Duality Definition and 282 Threads

In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
In mathematical contexts, duality has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f: V → W its dual f∗: W∗ → V∗.

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  1. F

    More Linear Programming - DUality

    Homework Statement Consider the following LOP P: Max z = 3x_1 + x_2 -\frac{1}{2}x_3 s.t. 2x_1 + x_2 + x_3 \leq 8 4x_1 + x_2 - x_3 \leq 10 x_1, x_2, x_3 \geq 0 1) Find a primal solution x and its objective value z = z(x) 2) What is the LOP D that is Dual to P? 3) Find a dual feasible...
  2. Vorde

    Wave-Particle Duality: Is Matter Both?

    I was originally taught (and believed) that all matter was neither a wave nor a particle but in fact exhibited properties of both. However after reading QED (by Feynman) and The Cosmic Code (I forget who wrote that), both seemed to say (Feynman directly and The Cosmic Code more subtlety) that...
  3. A. Neumaier

    An elementary view of duality?

    Are there simple analogues of AdS/CFT duality that are understandable without any knowledge of string theory?
  4. U

    Wave-particle duality for the electron

    Hello again! So this time I have a few questions about the wave nature of the electron. So I've read that an electron can behave both as a wave and as a particle. My questions are regarding the electron as a wave. 1) When does the electron behave as a wave? Is it a transverse or longitudinal...
  5. H

    Order-Chaos Duality: Human Perception

    Or is it really a duality - or just an another way for human kind to percieve things...?
  6. V

    The Myth of Wave-Particle Duality

    Ballentine mentioned in "The Statistical Interpretation of Quantum Mechanics" published in 1970 in the Review of Modern Physics the following which allegedly showed that the wave-like behavior of a particle if some kind of illusion (now after 40 years, are there no experiments that can refute...
  7. H

    Exploring Wave-Particle Duality and its Impact on Determinism

    1. How do we know EM radiation is a wave? Also, how do we know that the magnetic wave is exactly orthogonal to the electronic wave? 2. In the into to A Brief History of Time, SH says that wave-particle duality refutes determinism. How is that conclusion made?
  8. A

    Conceptual question on wave-particle duality (electron microscope)

    according to wave particle duality particles such as electrons exhibit wave like properties. and the wavelngth is given by lambda=h/p, so increase momentum and get shorter lambda. This property is utilised in the electron microscope to get very short wavelengths and so have very high...
  9. A

    Conceptual question on wave-particle duality (electron microscope)

    according to wave particle duality particles such as electrons exhibit wave like properties. and the wavelngth is given by lambda=h/p, so increase momentum and get shorter lambda. This property is utilised in the electron microscope to get very short wavelengths and so have very high...
  10. S

    The Duality in Energy and Matter and it's Intuitiveness

    I am a undergrad modern physics major currently attending my first lab (freshman). I am currently doing my first lab. I was given the research of the following nature. I have made a point plan on how to carry it but I am seeking for expert opinion for my plans and approach. Homework Statement...
  11. Femme_physics

    Does the wave–particle duality apply only to electrons and photons?

    Does the wave–particle duality apply only to electrons and photons?
  12. B

    Complex conjugation and vector space duality

    Hi all. First of all I want to say that I am new here and I want to apologise if this is the wrong forum to post my question. :) I've just finished a sort of a crash course on functional analysis that left much more questions in me than it answered but one thing bothers me a lot. When we talk...
  13. N

    Rainbows and wave-particle duality

    Correct me if I'm wrong. The wave function of a particle collapses when you observe/measure it? Because that you essentially disturb it. Right? So how is it that we see rainbows then if it's wave function collapses as we observe it? And how is it that we are able to split white light into...
  14. B

    Wave/particle duality in art and the world as a general pattern

    As a professional novelist, I see wave particle duality in all things including arts. For instance a book written in the first person (I) is the particle viewpoint just as the 3rd person represents one bifurcation of the wave function viewpoint. I see wave particle duality everywhere. Take...
  15. J

    Relativity of Simultaneity and wave/particle duality.

    Why is it that the relativity of simultaneity isn't used to explain wave/particle duality? RoS requires that matter be in different places according to the speed of the observer. All observers are equally correct. This would seem to require that particles posses a wave characteristic...
  16. A

    Understanding the Wave-Particle Duality: Exploring the Double Slit Experiment

    The basis of the well known experiment is to fire one photon at a double slit 'wall' and measure the wave pattern produced on a screen behind it. Now as is evident, one particle is seen to act like a wave, which is a fundamental contradiction. I have couple of fundamental questions...
  17. N

    Wave/particle duality - electrons and photons

    I understand that: 1. Electrons can interact with each other in both particle-like and wave-like ways (eg. can exhibit wavelike interference in double slit experiment) 2. Photons can interact with each other in both particle-like and wave-like ways 3. Electrons and photons can interact...
  18. MTd2

    State-Observable Duality (John Baez series)

    "I’m writing a paper called “Division algebras and quantum theory”, which is mainly about how quantum theory can be formulated using either the real numbers ℝ, the complex numbers ℂ, or the quaternions ℍ — and how these three versions are not really separate alternatives (as people often seem to...
  19. A

    How can the distance duality relation be derived in a model-independent way?

