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http://pediaview.com/openpedia/Electrostatic_force | # Electrostatic force
Electromagnetism
• Coulomb's law
• Gauss's law
Scientists
Coulomb's law or Coulomb's inverse-square law is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. Coulomb's law has been tested heavily and all observations are consistent with the law.
## History
Charles Augustin de Coulomb
Early investigators who suspected that the electrical force diminished with distance as the gravitational force did (i.e., as the inverse square of the distance) included Daniel Bernoulli [1] and Alessandro Volta, both of whom measured the force between plates of a capacitor, and Aepinus who supposed the inverse-square law in 1758.[2]
Based on experiments with charged spheres, Joseph Priestley of England was among the first to propose that electrical force followed an inverse-square law, similar to Newton's law of universal gravitation. However, he did not generalize or elaborate on this.[3] In 1767, he conjectured that the force between charges varied as the inverse square of the distance, saying:[4][5]
Coulomb’s torsion balance
May we not infer from this experiment, that the attraction of electricity is subject to the same laws with that of gravitation, and is therefore according to the squares of the distances; since it is easily demonstrated that, were the earth in the form of a shell, a body in the inside of it would not be attracted to one side more than another?
In 1769, Scottish physicist John Robison announced that, according to his measurements, the force of repulsion between two spheres with charges of the same sign varied as x-2.06.[6][7]
The dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, in the early 1770s by Henry Cavendish of England.
Finally, in 1785, the French physicist Charles Augustin de Coulomb published his first three reports of electricity and magnetism where he stated his law. This publication was essential to the development of the theory of electromagnetism.[8] He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
## The law
Coulomb's law states that the magnitude of the Electrostatics force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distances between them.[8]
If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.
The scalar and vector forms of the mathematical equation are
$|\boldsymbol{F}|=k_e{|q_1q_2|\over r^2}$ and $\boldsymbol{F}=k_e{q_1q_2\boldsymbol{\hat{r}_{21}}\over r_{21}^2}$ , respectively (here, $\boldsymbol{\hat{r}_{21}} = \frac{\boldsymbol{r_{21}}}{|\mathbf{r_{21}}|}$).
### Units
Electromagnetic theory is usually expressed using the standard SI units for the force F, quantity of charge q, and radial distance r, and where k e = 1 ⁄ 4πε 0 ε . Where ε 0 is the permittivity of free space and ε is the relative permittivity of the material in which the charges are immersed. Coulomb's law and Coulomb's constant can also be interpreted in terms of atomic units, or electrostatic units or Gaussian units. In atomic units the force is expressed in Hartrees per Bohr radius, the charge in terms of the elementary charge, and the distances in terms of the Bohr radius. In electrostatic units and Gaussian units, the unit charge (esu or statcoulomb) is defined in such a way that the Coulomb constant k disappears because it has the value of one and becomes dimensionless. The standard SI units will be used below.
The SI derived units for the electric field are volts per meter, newtons per coulomb, and teslas meters per second.
### An electric field
If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.
The magnitude of the electric field force, E in vacuum, is invertible from Coulomb's law. Since E = F ⁄ Q it follows from the Coulomb's law that the magnitude of the electric field E created by a single point charge q at a certain distance r in vacuum is given by:
$|\boldsymbol{E}|={1\over4\pi\varepsilon_0}{|q|\over r^2}$.
An electric field is a vector field which associates to each point of the space the Coulomb force that will experience a test unity charge. Given the electric field, the strength and direction of a force F on a quantity charge q in an electric field E is determined by the electric field. For a positive charge, the direction of the electric field points along lines directed radially away from the location of the point charge, while the direction is towards for a negative charge.
### Coulomb's constant
Main article: Coulomb's constant
Coulomb's constant (denoted k e ) is a proportionality factor also called the electric force constant or electrostatic constant, hence the subscript e, that appears in Coulomb's law as well as in other electric-related formulas.
The exact value of Coulomb's constant k e comes from three of the fundamental, invariant quantities that define free space in the SI system: the speed of light c 0 , magnetic permeability μ 0 , and electric permittivity ε 0 , related by Maxwell as
$\frac{1}{\mu_0\varepsilon_0}=c_0^2.$
Because of the way the SI base unit system made the natural units for electromagnetism, the speed of light in vacuum c 0 is 299,792,458 m s−1, the magnetic permeability μ 0 of free space is 4π·10−7H m−1, and the electric permittivity of free space is ε 0 = 1 ⁄ (μ 0 c 2
0
) ≈ 8.85418782×10−12 F·m−1
,[9] so that[10]
$\begin{align} k_e &= \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}=c_0^2\cdot10^{-7}\mathrm{H\ m}^{-1}\\ &= 8.987\ 551\ 787\ 368\ 176\ 4\cdot10^9\mathrm{N\ m^2\ C}^{-2}. \end{align}$
### Conditions for validity
There are two conditions to be fulfilled for the validity of Coulomb’s law:
1. The charges considered must be point charges.
2. They should be stationary with respect to each other.
## Scalar form
The absolute value of the force F between two point charges q and Q relates to the distance between the point charges and to the simple product of their charges by Coulomb's law. The diagram shows that like charges repel each other, and opposite charges attract each other.
The scalar form of Coulomb's Law relates the magnitude and sign of the electrostatic force F, acting simultaneously on two point charges q 1 and q 2 :
$|\boldsymbol{F}|=k_e{|q_1q_2|\over r^2}$
where r is the separation distance and k e is Coulomb's constant. If the product q 1 q 2 is positive, the force between them is repulsive; if q 1 q 2 is negative, the force between them is attractive.[11]
The law of superposition allows this law to be extended to include any number of point charges, to derive the force on any one point charge by a vector addition of these individual forces acting alone on that point charge. The resulting force vector happens to be parallel to the electric field vector at that point, with that point charge removed.
## Vector form
In the image, the vector F 1 is the force experienced by q 1 , and the vector F 2 is the force experienced by q 2 . When q 1 q 2 > 0 the forces are repulsive (as in the image) and when q 1 q 2 < 0 the forces are attractive (opposite to the image). Their magnitudes will always be equal.
Coulomb's law states that the force $\boldsymbol{F}$ on a charge, $q_1$ at position $\boldsymbol{r_1}$, experiencing an electric field due to the presence of another charge, $q_2$ at position $\boldsymbol{r_2}$ in vacuum is:
$\boldsymbol{F}={q_1q_2\over4\pi\varepsilon_0}{(\boldsymbol{r_1-r_2})\over|\boldsymbol{r_1-r_2}|^3}={q_1q_2\over4\pi\varepsilon_0}{\boldsymbol{\hat{r}_{21}}\over r_{21}^2},$
where $\boldsymbol{r_{21}}=\boldsymbol{r_1-r_2}$ and $\varepsilon_0$ is the electric constant. This is simply the scalar definition of Coulomb's law with the direction given by the unit vector, $\boldsymbol{\hat{r}_{21}}$, parallel with the line from charge $q_2$ to charge $q_1$.[12]
If both charges have the same sign (like charges) then the product $q_1q_2$ is positive and the direction of the force on $q_1$ is given by $\boldsymbol{\hat{r}_{21}}$; the charges repel each other. If the charges have opposite signs then the product $q_1q_2$ is negative and the direction of the force on $q_1$ is given by $-\boldsymbol{\hat{r}_{21}}$; the charges attract each other.
### System of discrete charges
The principle of linear superposition may be used to calculate the force on a small test charge, $q$ at position $\boldsymbol{r}$, due to a system of $N$ discrete charges in vacuum:
$\boldsymbol{F(r)}={q\over4\pi\varepsilon_0}\sum_{i=1}^Nq_i{\boldsymbol{r-r_i}\over|\boldsymbol{r-r_i}|^3}={q\over4\pi\varepsilon_0}\sum_{i=1}^Nq_i{\boldsymbol{\widehat{R_i}}\over|\boldsymbol{R_i}|^2},$
where $q_i$ and $\boldsymbol{r_i}$ are the magnitude and position respectively of the $i^{th}$ charge, $\boldsymbol{\widehat{R_i}}$ is a unit vector in the direction of $\boldsymbol{R}_{i} = \boldsymbol{r} - \boldsymbol{r}_i$ (a vector pointing from charges $q_i$ to $q$).[12]
### Continuous charge distribution
For a charge distribution an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge $dq$.
For a linear charge distribution (a good approximation for charge in a wire) where $\lambda(\boldsymbol{r'})$ gives the charge per unit length at position $\boldsymbol{r'}$, and $dl'$ is an infinitesimal element of length,
$dq = \lambda(\boldsymbol{r'})dl'$.[13]
For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where $\sigma(\boldsymbol{r'})$ gives the charge per unit area at position $\boldsymbol{r'}$, and $dA'$ is an infinitesimal element of area,
$dq = \sigma(\boldsymbol{r'})\,dA'.$
For a volume charge distribution (such as charge within a bulk metal) where $\rho(\boldsymbol{r'})$ gives the charge per unit volume at position $\boldsymbol{r'}$, and $dV'$ is an infinitesimal element of volume,
$dq = \rho(\boldsymbol{r'})\,dV'.$[12]
The force on a small test charge $q'$ at position $\boldsymbol{r}$ in vacuum is given by
$\boldsymbol{F} = {q'\over 4\pi\varepsilon_0}\int dq {\boldsymbol{r} - \boldsymbol{r'} \over |\boldsymbol{r} - \boldsymbol{r'}|^3}.$
## Experimental verification of Coulomb's law
Experiment to verify Coulomb's law.
It is possible to verify Coulomb's law with a simple experiment. Let's consider two small spheres of mass m and same-sign charge q, hanging from two ropes of negligible mass and length l. The forces acting on each sphere are three: the weight mg, the rope tension T and the electric force F.
