What is Dual: Definition and 372 Discussions

In geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedra – the (convex) Platonic solids and (star) Kepler–Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual. The dual of an isogonal polyhedron (one in which any two vertices are equivalent) is an isohedral polyhedron (one in which any two faces are equivalent), and vice-versa. The dual of an isotoxal polyhedron (one in which any two edges are equivalent) is also an isotoxal polyhedron.
Duality is closely related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

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1. ESA's Dual Stage 4 Grid Ion propulsion

I have found remarkably little information about ESA's Ion Drive in collaboration with Australia: https://en.wikipedia.org/wiki/Dual-Stage_4-Grid https://www.esa.int/gsp/ACT/doc/PRO/ACT-RPR-PRO-IAC2006-DS4G-C4.4.7.pdf Reference...
2. Is the Dual-Stage 4 Grid Ion Drive the Key to Faster Space Travel?

My post was removed originally about a designed based on this new engine. I don't know what specific forum rule I broke, I read the entire thing. But I am posting a new threat to ask about the validity of this specific technology. I am asking because there seems to be zero media coverage of it...
3. I Observer effect in the Dual Slit experiment

How detectors/observers in front of each slit works , does it somehow disturb with electron/photon? Does photon/electron must pass through detector/observer, so maybe his internal parts change final result on screen into just two lines?
4. A Maxwell theory invariant under dual field strength tensor application

Hello everybody! I know in classical field theory adding in the Lagrangian density a term of the form Fαβ (*F)αβ (where by * we denote the dual of the field strength tensor) does not change the EOM, since this corresponds to adding a total derivative term to the action. However when computing...
5. I Why is the dual of Z^n again Z^n ?

Hello, how can one proof that the dual of ##\mathbb{Z}^n## is ##\mathbb{Z}^n##? My idea: The definition of a dual lattice says, that it is as set of all lattice vectors ##x \in span(\Lambda)## such that ##\langle x , y \rangle## is an integer. When we now consider ##\mathbb{Z}^n## we see that...
6. I GM tube and dual probe scintillator

Recently I measured a range of sources with two detectors comparing their efficiency. One of the sources used (C-14) was a pure beta emitter and it made me wonder. If a radioactive source decays by beta then gamma, will one decay register as two counts as two radioactive particles are produced...
7. I How can the dual tensors derivation be achieved using rotation matrices?

Hello, I'm reading Group Theory in a nutshell for physicist by A Zee. When he introduced Dual tensors (pp 192), he made a claim with a light hint, and I have had great trouble deriving this claim, any help would be appreciated - Let ##R \in SO(N)## be an ##N##-dimensional rotation, then the...
8. A Anti-self Dual Part (2,2) Riemann Curvature Tensor

i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.
9. Optimization: Dual for L1 norm minimization with equality constraint

Hi, I was reading through some notes on standard problems and their corresponding dual problems. I came across the L2 norm minimization for an equality constraint, and then I thought how one might formulate the dual problem if we had an L1-norm instead. Question: Consider the following...
10. Convex Optimization: Dual Function Definition

Hi, I was working through the following problem and I am getting confused with the solution's definition of the dual. Problem: Given the optimization problem: minimize ## x^2 + 1 ## s.t. ## (x - 2) (x - 4) \leq 0 ## Attempt: I can define the Lagrangian as: L(x, \lambda) = (x^2 + 1) + \lambda...
11. I How to calculate the HP/torque required to rotate a dual sprocket/chain system?

Hi, I am trying to determine the HP motor/torque that will be required to rotate the sprocket/chain system in the image below from a resting position. The sprockets will be at rest, rotate 5 degrees, come to a stop for 5 minutes, and then rotate again 5 degrees, repeated. The speed of...
12. Optimization: Formulation of the dual of a semi-definite program (SDP)

Hi, I was working through the following optimization problem, and am getting stuck on how to get to the dual problem that is being presented. Question: Find the dual problem for the semidefinite primal problem below: min_{X} tr(C^T X) \text{subject to} AX = B X \succeq 0 (the answer is...
13. Optimization: How to find the dual problem?

