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Topic: Duality order theory

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Duality (mathematics)

Hasse diagram

Pontryagin duality

Boolean algebra

List of Boolean algebra topics

Modal logic

	Quantum Theory - MSN Encarta
		Quantum theory is more general than classical physics, and in principle, it could be used to predict the behavior of any physical, chemical, or biological system.
		The forces that quantum theory describes are the electrical, the magnetic, the weak, and the strong.
		One of the striking differences between quantum theory and classical physics is that quantum theory describes energy and matter both as waves and as particles.
	encarta.msn.com /encyclopedia_761559884/Quantum_Theory.html (2434 words)

	Duality (order theory) - Wikipedia, the free encyclopedia
		In the mathematical area of order theory, every partially ordered set P gives rise to a dual (or opposite) partially ordered set which is often denoted by P
		If a given statement is valid for all partially ordered sets, then its dual statement, obtained by inverting the direction of all order relations and by dualizing all order theoretic definitions involved, is also valid for all partially ordered sets.
		Note that the consideration of dual orders is so fundamental that it often occurs implicitly when writing ≥ for the dual order of ≤ without giving any prior definition of this "new" symbol.
	en.wikipedia.org /wiki/Duality_(order_theory) (336 words)

	Why strings?
		Quantum electrodynamics was a renormalizable theory, and by the 19402, this was regarded as a solved relativistic quantum theory.
		The dual model was never that successful at describing particles, but it was understood by 1970 that the dual models were actually quantum theories of relativistic vibrating strings and displayed very intriguing mathematical behavior.
		The possible advantage of string theory is that the analog of a Feynman diagram in string theory is a two-dimensional smooth surface, and the loop integrals over such a smooth surface lack the zero-distance, infinite momentum problems of the integrals over particle loops.
	superstringtheory.com /basics/basic3a.html (959 words)

	Symbolism.Org: Symbolism of Popular Culture: Duality
		Although duality is the origination of opposition, Jung notes that the polarity is often arranged in a system of four as a quaternio or quaternity.
		The vertical duality in space is between the context of the film represented by its setting in time and place and the content of the film represented by its characters and their dialogue, actions and objects.
		The horizontal duality in time is between the beginning of the film and the end of the film and is symbolized by the psychological state of the hero at the beginning of the film and his or her state at the end of the film.
	www.symbolism.org /writing/books/spc/duality/page2.html (2090 words)

	M-theory
		The theory requires mathematical tools which have yet to be invented in order to be fully understood.
		M-Theory in various geometric backgrounds is associated with the different superstring theories (in different geometric backgrounds), and these limits are related to each other by the principle of duality.
		The T duality infers the existence of open strings with positions fixed in the dimensions that are T-transformed.
	www.ebroadcast.com.au /lookup/encyclopedia/m_/M_Theory.html (790 words)

	paperaddressDuality and Dual States: Glossary Definition
		It is a property belonging to the branch of algebra known as lattice theory, which is involved with the concepts of order and structure common to different mathematical systems.
		Sometimes the language of a statement must be modified in order that the corresponding dual statement be clear; the dual of the statement "Two lines intersect in a point" is vague, while the dual of "Two lines determine a point" is clear.
		Duality, a pervasive property of algebraic structures, holds that two operations or concepts are interchangeable, all results holding in one formulation also holding in the other, the dual formulation.
	www.math.vt.edu /people/gao/duality/duality.html (3144 words)

	Leivant. Higher Order Logic. (Site not responding. Last check: 2007-10-20)
		Finite order logic (omega-order logic, or type theory) [Church 1940] is introduced in the form of a relational variant [Schutte 1960] which is essentially Andrews F-omega.
		This is because second order truth of wff's in the language of PA in the standard model is reducible to validity of certain (effectively recognizable) second order wff's (using the fact that we can characterize the standard model in second order).
		In first order, we have Fraisse's Theorem which states that properties of a model are first order definable iff "it can be recognized by a computation with a finite number of alternations between existential (nondeterministic) and universal (co-nondeterministic) guesses" [compare this to "finitely isomorphic" iff elementary equivalence for finite symbol sets].
	www.andrew.cmu.edu /user/cebrown/notes/leivant.html (4896 words)

	Universal Theory - String Theory (Site not responding. Last check: 2007-10-20)
		String Theories are proposed to resolve the inconsistencies between GTR and Quantum Mechanics.
		In 1995 in order to explain why there are five different models of string theory which are contradicting each other and at the same time each claimed to be the theory of everything, they introduced M-theory and took the liberty to add the 7th.
		To me the assumption of one dimensional string particle (extension of mass proper to unseen extra dimensions) and suggesting the existence of a super-space (which is still a space-like entity) beyond the familiar space-time is the frantic attempt of cliff settlers to keep their ground and not to pass the edge.
	www.universaltheory.org /string_theory.htm (2415 words)