In the equilibrium state:
$T \ \sin \theta_1 =F_1 \,\!$
()
and:
$T \ \cos \theta_1 =mg \,\!$
()
Dividing (1) over (2):
$\frac {\sin \theta_1}{\cos \theta_1 }= \frac {F_1}{mg}\Rightarrow F_1= mg \tan \theta_1$
()
Being $L_1 \,\!$ the distance between the charged spheres; the repulsion force between them $F_1 \,\!$, assuming Coulomb's law is correct, is equal to
$F_1 = \frac{q^2}{4 \pi \epsilon_0 L_1^2}$
()
so:
$\frac{q^2}{4 \pi \epsilon_0 L_1^2}=mg \tan \theta_1 \,\!$
()
If we now decharge one of the spheres, and we put it in contact with the charged sphere, each one of them adquiers a charge q/2. In the equilibrium state, the distance between the charges will be $L_2<L_1 \,\!$ and the repulsion force between them will be:
$F_2 = \frac{{(q/2)}^2}{4 \pi \epsilon_0 L_2^2}=\frac{q^2/4}{4 \pi \epsilon_0 L_2^2} \,\!$
()
We know that $F_2= mg. \tan \theta_2 \,\!$. And:
$\frac{\frac{q^2}{4}}{4 \pi \epsilon_0 L_2^2}=mg. \tan \theta_2$
Dividing (3) over (4), we get:
$\frac{\left( \cfrac{q^2}{4 \pi \epsilon_0 L_1^2} \right)}{\left(\cfrac{q^2/4}{4 \pi \epsilon_0 L_2^2}\right)}= \frac{mg \tan \theta_1}{mg \tan \theta_2} \Longrightarrow 4 {\left ( \frac {L_2}{L_1} \right ) }^2= \frac{ \tan \theta_1}{ \tan \theta_2}$
()
Measuring the angles $\theta_1 \,\!$ and $\theta_2 \,\!$ and the distance bethween the charges $L_1 \,\!$ and $L_2 \,\!$ is possible to verify that the equality is true, taking into account the experimental error. In practice, angles can be difficult to measure, so if the length of the ropes is sufficiently big, the angles will be enough small to make the following aproximation:
$\tan \theta \approx \sin \theta= \frac{\frac{L}{2}}{l}=\frac{L}{2l}\Longrightarrow\frac{ \tan \theta_1}{ \tan \theta_2}\approx \frac{\frac{L_1}{2l}}{\frac{L_2}{2l}}$
()
Using this approximation, the relationship (6) turns into this much more simple expression:
$\frac{\frac{L_1}{2l}}{\frac{L_2}{2l}}\approx 4 {\left ( \frac {L_2}{L_1} \right ) }^2 \Longrightarrow \,\!$ $\frac{L_1}{L_2}\approx 4 {\left ( \frac {L_2}{L_1} \right ) }^2\Longrightarrow \frac{L_1}{L_2}\approx\sqrt[3]{4} \,\!$
()
In this way, the verification is limited to measuring the distance between the charges and check that the division approximates the theoretical value.
## Electrostatic approximation
In either formulation, Coulomb’s law is fully accurate only when the objects are stationary, and remains approximately correct only for slow movement. These conditions are collectively known as the electrostatic approximation. When movement takes place, magnetic fields that alter the force on the two objects are produced. The magnetic interaction between moving charges may be thought of as a manifestation of the force from the electrostatic field but with Einstein’s theory of relativity taken into consideration. Other theories like Weber electrodynamics predict other velocity-dependent corrections to Coulomb's law.
### Atomic forces
Coulomb's law holds even within the atoms, correctly describing the force between the positively charged nucleus and each of the negatively charged electrons. This simple law also correctly accounts for the forces that bind atoms together to form molecules and for the forces that bind atoms and molecules together to form solids and liquids.
Generally, as the distance between ions increases, the energy of attraction approaches zero and ionic bonding is less favorable. As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable.
## Table of derived quantities (relevant for vacuum only)
At/on 1 by 2 Particle property Relationship Field property
Vector quantity
Force
$\boldsymbol{F_{21}}={q_1q_2\over4\pi\varepsilon_0}{\boldsymbol{\widehat{R_{21}}}\over|\boldsymbol{R_{21}}|^2}$
$\boldsymbol{F_{21}}=q_1\boldsymbol{E_{21}}$
Electric field
$\boldsymbol{E_{21}}={q_2\over4\pi\varepsilon_0}{\boldsymbol{\widehat{R_{21}}}\over|\boldsymbol{R_{21}}|^2}$
Relationship $\boldsymbol{F_{21}}=-\nabla U_{21}$ $\boldsymbol{E_{21}}=-\nabla V_{21}$
Scalar quantity
Electric energy
$U_{21}={1\over4\pi\varepsilon_0}{q_1q_2\over|\boldsymbol{R_{21}}|}$
$U_{21}=q_1V_{21}$
Electric potential
$V_{21}={1\over4\pi\varepsilon_0}{q_2\over|\boldsymbol{R_{21}}|}$
## See also
• Biot–Savart law
• Method of image charges
• Electromagnetic force
• Molecular modelling
• Static forces and virtual-particle exchange
• Darwin Lagrangian
• Newton's Law of Universal Gravitation, which uses a similar structure, but for mass instead of charge.
## Notes
1. see: Abel Socin (1760) Acta Helvetiсa, vol. 4, pages 224-225.
2. See: J.L. Heilbron, Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics (Los Angeles, California: University of California Press, 1979), pages 460-462, and 464 (including footnote 44)
3. Schofield (1997), 144–56.
4. Joseph Priestley, The History and Present State of Electricity, with Original Experiments (London, England: 1767), page 732:
May we not infer from this experiment, that the attraction of electricity is subject to the same laws with that of gravitation, and is therefore according to the squares of the distances; since it is easily demonstrated, that were the earth in the form of a shell, a body in the inside of it would not be attracted to one side more than another?
5. Robert S. Elliott (1999). Electromagnetics: History, Theory, and Applications. ISBN 978-0-7803-5384-8 [Amazon-US | Amazon-UK]
6. John Robison, A System of Mechanical Philosophy (London, England: John Murray, 1822), vol. 4. On page 68, the author states that in 1769 he announced his findings regarding the force between spheres of like charge. On page 73, the author states the force between spheres of like charge varies as x-2.06.
7. James Clerk Maxwell, ed., The Electrical Researches of the Honourable Henry Cavendish... (Cambridge, England: Cambridge University Press, 1879), pages 104-113: "Experiments on Electricity: Experimental determination of the law of electric force."
8. ^ a b In -- Coulomb (1785a) "Premier mémoire sur l’électricité et le magnétisme," Histoire de l’Académie Royale des Sciences, pages 569-577 -- Coulomb studied the repulsive force between bodies having electrical charges of the same sign:
Page 574 : Il résulte donc de ces trois essais, que l'action répulsive que les deux balles électrifées de la même nature d'électricité exercent l'une sur l'autre, suit la raison inverse du carré des distances.
Translation : It follows therefore from these three tests, that the repulsive force that the two balls --[that were] electrified with the same kind of electricity -- exert on each other, follows the inverse proportion of the square of the distance.
In -- Coulomb (1785b) "Second mémoire sur l’électricité et le magnétisme," Histoire de l’Académie Royale des Sciences, pages 578-611. -- Coulomb showed that oppositely charged bodies obey an inverse-square law of attraction.
9. ^ a b c
## References
• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X [Amazon-US | Amazon-UK].
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8 [Amazon-US | Amazon-UK].
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http://physics.stackexchange.com/questions/27434/why-cant-noncontextual-ontological-theories-have-stronger-correlations-than-com/27436 | # Why can't noncontextual ontological theories have stronger correlations than commutative theories?
EDIT: I found both answers to my question to be unsatisfactory. But I think this is because the question itself is unsatisfactory, so I reworded it in order to allow a good answer.
One take on contextuality is to develop an inequality on measurement outcomes that is satisfied for any ontological noncontextual theory, and see that it is violated by quantum mechanics.
Another take would be to assume an algebraic structure and see that if one restricts the observable algebra to be commutative, the expected values of certain operators are restricted to lie in a given range, whereas if we allow non-commutativity the range is greater.
These approaches coincide? I've seen plenty of works that assume that it does, but without discussing it; in particular there is this paper by Tsirelson that states (in the particular case of Bell inequalities) that both approaches are equivalent, but without proving it. Is it too obvious?
At first sight, an ontological noncontextual theory is much more general than some theory embedded in a C*-algebraic framework. Why then can't it generate stronger correlations than theories with commuting algebras of observables?
Can one find a more direct connection between non-commutativity and the violation of a contextual inequality?
-
## 4 Answers
Well, a reasonable approach to local hidden variables is to require commutativity of operators on different space-like separated systems. This is pretty straight forward to motivate, since otherwise you are essentially using non-local operators, which can be seen via the relevant transform instead as a non-local hidden variable theory with local operators.
You might want to check out Scott Aaronson's paper exploring the consequence of this when taken as an axiom together with some other desirable properties of hidden variable theories (Phys. Rev. A 71, 032325).
I'm not too sure it makes sense to talk about commutativity beyond this, since we care about post-measurement outcomes it is not clear that the domain of the operator should contain its image, and so multiplication, and hence the commutator, isn't necessarily defined.
-
I'm afraid you didn't understand my question. Commutativity of operators on different space-like separated system is respected by QM (or AQFT, for that matter). I'm talking about commutativity of the whole algebra. In fact it may not make sense to talk about commutativity in the strict sense; I've never seen anyone construct an operator algebra to model a hidden-variable theory. But if we only care about the measurement outcome, not the post-measurement state, we have commutativity in a trivial sense, by substituting the observable by its predefined outcome, which is just a real number. – Mateus Araújo Sep 20 '11 at 14:35
@Mateus: I think I did understand, which is what my last paragraph attempts to address. – Joe Fitzsimons Sep 20 '11 at 14:42
Non-commutativity of operators ensures that in general we can't construct a joint probability distribution over the observables that we model using those operators. In some states and for some choices of non-commuting operators we can construct joint probability distributions, for example the vacuum state and coherent states of a quantized simple harmonic oscillator generate a positive-definite Wigner function for position and momentum, which can be taken to be a probability distribution. Of course that possibility falls apart when one considers almost any superposition of coherent states, say. The Wigner function is not positive-definite in the general case, making the interpretation of the Wigner function as a probability distribution in the special cases quite tendentious.
Conversely, if we have a commutative algebra of operators we can construct a joint probability distribution over the any subset of the observables in any state over the algebra. One could take this property as a somewhat plausible definition of classicality.
For the technical basis of this, I like best two short papers, John Baez, Letters in Mathematical Physics 13 (1987) 135-136, and Lawrence J. LANDAU, PHYSICS LETTERS A, Volume 120, number 2 (1987), which put remarkably little interpretation in the way of the mathematics, but there is a substantial literature that has tried to get at this relationship in some sort of clear way.
A literature that gives an alternative way into the relationship between non-commutativity and measurement, and that focuses on the relationship between quantum theory and classical probability theory in a way that I find helpful, albeit not conclusive, is the positive-operator valued measure approach, which is well represented by the book by Paul Busch, Marian Grabowski, and Pekka J. Lahti, Operational Quantum Physics, Springer, 1995. Searching the literature or the ArXiv for anything more recent by any of these three authors will give you something enlightening to read. To my taste, Paul Busch is always worth reading.
As far as physicality is concerned, classical physics models measurements as not affecting other measurements, so that joint probability distributions over multiple measurements are possible. In the presence of any finite level of noise ---there always is noise, everywhere (only the thermal component of the noise goes away when one is close to absolute zero, the Lorentz invariant quantum component of the noise is not controllable)--- the uncontrolled nature of the noise is something that has to be accommodated by our models of our measurements. Quantum theory accommodates the non-trivial effects of joint measurements on each other by introducing non-commutativity of the operators that are used to model the measurements, whereas classical physics models the non-trivial effects of joint measurements on each other by modeling the measurement apparatus. Contextual models are precisely models that include the measurement apparatus, or the complete experimental apparatus, in the extreme case the whole universe, not just a putative measured system.
