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# Generalizing Heisenberg Uncertainty Priniciple

Writing the relationship between canonical momenta $\pi _i$ and canonical coordinates $x_i$

$$\pi _i =\text{ }\frac{\partial \mathcal{L}}{\partial \left(\frac{\partial x_i}{\partial t}\right)}$$

Then using the Lagrangian density for classical electrodynamics

$$\mathcal{L} = \frac{1}{2}\left(\epsilon _0E^2- \frac{1}{\mu _0}B^2\right)- \phi \rho _{\text{free}} + A\cdot J_{\text{free}} + \mathbb{E}\cdot \mathbb{P} + \mathbb{B}\cdot \mathbb{M}$$

Q1: Does it make sense to substitute the canonical scalar field $\phi$ (or electric scalar potential) for $\pi _i$ ; and $\phi \rho _{\text{free}}$ for canonical coordinates $x_i$. Such that these satisfy Heisenberg's Uncertainty relation?

Q2: Is there a generalized method for taking a Lagrangian and deriving conjugate variables, which then satisfy Heisenberg's Uncertainty relation?

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The fields in the electromagnetic Lagrangian satisfy the uncertainty principle with their conjugate momenta, but the result is a distributional uncertainty principle as appropriate for the distributional quantum fields.

From the Lagrangian, the canonical conjugate momentum to the field $A$ is the electric field $E$. If you take a gauge choice where A is purely spatial ($\phi=0$ gauge), you get the Bohr-Rosenfeld uncertainty relation between a measurement of A and a measurement of E in a region.

The commutation relation for the quantum field is

$$[A_i(x),E_j(y)] = \delta(x-y)\delta_{ij}$$

And the resulting uncertainty principle comes from smearing both fields using test functions. Define

$$A^f_i = \int f(x) A_i(x) dx$$ $$E^g_j = \int g(y) E_j(y) dy$$

Where f,g are positive $C^\infty$ bump functions, then

$$[A^f, E^g] = \delta_{ij} \int dx f(x)g(x)$$

This leads to a field theoretic uncertainty principle

$$\delta A_i^f \delta E_i^g \ge (\int dx f(x)g(x))$$

This uncertainty principle demands that the uncertainty products diverge in a particular way as the region becomes small, and Bohr and Roesenfeld verify that the physical uncertainty must be present when you use small charged quantum pith-balls to measure the value of E and A as best you can. This is automatic from a Lagrangian formulation of quantum mechanics, since you derive the commutation relation from the Lagrangian in a bosonic path integral. So this is one of those things that became trivial after Feynman's Lagrangian formulation became standard. It was controversial in Bohr and Rosenfeld's time, because field quantization made photons.

For a general bosonic Lagrangian, the canonical commutators

$$[\phi(x),\pi(y)] = \delta(x-y)$$

Where $\pi(x) = {\partial L\over \partial \dot{\phi} }$ leads to an uncertainty relation between $\phi$ and $\pi$ in the exact same way. These field uncertainty relations are in every way analogous to the usual momentum-position uncertainties in ordinary quantum mechanics.

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I think this is the right method, but I'm still tracking down the Bohr-Rosenfeld uncertainty relation and trying to understand. Thanks Ron. Just so you know, I'm looking for a generalized method for taking a Lagrangian and deriving conjugate variables, which then satisfy Heisenberg's Uncertainty relation? –  metzgeer Apr 28 '12 at 12:01
@metzgeer: When the Lagrangian is quadratic in momentum, these are the path integration variables and the classical conjugate momenta. The Bohr-Rosenfeld paper is from 1933, and is a perfectly clear classic. –  Ron Maimon Apr 28 '12 at 14:27

It doesn't really make sense to do this, for a few reasons: first, classical field theory has no concept of an uncertainty. It's not even clear what $\sigma_\phi$ and $\sigma_\pi$ would mean in that case. Besides, these quantities are necessarily going to commute. The uncertainty principle relates the product of the variances to the commutator, so the only result you'll get is that $\sigma_\phi\sigma_\pi \ge 0$. It's completely trivial. (If you're dealing with matrix-valued classical fields, then I guess you could get a nonzero commutator, but you still have the issue of why you're computing the $\sigma$s.)

It might make more sense in quantum field theory, where you could try to apply the generalized uncertainty relation,

$$\sigma_{\phi_i}\sigma_{\pi^{j\mu}} \ge \left|\frac{1}{2i}\langle[\phi_i,\pi^{j\mu}]\rangle\right|$$

There's a problem, though: the thing on the right (the commutator) is a field, which has a potentially different value at every point in spacetime, whereas the thing on the left is just a number. So this doesn't really work. It's only in basic quantum mechanics (or for only special operators), where the operator fields $\phi$ and $\pi$ are constant over all of space, that you can ignore the position dependence on the right.

