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Topic: Dimension theory

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Planck scale

Gauge theory

Power series

Physical constant

	PlanetMath: dimension
		The word dimension in mathematics has many definitions, but all of them are trying to quantify our intuition that, for example, a sheet of paper has somehow one less dimension than a stack of papers.
		One common way to define dimension is through some notion of a number of independent quantities needed to describe an element of an object.
		From the point of view of set theory there are as many real numbers as pairs of real numbers since there is a bijection from real numbers to pairs of real numbers.
	planetmath.org /encyclopedia/Dimension3.html (513 words)

	M-theory - Wikipedia, the free encyclopedia
		There were separate theories for electricity and magnetism, and although they were known to be related, it was not clear until Maxwell published his equations just what the relationship was.) Witten has suggested that a general formulation will probably require the development of new mathematical language.
		It was shown in the early 1990s that the various superstring theories were related by dualities, which allowed physicists to relate the description of an object in one string theory to the description of a different object in another theory.
		The Type IIA string theory and the Type IIB string theory are connected by T-duality; this means, essentially, that the IIA string theory description of a circle of radius R is exactly the same as the IIB description of a circle of radius 1/R. This is a profound result.
	en.wikipedia.org /wiki/M-theory (1574 words)

	String theory - Wikipedia, the free encyclopedia
		String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that are the basis of the Standard Model of particle physics.
		Investigating how a string theory may include fermions in its spectrum led to supersymmetry, a mathematical relation between bosons and fermions which is now an independent area of study.
		The reason for the unobservability of the fifth dimension (its compactness) was suggested by the Swedish physicist Oskar Klein in 1926.
	en.wikipedia.org /wiki/String_theory (2846 words)

	string theory
		String theories are classified according to whether or not the strings are required to be closed loops, and whether or not the particle spectrum includes fermions.
		In order to include fermions in string theory, there must be a special kind of symmetry called supersymmetry, which means that for every boson (a particle, of integral spin, that transmits a force) there is a corresponding fermion (a particle, of half-integral spin, that makes up matter).
		Now, if string theory is to work as a theory of quantum gravity, then the average length of a string has to be in the ballpark of the distance over which the quantization of spacetime –; the granularity of space and time – becomes noticeable.
	www.daviddarling.info /encyclopedia/S/stringtheory.html (2299 words)

	String theory - Facts, Information, and Encyclopedia Reference article
		String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles) that are the basis of the Standard Model of particle physics.
		The reason for the unobservability of the fifth dimension (its compactness) was suggested by the Swedish physicist Oskar Klein in 1926.
		Since string theory may not be tested in the foreseeable future, some scientists have asked if it even deserves to be called a scientific theory: it is not yet a falsifiable theory in the sense of Popper.
	www.startsurfing.com /encyclopedia/s/t/r/String_theory.html (2690 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-09)
		In the later thirties, the dimension theory of metrizable spaces with a countable base was constructed, and by the start of the sixties the dimension theory of arbitrary metrizable spaces was finished.
		Dimension theory is most meaningful, first, for the class of metric spaces with a countable base, and, secondly, for the class of all metric spaces.
		Several of the dimensions most used in algebra and ring theory may be defined on the lattice of submodules of some module, globalizing the definition by considering the supremum (or a similar invariant) of the dimension of all modules (perhaps restricting to a certain class of modules).
	eom.springer.de /D/d032450.htm (2461 words)

	Imagining the Tenth Dimension - A Book by Rob Bryanton
		When we imagined the fourth dimension, it was as if we were treating the entirety of three-dimensional space in a particular state as a single point, and drawing a fourth-dimensional line to another point representing space as it is in a different state.
		Then in the seventh dimension, we treated all of the possible timelines which could be generated from our big bang as if this were a single point, and imagined drawing a line to a point representing all of the possible timelines for a completely different universe.
		His theory explains that an observer is not necessary because all possible states do actually exist simultaneously, and the act of observation is not “collapsing” a particle, it is merely “observing” a part of a wave in one of its simultaneously existing states.
	www.tenthdimension.com /flash2.php (3274 words)

	Metric dimension (graph theory) - Wikipedia, the free encyclopedia
		In graph theory, the metric dimension of a graph G is the minimum cardinality of a resolving set for G.
		The metric dimension of G is the minimum cardinality of a resolving set for G.
		Harary and R. Melter, On the metric dimension of a graph.
	en.wikipedia.org /wiki/Metric_dimension_(graph_theory) (215 words)

	Fourth dimension atomic theory
		This modification occurs within the fourth dimension as the atom is constantly being accelerated, as it is forced to change its physical direction from the spiral galaxy and orbital acceleration velocity vectors.
		Because the Dovada research team works with a fourth dimension atomic model, many of the physical constants, similar to the Fine Structure Constant, are now very easily defined from first principles directly from the atoms spiral galaxy and orbital acceleration vectors and the reason for their very existence is now fully documented and understood.
		To comprehend the fourth dimension is quite easy if you look at it from this simple perspective, the first three dimensions are width, length and height, these are effectively instantaneous values, but when measured within the SI system measurements are related to a time period (one second).
	www.dovada.com /research/technology.htm (1363 words)