That's somewhat bashed out. Hope someone finds it congenial.
-
I don't think there is a more direct connection between noncontextuality and noncommutativity for a couple of reasons. Firstly, there are noncommutaive sets of observables and states that can be simulated by a noncontextual model. Think of the Kochen-Specker model for a qubit for example. Secondly, to really answer the question of what a violation of some inequality means, you should not assume that the data you collect in the experiment is necessarily produced by quantum theory (in particular, we don't do this for Bell's inequalities). Now, there are plenty of operational theories that are contextual (in Rob Spekkens sense) but that do not have a C*-algebraic structure, e.g. the theory wherein the state space is a square. Unless you can define what it means for measurements in these theories to be "commutative", which seems unlikely because they do not have an algebraic structure, then it is clear that the relationship between commutativity and noncontextuality breaks down in this context.
-
Good point. It would be interesting to find an operational definition of commutativity. But allow me to be picky: I think my answer is clear on relating commutative theories and noncontextual models; what it lacks is a relation between non-commutativity and contextuality. – Mateus Araújo Nov 7 '11 at 16:02
They can't, because ontological noncontextual theories are not more general than commuting subsets of quantum mechanics. In a nuthsell, commutative quantum mechanics is just classical probability theory, and the question whether there exists an ontological noncontextual model for quantum mechanics is precisely the question if it can be reduced to classical probability theory.
To see this, one needs Spekkens' operational definition of contextuality: an ontological noncontextual model is one where each state $\rho$ is represented by a probability distribution $\mu_\rho(\lambda)$ on an ontological space $\Lambda$, and each POVM $\{E^k\}$ by a probability distribution $\xi_{E^k}(λ)$. Then the probability of outcome $k$ will be given by $$\int \mathrm{d}λ\, \mu_\rho(λ) \xi_{E^k}(λ).$$
Now, if all $E^k$ commute, I can write $ρ$ in a basis in which they are diagonal. Then $$\mathrm{tr} (\rho E^k) = \sum_n \rho_{nn} E^k_{nn},$$ that is, the only part of $\rho$ who'll play a part in the expected value is its main diagonal, and that's just a probability distribution. If we do the identifications $\lambda \mapsto n$, $μ_ρ(λ) \mapsto ρ_{nn}$, and $\xi_{E^k}(λ) \mapsto E^k_{nn}$, we have an embedding of an ontological noncontextual model into commutative quantum mechanics.
In the particular case of nonlocality, one could see directly that a diagonal $\rho$ is always separable, and thus admits a local ontological model. The converse path, to construct a separable $\rho$ and commutative algebra from a local ontological model is essentially the same as above.
That said, I don't think my last question is answered; I'd still like to see a more direct, physical connection between non-commutativity and contextuality.
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http://www.citizendia.org/Mathematics | "Maths" and "Math" redirect here. For other uses of "Mathematics" or "Math", see Mathematics (disambiguation) and Math (disambiguation).
Euclid, Greek mathematician, 3rd century BC, as imagined by Raphael in this detail from The School of Athens. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Raphael Sanzio, usually known by his first name alone (in Italian Raffaello) (April 6 or March 28 1483 – April 6 1520 was an Italian painter and The School of Athens, or it Scuola di Atene in Italian, is one of the most famous Paintings by the Italian Renaissance artist [1]
Mathematics is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Quantity is a kind of property which exists as magnitude or multitude Structure is a fundamental and sometimes Intangible notion covering the Recognition, Observation, nature, and Stability of Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another Benjamin Peirce called it "the science that draws necessary conclusions". Benjamin Peirce (ˈpɜrs purse) April 4, 1809 – October 6, 1880) was an American Mathematician who [2] Other practitioners of mathematics maintain that mathematics is the science of pattern, and that mathematicians seek out patterns whether found in numbers, space, science, computers, imaginary abstractions, or elsewhere. A mathematician is a person whose primary area of study and research is the field of Mathematics. [3][4] Mathematicians explore such concepts, aiming to formulate new conjectures and establish their truth by rigorous deduction from appropriately chosen axioms and definitions. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse Deductive reasoning is Reasoning which uses deductive Arguments to move from given statements ( Premises to Conclusions which must be true if the In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject A definition is a statement of the meaning of a Word or Phrase. [5]
Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Abstraction in Mathematics is the process of extracting the underlying essence of a mathematical concept removing any dependence on real world objects with which it might originally Logic is the study of the principles of valid demonstration and Inference. Reasoning is the cognitive process of looking for Reasons for beliefs conclusions actions or feelings Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting A calculation is a deliberate process for transforming one or more inputs into one or more results with variable change Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as The shape ( OE sceap Eng created thing) of an object located in some space refers to the part of space occupied by the object as determined In Physics, motion means a constant change in the location of a body Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in the ancient Egyptian, Mesopotamian, Indian, Chinese, Greek and Islamic worlds. Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Mathematics in China emerged independently by the 11th century BC Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day. The Renaissance (from French Renaissance, meaning "rebirth" Italian: Rinascimento, from re- "again" and nascere The Timeline below shows the date of publication of major scientific theories and discoveries along with the discoverer [6]
Today, mathematics is used throughout the world in many fields, including natural science, engineering, medicine, and the social sciences such as economics. In Science, the term natural science refers to a naturalistic approach to the study of the Universe, which is understood as obeying rules or law of Engineering is the Discipline and Profession of applying technical and scientific Knowledge and Medicine is the art and science of healing It encompasses a range of Health care practices evolved to maintain and restore Human Health by the The social sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals including Anthropology, Communication studies Economics is the social science that studies the production distribution, and consumption of goods and services. Applied mathematics, the application of mathematics to such fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although applications for what began as pure mathematics are often discovered later. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application [7]
## Etymology
The word "mathematics" (Greek: μαθηματικά or mathēmatiká) comes from the Greek μάθημα (máthēma), which means learning, study, science, and additionally came to have the narrower and more technical meaning "mathematical study", even in Classical times. The Ancient Greek language is the historical stage in the development of the Hellenic language family spanning the Archaic (c Its adjective is μαθηματικός (mathēmatikós), related to learning, or studious, which likewise further came to mean mathematical. In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), in Latin ars mathematica, meant the mathematical art. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle, and meaning roughly "all things mathematical". English is a West Germanic language originating in England and is the First language for most people in the United Kingdom, the United States French ( français,) is a Romance language spoken around the world by 118 million people as a native language and by about 180 to 260 million people Marcus Tullius Cicero ( Classical Latin ˈkikeroː usually ˈsɪsərəʊ in English January 3, 106 BC &ndash December 7, 43 BC was a Roman Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. [8] In English, however, the noun mathematics takes singular verb forms. It is often shortened to math in English-speaking North America and maths elsewhere.
## History
A quipu, a counting device used by the Inca. Quipu or khipu (sometimes called talking knots) were recording devices used in the Inca Empire and its predecessor societies in the Andean The Inca Empire (or Inka Empire) was the largest empire in Pre-Columbian America.
Main article: History of mathematics
The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The area of study known as the history of mathematics is primarily an investigation into the origin of new discoveries in Mathematics and to a lesser extent an investigation --> Abstraction is the process or result of generalization by reducing the information The first abstraction was probably that of numbers. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. The realization that two apples and two oranges have something in common was a breakthrough in human thought. In addition to recognizing how to count physical objects, prehistoric peoples also recognized how to count abstract quantities, like time — days, seasons, years. Counting is the mathematical action of repeatedly adding (or subtracting one usually to find out how many objects there are or to set aside a desired number of objects (starting Stone Age Paleolithic See also Paleolithic, Recent African Origin, Early Homo sapiens, Early human migrations "Paleolithic" For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of A day (symbol d is a unit of Time equivalent to 24 Hours and the duration of a single Rotation of planet Earth with respect to the A season is one of the major divisions of the Year, generally based on yearly periodic changes in Weather. A year (from Old English gēr) is the time between two recurrences of an event related to the Orbit of the Earth around the Sun Arithmetic (addition, subtraction, multiplication and division), naturally followed. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication.
Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called quipu used by the Inca to store numerical data. A tally (or tally stick) was an ancient memory aid device to record and document numbers quantities or even messages Quipu or khipu (sometimes called talking knots) were recording devices used in the Inca Empire and its predecessor societies in the Andean Numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus. A numeral system (or system of numeration) is a Mathematical notation for representing numbers of a given set by symbols in a consistent manner The Rhind Mathematical Papyrus (RMP (also designated as papyrus British Museum 10057 and pBM 10058 is named after Alexander Henry Rhind, a Scottish The Indus Valley civilization developed the modern decimal system, including the concept of zero. The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin
From the beginnings of recorded history, the major disciplines within mathematics arose out of the need to do calculations relating to taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. The Pre-Columbian Maya civilization used a Vigesimal ( base - twenty) Numeral system. Commerce is a division of trade or production which deals with the exchange of goods and services from producer to final consumer Land measurement is the general concept describing the application and theory of Measurement of land. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study These needs can be roughly related to the broad subdivision of mathematics into the studies of quantity, structure, space, and change.
Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1. Bulletin of the American Mathematical Society (often abbreviated as Bull Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many 9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true "[9]
## Inspiration, pure and applied mathematics, and aesthetics
Sir Isaac Newton (1643-1727), an inventor of infinitesimal calculus. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements An inventor is a person who creates or discovers a new method form device or other useful means Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives
Main article: Mathematical beauty
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. Many Mathematicians derive aesthetic pleasure from their work and from Mathematics in general At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Commerce is a division of trade or production which deals with the exchange of goods and services from producer to final consumer Land measurement is the general concept describing the application and theory of Measurement of land. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study For example, Richard Feynman invented the Feynman path integral using a combination of mathematical reasoning and physical insight, and today's string theory continues to inspire new mathematics. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. The remarkable fact that even the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics. Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a In 1960 the Physicist Eugene Wigner published an article titled " The Unreasonable Effectiveness of Mathematics in the Natural Sciences " arguing that the "
As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Operations Research (OR in North America South Africa and Australia and Operational Research in Europe is an interdisciplinary branch of applied Mathematics and Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their
For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Aesthetics or esthetics ( also spelled æsthetics) is commonly known as the study of sensory or sensori-emotional values sometimes called NOTICE TO WOULD-BE-ROMEOS*************** Simplicity and generality are valued. Simplicity is the property condition or quality of being simple or un-combined There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Godfrey Harold Hardy FRS ( February 7, 1877 Cranleigh, Surrey, England &ndash December 1, 1947 A Mathematician's Apology is a 1940 essay by British mathematician G Mathematicians often strive to find proofs of theorems that are particularly elegant, a quest Paul Erdős often referred to as finding proofs from "The Book" in which God had written down his favorite proofs. Paul Erdős ( Hungarian: Erdős Pál, in English occasionally Paul Erdos or Paul Erdös, March 26, 1913 &ndash The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. Recreational mathematics is an umbrella term referring to Mathematical puzzles and Mathematical games.