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Why are you assuming the question is about classical fields? The OP didn't say this. –  Ron Maimon Apr 28 '12 at 2:03
Yes, he did say it was about classical electrodynamics. –  David Z Apr 28 '12 at 2:31
He just said he is using the Lagrangian for classical EM, this doesn't mean he isn't doing quantum mechanics, since the quantum Lagrangian is the same as the classical Lagrangian. The question makes no sense outside of QM. –  Ron Maimon Apr 28 '12 at 3:33
@metzgeer: You can't avoid the vector potential in EM--- the E and B formulation doesn't work because the phase change for a particle path is by the integral of A along the path. The gauge invariance is a headache that must be dealt with. If you want a source-free E,B formulation, you can do that, but it requires changing the action for a different path integral variables. The path-integral variable for the standard action is A, and you need to gauge fix it to make it correct (but you can do this in several ways without learning too much). The path integral is necessary for doing this right. –  Ron Maimon Apr 28 '12 at 6:54
@metzgeer: The "generalized uncertainty" is just canonical commutators for a general Lagrangian. For Path-Integral systems (Lagrangians at most quadratic in p) this is given by the classical fields (Path Integral variables) and their conjugate momenta. The derivation of the canonical commutation is on Wikipedia's Path Integral page, and it implies the uncertainty. The classical theory has unit Poisson brackets between any two canonical pair, but quantumly the commutators get corrections. –  Ron Maimon Apr 28 '12 at 14:31

It seems that OP's generalized Heisenberg uncertainty principle as formulated in the second subquestion(v5) is nothing but canonical quantization of an arbitrary classical Lagrangian field theory.

Formally, at the classical level, this involves performing a Dirac-Bergmann analysis to obtain a Hamiltonian formulation via a (not necessarily regular) Legendre transformation. This may$^{\dagger}$ introduce first class and second class constraints. In case of first class constraints, this indicates that the system possesses gauge symmetry. In case of second class constraints, the canonical Poisson bracket should be replace with the Dirac bracket.

Quantum mechanically, at the leading order in $\hbar$, the bracket should be replaced with $(i\hbar)^{-1}$ times the corresponding commutator, in accordance with the quantum mechanical correspondence principle, thereby leading to (generalized) Heisenberg uncertainty relations. At subleading orders in $\hbar$, the issue of operator ordering ambiguities arises.

Related topics are geometric and deformation quantization.

Reference:

1. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994.

--

$^{\dagger}$ E.g., in the Hamiltonian formulation of classical electrodynamics, Gauss' law becomes a first class constraint.

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For the special case of EM (as opposed to non-Abelian gauge fields) Henneaux and Teitelboim is overkill. The methods there are a little annoying to learn, because they don't provide the path integral dictionary, you have to make it yourself, and translating first and second class constraints to Path Integral is something that isn't presented in the literature. The generalized uncertainty relation is the distributional one I presented, and it depends on the size of the smearing function, requiring distributional fields. This can be understood in a simple gauge, like Dirac's. –  Ron Maimon Apr 30 '12 at 4:59


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# 1. Series 35. Year

#### Select year

Upload deadline: 12th October 2021 11:59:59 PM (local time in Czech Republic)

### 1. cars

Two cars start to move from the same point at the same time with velocities $v_1 = 100 \mathrm{km\cdot h^{-1}}$ and $v_2 = 60 \mathrm{km\cdot h^{-1}}$. Is it possible for the cars to move away from each other at any of the following velocities? If so, sketch the situations. \begin{align*} v_a &= 160 \mathrm{km\cdot h^{-1}} , & v_b &= 40 \mathrm{km\cdot h^{-1}} , \\ v_c &= 30 \mathrm{km\cdot h^{-1}} , & v_d &= 90 \mathrm{km\cdot h^{-1}} \end {align*}

### 2. two-second rule

The two-second rule is a driving principle which states that a safe time distance between two vehicles is at least two seconds long. Suppose a traffic junction where a $n_1$-lane road changes into a $n_2$-lane one. The maximum allowed speed in the first section is $v_1$. What is the lowest possible maximum speed $v_2$ that can be allowed in the second section so that there is no traffic jam and everyone can follow the two-second rule? The average length of a car is $l$ and it can change its speed in leaps.

### 3. to stop on skates

Skaters can stop using the „parallel slide“ method, in which they turn the blades of both skates perpendicular to the direction of movement, which significantly increases the friction with the ice. During this, the skater must tilt by the angle $\phi = 35 \mathrm{\dg }$ from the vertical direction, so he doesn't fall. Assume that he weighs $m = 70 \mathrm{kg}$ and that he is $H = 1{,}8 \mathrm{m}$ high (including the skates), with the center of gravity at a height of $h = 1{,}1 \mathrm{m}$ above the ice. Calculate the distance in which he stops from the initial speed $v_0 = 15 \mathrm{km\cdot h^{-1}}$.

### 4. fall to the seabed

A cylindrical capsule (Puddle Jumper) with a diameter $d = 4 \mathrm{m}$, a length $l = 10 \mathrm{m}$ and with a watertight partition in the middle of its length is submerged below the ocean surface and falls to the seabed at a speed of $v = 20 \mathrm{ft\cdot min^{-1}}$. At the depth $h = 1~200 ft$, the glass on the front base breaks and the corresponding half of the capsule is filled with water. At what speed will it fall now? How long will it take for the capsule to sink to the bottom at the depth $H=3~000 ft$? Assume that the walls of the capsule are very thin against its dimensions.

