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Topic: Duality (mathematics)


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In the News (Tue 19 Mar 19)

  
  Duality (mathematics) - Wikipedia, the free encyclopedia
In another group of dualities, the objects of one theory are translated into objects of another theory and the morphisms between objects in the first theory are translated into morphisms in the second theory, but with direction reversed.
Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed.
Dual graph is a concept in graph theory.
en.wikipedia.org /wiki/Duality_(mathematics)   (315 words)

  
 Duality - Wikipedia, the free encyclopedia
In dualism (philosophy of mind), the body and mind are considered to be irreducibly distinct.
In analytical psychology, duality is an archetype, that is to say, one of the deep powerful symbols of the functioning of the psyche.
In alchemy, duality is a dynamic principle of opposing or complementary elements or spirits.
en.wikipedia.org /wiki/Duality   (271 words)

  
 Encyclopedia: Duality (mathematics)
In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...
In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves.
Coherent duality in mathematics refers to a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the local theory.
www.nationmaster.com /encyclopedia/Duality-(mathematics)   (1058 words)

  
 [No title]
Thus the dual to the annuity capitalization formula is the fact that the constant income stream of r is the equivalent of the capital of $1.
This series-parallel duality principle is analogous to the Boolean duality principle, which states that any theorem of Boolean algebra remains valid under the interchange of conjunction and disjunction (including the interchange of the null element and unit).
The "inverse" or dual of a convex function f(x) is its "convex conjugate": EMBED Equation  The derivative of the convex conjugate is the inverse of the derivative in the SP algebra of monotonic increasing functions.
www.ellerman.org /Davids-Stuff/Maths/sp_math.doc   (4901 words)

  
 Inconsistent Mathematics
Inconsistent mathematics is the study of the mathematical theories that result when classical mathematical axioms are asserted within the framework of a (non-classical) logic which can tolerate the presence of a contradiction without turning every sentence into a theorem.
Gödel's contribution to the philosophy of mathematics was to show that the first three of these can be rigorously expressed in arithmetical theories, albeit in theories which are either inconsistent or incomplete.
Projective geometry is a mathematical theory which is interesting because we are creatures with an eye, since it explains why it is that things look the way they do in perspective.
plato.stanford.edu /entries/mathematics-inconsistent   (2006 words)

  
 Duality (mathematics) -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-11-03)
The prototypical example here is the (Click link for more info and facts about duality in projective geometry) duality in projective geometry: given any theorem in plane projective geometry, exchanging the terms "point" and "line" everywhere results in a new, equally valid theorem.
In another group of dualities, the objects of one theory are translated into objects of another theory and the (Click link for more info and facts about morphism) morphisms between objects in the first theory are translated into morphisms in the second theory, but with direction reversed.
Theorems showing that certain objects of interest are the (Click link for more info and facts about dual space) dual spaces (in the sense of linear algebra) of other objects of interest are also often called dualities.
www.absoluteastronomy.com /encyclopedia/D/Du/Duality_(mathematics).htm   (544 words)

  
 S. Majid On the Nature of Physics
Such duality considerations are certainly evident in some form in the context of quantum theory and gravity.
Dual to this is the notion of co-Heyting algebra and co-intuitionistic logic in which one drops the axiom that the intersection of a proposition and its negation is empty.
Another example is for a particle on a circle, where the allowed momentum modes are labeled by the dual group Z. Fourier transform interchanges the roles of the group and its dual.
www.maths.qmul.ac.uk /~majid/pessay.html   (1498 words)

  
 Dr. Sergey I. Repin   (Site not responding. Last check: 2007-11-03)
Mathematical and numerical analysis of problems in plasticity and nonlinear elasticity.
Mathematical and numerical analysis of discontinuous solutions in plasticity theory.
The solutions of problems in the mathematical theory of plasticity with discontinuities in the displacement fields.
amd.stu.neva.ru /repin/repin.htm   (606 words)

  
 Curriculum Vitae of
Proceedings of IUTAM Symposium on Complementarity, Duality and Symmetry in Nonlinear Mechanics, D.Y. Gao (eds).
Proceedings of IUTAM Symposium on Complementarity, Duality and Symmetry in Nonlinear Mechanics.
Gao, D.Y., Canonical duality theory: Connection between nonconvex mechanics and global optimization, to appear in Complementarity, Duality, and Global Optimization, Gao, D.Y. and Sherali, H. (ed), Springer, 2006.
www.math.vt.edu /people/gao/papers/list.htm   (1765 words)

  
 ipedia.com: Duality (physics) Article   (Site not responding. Last check: 2007-11-03)
In modern physics, duality most often refers to the paradigm underlying quantum mechanics, according to which matter or energy can exhibit properties associated with wave physics as well as classical particle mechanics.
This duality also implies that light and electromagnetic radiation, which in classical physics are considered to be wave phenomena, can exhibit particle-like properties.
Other fields where duality is encountered in this sense are statistical mechanics and string theory.
www.ipedia.com /duality__physics_.html   (260 words)