## Notation, language, and rigor
The infinity symbol ∞ in several typefaces. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness
Main article: Mathematical notation
Most of the mathematical notation in use today was not invented until the 16th century. See also Table of mathematical symbols Mathematical notation is used in Mathematics, and throughout the Physical sciences, Engineering [10] Before that, mathematics was written out in words, a painstaking process that limited mathematical discovery. In the 18th century, Euler was responsible for many of the notations in use today. The 18th century lasted from 1701 to 1800 in the Gregorian calendar, in accordance with the Anno Domini / Common Era numbering system Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is extremely compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.
Mathematical language also is hard for beginners. A language is a dynamic set of visual auditory or tactile Symbols of Communication and the elements used to manipulate them Words such as or and only have more precise meanings than in everyday speech. Also confusing to beginners, words such as open and field have been given specialized mathematical meanings. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Mathematical jargon includes technical terms such as homeomorphism and integrable. The Language of mathematics has a vast Vocabulary of specialist and technical terms Topological equivalence redirects here see also Topological equivalence (dynamical systems. But there is a reason for special notation and technical jargon: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".
Rigor is fundamentally a matter of mathematical proof. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements [11] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Today, mathematicians continue to argue among themselves about computer-assisted proofs. A computer-assisted proof is a Mathematical proof that has been at least partially generated by computer Since large computations are hard to verify, such proofs may not be sufficiently rigorous. [12] Axioms in traditional thought were "self-evident truths", but that conception is problematic. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. Symbolic logic is the area of Mathematics which studies the purely formal properties of strings of symbols In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Hilbert's program, formulated by German mathematician David Hilbert in the 1920s was to formalize all existing theories to a finite complete set of axioms and provide In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most In Mathematical logic, a sentence &sigma is called independent of a given first-order theory T if T neither proves nor In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. [13]
## Mathematics as science
Carl Friedrich Gauss, himself known as the "prince of mathematicians", referred to mathematics as "the Queen of the Sciences". Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German
Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German [14] In the original Latin Regina Scientiarum, as well as in German Königin der Wissenschaften, the word corresponding to science means (field of) knowledge. The German language (de ''Deutsch'') is a West Germanic language and one of the world's major languages. Indeed, this is also the original meaning in English, and there is no doubt that mathematics is in this sense a science. The specialization restricting the meaning to natural science is of later date. If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application Albert Einstein has stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical "[15]
Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper. Falsifiability (or "refutability" is the logical possibility that an assertion can be shown false by an observation or a physical experiment Sir Karl Raimund Popper ( July 28 1902 &ndash September 17 1994) was an Austrian and British Philosopher and a professor However, in the 1930s important work in mathematical logic showed that mathematics cannot be reduced to logic, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently. "[16] Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself. Imre Lakatos ( November 9, 1922 – February 2, 1974) was a Philosopher of mathematics and science,
An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. John Michael Ziman ( May 16, 1925 - January 2, 2005) was a Physicist and a humanist who worked in the area of Condensed [17] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Intuition is apparent ability to acquire knowledge without a clear inference or the use of reason In scientific inquiry an experiment ( Latin: Ex- periri, "to try out" is a method of investigating particular types of research questions or In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not use the scientific method. For the mathematical journal of the same name see Experimental Mathematics (journal Experimental mathematics is an approach to mathematics in which Scientific method refers to bodies of Techniques for investigating phenomena In his 2002 book A New Kind of Science, Stephen Wolfram argues that computational mathematics deserves to be explored empirically as a scientific field in its own right. A New Kind of Science is a Controversial book by Stephen Wolfram, published in 2002 Stephen Wolfram (born August 29, 1959 in London) is a British Physicist, Mathematician and Businessman known for his
The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. The term liberal arts refers to a particular type of educational Curriculum broadly defined as a Classical education. Engineering is the Discipline and Profession of applying technical and scientific Knowledge and One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. A university is an institution of Higher education and Research, which grants Academic degrees in a variety of subjects In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics.
Mathematical awards are generally kept separate from their equivalents in science. The most prestigious award in mathematics is the Fields Medal,[18][19] established in 1936 and now awarded every 4 years. The Fields Medal is a prize awarded to two three or four Mathematicians not over 40 years of age at each International Congress of the International Mathematical It is often considered, misleadingly, the equivalent of science's Nobel Prizes. The Nobel Prize (Nobelpriset (Nobelprisen is a Swedish prize established in the 1895 will of Swedish chemist Alfred Nobel; it was first awarded in Peace, Literature The Wolf Prize in Mathematics, instituted in 1979, recognizes lifetime achievement, and another major international award, the Abel Prize, was introduced in 2003. The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding Mathematicians The prize is named after Norwegian These are awarded for a particular body of work, which may be innovation, or resolution of an outstanding problem in an established field. A famous list of 23 such open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. Hilbert's problems are a list of twenty-three problems in Mathematics put forth by German Mathematician David Hilbert at the Paris David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. The Millennium Prize Problems are seven problems in Mathematics that were stated by the Clay Mathematics Institute in 2000 Solution of each of these problems carries a \$1 million reward, and only one (the Riemann hypothesis) is duplicated in Hilbert's problems. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved
## Fields of mathematics
An abacus, a simple calculating tool used since ancient times
As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. An abacus, also called a counting frame, is a calculating tool used primarily by Asians for performing arithmetic processes Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i. e. , arithmetic, algebra, geometry, and analysis). Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Analysis has its beginnings in the rigorous formulation of Calculus. In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Applied mathematics is a branch of Mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains Uncertainty is a term used in subtly different ways in a number of fields including Philosophy, Statistics, Economics, Finance, Insurance
### Quantity
The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. A number is an Abstract object, tokens of which are Symbols used in Counting and measuring. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone The deeper properties of integers are studied in number theory, whence such popular results as Fermat's last theorem. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Number theory also holds two widely-considered unsolved problems: the twin prime conjecture and Goldbach's conjecture. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics.
As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions"). In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object These, in turn, are contained within the real numbers, which are used to represent continuous quantities. In Mathematics, the real numbers may be described informally in several different ways Real numbers are generalized to complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted These are the first steps of a hierarchy of numbers that goes on to include quarternions and octonions. Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of counting to infinity. Transfinite numbers are Cardinal numbers or Ordinal numbers that are larger than all finite numbers yet not necessarily absolutely infinite. Another area of study is size, which leads to the cardinal numbers and then to another conception of infinity: the aleph numbers, which allow meaningful comparison of the size of infinitely large sets. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.
| | | | | |
|----------------------------------------------------------------------------|----------------------------------------------------------------------------|----------------------------------------------------------------------------|----------------------------------------------------------------------------|----------------------------------------------------------------------------|
| $1, 2, 3\,\!$ | $-2, -1, 0, 1, 2\,\!$ | $-2, \frac{2}{3}, 1.21\,\!$ | $-e, \sqrt{2}, 3, \pi\,\!$ | $2, i, -2+3i, 2e^{i\frac{4\pi}{3}}\,\!$ |
| Natural numbers | Integers | Rational numbers | Real numbers | Complex numbers |
### Structure
Many mathematical objects, such as sets of numbers and functions, exhibit internal structure. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The structural properties of these objects are investigated in the study of groups, rings, fields and other abstract systems, which are themselves such objects. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division This is the field of abstract algebra. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules An important concept here is that of vectors, generalized to vector spaces, and studied in linear algebra. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Linear algebra is the branch of Mathematics concerned with The study of vectors combines three of the fundamental areas of mathematics: quantity, structure, and space. Vector calculus expands the field into a fourth fundamental area, that of change. Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner
### Space
The study of space originates with geometry - in particular, Euclidean geometry. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Order theory is a branch of Mathematics that studies various kinds of Binary relations that capture the intuitive notion of ordering providing a framework for saying Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Trigonometry combines space and numbers, and encompasses the well-known Pythagorean theorem. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Within differential geometry are the concepts of fiber bundles and calculus on manifolds. In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the Lie groups are used to study space, structure, and change. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group Topology in all its many ramifications may have been the greatest growth area in 20th century mathematics, and includes the long-standing Poincaré conjecture and the controversial four color theorem, whose only proof, by computer, has never been verified by a human. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country
### Change
Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a powerful tool to investigate it. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of A fractal is generally "a rough or fragmented geometric shape that can be split into parts each of which is (at least approximately a reduced-size copy of the whole" In Science, the term natural science refers to a naturalistic approach to the study of the Universe, which is understood as obeying rules or law of Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Functions arise here, as a central concept describing a changing quantity. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The rigorous study of real numbers and real-valued functions is known as real analysis, with complex analysis the equivalent field for the complex numbers. Real analysis is a branch of Mathematical analysis dealing with the set of Real numbers In particular it deals with the analytic properties of real Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of Mathematics investigating functions of Complex The Riemann hypothesis, one of the most fundamental open questions in mathematics, is drawn from complex analysis. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. For functional analysis as used in psychology see the Functional analysis (psychology article Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another One of many applications of functional analysis is quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that In Mathematics, a deterministic system is a system in which no Randomness is involved in the development of future states of the system
### Foundations and philosophy
In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed, as well as category theory which is still in development. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position In Mathematics, chaos theory describes the behavior of certain dynamical systems – that is systems whose state evolves with time – that may exhibit dynamics that Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets
Mathematical logic is concerned with setting mathematics on a rigid axiomatic framework, and studying the results of such a framework. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject As such, it is home to Gödel's second incompleteness theorem, perhaps the most widely celebrated result in logic, which (informally) implies that any formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist Gödel showed how to construct, whatever the given collection of number-theoretical axioms, a formal statement in the logic that is a true number-theoretical fact, but which does not follow from those axioms. Therefore no formal system is a true axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science. Recursion theory, also called computability theory, is a branch of Mathematical logic that originated in the 1930s with the study of Computable functions In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models Proof theory is a branch of Mathematical logic that represents proofs as formal Mathematical objects facilitating their analysis by mathematical techniques Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their
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|----------------------------------------------------------------------------|------------|-----------------|
| $p \Rightarrow q \,$ | | |
| Mathematical logic | Set theory | Category theory |
### Discrete mathematics
Discrete mathematics is the common name for the fields of mathematics most generally useful in theoretical computer science. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Discrete mathematics, also called finite mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the Theoretical computer science is the collection of topics of Computer science that focuses on the more abstract logical and mathematical aspects of Computing, such This includes computability theory, computational complexity theory, and information theory. In Computer science, computability theory is the branch of the Theory of computation that studies which problems are computationally solvable using different Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources Information theory is a branch of Applied mathematics and Electrical engineering involving the quantification of Information. Computability theory examines the limitations of various theoretical models of the computer, including the most powerful known model - the Turing machine. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware. Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence concepts such as compression and entropy. There are close parallels between the mathematical expressions for the thermodynamic Entropy, usually denoted by S, of a physical system in the Statistical thermodynamics
As a relatively new field, discrete mathematics has a number of fundamental open problems. The most famous of these is the "P=NP?" problem, one of the Millennium Prize Problems. The relationship between the Complexity classes P and NP is an unsolved question in Theoretical computer science. The Millennium Prize Problems are seven problems in Mathematics that were stated by the Clay Mathematics Institute in 2000 [20]
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|----------------------------------------------------------------------------|-----------------------|--------------|--------------|
| $\begin{matrix} (1,2,3) & (1,3,2) \\ (2,1,3) & (2,3,1) \\ (3,1,2) & (3,2,1) \end{matrix}$ | | | |
| Combinatorics | Theory of computation | Cryptography | Graph theory |
### Applied mathematics
Applied mathematics considers the use of abstract mathematical tools in solving concrete problems in the sciences, business, and other areas. Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects The theory of computation is the branch of Computer science that deals with whether and how efficiently problems can be solved on a Model of computation, using an Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects Science (from the Latin scientia, meaning " Knowledge " or "knowing" is the effort to discover, and increase human understanding A business (also called firm or an enterprise) is a legally recognized organizational entity designed to provide goods and/or services to An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Probability theory is the branch of Mathematics concerned with analysis of random phenomena Most experiments, surveys and observational studies require the informed use of statistics. (Many statisticians, however, do not consider themselves to be mathematicians, but rather part of an allied group. ) Numerical analysis investigates computational methods for efficiently solving a broad range of mathematical problems that are typically too large for human numerical capacity; it includes the study of rounding errors or other sources of error in computation. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) For the acrobatic movement roundoff see Roundoff. A round-off error, also called rounding error, is the difference between the
## Common misconceptions
Mathematics is not a closed intellectual system, in which everything has already been worked out. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Fluid mechanics is the study of how Fluids move and the Forces on them Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) In Mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function Probability is the likelihood or chance that something is the case or will happen Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Mathematical finance is the branch of Applied mathematics concerned with the Financial markets. Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, There is no shortage of open problems. Mathematicians publish many thousands of papers embodying new discoveries in mathematics every month.
