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Topic: Mathematical structure

	Mathematics - Wikipedia, the free encyclopedia
		Mathematics is often defined as the study of topics such as quantity, structure, space, and change.
		Nowadays, mathematics derives much inspiration from the natural sciences and it is not uncommon for new mathematics to be pioneered by physicists, although it may need to be recast into more rigorous language.
		Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
	en.wikipedia.org /wiki/Mathematics (2921 words)

	Mathematical structure
		Mathematics is often taught in a way that is boring and unnecessarily difficult.
		Formal mathematics is important because people understand it and connect it with every day life but it is usually taught as if such connections were irrelevant.
		It is that aspect of mathematics that has practical importance and computers are an accurate metaphor for a logically determined sequence of events in a potentially infinite universe.
	www.mtnmath.com /whatth/node17.html (255 words)

	Mathematical structure - Wikipedia, the free encyclopedia
		In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance.
		A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, and equivalence relations.
		As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a topological group.
	en.wikipedia.org /wiki/Mathematical_structure (244 words)

	Mathematical structure -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-26)
		As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a (additional info and facts about topological group) topological group.
		algebraic structure: there are operations of multiplication and addition that make it into a (A piece of land cleared of trees and usually enclosed) field.
		Its algebraic structure and topology make it into a (additional info and facts about Lie group) Lie group, a type of (additional info and facts about topological group) topological group.
	www.absoluteastronomy.com /encyclopedia/m/ma/mathematical_structure.htm (257 words)

	Understanding the Multiplicative Structure
		This paper focuses on the Invariance Structure — a substructure of the Semantic Structure — and the mathematical structure which consists of principles underlying the Invariance structure.
		Thus, because of the importance and ubiquity of the MCF in mathematics, it is an important challenge for mathematics educators, instructional psychologists, and curriculum developer, to study this field and understand children’s acquisition of the knowledge it involves, so that instruction that facilitates children’s construction of this knowledge can be devised.
		Second, these mathematical principles are believed to be the foundation for multiplicative and proportional reasoning; that is, these principles are the basic theorems in actions (ala Vergnaud, 1988), from which more complex theorem in actions can be constructed by children in solving advanced multiplicative and proportional reasoning problems.
	education.umn.edu /rationalnumberproject/90_1.html (2126 words)

	20th WCP: Semantic Realism: Why Mathematicians Mean What They Say
		In this paper I argue that if we distinguish between ontological realism (the claim that mathematical objects exist independently of their linguistic expression) and semantic realism (the claim that mathematical statements which talk about mathematical objects are meaningful), then we no longer have to choose between platonism and formalism.
		We say that category theory is the language of mathematical concepts and relations because it allows us to talk about their structure in terms of "objects" and "arrows", wherein such terms are taken as syntactic assemblages waiting for an interpretation of the appropriate sort to give them formulas meaning.
		We say that category theory is the language of mathematical theories and their relations because it allows us to talk about their structure in terms of "objects" and "functors", wherein such terms are, again, taken as syntactic assemblages waiting for an interpretation of the appropriate sort to give them formulas meaning.
	www.bu.edu /wcp/Papers/Math/MathLand.htm (3322 words)

	Mathematics and consciousness
		Mathematics is the study of all possible structures.
		Through mathematics we can extrapolate from the experience that is the essence of the human mind to what life and conscious experience may evolve into.
		In mathematics it is common to measure the power of a mathematical system by the level of self reflection or iteration that is definable within the system.
	www.mtnmath.com /whatth/node18.html (256 words)

	DIMACS Workshop on Discrete Mathematical Chemistry:Abstracts
		The structures of such cubic IPMS's decorated by trigonal carbon atoms are conveniently described by the octants of their unit cells, which are required to have three-fold symmetry.
		In cases when graph representation of the protein structure is still too detailed when compared to the structural sensitivity of the spectroscopic technique, the invariants of the graphs are generated as useful quantitative descriptors of protein structure that match the information content of the experimental data.
		Examples of descriptors of secondary structure segment topology and descriptors of tertiary fold type for proteins will be presented and the methods for reduction of the ambiguity of the information by comparison of set of descriptor compatible structures with the sequence based structural predictions will be discussed.
	dimacs.rutgers.edu /Workshops/Chemistry/abstracts.html (6003 words)

	[No title] (Site not responding. Last check: 2007-10-26)
		Mathematical structure in the form of lists of general properties is not a good starting point for instruction.
		Rather, students gain a sense of the structure of mathematics over an extended time period through the general accumulation of experience, as well as through more focused activities.
		In mathematics, just as with a building, students can develop an understanding and appreciation of its underlying structure independent of a knowledge of the corresponding technical vocabulary and symbolism.
	www.prel.org /work/ms/rsc/STANDA~1.97/m10d14.htm (176 words)