  
 Duality Principles in Nonconvex Systems
The term duality as used in our daily life means the sort of harmony of two opposite or complementary parts through which they integrate into a whole.
Duality in mathematics, roughly speaking, is a fundamental concept that underlies many aspects of extremum principles in natural systems.
With the aid of these potentially useful notations, common mathematical frameworks are constructed for many different systems, independently of their physical content, and the frameworks also apply to wider classes of problems in engineering and science.
www.math.vt.edu /people/gao/books/preface.html   (1545 words)

  
 Math
Category theory provides the framework to identify the concrete universals in mathematics, the concrete instances of a mathematical property that exemplify the property is such a perfect and paradigmatic way that all other instances have the property by virtue of participating in the concrete universal.
The duality in economics is convex duality, and it is shown that series-parallel duality is the "derivative" of convex duality.
The mathematical basis behind DEB (algebraic operations on ordered pairs of numbers) was developed in the nineteenth century by Sir William Rowan Hamilton as an abstract mathematical construction to deal with complex numbers and fractions.
www.ellerman.org /Davids-Stuff/Maths/Math.htm   (1808 words)

  
 Duality Article, Duality Information   (Site not responding. Last check: 2007-11-03)
In analytical psychology, duality isan archetype, that is to say, one of the deep powerful symbols of the functioning of the psyche.
In alchemy, duality is a dynamic principle of opposing orcomplementary elements or spirits.
In electronics, duality is the relationship between two devices or circuits that have mathematical descriptions that areidentical except that voltages in one correspond to currents in the other, and vice versa.
www.anoca.org /dual/different/duality.html   (295 words)

  
 CISNA - The Duality Principle   (Site not responding. Last check: 2007-11-03)
Duality should be considered as an axiom of the world in which we live.
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.
A summary of the ideas behind dual spaces, including several articles on TOE and the Mathematics of Duality, can be found in several past issues of Scientific American http://www.sciam.com/.
www.posttool.com /cisna/dual.html   (3891 words)

  
 Duals   (Site not responding. Last check: 2007-11-03)
The dual decisions are functionals, which could be specialized to classes defined by a finite number of parameters (which then become the dual variables).
The strong duality result for a particular (F,G,H) in F×G×H is equivalent to (x*, F, G, H) being a saddlepoint of T. Other theorems include the notion of one dual being stronger than another.
The Lagrangian dual is a strong dual for a convex program that satisfies Slater's interiority condition.
carbon.cudenver.edu /~hgreenbe/glossary/duals.html   (1508 words)

  
 wikien.info: Main_Page   (Site not responding. Last check: 2007-11-03)
A variety of different kinds of duality occurring in mathematics are listed at duality (mathematics).
Dual is a brand name for Hifi equipment, mostly known for vinyl record players.
In electronics, duality is the relationship between two devices or circuits that have mathematical descriptions that are identical except that voltages in one correspond to currents in the other, and vice versa.
www.hostingciamca.com /index.php?title=Dual   (301 words)

  
 Re: how mathematics is changed with Natural Numbers = finite-integers + infinite-integers (p-adics)
The history of mathematics will follow in a somewhat similar path that the history of physics followed concerning the debate and finally enlightenment and understanding of Duality.
Modern mathematics is ignorant of the Particle nature of mathematics which is the Adics.
Yet the history of mathematics is so very primitive that the mathematics community is not even aware of a Duality and has no clue as to the full realization of the particle-nature of mathematics.
www.jwgh.org /ark/articles/dualism.html   (469 words)

  
 Physics Virtual Bookshelf: Quantum Mechanics
A brief summary of wave-particle duality, from a first year physics course that uses minimal mathematics; the entire set of materials from the course is available by clicking here.
A brief summary of quantum interference and the uncertainty principle, from a first year physics course that uses minimal mathematics; the entire set of materials from the course is available by clicking here.
An overview of quantum mechanics, from a first year physics course that uses minimal mathematics; the entire set of materials from the course is available by clicking here.
www.upscale.utoronto.ca /GeneralInterest/QM.html   (429 words)

  
 tutORial: Virtual Duality   (Site not responding. Last check: 2007-11-03)
Duality Theory is an important topic in LP.
Please note that the Duality Machine is very conservative: it will not attempt to simplify the dual, say, by identifying superfluous constraints.
Maintained by: Moshe Sniedovich, Department of Mathematics and Statistics.
www.ifors.ms.unimelb.edu.au /tutorial/duality   (90 words)

  
 Duality Based on Galois Connection. Part I
In the paper, we investigate the duality of categories of complete lattices and maps preserving suprema or infima according to [15, p.
The duality is based on the concept of the Galois connection.
Duality of Subcategories of {\it INF} and {\it SUP}
www.mizar.org /JFM/Vol13/waybel34.html   (128 words)