Mathematics is not numerology, nor is it accountancy; nor is it restricted to arithmetic. Numerology is any of many Systems Traditions or Beliefs in a mystical or Esoteric relationship between Numbers and physical Accountancy or accounting is the measurement statement or provision of assurance about financial information primarily used by Lenders managers, Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone
Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. Pseudomathematics is a form of Mathematics -like activity that does not work within the framework definitions rules or rigor of formal mathematical models It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. Pseudoscience is defined as a body of knowledge methodology belief or practice that is claimed to be Scientific or made to appear scientific but does not adhere to the The misconceptions involved are normally based on:
• misunderstanding of the implications of mathematical rigor;
• attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, often in the belief that the journal is biased against the author;
• lack of familiarity with, and therefore underestimation of, the existing literature. Rigour or rigor (see spelling differences) has a number of meanings in relation to intellectual life and discourse A mathematical journal is a Scientific journal which publishes exclusively (or almost exclusively mathematical papers. An academic journal is a peer-reviewed Periodical in which scholarship relating to a particular Academic discipline is published Peer review (also known as refereeing) is the process of subjecting an author's scholarly work research or Ideas to the scrutiny of others who are
The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. Kurt Heegner (1893–1965 was a German private scholar from Berlin, who specialized in Radio Engineering and Mathematics. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the Marin Mersenne, Marin Mersennus or le Père Mersenne ( September 8, 1588 &ndash September 1, 1648) was
### Mathematics and physical reality
Mathematical concepts and theorems need not correspond to anything in the physical world. Insofar as a correspondence does exist, while mathematicians and physicists may select axioms and postulates that seem reasonable and intuitive, it is not necessary for the basic assumptions within an axiomatic system to be true in an empirical or physical sense. Thus, while many axiom systems are derived from our perceptions and experiments, they are not dependent on them. In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems
For example, we could say that the physical concept of two apples may be accurately modeled by the natural number 2. Note The term model has a different meaning in Model theory, a branch of Mathematical logic. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an On the other hand, we could also say that the natural numbers are not an accurate model because there is no standard "unit" apple and no two apples are exactly alike. The modeling idea is further complicated by the possibility of fractional or partial apples. In Mathematics, a fraction (from the Latin fractus, broken is a concept of a proportional relation between an object part and the object So while it may be instructive to visualize the axiomatic definition of the natural numbers as collections of apples, the definition itself is not dependent upon nor derived from any actual physical entities.
Nevertheless, mathematics remains extremely useful for solving real-world problems. This fact led physicist Eugene Wigner to write an article titled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". Eugene Paul "EP" Wigner ( Hungarian Wigner Pál Jenő) ( November 17, 1902 &ndash January 1, 1995) was a In 1960 the Physicist Eugene Wigner published an article titled " The Unreasonable Effectiveness of Mathematics in the Natural Sciences " arguing that the
## Notes
1. ^ No likeness or description of Euclid's physical appearance made during his lifetime survived antiquity. Mathematics is the search for fundamental truths in pattern quantity and change Wikipedia talkFeatured lists#Proposed change to all featured lists for an explanation of this and other inclusion tags below -->This article itemizes the various Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. Mathematics education is a term that refers both to the practice of Teaching and Learning Mathematics, as well as to a field of scholarly Research This article is about using Mathematics to study the inner-workings of Multiplayer games which on the surface may not appear mathematical at all Note The term model has a different meaning in Model theory, a branch of Mathematical logic. A mathematical problem is a problem that is amenable to being analyzed and possibly solved with the methods of Mathematics. Mathematics competitions or mathematical olympiads are competitive events where participants write a mathematics test Dyscalculia is a type of specific learning disability (SLD involving innate difficulty in learning or comprehending Mathematics. Therefore, Euclid's depiction in works of art depends on the artist's imagination (see Euclid). Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry
2. ^ Peirce, p. 97
3. ^ Steen, L.A. (April 29, 1988). Lynn Arthur Steen is an American Mathematician who is Professor of Mathematics at St The Science of Patterns. Science, 240: 611–616. Science is the Academic journal of the American Association for the Advancement of Science and is considered one of the world's most prestigious Scientific and summarized at Association for Supervision and Curriculum Development.
4. ^ Devlin, Keith, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 9780716750475
5. ^ Jourdain
6. ^ Eves
7. ^ Peterson
8. ^ The Oxford Dictionary of English Etymology, Oxford English Dictionary
9. ^ Sevryuk
10. ^ Earliest Uses of Various Mathematical Symbols (Contains many further references)
11. ^ See false proof for simple examples of what can go wrong in a formal proof. Keith J Devlin is an English Mathematician and Writer. He currently is Executive Director of Stanford University 's Center for the Study The Oxford Dictionary of English Etymology is a notable Etymological dictionary of the English language, published by Oxford University Press The Oxford English Dictionary ( OED) published by the Oxford University Press (OUP is a comprehensive Dictionary of the English In Mathematics, there are a variety of spurious proofs of obvious Contradictions Although the proofs are flawed the errors usually by design are comparatively subtle The history of the Four Color Theorem contains examples of false proofs accepted by other mathematicians. The four color theorem (also known as the four color map theorem) states that given any plane separated into regions such as a political map of the states of a country
12. ^ Ivars Peterson, The Mathematical Tourist, Freeman, 1988, ISBN 0-7167-1953-3. p. 4 "A few complain that the computer program can't be verified properly," (in reference to the Haken-Apple proof of the Four Color Theorem).
13. ^ Patrick Suppes, Axiomatic Set Theory, Dover, 1972, ISBN 0-486-61630-4. p. 1, "Among the many branches of modern mathematics set theory occupies a unique place: with a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects. "
14. ^ Waltershausen
15. ^ Einstein, p. 28. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?" He, too, is concerned with The Unreasonable Effectiveness of Mathematics in the Natural Sciences. In 1960 the Physicist Eugene Wigner published an article titled " The Unreasonable Effectiveness of Mathematics in the Natural Sciences " arguing that the
16. ^ Popper 1995, p. 56
17. ^ Ziman
18. ^ "The Fields Medal is now indisputably the best known and most influential award in mathematics. " Monastyrsky
19. ^ Riehm
20. ^ Clay Mathematics Institute P=NP
## References
• Benson, Donald C. , The Moment of Proof: Mathematical Epiphanies, Oxford University Press, USA; New Ed edition (December 14, 2000). ISBN 0-19-513919-4.
• Boyer, Carl B., A History of Mathematics, Wiley; 2 edition (March 6, 1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he ISBN 0-471-54397-7. — A concise history of mathematics from the Concept of Number to contemporary Mathematics.
• Courant, R. and H. Robbins, What Is Mathematics? : An Elementary Approach to Ideas and Methods, Oxford University Press, USA; 2 edition (July 18, 1996). ISBN 0-19-510519-2.
• Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. For other persons named Philip Davis see Philip Davis (disambiguation. Reuben Hersh (born 1927 is an American Mathematician and Academic, best known for his writings on the nature practice and social impact of mathematics The Mathematical Experience is a 1981 book by Philip J Davis and Reuben Hersh that discusses the practice of modern Mathematics from a Mariner Books; Reprint edition (January 14, 1999). ISBN 0-395-92968-7. — A gentle introduction to the world of mathematics.
• Einstein, Albert (1923). Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical "Sidelights on Relativity (Geometry and Experience)". P. Dutton. , Co.
• Eves, Howard, An Introduction to the History of Mathematics, Sixth Edition, Saunders, 1990, ISBN 0-03-029558-0.
• Gullberg, Jan, Mathematics—From the Birth of Numbers. W. W. Norton & Company; 1st edition (October 1997). ISBN 0-393-04002-X. — An encyclopedic overview of mathematics presented in clear, simple language.
• Hazewinkel, Michiel (ed. ), Encyclopaedia of Mathematics. The Encyclopaedia of Mathematics is a large reference work in Mathematics. Kluwer Academic Publishers 2000. — A translated and expanded version of a Soviet mathematics encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM, and online [1].
• Jourdain, Philip E. B. , The Nature of Mathematics, in The World of Mathematics, James R. Newman, editor, Dover, 2003, ISBN 0-486-43268-8.
• Kline, Morris, Mathematical Thought from Ancient to Modern Times, Oxford University Press, USA; Paperback edition (March 1, 1990). Morris Kline ( May 1, 1908 – June 10, 1992) was a Professor of Mathematics, a writer on the history, philosophy ISBN 0-19-506135-7.