	CISNA - The Duality Principle
		If students were to be presented with the structure of duality, and shown how it applies to each of us in our daily lives before they began studying the theories of any scientific discipline, they would have the opportunity to incorporate this knowledge into their understanding of the disciplines.
		It is a geometrical theory which postulates that the presence of mass and energy causes spacetime to be curved, and this curvature affects the path of free particles and even deflects the path of light.
		String theory is a theoretical framework for constructing a unified theory which includes both the microscopic forces of the atoms and macroscopic gravitational forces.
	www.posttool.com /cisna/dual.html (3891 words)

	Duality symmetries in string theory
		In case of quantum field theories this would require that each of the two theories correspond to a renormalizable local quantum field theory (note 9), whereas in the case of string theory, this would require that each of the two theories is governed by simple interactions of the kind described in the previous article.
		The order parameter is the analog of the field associated with an elementary particle, whereas the disorder parameter is the analog of the field associated with a composite particle.
		Historically string theory started as an alternative to this reductionist viewpoint in which all particles were considered to be on equal footing, and the interaction between these particles were supposed to be governed by the requirement of consistency rather than as a result of interaction among some elementary constituents.
	www.ias.ac.in /currsci/dec251999/articles20.htm (8398 words)

	Theories of Everything
		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.
		Duality between strings and fivebranes still remained conjectural, however, because of the problem of quantizing five-branes.
		In supersymmetric theories, one finds that the strengths of the strong, weak and electromagnetic forces all converge at an energy E of 10^16 giga electron volts.
	www.dhushara.com /book/quantcos/fs/fstr.htm (4213 words)

	de Broglie Waves
		Relatively straightforward tests are offered by diffraction and interference - if a beam of such ``particles'' were shone at a diffraction grating and a diffraction pattern of a series of light and dark fringes results, then one would be forced to adopt the wave picture for this phenomena.
		We recall that for a good diffraction pattern to result the size of the diffraction slits should be of the same order as the wavelength of the light used.
		Thus, it is possible to verify the wave nature of electrons in such diffraction experiments, and indeed this property is the principle behind the relatively common electron microscope.
	theory.uwinnipeg.ca /physics/quant/node6.html (284 words)

	Duality Principles in Nonconvex Systems
		The aim of this book is to give, within a unified framework, a self-contained comprehensive presentation of the mathematical theory of duality for general non-convex and non-smooth systems, and the author hopes that it will provide a smooth, stimulating and provocative blend of different flavors of mechanics.
		In the present day, the theory of duality has become a vast subject, especially due to the modern work in optimization, game theory, economic science, theoretical physics and chemistry, mathematical programming, variational analysis, nonconvex-nonsmooth analysis and control, critical point theory and in many other areas.
		It is intended that the present book will fill the (duality) gap between the mathematical and engineering sciences by providing a systematic exposition of duality theory as a self-contained theoretical system with substantial applications in physics, continuum mechanics, large deformation structures, mathematical programming, partial differential equations and geometry.
	www.math.vt.edu /people/gao/books/preface.html (1545 words)

	info-theory.nb (Site not responding. Last check: 2007-10-20)
		Information theory provides us with a formula for determining the number of bits required in an optimal code even when we don't know the code.
		The key to gaining an intuitive understanding of the -P log P formula for calculating information content is to see the duality between the number of messages to be encoded and their probabilities.
		Since the actual distribution we're trying to encode is not uniform, we take the weighted average of the estimated information content of each value (heads or tails, in the case of a coin), weighted by the probability P of that value occuring.
	www-2.cs.cmu.edu /~dst/Tutorials/Info-Theory (2126 words)

	Grand Unified Theory
		The theory formerly known as superstrings (M theory) unifies all 4 forces, is background free, and the math all works.
		String theory in physics motivated/required development of some brand new pure mathematics (the details of which I'm not acquainted with); this has been widely cited as an inversion of the popular notion that physics is only about applying pure math that mathematicians came up with a long time ago.
		Osvaldo Domann, "[http://www.odomann.com The theory of longitudinal and transversal rotational momentum]" A mathematical theory that unites all physical forces and eliminates the wave/particle duality.
	c2.com /cgi/wiki?GrandUnifiedTheory (1480 words)

	The Dual Nature of Light as Reflected in the Nobel Archive
		The wave theory is also able to account for the fact that the edges of a shadow are not quite sharp.
		The mathematical theory of electromagnetism by James Clerk Maxwell, set up in 1864, led to the view that light is of electromagnetic nature, propagating as a wave from the source to the receiver.
		Paul Dirac published in 1927 a mathematical theory for the interaction between electromagnetic fields, such as light or X-rays, and charged particles with both aspects of light inbedded - it was a theory of quantised fields.
	nobelprize.org /nobel_prizes/physics/articles/ekspong (3614 words)

	How are string theories related?
		String theories feature two kinds of perturbative expansions: an expansion in powers of the string parameter a' in the conformal field theory on the two-dimensional string worldsheet, and a quantum loop expansion for string scattering amplitudes in d-dimensional spacetime.
		This duality allows an understanding of the strong coupling limit of the theory that would not be possible by any other means.
		At very strong coupling, heterotic SO(32) string theory has excitations that are open strings, but these open strings are highly unstable in the weakly coupled limit of the theory, which is the limit in which heterotic string theory is commonly understood.
	superstringtheory.com /basics/basic6a1.html (676 words)