### 5. mechanically (un)stable capacitor

Assume a charged parallel-plate capacitor in a horizontal position. One of its plates is fixed and the other levitates directly below it in an equilibrium position. The lower plate is not mechanically fixed in its place. What is the capacitance of the capacitor depending on the voltage applied? Is the capacitor mechanically stable?

### P. so hot

You may have noticed that not all volcanos on Earth have the same „universal“ shape – they differ from each other. For example, compare the photos of the Hawaiian volcano Mauna Loa and the Italian Vesuvio. They differ not only in the steepness of their walls but also in the style of eruptions. Both of these properties are related to the viscosity of magma. Discuss the effect of the viscosity of magma on the style and dangerousness of eruptions. Is is related to the geographic location of the volcanoes?

### E. Is the pasta ready?

Measure the dependency of the time it takes for water to start boiling on its volume. Repeat the measurement several times for at least five different volumes. Pay attention to the consistency of the external conditions, especially the criterion you use for assessing when the water starts boiling and the initial temperature of the water, vessel and stove. Try to explain the resulting relation.

### S.

We are sorry. This type of task is not translated to English.

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# Why use $m_k$ to approximate $f_k$ in trust region method for optimization?

In Numerical Optimization book by Nocedal, page 19, The author describes trust region method as follow:

In the second algorithmic strategy, known as trust region, the information gathered about $f$ is used to construct a model function $m_k$ whose behavior near the current point $x_k$ is similar to that of the actual objective function $f$.

we find the candidate step $p$ by approximately solving the following subproblem: $$\min_p ~~m_k(x_k + p)$$ where $x_k + p$ lies inside the trust region.

The model $m_k$ in is usually defined to be a quadratic function of the form $$m_k(x_k + p) = f_k + p^\top \nabla f_k + \frac 1 2 p^\top B_k p$$

I have a naive question to ask: Specifically, I understand how to approximate $f$ but not why: we can use Taylor expansion to approximate, but I do not understand why we need approximate at fist place? Is that because we have a simpler system (quadratic form) that we can solve? Note, the approximation (quadratic form) may not be convex, as shown in the example in book page 20.


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# Mathematics Colloquia and Seminars

Period domains $D$ can be described as certain analytic open sets of flag varieties; due to the presence of monodromy, the period map of a family of algebraic varieties lands in a quotient $D/\Gamma$ by an arithmetic group. In the very special case when $D/\Gamma$ is itself algebraic, understanding the interaction between algebraic structures on the source and target of the uniformization $D\rightarrow D/\Gamma$ is a crucial component of the modern approach to the Andr\'e-Oort conjecture. We prove a version of the Ax-Schanuel conjecture for general period maps $X\rightarrow D/\Gamma$ which says that atypical algebraic relations between $X$ and $D$ are governed by Hodge loci. We will also discuss some geometric and arithmetic applications. This is joint work with J. Tsimerman.


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## G = C32⋊11SD16order 144 = 24·32

### 2nd semidirect product of C32 and SD16 acting via SD16/Q8=C2

Aliases: C12.18D6, C3211SD16, (C3×Q8)⋊3S3, Q82(C3⋊S3), (C3×C6).36D4, C324C85C2, (Q8×C32)⋊2C2, C12⋊S3.3C2, C33(Q82S3), C6.24(C3⋊D4), (C3×C12).14C22, C2.6(C327D4), C4.3(C2×C3⋊S3), SmallGroup(144,98)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C32⋊11SD16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C12⋊S3 — C32⋊11SD16
 Lower central C32 — C3×C6 — C3×C12 — C32⋊11SD16
 Upper central C1 — C2 — C4 — Q8

Generators and relations for C3211SD16
G = < a,b,c,d | a3=b3=c8=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=dbd=b-1, dcd=c3 >

Subgroups: 234 in 60 conjugacy classes, 27 normal (11 characteristic)
C1, C2, C2, C3 [×4], C4, C4, C22, S3 [×4], C6 [×4], C8, D4, Q8, C32, C12 [×4], C12 [×4], D6 [×4], SD16, C3⋊S3, C3×C6, C3⋊C8 [×4], D12 [×4], C3×Q8 [×4], C3×C12, C3×C12, C2×C3⋊S3, Q82S3 [×4], C324C8, C12⋊S3, Q8×C32, C3211SD16
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], SD16, C3⋊S3, C3⋊D4 [×4], C2×C3⋊S3, Q82S3 [×4], C327D4, C3211SD16