• Monastyrsky, Michael (2001). "Some Trends in Modern Mathematics and the Fields Medal". Canadian Mathematical Society. Retrieved on 2006-07-28. Year 2006 ( MMVI) was a Common year starting on Sunday of the Gregorian calendar. Events 1540 - Thomas Cromwell is executed at the order of Henry VIII of England on charges of Treason.
• Oxford English Dictionary, second edition, ed. The Oxford English Dictionary ( OED) published by the Oxford University Press (OUP is a comprehensive Dictionary of the English John Simpson and Edmund Weiner, Clarendon Press, 1989, ISBN 0-19-861186-2.
• The Oxford Dictionary of English Etymology, 1983 reprint. The Oxford Dictionary of English Etymology is a notable Etymological dictionary of the English language, published by Oxford University Press ISBN 0-19-861112-9.
• Pappas, Theoni, The Joy Of Mathematics, Wide World Publishing; Revised edition (June 1989). ISBN 0-933174-65-9.
• Peirce, Benjamin. "Linear Associative Algebra". American Journal of Mathematics (Vol. 4, No. 1/4. (1881). JSTOR. JSTOR (short for Journal Storage) is a United States -based online system for archiving Academic journals founded in 1995
• Peterson, Ivars, Mathematical Tourist, New and Updated Snapshots of Modern Mathematics, Owl Books, 2001, ISBN 0-8050-7159-8.
• Paulos, John Allen (1996). John Allen Paulos (born July 4, 1945) is a professor of Mathematics at Temple University in Philadelphia who has gained fame as a A Mathematician Reads the Newspaper. Anchor. ISBN 0-385-48254-X.
• Popper, Karl R. (1995). Sir Karl Raimund Popper ( July 28 1902 &ndash September 17 1994) was an Austrian and British Philosopher and a professor "On knowledge", In Search of a Better World: Lectures and Essays from Thirty Years. Routledge. ISBN 0-415-13548-6.
• Riehm, Carl (August 2002). "The Early History of the Fields Medal". Notices of the AMS 49 (7): 778-782. AMS.
• Sevryuk, Mikhail B. (January 2006). "Book Reviews" (PDF). Bulletin of the American Mathematical Society 43 (1): 101-109. Bulletin of the American Mathematical Society (often abbreviated as Bull
• Waltershausen, Wolfgang Sartorius von (1856, repr. 1965). Gauss zum Gedächtniss. Sändig Reprint Verlag H. R. Wohlwend. ISBN 3-253-01702-8.
• Ziman, J. M. , F. R. S. (1968). "Public Knowledge:An essay concerning the social dimension of science". | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 7, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9336841106414795, "perplexity_flag": "middle"} |
http://en.wikipedia.org/wiki/Cyclomatic_complexity | Cyclomatic complexity
Cyclomatic complexity (or conditional complexity) is a software metric (measurement). It was developed by Thomas J. McCabe, Sr. in 1976 and is used to indicate the complexity of a program. It directly measures the number of linearly independent paths through a program's source code. The concept, although not the method, is somewhat similar to that of general text complexity measured by the Flesch-Kincaid Readability Test.
Cyclomatic complexity is computed using the control flow graph of the program: the nodes of the graph correspond to indivisible groups of commands of a program, and a directed edge connects two nodes if the second command might be executed immediately after the first command. Cyclomatic complexity may also be applied to individual functions, modules, methods or classes within a program.
One testing strategy, called Basis Path Testing by McCabe who first proposed it, is to test each linearly independent path through the program; in this case, the number of test cases will equal the cyclomatic complexity of the program.[1]
Description
A control flow graph of a simple program. The program begins executing at the red node, then enters a loop (group of three nodes immediately below the red node). On exiting the loop, there is a conditional statement (group below the loop), and finally the program exits at the blue node. For this graph, E = 9, N = 8 and P = 1, so the cyclomatic complexity of the program is 9 - 8 + (2*1) = 3.
The cyclomatic complexity of a section of source code is the count of the number of linearly independent paths through the source code. For instance, if the source code contained no decision points such as IF statements or FOR loops, the complexity would be 1, since there is only a single path through the code. If the code had a single IF statement containing a single condition there would be two paths through the code, one path where the IF statement is evaluated as TRUE and one path where the IF statement is evaluated as FALSE.
Mathematically, the cyclomatic complexity of a structured program[note 1] is defined with reference to the control flow graph of the program, a directed graph containing the basic blocks of the program, with an edge between two basic blocks if control may pass from the first to the second. The complexity M is then defined as:[2]
M = E − N + 2P
where
E = the number of edges of the graph
N = the number of nodes of the graph
P = the number of connected components (exit nodes).
The same function as above, shown as a strongly connected control flow graph, for calculation via the alternative method. For this graph, E = 10, N = 8 and P = 1, so the cyclomatic complexity of the program is 10 - 8 + 1 = 3.
An alternative formulation is to use a graph in which each exit point is connected back to the entry point. In this case, the graph is said to be strongly connected, and the cyclomatic complexity of the program is equal to the cyclomatic number of its graph (also known as the first Betti number), which is defined as:[2]
M = E − N + P
This may be seen as calculating the number of linearly independent cycles that exist in the graph, i.e. those cycles that do not contain other cycles within themselves. Note that because each exit point loops back to the entry point, there is at least one such cycle for each exit point.
For a single program (or subroutine or method), P is always equal to 1. Cyclomatic complexity may, however, be applied to several such programs or subprograms at the same time (e.g., to all of the methods in a class), and in these cases P will be equal to the number of programs in question, as each subprogram will appear as a disconnected subset of the graph.
It can be shown that the cyclomatic complexity of any structured program with only one entrance point and one exit point is equal to the number of decision points (i.e., 'if' statements or conditional loops) contained in that program plus one.[2][3]
Cyclomatic complexity may be extended to a program with multiple exit points; in this case it is equal to:
π - s + 2
where π is the number of decision points in the program, and s is the number of exit points.[3][4]
Formal definition
An even subgraph of a graph (also known as an Eulerian subgraph) is one where every vertex is incident with an even number of edges; such subgraphs are unions of cycles and isolated vertices. In the following, even subgraphs will be identified with their edge sets, which is equivalent to only considering those even subgraphs which contain all vertices of the full graph.
The set of all even subgraphs of a graph is closed under symmetric difference, and may thus be viewed as a vector space over GF(2); this vector space is called the cycle space of the graph. The cyclomatic number of the graph is defined as the dimension of this space. Since GF(2) has two elements and the cycle space is necessarily finite, the cyclomatic number is also equal to the 2-logarithm of the number of elements in the cycle space.
A basis for the cycle space is easily constructed by first fixing a maximal spanning forest of the graph, and then considering the cycles formed by one edge not in the forest and the path in the forest connecting the endpoints of that edge; these cycles constitute a basis for the cycle space. Hence, the cyclomatic number also equals the number of edges not in a maximal spanning forest of a graph. Since the number of edges in a maximal spanning forest of a graph is equal to the number of vertices minus the number of components, the formula $E-N+P$ above for the cyclomatic number follows.[5]
For the more topologically inclined, cyclomatic complexity can alternatively be defined as a relative Betti number, the size of a relative homology group:
$M := b_1(G,t) := \operatorname{rank}\,H_1(G,t)$
which is read as “the first homology of the graph G, relative to the terminal nodes t”. This is a technical way of saying “the number of linearly independent paths through the flow graph from an entry to an exit”, where:
• “linearly independent” corresponds to homology, and means one does not double-count backtracking;
• “paths” corresponds to first homology: a path is a 1-dimensional object;
• “relative” means the path must begin and end at an entry or exit point.
This corresponds to the intuitive notion of cyclomatic complexity, and can be calculated as above.
Alternatively, one can compute this via absolute Betti number (absolute homology – not relative) by identifying (gluing together) all terminal nodes on a given component (or equivalently, draw paths connecting the exits to the entrance), in which case (calling the new, augmented graph $\tilde G$, which is ), one obtains:
$M = b_1(\tilde G) = \operatorname{rank}\,H_1(\tilde G)$
It can also be computed via homotopy. If one considers the control flow graph as a 1-dimensional CW complex called $X$, then the fundamental group of $X$ will be $\pi_1(X) = \Z^n$. The value of $n+1$ is the cyclomatic complexity. The fundamental group counts how many loops there are through the graph, up to homotopy, and hence aligns with what we would intuitively expect.
This corresponds to the characterization of cyclomatic complexity as “number of loops plus number of components”.
Etymology / Naming
The name Cyclomatic Complexity presents some confusion, as this metric does not only count cycles (loops) in the program. Instead, the name refers to the number of different cycles in the program control flow graph, after having added an imagined branch back from the exit node to the entry node.[2]
A better name for popular usage would be Conditional Complexity, as "it has been found to be more convenient to count conditions instead of predicates when calculating complexity".[6]
Applications
Limiting complexity during development
One of McCabe's original applications was to limit the complexity of routines during program development; he recommended that programmers should count the complexity of the modules they are developing, and split them into smaller modules whenever the cyclomatic complexity of the module exceeded 10.[2] This practice was adopted by the NIST Structured Testing methodology, with an observation that since McCabe's original publication, the figure of 10 had received substantial corroborating evidence, but that in some circumstances it may be appropriate to relax the restriction and permit modules with a complexity as high as 15. As the methodology acknowledged that there were occasional reasons for going beyond the agreed-upon limit, it phrased its recommendation as: "For each module, either limit cyclomatic complexity to [the agreed-upon limit] or provide a written explanation of why the limit was exceeded."[7]
Implications for Software Testing
Another application of cyclomatic complexity is in determining the number of test cases that are necessary to achieve thorough test coverage of a particular module.
It is useful because of two properties of the cyclomatic complexity, M, for a specific module:
• M is an upper bound for the number of test cases that are necessary to achieve a complete branch coverage.
• M is a lower bound for the number of paths through the control flow graph (CFG). Assuming each test case takes one path, the number of cases needed to achieve path coverage is equal to the number of paths that can actually be taken. But some paths may be impossible, so although the number of paths through the CFG is clearly an upper bound on the number of test cases needed for path coverage, this latter number (of possible paths) is sometimes less than M.
All three of the above numbers may be equal: branch coverage $\leq$ cyclomatic complexity $\leq$ number of paths.
For example, consider a program that consists of two sequential if-then-else statements.
```if( c1() )
f1();
else
f2();
if( c2() )
f3();
else
f4();
```
The control flow graph of the source code above; the red circle is the entry point of the function, and the blue circle is the exit point. The exit has been connected to the entry to make the graph strongly connected.
In this example, two test cases are sufficient to achieve a complete branch coverage, while four are necessary for complete path coverage. The cyclomatic complexity of the program is 3 (as the strongly connected graph for the program contains 9 edges, 7 nodes and 1 connected component) (9-7+1).
In general, in order to fully test a module all execution paths through the module should be exercised. This implies a module with a high complexity number requires more testing effort than a module with a lower value since the higher complexity number indicates more pathways through the code. This also implies that a module with higher complexity is more difficult for a programmer to understand since the programmer must understand the different pathways and the results of those pathways.