	Google World Order (conspiracy theory) - Uncyclopedia
		Elements of the theory are present in the populism of the nineteenth century.
		The theories in their present form can be traced to the collapse of Soviet communism due to Gorbachev's ineffective search algorythms and President George H. Bush's speech of 11 September 1990.
		Rainbow helicopters are part of a conspiracy theory, especially prevalent among the crazy redneck movement, that claims that special unmarked "rainbow-coloured" helicopters are being used now in secret military operations and are going to be used by secret agents of the Google World Order to implement the Google World Order.
	uncyclopedia.org /wiki/Google_World_Order_(conspiracy_theory) (2308 words)

	PlanetMath: Serre duality
		While Serre duality is not in a strict sense a generalization of Poincaré duality, they are philosophically similar, and both fit into a larger pattern on duality results.
		Cross-references: duality, Poincaré duality, strict, locally free, nonsingular, sheaf, coherent sheaf, pairing, perfect, field, algebraically closed, projective varieties, dimension, schemes
		This is version 7 of Serre duality, born on 2003-08-15, modified 2006-08-11.
	planetmath.org /encyclopedia/SerreDuality.html (134 words)

	Sankara: Readings
		Therefore, noble aspirants for release are intent only on meditating thereon; they have no concern for any theory of creation, which is futile.
		Nor is it that the magic produced by the magician exists in fact and is then removed when the veil is lifted off the sight of the onlookers.
		Even thus, here, the duality which is called the world is mere illusion.
	nonduality.com /sankara.htm (1432 words)

	[No title]
		In order to counterbalance the(A) tendency towards these extremes, a view of science is needed which(THAT) alerts us to the boundaries beyond which, if we trespass, leads to the same said abuses(LEADS TO THESE SAME ABUSES).
		In biology the functions of chlorophyll and hemoglobin, inhaling and exhaling, muscle contraction and extension, osteoblasts and osteoclasts, lymphatic and myeloid elements are known to have opposite functions.
		Shoaf, R. Milton, Poet of Duality: A Study of Semiosis in the Poetry and the Prose.
	www.isna.net /uploads/media/Sabah_Karam_-_Duality_Principle.doc (3233 words)

	Universal Law of Duality
		Because duality and polarity are universally exhibited in all of creation, parallels can be seen between differing cultural and religious traditions.
		According to the prevailing Big Bang Theory, before our universe came into being, everything was contracted into a super-dense primordial matter that contained all space and time.
		In this regard, fl holes are said to suck all energy and matter into an infinitely dense center in which the gravitational force is so great that light is not able to escape from it.
	www.som.org /2laws/universallaws/duality.htm (1845 words)

	Nondual Perspectives: John Davis: Transpersonal Psychology
		Many psychologists are promoting a shift away from psychology as a natural science and toward psychology’s roots in philosophy.
		In social psychology, developmental psychology, personality theory, and to a lesser extent in clinical psychology, this shift is being tagged ‘postmodern’.
		We move easily from experience to theory and back again, each balancing, supporting, and completing the other.
	nonduality.com /tp.htm (4897 words)

	Binary Code & The Duality of Man Theory - The Lost Blog
		The point about duality and everyone having some good and some evil in them definitely has a strong connection with the show.
		No one is going to lose sleep over whether or not this theory is the real intention at the heart of the producers and writers of the show.
		But true fans of the show know and appreciate that there is a lot more to it in terms of the “deeper” topics it tries to explore–in much the same way that Animal Farm is not just a book about talking pigs trying to take over a farm.
	www.lostblog.net /lost/tv/show/binary-code-the-duality-of-man-theory (4143 words)

	Wikinfo \| M-theory
		M-theory is a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together.
		According to Dr. Edward Witten, who proposed the theory, mathematical tools which have yet to be invented are needed in order to fully understand it.
		Images, some of which are used under the doctrine of Fair use or used with permission, may not be available.
	wikinfo.org /wiki.php?title=M-theory (912 words)

	Wave-particle duality Summary
		Wave-particle duality is a term used to describe the modeling of nature by both wave theory and particle theory.
		Revival of the Wave Theory of Light in the Early Nineteenth Century Summary
		In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles.
	www.bookrags.com /Wave-particle_duality (200 words)

	More on M-theory
		M-Theory in various geometric backgrounds is associated with the different superstring theories (in different geometric backgrounds), and these limits are related to each other by the principle of dualism.
		* The type I theory is special in that it is based on unoriented open and closed strings.
		One interpretation of this result is that the 11th dimension was always present but invisible because the radius of the 11th dimension is proportional to the string coupling constant and the traditional perturbative string theory presumes it to be infinitesimal.
	www.artilifes.com /m-theory.htm (951 words)

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