	Eleventh Dimension
		This is an elegant theory that describes the macroscopic world of fl holes, quasars and the big bang.
		The form of quantum theory that goes furthest in describing matter and its interactions is the Standard Model, which is based on a bizarre bestiary of particles such as quarks, leptons and bosons (see Diagram).
		In superstring theory, the subatomic particles we see in nature are nothing more than different resonances of the vibrating superstrings, in the same way that different musical notes emanate from the different modes of vibration of a violin string.
	www.fortunecity.com /emachines/e11/86/dimens.html (3039 words)

	Theory Continued Page 4
		Calabi-Yau manifolds appear in the theory because in passing from the 10-dimensional space time to a physically realistic description in four dimension, string theory requires that the additional 6-dimensional space is to be a Calabi-Yau manifold.
		Perturbation theory is a very useful tool for studying the physics at weak coupling, and most of our current understanding of particle physics and string theory is based on it.
		Their belief was consistent with quantum theory, since the mathematics of quantum theory does not depend on the existence of particle substance or charge substance.
	www.superstringtheory.fanspace.com /contact.html (5607 words)

	Hausdorff dimension -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-09)
		However, topological dimension behaves in quite unexpected ways on certain highly irregular sets such as ((mathematics) a geometric pattern that is repeated at every scale and so cannot be represented by classical geometry) fractals.
		To define the Hausdorff dimension for X, we first consider the number N(r) of (Round object that is hit or thrown or kicked in games) balls of radius at most r required to cover X completely.
		For example (additional info and facts about box-counting dimension) box-counting dimension, generalises the idea of counting the squares of (Paper that has lines to permit drawing graphs) graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller.
	www.absoluteastronomy.com /encyclopedia/h/ha/hausdorff_dimension.htm (1882 words)

	Theories of Everything
		When the extra dimension curls into a circle, M-theory yieolds a typeIIA superstring, which is further related by a duality to the type IIB superstring.
		Until recently, the best hope for a theory that would unite gravity with quantum mechanics and describe all physical phenomena was based on strings: one-dimensional objects whose modes of vibration represent the elementary particles.
		A p-brane of dimension 0 is a particle, that of dimension 1 is a string and that of dimension 2 is a sheet or bubble.
	www.dhushara.com /book/quantcos/fs/fstr.htm (4213 words)

	NOVA \| The Elegant Universe \| Watch the Program \| PBS
		String theory is radically changing our ideas about the nature of space, opening up the possibility that extra dimensions, rips in the fabric of space, and parallel universes actually exist.
		The extra dimension of space required to unify string theory suggests that we may be trapped on just one tiny slice of a higher-dimensional universe.
		By the mid-1980s physicists had developed five different versions of string theory, raising the question of whether it would prove to be a theory of everything or a theory of nothing.
	www.pbs.org /wgbh/nova/elegant/program.html (715 words)

	Fractal Theory and Dimension
		Dimension is a term used to measure the size of a set.
		But the value of their dimension does not necessarily need to be an integer.
		There are two main types of dimension that we measure: box dimension and topological dimension.
	hypatia.math.uri.edu /~kulenm/honprsp02/dimension.html (518 words)

	Amazon.com: Dimension theory (Princeton mathematical series): Books: Witold Hurewicz (Site not responding. Last check: 2007-10-09)
		The `sum theorem' for dimension n is proven, which says that a space which is the countable union of closed sets of dimension less than or equal to n also has dimension less than or equal to n.
		This allows a characterization of dimension in terms of the extensions of mappings into spheres, namely that a space has dimension less than or equal to n if and only if for every closed set and mapping from this closed set into the n-sphere, there is an extension of this mapping to the whole space.
		In chapter 7 the author relates dimension theory to measure theory, and proves that a space has dimension less than or equal to n if and only if it is homeomorphic to a subset of the (2n+1)-dimensional cube whose (n+1)-dimensional measure is zero.
	www.amazon.com /Dimension-theory-Princeton-mathematical-Hurewicz/dp/B0007E46WC (1649 words)

	Fractal Theory and Dimension
		Dimension is a term used to measure the size of a set.
		But the value of their dimension does not necessarily need to be an integer.
		There are two main types of dimension that we measure: box dimension and topological dimension.
	www.math.uri.edu /~kulenm/honprsp02/dimension.html (518 words)

	String Theory and the Human Mind
		This theory in brief says that instead of the basic particles of matter being point-like objects, they are very tiny vibrating strings and while occupying their place in the three-dimensional world, they are not confined to the three dimensions of space with which we are familiar.
		After the theory of quantum mechanics was developed by Schroedinger and Heisenberg in the 1920s, which could be proved in the laboratory using three-dimensional instruments, Einstein became a lonely man. He did not accept the uncertainty principle of quantum mechanics and that largely isolated him from the physics community.
		To understand that the fourth dimension is larger that the third dimension, we must realize that the spacetime continuum is composed of the physical reality of all moments of time from the big-bang onward, while our puny third dimension is only the current moment of time which passes into spacetime the instant after it occurs.
	www.wwitherspoon.org /StringTheory.htm (10731 words)