  
 Amazon.com: Metrics, Connections and Gluing Theorems (Cbms Regional Conference Series in Mathematics): Books: Clifford ...   (Site not responding. Last check: 2007-11-03)
Anti-self duality is rather special to 4-dimensions and the imposition of this condition on curvatures of connections on vector bundles and on curvatures of Riemannian metrics has resulted in some spectacular mathematics.
The question of the existence of metrics on the manifold for which the anti-self dual part vanishes is the subject of the book, with particular attention paid to complex vector bundles over the manifold.
The anti-self dual equations are consequently a set of algebraic equations for the curvature, which are equivalent, over an open set in the manifold, to a first-order differential equation involving a 1-form over this open set and taking values in the Lie algebra of complex 2-space.
www.amazon.com /exec/obidos/ASIN/0821803239/top100   (948 words)

  
 Applied Mathematics Colloquium Abstracts (Fall 2004)   (Site not responding. Last check: 2007-11-03)
The Fisher-Kolmogorov-Petrovsky-Piscunov (FKPP) equation is a classical model used to describe the evolution of a spatially distributed population with local logistic (growth-saturation) dynamics and diffusive spreading.
The correspondence is not in terms of a fluctuating hydrodynamic description for the reaction-diffusion model, but rather via the concept of `duality', an idea that has played a major role in the probabilistic analysis of interacting particle systems in recent decades.
The idea of duality will be discussed and used to derive an exact formula for the extinction probability of any initial configuration for the stochastic FKPP equation.
www.amath.unc.edu /Seminar/2004/Oct18.html   (171 words)

  
 Archimedes Plutonium   (Site not responding. Last check: 2007-11-03)
And that the world of mathematics needs to move into the direction of recognition that the Finite Integers/Counting Numbers/ Natural Numbers become Infinite Integers as long as there is an endless adding of 1.
For what the future of mathematics (a subdepartment of physics) will be-- is that there exists 3 and only 3 types of Numbers and there exists 3 and only 3 types of Geometries and those numbers each, each and every one of those Numbers is a unique point in 1 of those 3 geometries.
At present the mathematics of the math community is a hair twirling psycho-philosophical babble of Imprecise definitions with Axioms that are not even understood and used and misused for convenience sake.
www.iw.net /~a_plutonium/File108.html   (5426 words)

  
 Mathematics Colloquium: Zhu Abstract, Spring 2004
We show that circular coloring of graphs is closely related to scheduling problems in distributed computation and to other problems in computer science.
We discuss the duality between circular chromatic number and circular flow number of graphs embedded in surfaces and pseudo-surfaces.
We introduce the concept of bounded treewidth duality for H-coloring problems and prove that H-coloring problems with this property are polynomial-time solvable.
www.math.uiuc.edu /Colloquia/04SP/zhu_apr08-04.html   (407 words)

  
 Amazon.ca: Books: Duality Principles in Nonconvex Systems: Theory, Methods and Applications   (Site not responding. Last check: 2007-11-03)
Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this book is the first to provide, within a unified framework, a self-contained comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics.
It shows how to use formal language and duality methods to model natural phenomena, to construct intrinsic frameworks in different fields and to provide ideas, concepts and powerful methods for solving non-convex, non-smooth problems arising naturally in engineering and science.
The whole volume or selected chapters can also be recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the notions of optimization and variational methods are employed.
www.amazon.ca /exec/obidos/ASIN/0792361458   (480 words)

  
 Duality in Reverse Convex Optimization
Duality in Reverse Convex Optimization: SIAM Journal on Optimization Vol.
A duality theorem for the general problem of minimizing an extended real-valued convex function on a locally convex linear space under a reverse convex constraint is considered.
In the particular case of the distance to a reverse convex subset in a normed linear space, we recover as a corollary a duality theorem due to C. Franchetti and I. Singer [Boll.
epubs.siam.org /sam-bin/dbq/article/29585   (130 words)

  
 La Trobe University - staff - Bert Mond
Duality with Generalized Convexity (with R. Egudo), Journal of the Australian Mathematical Society, series B, Vol.
Duality for Variational Problems with Invexity (with S. Chandra and I. Husain), Journal of Mathematical Analysis and Applications, Vol.
Duality with Invexity for a Class of Nondifferentiable Static and Continuous Programming Problems (with I. Smart), Journal of Mathematical Analysis and Applications, Vol.
www.latrobe.edu.au /mathstats/staff/mond.html   (4102 words)

  
 Amazon.co.uk: Duality Principles in Nonconvex Systems: Theory, Methods and Applications (Nonconvex Optimization & ...   (Site not responding. Last check: 2007-11-03)
Motivated by practical problems in engineering and physics, drawing on a wide range of applied mathematical disciplines, this text provides a comprehensive mathematical theory of duality for general non-convex, non-smooth systems, with emphasis on methods and applications in engineering mechanics.
A sequential canonical dual transformation method for solving nonlinear problems is developed heuristically and illustrated by use of examples as well as extensive applications of nonlinear systems.
The whole volume or selected chapters can also be recommended as a text for both senior undergraduate and graduate courses in applied mathematics, mechanics, general engineering science and other areas in which the ideas of optimization and variational methods are employed.
amazon.co.uk /exec/obidos/ASIN/0792361458   (593 words)

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