induction on their size?
	pauillac.inria.fr /pipermail/coq-club/2004/001233.html (260 words)

	STL set + multiset (Site not responding. Last check: 2007-10-13)
		If the item is already present, there will be multiple copies in the multiset..
		Note that this is the smallest item in the multiset, because the multiset is always stored in sorted order.
		Note that this is the largest item in the multiset, because the multiset is always stored in sorted order.
	www.cs.duke.edu /courses/fall99/cps149s/resources/n10_3.html (457 words)

	A Logic Language based on GAMMA-like Multiset Rewriting - Ciancarini, Fogli, Gaspari (ResearchIndex)
		The language combines the ability of describing parallel programs made of multiset transformation rules as in GAMMA with the execution model of logic programming in a strongly typed framework as in Godel.
		We describe the design choices, the syntax and the semantics of the language, and a prototype implementation.
		Multiset Rewriting By Multiset Constraint Solving - Agostino Dovier Carla
	citeseer.ist.psu.edu /24955.html (643 words)

	Theory Multiset (Isabelle2005: October 2005)
		0::nat) ∈ ?multiset" by simp qed lemmas multiset_typedef [simp] = Abs_multiset_inverse Rep_multiset_inverse Rep_multiset and [simp] = Rep_multiset_inject [symmetric] constdefs Mempty :: "'a multiset" ("{#}") "{#} == Abs_multiset (λa.
		Rep_multiset M a - Rep_multiset N a)" Zero_multiset_def [simp]: "0 == {#}" size_def: "size M == setsum (count M) (set_of M)" constdefs multiset_inter :: "'a multiset => 'a multiset => 'a multiset" (infixl "#∩" 70) "multiset_inter A B ≡ A - (A - B)" text {* \medskip Preservation of the representing set @{term multiset}.
		M a + N a) ∈ multiset" apply (simp add: multiset_def) apply (drule (1) finite_UnI) apply (simp del: finite_Un add: Un_def) done lemma diff_preserves_multiset [simp]: "M ∈ multiset ==> (λa.
	www.cl.cam.ac.uk /Research/HVG/Isabelle/dist/library/HOL/Library/Multiset.html (2446 words)

	First-order, non-modal definition of 'bag/multiset'
		A bag, or multiset, is a set whose members are not necessarily distinct, i.e., it can contain duplicates.
		where z is called the power set of x, denoted P(x), then a multiset w is a set such that the cardinality of P(w) can be less than the cardinality of w.
		If every subset of a multiset is a set, not a multiset, do you then have a separate "sub-multiset" relation for comparison of two multisets?
	www.physicsforums.com /showthread.php?p=979433#post979433 (1202 words)

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