	Synthesis of Research on Problem Solving
		However, constructivism is consistent with current cognitive theories of problem solving and mathematical views of problem solving involving exploration, pattern finding, and mathematical thinking (36,15,20); thus we urge that teachers and teacher educators become familiar with constructivist views and evaluate these views for restructuring their approaches to teaching, learning, and research dealing with problem solving.
		Thus, it is argued, the beliefs of mathematics students, mathematics teachers, parents, policy makers, and the general public about the roles of problem solving in mathematics become prerequisite or co-requisite to developing problem solving.
		Schoenfeld (31) indicates that capable mathematics students when removed from the context of coursework have difficulty doing what may be considered elementary mathematics for their level of achievement.
	jwilson.coe.uga.edu /emt725/PSsyn/PSsyn.html (7645 words)

	Dante Alighieri @ Catharton Authors (Site not responding. Last check: 2007-10-26)
		Mathematics on the other hand is a universal language that can be understood by all.
		One of the most noteworthy discoveries in these mathematical patterns of Dante is the realizations that the three Sum-Digits exude the same pattern of cantos as found in the three volumes.
		As one begins to analyze these different mathematical schemas within the Commedia it is realized that these are not childish calculation, though that may be one's first impression (because of their simplistic nature), but rather they are highly sophisticated mathematical patterns obviously not meant for the average observer.
	www.catharton.com /authors/98a.htm (1441 words)

	Math Markup Language (Chapter 5)
		In many situations, notational structure and mathematical structure are closely related, so a sophisticated processing application may be able to heuristically infer mathematical meaning from notational structure.
		Though the details of mapping from mathematical structure to mathematical meaning are exceedingly complex, in practice, there is wide agreement about the conventional meaning of many basic mathematical constructs.
		In mathematics, by contrast, there are many more notational constructs and they often have fairly specific meaning to a reader; though a fraction may mean several different things in different contexts, it is a much more specific notational device than a paragraph.
	www.w3.org /TR/WD-math-980106/chapter5.html (2340 words)

	Tucson Unified School District
		Until the updates are posted, the crosswalk documents must be used to ensure the alignment of sample lesson plans to articulated standards.
		Mathematical Structure/Logic - M6-P2 - PO 1 - Determine if the converse of a given statement is true or false #1
		Mathematical Structure/Logic - M6-P5 - PO 1 - Determine whether a given algebraic expression and a possible form are equivalent #1
	instech.tusd.k12.az.us /focus/twelve/gr12math6.htm (410 words)

	VEHICLES FOR UNDERSTANDING AND DOING (Site not responding. Last check: 2007-10-26)
		The side face of the cube represents Vehicles for Understanding and Learning: mathematical structure, patterns and relations, computation and estimation, measurement and scale, and models.
		Mathematical models describe real-world problems and phenomena in science, social science, economics, architecture, and the like.
		Computers and other technological tools can be used to create a mathematical model for complex phenomena such as the prediction of food supply conditions around the world given certain conditions of supply and demand.
	www.coe.uga.edu /framework/chapters/part25.html (371 words)

	The ACELA Project: Aims and Plans
		In such cases a reader can be greatly helped when provided with an easily navigable presentation of the mathematical structure of that particular theory, which is basically a graph containing the definitions, lemmas, and theorems and their interrelations.
		For mathematics it is important that in an electronic book there is no reason for reader annotations to be limited by the size of the margin.
		It has been set up as a collection of co-operating subprojects (textbook, algorithms, kernel architecture, mathematical objects, proofs), in which each subproject has goals and deliverables that are useful and worth aiming for in their own right, and no subproject is critically dependent on the results of some other subproject.
	homepages.cwi.nl /~steven/acela (6394 words)

	Foundations of Mathematics. Mathematical Logic. By K.Podnieks
		Mathematics is the part of science you could continue to do if you woke up tomorrow and discovered the universe was gone.
		In fact, mathematics is a complicated system of interrelated theories each representing some significant mathematical structure (natural numbers, real numbers, sets, groups, fields, algebras, all kinds of spaces, graphs, categories, computability, all kinds of logic, etc.).
		Maslov could have put it as follows: most of a mathematician's working time is spent along the first dimension (working in a fixed mathematical theory, on a fixed mathematical structure), but, sometimes, he/she needs also moving along the second dimension (changing his/her theories/structures or, inventing new ones).
	www.ltn.lv /~podnieks (1167 words)