Hi, I am working on the following optimization problem, and am confused how to apply the Lagrangian in the following scenario: Question: Let us look at the following problem \min_{x \in \mathbb{R}_{+} ^{n}} \sum_{j=1}^{m} x_j log(x_j) \text{subject to} A \vec{x} \leq b \rightarrow A\vec{x}...
14. Managing Dual Linux OS and Partitions on PC

Hi, I am running on pc with 2 different Linux OS and the following partitions /dev/sda1 (Boot) /dev/sda2 (ArchLinux) /dev/sda3 (something empty) /dev/sdb1 (Ubuntu) /dev/sdb2 (EFI System partition) /dev/sdb3 (no name) Since I basically don't use ArchLinux, I wanted to uninstall it. I...

16. I Expressing Vectors of Dual Basis w/Metric Tensor

I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways: ##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...
17. I Exploring the Connection Between Dual Vectors and Covectors in Vector Spaces

I understand that a vector space is a set of objects closed under addition and scalar multiplication and satisfies several properties. A functional is a map that takes a vector and produces a scalar. A functional is also called a dual vector. A covector is an object which transforms via the...
18. B Dual slit with controlled electron trajectories

In the traditional single electron duel slit experiment, I assume a cathode emits electrons in an unfocused direction spreading across the dual slits like a flashlight beam, but one electron at a time. Electrons however can be finely focused and controlled using magnetic or electric fields...
19. A Explore Carroll's Theory on Dual Space and Real Numbers

"The dual space is the space of all linear maps from the original vector space to the real numbers." Spacetime and Geometry by Carroll. Dual space can be anything that maps a vector space (including matrix and all other vector spaces) to real numbers. So why do we picked only a vector as a...
20. Programs Dual Master Degrees - Worth it?

Does anyone here have a masters degree in engineering, and then a second one in systems engineering ? I'm considering it because I can get help with the costs. I do systems engineering on electrical systems. So I see some possible advantages to getting both. I already have a masters in...
21. Calculate the dual basis and tangent basis vectors

a) Since ##tan(x/x_0)## is not defined for ##x=\pm\pi/2\cdot x_0## I assume x must be in between those values therefore ##-\pi/2\cdot x_0 < x < \pi/2\cdot x_0## and y can be any real number. Is this the correct answer on a)? b) I can solve x and y for s and t which gives me ##y=y_0\cdot s## and...
22. E

B Simple question about the dual space

Rookie question; for a vector space ##V##, with basis ##v_1, v_2, \dots, v_n##, the dual space ##V^*## is the set of linear functionals ##\varphi: V \rightarrow \mathbb{R}##. Dual basis will satisfy ##\varphi^i(v_j) = \delta_{ij}##. Is the action of any dual vector on any vector always an inner...
23. Why the bra vector is said to belong in the dual space?

hi i was recently introduced to the Dirac notation and i guess i am following it really well , but can't get my head around the idea that the bra vector said to live in the dual space of the ket vectors , i know about linear transformation and the structure of the vector spaces , and i realize...
24. A Double Dual of Vector Space: Is V** Always Same as V?

Hi I believe I understand the concept of a vector space V and its dual V*. I also understand that for V finite dimensional, there is a natural isomorphism between V and V**. What I am struggling to understand is - Does this natural isomorphism mean that V** is always IDENTICAL to V (identical...
25. A Curvature Tensor for Dual Vectors

Good day all. Given that in Sean Carroll`s Lectures on GR he states that when calculating the covariant derivative of a 1-Form the Christoffel symbols have a negative sign as opposed to for the covariant derivative of a vector, would it be naive to think that, given the usual equation for the...
26. I Closure & Interior as Dual Notions .... Proving Willard Theorem 3.11 ...

I am reading Stephen Willard: General Topology ... ... and am studying Chapter 2: Topological Spaces and am currently focused on Section 3: Fundamental Concepts ... ... I need help in order to prove Theorem 3.11 Part 1-a using the duality relations between closure and interior ... ..The...

35. MHB The Dual Space and Covectors .... Browdwer, Theorem 12.2 and Corollaries ....

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 12: Multilinear Algebra and am specifically focused on Section 12.1: Vectors and Tensors ... I need help in fully understanding Corollary 12.4 to Theorem 12.2 ... ... Theorem...
36. MHB Dual Space .... Loring Tu, Section 3.1, page 19 .... ....

I am reading Loring W.Tu's book: "An Introduction to Manifolds" (Second Edition) ... I need help in order to fully understand Tu's section on the dual space ... ... In his section on the dual space, Tu writes the following: In the above text from Tu, just preliminary to Proposition 3.1 Tu...
37. MHB The Dual Space and Differential Forms .... ....