Unfortunately, it is not always practical to test all possible paths through a program. Considering the example above, each time an additional if-then-else statement is added, the number of possible paths doubles. As the program grew in this fashion, it would quickly reach the point where testing all of the paths was impractical.
One common testing strategy, espoused for example by the NIST Structured Testing methodology, is to use the cyclomatic complexity of a module to determine the number of white-box tests that are required to obtain sufficient coverage of the module. In almost all cases, according to such a methodology, a module should have at least as many tests as its cyclomatic complexity; in most cases, this number of tests is adequate to exercise all the relevant paths of the function.[7]
As an example of a function that requires more than simply branch coverage to test accurately, consider again the above function, but assume that to avoid a bug occurring, any code that calls either f1() or f3() must also call the other.[note 2] Assuming that the results of c1() and c2() are independent, that means that the function as presented above contains a bug. Branch coverage would allow us to test the method with just two tests, and one possible set of tests would be to test the following cases:
• c1() returns true and c2() returns true
• c1() returns false and c2() returns false
Neither of these cases exposes the bug. If, however, we use cyclomatic complexity to indicate the number of tests we require, the number increases to 3. We must therefore test one of the following paths:
• c1() returns true and c2() returns false
• c1() returns false and c2() returns true
Either of these tests will expose the bug.
Cohesion
One would also expect that a module with higher complexity would tend to have lower cohesion (less than functional cohesion) than a module with lower complexity. The possible correlation between higher complexity measure with a lower level of cohesion is predicated on a module with more decision points generally implementing more than a single well defined function. A 2005 study showed stronger correlations between complexity metrics and an expert assessment of cohesion in the classes studied than the correlation between the expert's assessment and metrics designed to calculate cohesion.[8]
Correlation to number of defects
A number of studies have investigated cyclomatic complexity's correlation to the number of defects contained in a function or method.[citation needed] Some[citation needed] studies find a positive correlation between cyclomatic complexity and defects: functions and methods that have the highest complexity tend to also contain the most defects, however the correlation between cyclomatic complexity and program size has been demonstrated many times and since program size is not a controllable feature of commercial software the usefulness of McCabes's number has been called to question. The essence of this observation is that larger programs (more complex programs as defined by McCabe's metric) tend to have more defects. Although this relation is provably true, it isn't commercially useful[9] . As a result the metric has not been accepted by commercial software development organizations.
Studies that controlled for program size (i.e., comparing modules that have different complexities but similar size, typically measured in lines of code) are generally less conclusive, with many finding no significant correlation, while others do find correlation. Some researchers who have studied the area question the validity of the methods used by the studies finding no correlation.[10]
Les Hatton claimed (Keynote at TAIC-PART 2008, Windsor, UK, Sept 2008) that McCabe's Cyclomatic Complexity number has the same predictive ability as lines of code.[11]
Notes
1. This is a fairly common type of condition; consider the possibility that f1 allocates some resource which f3 releases.
References
1.
2. McCabe (December 1976). "A Complexity Measure". IEEE Transactions on Software Engineering: 308–320. Template:Working link
3. ^ a b Belzer, Kent, Holzman and Williams (1992). Encyclopedia of Computer Science and Technology. CRC Press. pp. 367–368.
4. Harrison (October 1984). "Applying Mccabe's complexity measure to multiple-exit programs". Software: Practice and Experience (J Wiley & Sons).
5. Diestel, Reinhard (2000). Graph theory. Graduate texts in mathematics 173 (2 ed.). New York: Springer. ISBN 0-387-98976-5.
6. McCabe (December 1976). "A Complexity Measure". IEEE Transactions on Software Engineering: 315.
7. ^ a b
8. Stein et al; Cox, Glenn; Etzkorn, Letha (2005). "Exploring the relationship between cohesion and complexity". Journal of Computer Science 1 (2): 137–144. doi:10.3844/jcssp.2005.137.144.
9. G.S. Cherf (1992). "An Investigation of the Maintenance and Support Characteristics of Commercial Software". Journal of Software Quality (Springer-Verlag) 1 (3): 147–158. ISSN 1573-1367.
10. Kan (2003). Metrics and Models in Software Quality Engineering. Addison-Wesley. pp. 316–317. ISBN 0-201-72915-6.
11. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 10, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9325029850006104, "perplexity_flag": "middle"} |
http://nrich.maths.org/6495/solution?nomenu=1 | ## 'Weekly Challenge 28: the Right Volume' printed from http://nrich.maths.org/
### Show menu
If we suppose that the curve $y=f(x)$ is integrable then the volume so created will be
V = \int^1_0 \pi y^2 dx
To get a feel for the sort of curve we might need, first consider the special case $y=x$, which clearly passes through the two points.
Then,
V = \int^1_0 \pi x^2 = \pi\left[\frac{x^3}{3}\right]^1_0 = \frac{\pi}{3}
This is slightly larger than $1$, so we could consider a family of curves which at beneath $y=x$ with enough flexibility to all us to vary the final value of the volume. A simple choice is parabolas $y=Ax(x-1)$ for some multiplicative constant. These give
V = \int^1_0 \pi Ax^2(x-1)^2dx
Now that I see it, I'm not too keen on doing this integral, so I'm going to consider instead $y=A\sqrt{|x(x-1)|}$
This gives a volume
V = \int^1_0 \pi A x(1-x) dx = \pi A\left[\frac{x^2}{2}-\frac{x^3}{3}\right]^1_0 = \pi A \left[\frac{1}{2}-\frac{1}{3}\right] = \frac{\pi A}{6}
Thus, a curve which has the required properties is
y = \frac{6}{\pi}\sqrt{|x(x-1)|}
There are, of course, others! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8491252064704895, "perplexity_flag": "middle"} |
http://math.stackexchange.com/questions/26337/show-that-the-hermite-polynomials-form-a-basis-of-mathbbp-3 | # Show that the Hermite polynomials form a basis of $\mathbb{P}_3$
I have this question that I took a shot at but I am not very familiar with Hermite or Laguerre, this my first time running across these type of polynomials and need some help please.
(a) The first four Hermite polynomials are $1, 2t,-2+4t^2,$ and $-12t+8t^3$. These polynomials arise naturally in the study of certain important differential equations in mathematical physics. Show that the first four Hermite polynomials form a basis of $\mathbb{P}_3$.
(b) The first four Laguerre polynomials are $1, 1-t, 2-4t + t^2,$ and $6-18t + 9t^2- t^3$. Show that these four Laguerre polynomials form a basis of $\mathbb{P}_3$.
Results:
(a) The first four Hermite polynomials will be shown to form a basis of $\mathbb{P}_3$ by showing that they are linearly independent and that the number of polynomials equals the dimension of $\mathbb{P}_3$. Consider the following linear combination of the four Hermite polynomials:
$x(1) + y(2t) + z(-2+4t^2) + w(-12t + 8t^3) = at^3 + bt^2 + ct + d$
The first four Hermite polynomials will be shown to be linearly independent by showing that the only linear combination of them that produces the zero polynomial ($0t^3+0t^2+0t+0$) is the trivial combination of zero times each polynomial.
That is all I could come about thus far with it. Can anyone refine or correct this if this is the wrong approach to the problem.
-
## 2 Answers
Forget all this Hermite and Laguerre stuff. The fact is that any family of polynomials with all different degrees is linearly free. Hence any family of polynomials with degrees $0$, $1$, $\ldots$, $n$ is a basis of the vector space of polynomials of degree at most $n$ (the space you denote by $\mathbb{P}_n$).
A relatively more sophisticated way of saying the same thing is that any triangular matrix with no zero on its diagonal is invertible.
-
## Did you find this question interesting? Try our newsletter
The only other thing to note, that Hermite $H_i$ and Laguerre $L_i$ polynomials are orthogonal. In a sense that, $\int H_i(x) H_j(x) w(x) dx = 0$ when $j \neq i$, where $w(x) = e^{x^2}$, $\int L_i(x) L_j(x) v(x) dx = 0$ when $j \neq i$, where $v(x) = e^{-x}.$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 25, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9317505359649658, "perplexity_flag": "head"} |
http://www.physicsforums.com/showthread.php?p=1928756 | Physics Forums
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## Control theory: Laplace versus state space representation
I'm taking a course in control theory, and have been wondering for a while what the benefits are when you describe a system based on the Laplace method with transfer functions, compared to when you use the state space representation method. In particular, when using the Laplace method you are limited to a system where the initial conditions all of them have to be equal to zero. The following questions then:
1) Is it correct that with the state space representation, the initial conditions could be whatever, i.e., you are not limitied to a system where they equal zero?
2) If so, why all this fuzz about using the Laplace method? Why not always use the state space representation?
I'm most eager for an answer on question number 1 above...
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Quote by Mårten I'm taking a course in control theory, and have been wondering for a while what the benefits are when you describe a system based on the Laplace method with transfer functions, compared to when you use the state space representation method. In particular, when using the Laplace method you are limited to a system where the initial conditions all of them have to be equal to zero.
i do not think that is true. just as when you use the Laplace Transform (which is what i think you mean by the "Laplace method") to represent a linear differential equation with initial conditions, the same applies to representing a linear, time-invariant system.
the main difference between the transfer function representation of a linear system and the state-space representation is that the former is concerned only with the input-output relationship while the latter is also concerned with what is going on inside the box. the state-space representation is more general and you can have all sorts of different state-space representations that appear to have the same input-output relationship.
a consequence of this is what happens with pole-zero cancellation. you might think that you have a nice 2nd order, stable linear system judging from the input-output relationship, but internally there might be an internal pole that is unstable but canceled by a zero sitting right on top of it. so before things inside blow up (and become non-linear), everything looks nice on the outside while things are going to hell on the inside. you find out that things aren't so nice on the inside when some internal state (that is not observable at the outside) hits the rails.
Quote by rbj i do not think that is true. just as when you use the Laplace Transform (which is what i think you mean by the "Laplace method") to represent a linear differential equation with initial conditions, the same applies to representing a linear, time-invariant system.
Hm... I don't quite understand...
For instance, when using the Laplace transform on the following equation, we get:
$$y'' + 4y' + 2y = u, y(0) = 0, y'(0) = 0,$$
$$s^2Y(s) + 4sY(s) + 2Y(s) = U(s),$$
$$Y(s) = \frac{1}{s^2+4s+2}U(s).$$
Now this simple result wouldn't be possible, if it wasn't for the initial conditions y(0) = 0 and y'(0) = 0.
But when we have a state space representation, like the following
$$X'(t) = AX(t) + BU(t),$$
$$Y(t) = CX(t) + DU(t),$$
$$X(t_0) = E,$$
then I thought it was possible to choose the initial conditions X(t_0) to what ever, e.g.
$$X(t_0) = E = \{x_1(0),x_2(0),x_3(0)\}^T = \{2,7,6\}^T,$$
so I am not limited to initial conditions where all x_i(0) = 0 like in the Laplace transform case. Or am I thinking erroneously here?