	Read about Mathematical structure at WorldVillage Encyclopedia. Research Mathematical structure and learn about ... (Site not responding. Last check: 2007-10-26)
		In mathematics, a structure on a set is some additional mathematical objects that, loosely speaking, attach to the set, making it easier to visualize or work with.
		As another example, if a set both has a topology and is a group, and the two structures are related in a certain way, the set becomes a
		algebraic structure: there are operations of multiplication and addition that make it into a
	encyclopedia.worldvillage.com /s/b/Mathematical_structure (204 words)

	Max Tegmark's library: ``theory of everything''
		The only postulate in this theory is that all structures that exist mathematically exist also physically, by which we mean that in those complex enough to contain self-aware substructures (SASs), these SASs will subjectively perceive themselves as existing in a physically ``real'' world.
		The figure at the top shows a small part of the ``family tree'' of mathematical structures as described in the paper.
		Arrows that meet indicate the combination of structures - for instance, an algebra is a vector space that is also a ring, and a Lie group is a group that is also a manifold.
	space.mit.edu /home/tegmark/toe.html (671 words)

	objections » True Geometry (Site not responding. Last check: 2007-10-26)
		A presuposition of that question is that there is a mathematical geometry that is true of physical space.
		A mathematical geometry (better: mathematical geometric structure) is a collection of points satisfying certain axioms.
		The characterization I just gave of a mathematical geometric structure is likely far too crass, but I suspect that for present purposes it doesn’t matter.) Physical space is, well physical space.
	www.ibiblio.org /lenhart/objections/index.php?p=34 (520 words)

	Theomatic Structure
		However, theomatics is: (1) the first time in history where the principles according to which this structure operates have been discovered and made explicable, and (2) the fact that God did this can be proven in a scientific manner that is both repeatable and predictable.
		The entire word structure of the Bible, was designed by God in a unique way so that theomatics could work.
		In otherwords, God specifically structured the grammar of the Hebrew and Greek languages so that all of the number patterns could fit and flow together.
	www.pcez.com /~itrpdx/maindcs/structure.htm (1492 words)

	Xah: Algorithmic Mathematical Art
		Commonly found are illustration of famous plane-filling curves in mathematics, plant-growth modeling, or simplistic symmetric drawings for children as a demo of Logo.
		A visual artwork is mathematical if it appeals to mathematicians and exhibits a strong mathematical structure.
		Generating a algorithmic mathematical art by a algorithmic process can be likened to specifying a sequence by recursive formula or a group by generators and relations.
	www.xahlee.org /Periodic_dosage_dir/t1/20040113_cmaci_larcu.html (3394 words)

	ipedia.com: Wavefunction Article (Site not responding. Last check: 2007-10-26)
		In mathematical terms, such continuous orthonormal bases are referred to as diagonalizations, because mathematically they correspond to representing certain commutative algebras of operators as algebras of multiplication operators.
		Due to the commutation relationship of the position and momentum operators, for a system of spinless particles in euclidean space the r-space and k-space wavefunctions are Fourier transform pairs.
		The precise formulation of this last statement is rather subtle and is called the Stone-von Neumann theorem in the mathematical physics literature.
	www.ipedia.com /wavefunction.html (606 words)

	Distributed Object Computation Testbed (DOCT) (Site not responding. Last check: 2007-10-26)
		Document structure analysis is reduced to parsing OCR results and recreating document structure by performing fuzzy searches for standard phrases and some format analysis.
		The remaining structure analysis is handled indirectly by specialized processing for non-textual data types such as tables and mathematical equations whose structure and content is inherently visual in nature and is derivable by analysis once the data type is known.
		Document structure analysis for directly mapped elements is reduced to parsing OCR results and recreating document structure by performing fuzzy searches for standard phrases and some format analysis.
	www.sdsc.edu /DOCT/Publications/a3-3/a3-3.html (5010 words)

	Mathematical Miracle of the Quran. The original File Mathematical Miracle of the Quran, Authorized English Translation ...
		There are two major facets of the Quran's mathematical system: (1) The mathematical literary composition, and (2) The mathematical structure involving the numbers of suras and verses.
		Because of this comprehensive mathematical coding, the slightest distortion of the Quran's text or physical arrangement is immediately exposed.
		However, the Quran's mathematical system is not limited to the word "God;" it is extremely vast, extremely intricate, and totally comprehensive.
	www.submission.org /math-ap1.html (3765 words)

	Mathematical Structures in Computer Science (Site not responding. Last check: 2007-10-26)
		Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science.
		The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work.
		The journal welcomes applications to computing based on the use of specific mathematical structures (e.g.
	titles.cambridge.org /journals/journal_catalogue.asp?mnemonic=msc (218 words)

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