Character table of C3211SD16

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 8A 8B 12A 12B 12C 12D 12E 12F 12G 12H 12I 12J 12K 12L size 1 1 36 2 2 2 2 2 4 2 2 2 2 18 18 4 4 4 4 4 4 4 4 4 4 4 4 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ4 1 1 -1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 2 0 -1 -1 2 -1 2 2 -1 -1 -1 2 0 0 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ6 2 2 0 2 -1 -1 -1 2 2 -1 2 -1 -1 0 0 -1 -1 -1 -1 2 -1 -1 -1 2 -1 2 -1 orthogonal lifted from S3 ρ7 2 2 0 2 2 2 2 -2 0 2 2 2 2 0 0 -2 0 0 0 0 0 0 0 0 -2 -2 -2 orthogonal lifted from D4 ρ8 2 2 0 -1 -1 2 -1 2 -2 -1 -1 -1 2 0 0 2 -2 -2 1 1 1 1 1 1 -1 -1 -1 orthogonal lifted from D6 ρ9 2 2 0 -1 2 -1 -1 2 -2 -1 -1 2 -1 0 0 -1 1 1 1 1 -2 1 -2 1 -1 -1 2 orthogonal lifted from D6 ρ10 2 2 0 -1 -1 -1 2 2 2 2 -1 -1 -1 0 0 -1 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 orthogonal lifted from S3 ρ11 2 2 0 2 -1 -1 -1 2 -2 -1 2 -1 -1 0 0 -1 1 1 1 -2 1 1 1 -2 -1 2 -1 orthogonal lifted from D6 ρ12 2 2 0 -1 -1 -1 2 2 -2 2 -1 -1 -1 0 0 -1 1 1 -2 1 1 -2 1 1 2 -1 -1 orthogonal lifted from D6 ρ13 2 2 0 -1 2 -1 -1 2 2 -1 -1 2 -1 0 0 -1 -1 -1 -1 -1 2 -1 2 -1 -1 -1 2 orthogonal lifted from S3 ρ14 2 2 0 -1 2 -1 -1 -2 0 -1 -1 2 -1 0 0 1 √-3 -√-3 √-3 -√-3 0 -√-3 0 √-3 1 1 -2 complex lifted from C3⋊D4 ρ15 2 2 0 2 -1 -1 -1 -2 0 -1 2 -1 -1 0 0 1 -√-3 √-3 √-3 0 -√-3 -√-3 √-3 0 1 -2 1 complex lifted from C3⋊D4 ρ16 2 2 0 -1 -1 2 -1 -2 0 -1 -1 -1 2 0 0 -2 0 0 -√-3 -√-3 -√-3 √-3 √-3 √-3 1 1 1 complex lifted from C3⋊D4 ρ17 2 2 0 -1 2 -1 -1 -2 0 -1 -1 2 -1 0 0 1 -√-3 √-3 -√-3 √-3 0 √-3 0 -√-3 1 1 -2 complex lifted from C3⋊D4 ρ18 2 2 0 -1 -1 -1 2 -2 0 2 -1 -1 -1 0 0 1 √-3 -√-3 0 √-3 -√-3 0 √-3 -√-3 -2 1 1 complex lifted from C3⋊D4 ρ19 2 2 0 2 -1 -1 -1 -2 0 -1 2 -1 -1 0 0 1 √-3 -√-3 -√-3 0 √-3 √-3 -√-3 0 1 -2 1 complex lifted from C3⋊D4 ρ20 2 2 0 -1 -1 -1 2 -2 0 2 -1 -1 -1 0 0 1 -√-3 √-3 0 -√-3 √-3 0 -√-3 √-3 -2 1 1 complex lifted from C3⋊D4 ρ21 2 2 0 -1 -1 2 -1 -2 0 -1 -1 -1 2 0 0 -2 0 0 √-3 √-3 √-3 -√-3 -√-3 -√-3 1 1 1 complex lifted from C3⋊D4 ρ22 2 -2 0 2 2 2 2 0 0 -2 -2 -2 -2 -√-2 √-2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ23 2 -2 0 2 2 2 2 0 0 -2 -2 -2 -2 √-2 -√-2 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from SD16 ρ24 4 -4 0 -2 -2 4 -2 0 0 2 2 2 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ25 4 -4 0 -2 -2 -2 4 0 0 -4 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ26 4 -4 0 -2 4 -2 -2 0 0 2 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3 ρ27 4 -4 0 4 -2 -2 -2 0 0 2 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊2S3

Smallest permutation representation of C3211SD16
On 72 points
Generators in S72
(1 11 56)(2 49 12)(3 13 50)(4 51 14)(5 15 52)(6 53 16)(7 9 54)(8 55 10)(17 30 48)(18 41 31)(19 32 42)(20 43 25)(21 26 44)(22 45 27)(23 28 46)(24 47 29)(33 62 69)(34 70 63)(35 64 71)(36 72 57)(37 58 65)(38 66 59)(39 60 67)(40 68 61)
(1 22 34)(2 35 23)(3 24 36)(4 37 17)(5 18 38)(6 39 19)(7 20 40)(8 33 21)(9 43 68)(10 69 44)(11 45 70)(12 71 46)(13 47 72)(14 65 48)(15 41 66)(16 67 42)(25 61 54)(26 55 62)(27 63 56)(28 49 64)(29 57 50)(30 51 58)(31 59 52)(32 53 60)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)
(2 4)(3 7)(6 8)(9 50)(10 53)(11 56)(12 51)(13 54)(14 49)(15 52)(16 55)(17 35)(18 38)(19 33)(20 36)(21 39)(22 34)(23 37)(24 40)(25 72)(26 67)(27 70)(28 65)(29 68)(30 71)(31 66)(32 69)(41 59)(42 62)(43 57)(44 60)(45 63)(46 58)(47 61)(48 64)