    Hi , I need help from someone who has worked out the distance duality relation (between angular diameter distance and the luminosity distance).I am having trouble deriving it and the Etherington paper is a bit too tough for me where it was first discussed (1933). In one of the reviews its...
  20. G

    Calculating Electron Kinetic Energy, Velocity, Momentum, and Wavelength

    Homework Statement Some electrons are accelerated through a P.d. of 500 V. [e = 1.6x10^-19 C, mass of electron = 9.1x10^-31 Kg, h = 6.6x10^-34 Js.] 1. What is the kinetic energy of one of the electrons in eV? 2. What is the velocity of one of the electrons? 3. What is the...
  21. T

    Wave Particle Duality and its relation to the Wave Function

    Ok so from my understanding, the wave particle duality of matter is simply the fact that matter sometimes behaves as a wave and sometimes behaves as a particle. Ok, but I wonder if this has anything to do with the wavefunction. The wave function gives us the probability spread of matter at any...
  22. E

    Questions about wave-particle duality terms

    Hello, I have some questions, but I'm not posting on quantum mechanics because as written in the FAQ, wave-particle duality is not really the realm of quantum mechanics, but is only a way for us to imagine what's going on. Is it true that the uncertainty on the location \Deltax, is in fact...
  23. C

    Particle-Wave Duality Contention

    This problem borders with the question of either wave only or particle only descriptions of duality, and I think it could possibly even end up being a personal opinion of the (hopefully qualified) repliers. In a normal atom, are there electrons orbiting around the nucleus? [This seems like a...
  24. G

    Particle/wave duality: physical reality? Or really physical crackpottery?

    I, as with Albert Einstein, shall always side with Issac Newton; that is to say, that light is corpuscular. There is no such thing as a light 'wave'. Sure, Thomas Young threw the world of physics a curveball in 1802, backed by Huygens and his somewhat sound reasoning (for the time). But who...
  25. Z

    Particle-wave duality of sound waves

    I am extremely confused when it comes to particle-wave duality outside of the electromagnetic spectrum and quantum world. Are sound waves strictly waves or do they have "particle"-like aspects? I understand that every particle has its own wave. Does this apply to every wave having a...
  26. W

    Exploring Duality between Vector Spaces

    Hello all, Recently I have read something about duality between vector spaces, however my intuition towards this is not clear. Wish someone can give me a hint. Recall the definition of a dual pair is a 3-tuple (X,Y,\langle \cdot , \cdot \rangle) , so essentially duality between vector...
  27. H

    Wave-partical duality and Huygens–Fresnel principle

    Hi all! Couple days ago my friend asked me about the following. He said: [PLAIN]http://img685.imageshack.us/img685/2886/dualism.png I have explained it to him in this way. The meaning of the quantum theory of light is NOT that we consider the light as being a gas of particles with...
  28. M

    Using wave\particle duality to predict future event?

    This is going to be a little weird so please bear with me for a moment. Based on the results of experiments like the delayed choice quantum eraser it would seem that you could use one of these experiments to determine the results of a future random event. Say for example you ran a series of...
  29. MTd2

    CDM/MOND duality through Holography.

    I didn't notice this paper, until it was featured today on TR: http://www.technologyreview.com/blog/arxiv/25214/?ref=rss http://arxiv.org/abs/1005.3537 MONDian Dark Matter Chiu Man Ho, Djordje Minic, Y. Jack Ng (Submitted on 19 May 2010) We provide a holographic dual description...
  30. N

    Wave-Particle Duality (explained?)

    Hi All, I'm nk and new to your forum! Pl., excuse my editing skills just this post. I'll catch on fast-promise! It is known that "In physics and chemistry, wave–particle duality is the concept that all matter exhibits both wave-like and particle-like properties."-Reference...
  31. H

    Exploring Particle-Wave Duality: Understanding the Double Slit Experiment

    I'm thinking about the wave/particle duality and I have a few questions about it. Question: How do we know a particle is also a wave if we can't measure it as one? Answer: The double slit experiment, for example, allows us to INFERE the wave nature of particles because of the way multiple...
  32. J

    Exploring Wave-Particle Duality and the Diffraction Pattern

    Hi, I’ve got a general question about wave-particle duality, and the experiments done with slits. When a laser is beamed through a small slit, we get a diffraction pattern. This pattern is interesting because it contains dark bands. When it comes to explaining the existence of...
  33. E