I am reading the book: Multivariable Mathematics by Theodore Shifrin ... and am focused on Chapter 8, Section 2, Differential Forms ... I need some help in order to fully understand some statements of Shifrin at the start of Chapter 8, Section 2 on the dual space ... The relevant text from...
38. I Why are scalars and dual vectors 0- and 1-forms?

I am told: "A differential p-form is a completely antisymmetric (0,p) tensor. Thus scalars are automatically 0-forms and dual vectors (one downstairs index) are one-forms." Since an antisymmetric tensor is one where if one swaps any pair of indices the value of the component changes sign and 1)...
39. MHB Dual Spaces .... Friedberg et al, Example 4, Section 2.6

I am reading the book: "Linear Algebra" by Stephen Friedberg, Arnold Insel, and Lawrence Spence ... and am currently focused on Section 2.6: Dual Spaces ... ... I need help with an aspect of Example 4, Section 2.6 ... Example 4, Section 2.6 reads as follows: (see below for details of Section...
40. I Photon detector on a dual slit experiment

My experiment is to place a detector on one of the slits in the dual slit experiment so That it would record or not the passage of the photon, and then reset the detector to its base state, Erasing the result. In such a case, would the interference pattern be destroyed simply because the...
41. Do I Need to Take Any Special Steps to Format a Dual Boot System?

I have a dual boot laptop: Windows 7 and Ubuntu. I would like to format my laptop, and install a fresh copy of Windows 7, and install the new version Ubuntu 18.04.1 LTS. My question is: do I need to do any thing special? Usually, when I start my laptop I press one of the F keys (I forgot which...
42. Equations of motion for a dual mass-spring-damper

Homework Statement We have been given the following mass-spring-damper diagram, and are asked to derive the equations of motion. The positions of the two masses are given as q1 and q2. The Attempt at a Solution I began by drawing free-body diagrams for each mass. Then I set up the...
43. Finite dual disk capacitor: estimating charge distribution

Homework Statement Working through Purcell (among others) as fun applied math/math modeling refresher. But, I have struggled all week in establishing from first principles that the potential/field/distribution for a configuration of two capacitive disks of radius 1 and separation s along the...
44. Programs Should I do a 'dual' masters degree?

At my university they offer a program where you can pick two science disciplines and do a sort of 'half and half' masters degree. My B.S. is in Physics, and I somewhat want to do material science, which also directly relates to my current job and undergrad focus, but both mathematics and...
45. I Connection forms and dual 1-forms for cylindrical coordinate

I ran across exercise 2.8.4 in Oneill's Elementary Differential Geometry. It says "Given a frame field ##E_1## and ##E_2## on ##R^2## there is an angle function ##\psi## such that ##E_1=\cos(\psi)U_1+\sin(\psi)U_2##, ##E_2=-\sin(\psi)U_1+\cos(\psi)U2## (where ##U_1##, ##U_2##, ##U_3## are the...
46. A Proving the Dual of Schanuel's Lemma

Given: the short exact sequences 0 → M → E → K → 0 and 0 → M → E' → K' → 0 where M is a left R-module and E and E' are injective left R-modules. Prove: E ⊕ K' ≅ E' ⊕ K. First, let f be the morphism represented by M → E and g be the morphism represented by M → E'. Therefore we can construct a...
47. Geometric Version of Maxwell Equation related to Tensor Dual

Homework Statement From Misner, Thorne and Wheeler's text Gravitation (MTW), exercise 3.15: Show that, if F is the EM field tensor, then ##\nabla \cdot *F## is a geometric, frame-independent version of the Maxwell equation...
48. I Exploring Dual Vector Relationships: f(\vec{x}) and \mbox{d}f_{\vec{x}}(\vec{y})

f(\vec{x}+\epsilon \vec{y})-f(\vec{x})=\epsilon \mbox{d}f_{\vec{x}}(\vec{y})+O(\epsilon^2). Is ##\mbox{d}f_{\vec{x}}(\vec{y})## dual vector and why? Is it because ##\mbox{d}## is linear transformation? Also why equality f(\vec{x}+\epsilon \vec{y})-f(\vec{x})=\epsilon...
49. B How to find the dual basis vector for the following

ei=i+j+2vk , how to find the dual basis vector if the above is a natural base?
50. Exploring the Concept of Dual: Truth Table Homework

Homework Statement What is dual? Homework EquationsThe Attempt at a Solution Not on net. Meaning of dual i don't know and can't find.