G:=sub<Sym(72)| (1,11,56)(2,49,12)(3,13,50)(4,51,14)(5,15,52)(6,53,16)(7,9,54)(8,55,10)(17,30,48)(18,41,31)(19,32,42)(20,43,25)(21,26,44)(22,45,27)(23,28,46)(24,47,29)(33,62,69)(34,70,63)(35,64,71)(36,72,57)(37,58,65)(38,66,59)(39,60,67)(40,68,61), (1,22,34)(2,35,23)(3,24,36)(4,37,17)(5,18,38)(6,39,19)(7,20,40)(8,33,21)(9,43,68)(10,69,44)(11,45,70)(12,71,46)(13,47,72)(14,65,48)(15,41,66)(16,67,42)(25,61,54)(26,55,62)(27,63,56)(28,49,64)(29,57,50)(30,51,58)(31,59,52)(32,53,60), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(9,50)(10,53)(11,56)(12,51)(13,54)(14,49)(15,52)(16,55)(17,35)(18,38)(19,33)(20,36)(21,39)(22,34)(23,37)(24,40)(25,72)(26,67)(27,70)(28,65)(29,68)(30,71)(31,66)(32,69)(41,59)(42,62)(43,57)(44,60)(45,63)(46,58)(47,61)(48,64)>;

G:=Group( (1,11,56)(2,49,12)(3,13,50)(4,51,14)(5,15,52)(6,53,16)(7,9,54)(8,55,10)(17,30,48)(18,41,31)(19,32,42)(20,43,25)(21,26,44)(22,45,27)(23,28,46)(24,47,29)(33,62,69)(34,70,63)(35,64,71)(36,72,57)(37,58,65)(38,66,59)(39,60,67)(40,68,61), (1,22,34)(2,35,23)(3,24,36)(4,37,17)(5,18,38)(6,39,19)(7,20,40)(8,33,21)(9,43,68)(10,69,44)(11,45,70)(12,71,46)(13,47,72)(14,65,48)(15,41,66)(16,67,42)(25,61,54)(26,55,62)(27,63,56)(28,49,64)(29,57,50)(30,51,58)(31,59,52)(32,53,60), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72), (2,4)(3,7)(6,8)(9,50)(10,53)(11,56)(12,51)(13,54)(14,49)(15,52)(16,55)(17,35)(18,38)(19,33)(20,36)(21,39)(22,34)(23,37)(24,40)(25,72)(26,67)(27,70)(28,65)(29,68)(30,71)(31,66)(32,69)(41,59)(42,62)(43,57)(44,60)(45,63)(46,58)(47,61)(48,64) );

G=PermutationGroup([(1,11,56),(2,49,12),(3,13,50),(4,51,14),(5,15,52),(6,53,16),(7,9,54),(8,55,10),(17,30,48),(18,41,31),(19,32,42),(20,43,25),(21,26,44),(22,45,27),(23,28,46),(24,47,29),(33,62,69),(34,70,63),(35,64,71),(36,72,57),(37,58,65),(38,66,59),(39,60,67),(40,68,61)], [(1,22,34),(2,35,23),(3,24,36),(4,37,17),(5,18,38),(6,39,19),(7,20,40),(8,33,21),(9,43,68),(10,69,44),(11,45,70),(12,71,46),(13,47,72),(14,65,48),(15,41,66),(16,67,42),(25,61,54),(26,55,62),(27,63,56),(28,49,64),(29,57,50),(30,51,58),(31,59,52),(32,53,60)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72)], [(2,4),(3,7),(6,8),(9,50),(10,53),(11,56),(12,51),(13,54),(14,49),(15,52),(16,55),(17,35),(18,38),(19,33),(20,36),(21,39),(22,34),(23,37),(24,40),(25,72),(26,67),(27,70),(28,65),(29,68),(30,71),(31,66),(32,69),(41,59),(42,62),(43,57),(44,60),(45,63),(46,58),(47,61),(48,64)])

C3211SD16 is a maximal subgroup of
S3×Q82S3  D126D6  Dic6.22D6  D12.13D6  SD16×C3⋊S3  C247D6  C24.35D6  C24.28D6  C62.134D4  C62.73D4  C62.74D4  He310SD16  C36.20D6  C32.3GL2(𝔽3)  C3313SD16  C3315SD16  C3327SD16  C325GL2(𝔽3)
C3211SD16 is a maximal quotient of
C12.10Dic6  C62.113D4  C62.117D4  C36.20D6  He311SD16  C3313SD16  C3315SD16  C3327SD16

Matrix representation of C3211SD16 in GL6(𝔽73)

 0 1 0 0 0 0 72 72 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 43 13 0 0 0 0 43 30 0 0 0 0 0 0 61 55 0 0 0 0 4 0
,
 1 0 0 0 0 0 72 72 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 48 72

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,43,43,0,0,0,0,13,30,0,0,0,0,0,0,61,4,0,0,0,0,55,0],[1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,48,0,0,0,0,0,72] >;

C3211SD16 in GAP, Magma, Sage, TeX

C_3^2\rtimes_{11}{\rm SD}_{16}
% in TeX

G:=Group("C3^2:11SD16");
// GroupNames label

G:=SmallGroup(144,98);
// by ID

G=gap.SmallGroup(144,98);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,73,55,218,116,50,964,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^3>;
// generators/relations

Export

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text:
How do you find the domain of f(x) =2sqrt(3x + 4) - 5?