    Light wave/particle duality resolution

    Just two days ago I attended a lecture held by a professor at UiO who discussed the need to resolve the wave/particle duality model of light. He was unsatisfied by the deterministic approach to the problem (Bohr's principle). In general, he viewed the photon model as poorly defined and poorly...
  34. G

    Wave Particle Duality: Exploring Matter & Particle Properties

    What do we actualy mean a matter or particle having both wave and matter properties, and do the electrons occilate that means it has both up and down with the translational motion around the atom!
  35. Vectronix

    The Understanding of Particle-Wave Duality (Can it be fully understood?)

    Hello. Why would a double-slit experiment for things like bullets being fired from a gun that shoots in random directions produce a probability distribution like those of objects or particles, but the double-slit experiment for an electron gun produce a wave-like interference pattern? I read...
  36. T

    Wave–particle duality theory question

    Note this is more of a coursework understanding question rather than a specific homework question. Homework Statement I have been reading about matter waves and de Broglie relations which suggest that electrons can act as waves. From wikipedia (http://en.wikipedia.org/wiki/Matter_wave) it...
  37. 5

    What is meant by ths wave-particle duality of electrons?

    What is meant by ths wave-particle duality of electrons?
  38. T

    Wave-Particle Duality: Exploring Quantum Physics

    Hi! I have been reading a little about quantum physics (I am in no way a physicist or scientist), and am trying to get my head round Wave / Particle duality. The only way I can visualise it is similar to the way to dsp / computer audio; a stream of particles oscillating in space. Particles...
  39. N

    Brain-Computer Interface & Spinoza's Philosophy

    So if we were to upload our minds into a brain computer interface one day, what implications does this have on Spinoza's philosophy that the mind and body are inseparable?
  40. M

    Wave-Particle Duality: Physics Implications

    I was just wondering about, after I had read about this topic. Which consequences did it have on physics that we can consider light as waves in some experiments and as particles (photons, quanta) in other experiments. And maybe even if all matter can have the same properties as light?
  41. D

    Wave Particle Duality: Magnifying Power of Electron Microscope

    Homework Statement Microscopes are inherently limited by the wavelength of the light used. How much smaller (in order of magnitude) can we "see" using an electron microscope whose electrons have been accelerated through a potential difference of 50,000V than using red light(500nm)...
  42. Q

    Wave Particle Duality: Diffracting Particles Explained

    Ok, if a particle is a wave and can diffract (e.g. through graphite sheet), I mean does the particle split into two or something, or does the particle somehow 'transform' into a wave? I am always inquisitive about this question.
  43. W

    Alexander Duality & Cup Product: Commuting?

    does Alexander duality commute with cup product?
  44. A

    Wave particle duality as applied to photon

    so basically a wave particle duality principle state that sometimes a "matter" would behave as a particle and a wave a another time. The was to reconcile this is through QM's statistical interpretation and the math construct is very obvious that such proposed matter is indeed a wave-particle in...
  45. Demystifier

    Wave-particle duality of phonons?

    As every condensed-matter physicist knows, a phonon is a quantum of sound (or more precisely, of oscillations in a condensed-matter material). Clearly, sound is a wave. But quantum theory teaches us that, in a certain sense, sound is also a particle. Here I want to understand how far the view of...
  46. M

    Physics Faq clarifiaction - light duality

    There's a very useful FAQ entitled IS LIGHT A WAVE OR A PARTICLE? It carefully warns against confusing a "particle" of light with a regular particle. I have a funny feeling I may be doing this... hence the need for a clarification! When considering light as a wave; should I consider that...
  47. M

    Duality Pairing and Functionals

    Hello all, does anybody know what means duality pairing in connection with functional. For example limE\rightarrow0\frac{\partial}{\partialE}F(u+Ev)=<DF(u),v>. Where F is functional F:K\rightarrowR. Thank You for answers.
  48. M

    Relating to Duality in Vector Spaces

    This subject came up in some notes on linear algebra I'm reading and I don't get it. Please help me understand. -- First, the relevant background and notation relating to my question: Let S be a nonempty set and F be a field. Denote by l(S) the family of all F-valued functions on S and...
  49. A

    Light is known to have a wave-particle duality

    light is known to have a wave-particle duality which of the 2 explain lights behavior best and which one is widely accepted?
  50. E

    Duality of Electron: Mass, Charge and Wave Mechanics

    Hi, Originally this was going to be about duality of electron but after deeper study I decided against it. This is what I understand thus far -- please feel free to correct me! The electron is the fundamental stuff that expresses both mass and charge. The charge of the electron is opposite...
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