	BBC - Science & Nature - Horizon - Parallel Universes - Transcript
		The theory is breathtaking and it has an extraordinary conclusion: that the Universe we live in is not the only one.
		M Theory was about to come up with a suitably outrageous answer and parallel universes would be at the very heart of it.
		M Theory may really be able to explain everything in the Universe, but the victory will be bittersweet, for at the end of its long quest, science has discovered that the Universe it's thought to explain may be nothing special.
	www.bbc.co.uk /science/horizon/2001/parallelunitrans.shtml (5261 words)

	The Current Science of Time Travel
		But the discussion has led scientists to divide into the two popular theories related to time travel which we have here found to be insufficient: the fixed time theory and the parallel dimension theory.
		All such proof of other dimensions gives us is the possibility that we might be able to travel to such dimensions, and the related possibility that we might be able to travel to the past or future of such dimensions--none of which is clearly indicated by the present theory.
		The Parallel Dimension theory does not answer that at all, and while it may be very interesting in terms of imagined science fiction approaches to time travel, it is not adequate for the current wormhole theory.
	www.mjyoung.net /time/science.html (1703 words)

	The Two Brothers: Why Parallel Dimension Theory Is Not Time Travel
		The parallel and divergent dimensions theories are repeatedly presented as "this is how time travel really works", and the response is given that this is a very interesting theory--but it's not time travel.
		Parallel Dimension theory does this by saying you aren't in your own past; that in essence says that when you travel to the past, you don't travel to the past but actually do something else.
		Most theorists see only the two theories, and think it is a choice between being unable to control your own actions on the one hand (a consequence of the Fixed Time theory) and being unable to reach your own past on the other (the meaning of the Parallel Dimensions theory).
	www.mjyoung.net /time/brothers.html (1961 words)

	The String Theory//10th Dimension
		Probing stuff at the plank length where the hidden 7 spatial dimensions that the theory proposes are hidden isn't feasible.
		The second dimension is perpendicular to the first one, the third dimension is perpendicular to the second dimension.
		Now, the fourth dimension is simply perpendicular to the third dimension, the fifth perpendicular to the fourth, and so on....
	www.teamliquid.net /forum/viewmessage.php?topic_id=47304 (1024 words)

	sciforums.com - Four spatial dimension "Shadows" theory
		In this model all forms of energy are the result of a curvature in the third spatial dimension with respect to the fourth spatial dimension.
		Shadows postulates the quantized energy associated with photons is the result of the curvature caused by the peaks and valleys of a “matterenergy” wave moving on in the surface of three dimensional space with respect to the fourth spatial dimension.
		Imagine the two-dimensional “surface” of a three dimensional sphere to be analogous to the “spherical surface” of a dimensional sphere with respect to the fourth spatial dimension.
	www.sciforums.com /showthread.php?p=198735 (2628 words)

	Ontolingua Theory STANDARD-UNITS
		This theory defines a set of basic units of measure, a set of fundamental dimensions and a few others.
		It is intended that this theory represent enough information to convert among any pair of units of the same dimension that are either defined as basic units here or built up from the basic units using the composition operators * and EXPT.
		The physical dimension of force is defined as mass times length over time squared.
	www-ksl.stanford.edu /htw/dme/thermal-kb-tour/standard-units.html (623 words)

	ipedia.com: Hausdorff dimension Article (Site not responding. Last check: 2007-10-09)
		The Hausdorff dimension agrees with the ordinary (topological) dimension on "well-behaved sets", but it is applicable to many more sets and is not always a natural number.
		The Hausdorff dimension should not be confused with the (similar) box-counting dimension.
		The Hausdorff dimension is a well-defined extended real number for any set E and we always have 0 ≤ d(E) ≤ ∞.
	www.ipedia.com /hausdorff_dimension.html (750 words)

	Fractal Dimension Theory & Application (Site not responding. Last check: 2007-10-09)
		Despite such criticisms, fractal theory is being constantly applied to varied fields, and led to improvements in turbulence modeling, ecological predictions, and chaos theory (see Gleick, J.) Fractal theory is not inherently based on graphics dumps, but rather on comparing the "roughness" of shapes.
		Chaos theory itself is far beyond the scope of this paper, and is mentioned only to reference applications of the fractal mathematics.
		Fractal maths, and one of its offspring, chaos theory, have revolutionized the way that scientists and engineers have approached various problems in many different fields, whether projecting spread of epidemics within communities, or saving petrochemical companies millions of dollars to be able to minimize turbulence, and lost energy due to drag in pipelines.
	www.timothyward.com /dox/ccsu/ew362rp.htm (2531 words)

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