Domain: $x \ge - \frac{4}{3}$ , Range: $f \left(x\right) \ge - 5$
Domain (possible value of x) : $3 x + 4 \ge 0 \mathmr{and} 3 x \ge - 4 \mathmr{and} x \ge - \frac{4}{3}$ since under root can not be negative number.
Range (possible value of f(x)) : $f \left(x\right) > = 2 \cdot 0 - 5 \ge - 5$
Domain: $x \ge - \frac{4}{3}$ , Range: $f \left(x\right) \ge - 5$


text:
Definitions

# Naive set theory

Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and the everyday usage of set theory concepts in most contemporary mathematics.

Sets are of great importance in mathematics; in fact, in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory can be seen as a stepping-stone to more formal treatments, and suffices for many purposes.

## Requirements

A naive theory is a non-formalized theory, that is to say a theory that uses a natural language to describe sets. The words and, or, if ... then, not, for some, for every are not subject to rigorous definition. It is useful to study sets naively at an early stage of mathematics in order to develop facility for working with them. Furthermore, a firm grasp of set theoretical concepts from a naive standpoint is important as a first stage in understanding the motivation for the formal axioms of set theory.

This article develops a naive theory. Sets are defined informally and a few of their properties are investigated. Links in this article to specific axioms of set theory describe some of the relationships between the informal discussion here and the formal axiomatization of set theory, but no attempt is made to justify every statement on such a basis. The first development of set theory was a naive set theory. It was created at the end of the 19th century by Georg Cantor in order to allow mathematicians to work with infinite sets consistently.

As it turned out, assuming that one could perform any operations on sets without restriction led to paradoxes such as Russell's paradox or Berry's paradox. In response, axiomatic set theory was developed to determine precisely what operations were allowed and when. Today, when mathematicians talk about "set theory" as a field, they usually mean axiomatic set theory. Informal applications of set theory in other fields are sometimes referred to as applications of "naive set theory", but usually are understood to be justifiable in terms of an axiomatic system (normally the Zermelo–Fraenkel set theory). Some believe that Georg Cantor's set theory was not actually implicated in the paradoxes (this is a matter which continues to be discussed). He was aware of some of them and did not appear to believe that they discredited his theory. It is hard to be sure of this because he did not give an axiomatization. Frege did explicitly axiomatize a theory, in which the formalized version of naive set theory can be interpreted, and it is this formal theory which Bertrand Russell actually addressed when he presented his paradox.

These early attempts therefore led to inconsistency. A naive set theory is not necessarily inconsistent, if it correctly specifies the sets allowed to be considered. This can be done by the means of definitions, which are implicit axioms. It can be done by systematically making explicit all the axioms, as in the case of the well-known book Naive Set Theory by Paul Halmos, which is actually a somewhat (not all that) informal presentation of the usual axiomatic Zermelo–Fraenkel set theory. It is 'naive' in that the language and notations are those of ordinary informal mathematics, and in that it doesn't deal with consistency or completeness of the axiom system.

## Sets, membership and equality

In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.

If x is a member of A, then it is also said that x belongs to A, or that x is in A. In this case, we write x ∈ A. (The symbol ∈ is a derivation from the Greek letter epsilon, "ε", introduced by Peano in 1888.) The symbol ∉ is sometimes used to write x ∉ A, meaning "x is not in A".

Two sets A and B are defined to be equal when they have precisely the same elements, that is, if every element of A is an element of B and every element of B is an element of A. (See axiom of extensionality.) Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6. If the sets A and B are equal, this is denoted symbolically as A = B (as usual).

We also allow for an empty set, often denoted Ø and sometimes $\left\{\right\}$: a set without any members at all. Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty set.) Note that $empty ne \left\{0\right\} ne \left\{empty\right\}$.

## Specifying sets

The simplest way to describe a set is to list its elements between curly braces (known as defining a set extensionally). Thus {1,2} denotes the set whose only elements are 1 and 2. (See axiom of pairing.) Note the following points:

• Order of elements is immaterial; for example, {1,2} = {2,1}.
• Repetition (multiplicity) of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}.

(These are consequences of the definition of equality in the previous section.)

This notation can be informally abused by saying something like {dogs} to indicate the set of all dogs, but this example would usually be read by mathematicians as "the set containing the single element dogs".

An extreme (but correct) example of this notation is {}, which denotes the empty set.

We can also use the notation {x : P(x)}, or sometimes {x | P(x)}, to denote the set containing all objects for which the condition P holds (known as defining a set intensionally). For example, {x : x is a real number} denotes the set of real numbers, {x : x has blonde hair} denotes the set of everything with blonde hair, and {x : x is a dog} denotes the set of all dogs.

This notation is called set-builder notation (or "set comprehension", particularly in the context of Functional programming). Some variants of set builder notation are:

• {x ∈ A : P(x)} denotes the set of all x that are already members of A such that the condition P holds for x. For example, if Z is the set of integers, then {x ∈ Z : x is even} is the set of all even integers. (See axiom of specification.)
• {F(x) : x ∈ A} denotes the set of all objects obtained by putting members of the set A into the formula F. For example, {2x : x ∈ Z} is again the set of all even integers. (See axiom of replacement.)
• {F(x) : P(x)} is the most general form of set builder notation. For example, {x's owner : x is a dog} is the set of all dog owners.

## Subsets

Given two sets A and B we say that A is a subset of B if every element of A is also an element of B. Notice that in particular, B is a subset of itself; a subset of B that isn't equal to B is called a proper subset.

If A is a subset of B, then one can also say that B is a superset of A, that A is contained in B, or that B contains A. In symbols, A ⊆ B means that A is a subset of B, and B ⊇ A means that B is a superset of A. Some authors use the symbols "⊂" and "⊃" for subsets, and others use these symbols only for proper subsets. For clarity, one can explicitly use the symbols "$subsetneq$" and "$supsetneq$" to indicate non-equality. In this encyclopedia, "⊆" and "⊇" are used for subsets while "⊂" and "⊃" are reserved for proper subsets.

As an illustration, let R be the set of real numbers, let Z be the set of integers, let O be the set of odd integers, and let P be the set of current or former U.S. Presidents. Then O is a subset of Z, Z is a subset of R, and (hence) O is a subset of R, where in all cases subset may even be read as proper subset. Note that not all sets are comparable in this way. For example, it is not the case either that R is a subset of P nor that P is a subset of R.

It follows immediately from the definition of equality of sets above that, given two sets A and B, A = B iff A ⊆ B and B ⊆ A. In fact this is often given as the definition of equality. Usually when trying to prove that two sets are equal, one aims to show these two inclusions. Note that the empty set is a subset of every set (the statement that all elements of the empty set are also members of any set A is vacuously true).

The set of all subsets of a given set A is called the power set of A and is denoted by $2^A$ or $P\left(A\right)$; the "P" is sometimes in a fancy font. If the set A has n elements, then $P\left(A\right)$ will have $2^n$ elements.

## Universal sets and absolute complements

In certain contexts we may consider all sets under consideration as being subsets of some given universal set. For instance, if we are investigating properties of the real numbers R (and subsets of R), then we may take R as our universal set. A universal set is only temporarily defined by the context; there is no such thing as a "universal" universal set, "the set of everything" (see Paradoxes below).

Given a universal set U and a subset A of U, we may define the complement of A (in U) as

AC := {x ∈ U : x ∉ A}.
In other words, AC ("A-complement"; sometimes simply A, "A-prime" ) is the set of all members of U' which are not members of A. Thus with R, Z and O defined as in the section on subsets, if Z is the universal set, then OC is the set of even integers, while if R is the universal set, then OC is the set of all real numbers that are either even integers or not integers at all.

## Unions, intersections, and relative complements

Given two sets A and B, we may construct their union. This is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B.

The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B.

Finally, the relative complement of B relative to A, also known as the set theoretic difference of A and B, is the set of all objects that belong to A but not to B. It is written as A  B or A − B. Symbolically, these are respectively

A ∪ B := {x : (x ∈ Aor (x ∈ B)};
A ∩ B := {x : (x ∈ Aand (x ∈ B)} = {x ∈ A : x ∈ B} = {x ∈ B : x ∈ A};
A  B := {x : (x ∈ A) and not (x ∈ B) } = {x ∈ A : not (x ∈ B)}.

Notice that A doesn't have to be a subset of B for B  A to make sense; this is the difference between the relative complement and the absolute complement from the previous section.

To illustrate these ideas, let A be the set of left-handed people, and let B be the set of people with blond hair. Then A ∩ B is the set of all left-handed blond-haired people, while A ∪ B is the set of all people who are left-handed or blond-haired or both. A  B, on the other hand, is the set of all people that are left-handed but not blond-haired, while B  A is the set of all people who have blond hair but aren't left-handed.

Now let E be the set of all human beings, and let F be the set of all living things over 1000 years old. What is E ∩ F in this case? No human being is over 1000 years old, so E ∩ F must be the empty set {}.

For any set A, the power set $P\left(A\right)$ is a Boolean algebra under the operations of union and intersection.

## Ordered pairs and Cartesian products

Intuitively, an ordered pair is simply a collection of two objects such that one can be distinguished as the first element and the other as the second element, and having the fundamental property that, two ordered pairs are equal if and only if their first elements are equal and their second elements are equal.

Formally, an ordered pair with first coordinate a, and second coordinate b, usually denoted by (a, b), is defined as the set {{a}, {a, b}}.

It follows that, two ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d.

Alternatively, an ordered pair can be formally thought of as a set {a,b} with a total order.

(The notation (a, b) is also used to denote an open interval on the real number line, but the context should make it clear which meaning is intended. Otherwise, the notation ]a, b[may be used to denote the open interval whereas (a, b) is used for the ordered pair).

If A and B are sets, then the Cartesian product (or simply product) is defined to be:

A × B = {(a,b) : a is in A and b is in B}.
That is, A × B is the set of all ordered pairs whose first coordinate is an element of A and whose second coordinate is an element of B.

We can extend this definition to a set A × B × C of ordered triples, and more generally to sets of ordered n-tuples for any positive integer n. It is even possible to define infinite Cartesian products, but to do this we need a more recondite definition of the product.

Cartesian products were first developed by René Descartes in the context of analytic geometry. If R denotes the set of all real numbers, then R2 := R × R represents the Euclidean plane and R3 := R × R × R represents three-dimensional Euclidean space.

## Some important sets

Note: In this section, a, b, and c are natural numbers, and r and s are real numbers.

1. Natural numbers are used for counting. A blackboard bold capital N ($mathbb\left\{N\right\}$) often represents this set.
2. Integers appear as solutions for x in equations like x + a = b. A blackboard bold capital Z ($mathbb\left\{Z\right\}$) often represents this set (from the German Zahlen, meaning numbers).
3. Rational numbers appear as solutions to equations like a + bx = c. A blackboard bold capital Q ($mathbb\left\{Q\right\}$) often represents this set (for quotient, because R is used for the set of real numbers).
4. Algebraic numbers appear as solutions to polynomial equations (with integer coefficients) and may involve radicals and certain other irrational numbers. A blackboard bold capital A ($mathbb\left\{A\right\}$) or a Q with an overline ($overline\left\{mathbb\left\{Q\right\}\right\}$) often represents this set. The overline denotes the operation of algebraic closure.
5. Real numbers represent the "real line" and include all numbers that can be approximated by rationals. These numbers may be rational or algebraic but may also be transcendental numbers, which cannot appear as solutions to polynomial equations with rational coefficients. A blackboard bold capital R ($mathbb\left\{R\right\}$) often represents this set.
6. Complex numbers are sums of a real and an imaginary number: r + si. Here both r and s can equal zero; thus, the set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers, which form an algebraic closure for the set of real numbers, meaning that every polynomial with coefficients in $mathbb\left\{R\right\}$ has at least one root in this set. A blackboard bold capital C ($mathbb\left\{C\right\}$) often represents this set. Note that since a number r + si can be identified with a point (r, s) in the plane, C is basically "the same" as the Cartesian product R×R ("the same" meaning that any point in one determines a unique point in the other and for the result of calculations it doesn't matter which one is used for the calculation).

We referred earlier to the need for a formal, axiomatic approach. What problems arise in the treatment we have given? The problems relate to the formation of sets. One's first intuition might be that we can form any sets we want, but this view leads to inconsistencies. For any set x we can ask whether x is a member of itself. Define
Z = {x : x is not a member of x}.
Now for the problem: is Z a member of Z? If yes, then by the defining quality of Z, Z is not a member of itself, i.e., Z is not a member of Z. This forces us to declare that Z is not a member of Z. Then Z is not a member of itself and so, again by definition of Z, Z is a member of Z. Thus both options lead us to a contradiction and we have an inconsistent theory. More succinctly, one says that Z is a member of Z if and only if Z is not a member of Z. Axiomatic developments place restrictions on the sort of sets we are allowed to form and thus prevent problems like our set Z from arising. This particular paradox is Russell's paradox.

The penalty is that one must take more care with one's development, as one must in any rigorous mathematical argument. In particular, it is problematic to speak of a set of everything, or to be (possibly) a bit less ambitious, even a set of all sets. In fact, in the standard axiomatisation of set theory, there is no set of all sets. In areas of mathematics that seem to require a set of all sets (such as category theory), one can sometimes make do with a universal set so large that all of ordinary mathematics can be done within it (see universe). Alternatively, one can make use of proper classes. Or, one can use a different axiomatisation of set theory, such as W. V. Quine's New Foundations, which allows for a set of all sets and avoids Russell's paradox in another way. The exact resolution employed rarely makes an ultimate difference.

## References

• Halmos, P.R., Naive Set Theory, D. Van Nostrand Company, Princeton, NJ, 1960. Reprinted, Springer-Verlag, New York, NY, 1974, ISBN 0-387-90092-6.
• Bourbaki, N., Elements of the History of Mathematics, John Meldrum (trans.), Springer-Velag, Berlin, Germany, 1994.
• Devlin, K.J., The Joy of Sets: Fundamentals of Contemporary Set Theory, 2nd edition, Springer-Verlag, New York, NY, 1993.
• van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.
• Kelley, J.L., General Topology, Van Nostrand Reinhold, New York, NY, 1955.


text:
### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### 14 Divisors

What is the smallest number with exactly 14 divisors?

### Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

# Factorised Factorial

##### Stage: 3 Short Challenge Level:
$n!$ is divisible by $5^3$ so $n!$ must be at least $15$. But $15!$ is only divisible by $2^{11}$ so $n$ is not $15$. $n!$ is not divisible by $17$ so $n$ is less than $17$. Hence $n=16$.

This problem is taken from the UKMT Mathematical Challenges.
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