## Control theory: Laplace versus state space representation
Quote by Mårten Hm... I don't quite understand... For instance, when using the Laplace transform on the following equation, we get: $$y'' + 4y' + 2y = u, y(0) = 0, y'(0) = 0,$$ $$s^2Y(s) + 4sY(s) + 2Y(s) = U(s),$$ $$Y(s) = \frac{1}{s^2+4s+2}U(s).$$ Now this simple result wouldn't be possible, if it wasn't for the initial conditions y(0) = 0 and y'(0) = 0.
but this
$$s^2 Y(s) + 4s Y(s) + 2 Y(s) = U(s)$$
is not quite complete, in the general case. to go from
$$y'' + 4y' + 2y = u$$
it should be
$$(s^2 Y(s) - s y(0)) - y'(0)) + 4(s Y(s) - y(0)) + 2 Y(s) = U(s)$$
that's the "Laplace method".
Quote by rbj but this $$s^2 Y(s) + 4s Y(s) + 2 Y(s) = U(s)$$ is not quite complete, in the general case. to go from $$y'' + 4y' + 2y = u$$ it should be $$(s^2 Y(s) - s y(0)) - y'(0)) + 4(s Y(s) - y(0)) + 2 Y(s) = U(s)$$ that's the "Laplace method".
But, hm... In my control theory text book, it says that when using Laplace transforms in order to more easily handle ODEs, you always assume that $y^{(n)}(0)=0$ for all n. This in order to get the manageable form $Y(s) = G(s)U(s)$. All the calculations in my book assumes this. As I understood it, if you not assume this, the whole point with using Laplace transforms is lost, because then you cannot easily multiply the different boxes around the feedback loop for instance.
But what I haven't understood yet, is whether the state space representation method is limited to this as well or not (that all the initial values, each of them, are set to 0)? Are you instead allowed to use an initial value state vector X(t_0), as for instance, the one I used above in an earlier message?
Quote by Mårten But, hm... In my control theory text book, it says that when using Laplace transforms in order to more easily handle ODEs, you always assume that $y^{(n)}(0)=0$ for all n. This in order to get the manageable form $Y(s) = G(s)U(s)$. All the calculations in my book assumes this.
well, sure. otherwise you get $Y(s) = G(s)U(s)$ + some other stuff.
As I understood it, if you not assume this, the whole point with using Laplace transforms is lost, because then you cannot easily multiply the different boxes around the feedback loop for instance. But what I haven't understood yet, is whether the state space representation method is limited to this as well or not (that all the initial values, each of them, are set to 0)? Are you instead allowed to use an initial value state vector X(t_0), as for instance, the one I used above in an earlier message?
sure, if you want to represent the (transformed) output as only the transfer function times a single input, then you have to assume that the system is completely "relaxed" at time 0. but you can model a system in terms of only the input and output (and their initial conditions) with the Laplace Transforms of the input and output. but it won't be a simple transfer function.
the main difference between this and the state-space model is that the state-space model is trying to model what is going on inside the box. it is more general than the simple input-output description of the system.
Quote by rbj but you can model a system in terms of only the input and output (and their initial conditions) with the Laplace Transforms of the input and output. but it won't be a simple transfer function.
Can you make some kind of distinction between simple and complicated transfer function?
Quote by misgfool Can you make some kind of distinction between simple and complicated transfer function?
in the present context,
$$(s^2 Y(s) - s y(0)) - y'(0)) + 4(s Y(s) - y(0)) + 2 Y(s) = U(s)$$
i meant what would happen if you solved for Y(s). it is not only a function of U(s), but also a function of the two initial conditions.
$$Y(s) = \frac{1}{s^2 + 4s + 2}U(s) \ + \ \frac{s+4}{s^2 + 4s + 2} y(0) \ + \ \frac{1}{s^2 + 4s + 2} y'(0)$$
the simple one (assuming a completely relaxed system at t=0) would be
$$Y(s) = \frac{1}{s^2 + 4s + 2}U(s)$$
or
$$\frac{Y(s)}{U(s)} \equiv G(s) = \frac{1}{s^2 + 4s + 2}$$
Quote by rbj the main difference between this and the state-space model is that the state-space model is trying to model what is going on inside the box. it is more general than the simple input-output description of the system.
Okey, I think I got that, it also models what happens inside the box as you say.
But, still, I haven't got any answer on the question about the limitations on the initial values in the state space representations. There are no such limitations in the state space representation, are there? You can choose the initial values to whatever you like, can you?
Quote by Mårten But, still, I haven't got any answer on the question about the limitations on the initial values in the state space representations. There are no such limitations in the state space representation, are there? You can choose the initial values to whatever you like, can you?
and you can't with the input-output model?
with either model, you can put in whatever initial values you want. but the input-output model only allows you to put in initial values for the output(s) and the various derivatives of the output (up to the order of the system). since the input-output model doesn't even think about the internal states, then i guess you can't choose initial values of internal states.
Quote by rbj with either model, you can put in whatever initial values you want. but the input-output model only allows you to put in initial values for the output(s) and the various derivatives of the output (up to the order of the system). since the input-output model doesn't even think about the internal states, then i guess you can't choose initial values of internal states.
Okey, I got it!
Next thing then is: With the state space model, you have initial values for the internal states, that is X(t_0). But what about for the output, Y(t)? There's no need for initial values there? Sorry, if this is obvious...
Quote by Mårten With the state space model, you have initial values for the internal states, that is x(t_0). But what about for the output, y(t)? There's no need for initial values there?
try to leave the variables with caps for transformed signals and small case for time-domain. y(t) not Y(t) and Y(s) (or in discrete-time Z Transform, Y(z)). it just keeps things clear.
those initial conditions (for y(t)) are fully determined by the initial conditions for the states in X(t). they cannot be independently specified. and with the state-space model, with all of those states (an equal number of states to the order of the system), you need not and will not have initial conditions for higher derivatives (like you did for y'(0)). just having initial conditions for every element in the X(0) vector is good enough.
Before we go on - thank you very much for this personal class I get here in control theory! It helps me a lot!
Quote by rbj try to leave the variables with caps for transformed signals and small case for time-domain. y(t) not Y(t) and Y(s) (or in discrete-time Z Transform, Y(z)). it just keeps things clear.
Okey, sorry. That was to signal that they could be matrices or vectors. Bold face may be better...
Quote by rbj those initial conditions (for y(t)) are fully determined by the initial conditions for the states in X(t). they cannot be independently specified. and with the state-space model, with all of those states (an equal number of states to the order of the system), you need not and will not have initial conditions for higher derivatives (like you did for y'(0)). just having initial conditions for every element in the X(0) vector is good enough.
Okey, I think I got it there. y(0) would just have been a transformation (made by the C-matrix) of the vector x(0).
Then comes the ultimate question: Why don't always use the state-space model in preference to the Laplace transform model? What benefits does the Laplace transform model have, which the state-space model doesn't have?
Quote by Mårten Before we go on - thank you very much for this personal class I get here in control theory! It helps me a lot! Okey, sorry. That was to signal that they could be matrices or vectors. Bold face may be better... Okey, I think I got it there. y(0) would just have been a transformation (made by the C-matrix) of the vector x(0).
yeah, and it would be nice if i practice what i preach. i should have said x(t) for the state vector instead of X(t).
Then comes the ultimate question: Why don't always use the state-space model in preference to the Laplace transform model? What benefits does the Laplace transform model have, which the state-space model doesn't have?
simplicity. if a system is known to be linear and time-invariant, and if you don't care about what's going on inside the black box, but only on how the system interacts (via its inputs and outputs) with the rest of the world that it is connected to, the input-output transfer function description is all that you need. if you have an Nth-order, single-input, single-output system, then 2N+1 numbers fully describe your system, from an input-output POV. with the state variable description, an Nth-order system (single input and output) has N2 numbers, just for the A matrix. and 2N+1 numbers for the B, C, and D matrices. so there are many different state-variable systems that have the same transfer function. all of these different systems behave identically with the outside world until some internal state saturates or goes non-linear and that's where they are modeled differently in practice. you can even have internal state(s) blow up and not even know it (if the system is "not completely observable") until the state(s) that are unstable hit the "rails", the maximum values they can take before clipping or some other non-linearity. when something like this happens, there is pole-zero cancellation, as far as the input-output description is concerned. so maybe some zero killed the pole and they both disappeared in the transfer function, G(s), but inside that bad pole still exists.
Sorry for late reply, been away for a while...
Quote by rbj simplicity. if a system is known to be linear and time-invariant, and if you don't care about what's going on inside the black box, but only on how the system interacts (via its inputs and outputs) with the rest of the world that it is connected to, the input-output transfer function description is all that you need. if you have an Nth-order, single-input, single-output system, then 2N+1 numbers fully describe your system, from an input-output POV. with the state variable description, an Nth-order system (single input and output) has N2 numbers, just for the A matrix. and 2N+1 numbers for the B, C, and D matrices
Hm... Still something here that confuses me. Imagine that you have a system described by the ODE $y'' + 4y' + 2y = u$. It's possible to describe this system with a transfer function $G(s) = \frac{1}{s^2+4s+2}$. It's also possible to describe this system with a state space representation, constructing the A-matrix and so on. Then the following obscurities:
1) With the state space representation the system is described by N2 numbers in the A-matrix. But most of the numbers in the A-matrix are just zeros and ones. So it seems that the number of significant numbers in the A-matrix are just 2N+1. So then the state space representation is not so much more complicated?
2) If a system is described by this ODE above, how could there be any internal information that is being revealed if putting up this system on a state space representation, compared to when using the transfer function to describe it?
Quote by Mårten Sorry for late reply, been away for a while...
Hm... Still something here that confuses me. Imagine that you have a system described by the ODE $y'' + 4y' + 2y = u$. It's possible to describe this system with a transfer function $G(s) = \frac{1}{s^2+4s+2}$. It's also possible to describe this system with a state space representation, constructing the A-matrix and so on.
this is true, but while there is really only one G(s) that accurately represents what the ODE says, there are many (and infinite number) of state space representations that will have the same transfer function, G(s). some are completely controllable, some are completely observable, some are both, some are neither completely controllable nor completely observable.
Then the following obscurities: 1) With the state space representation the system is described by N2 numbers in the A-matrix. But most of the numbers in the A-matrix are just zeros and ones. So it seems that the number of significant numbers in the A-matrix are just 2N+1. So then the state space representation is not so much more complicated?
it's that there are many different state space representations (all with different A, B, and C matrices, but i think the D matrix is the same) that have the same effect as far as input and output are concerned. put these different state-variable systems (with identical transfer functions) in a box and draw out the inputs and outputs to the rest of the world and the rest of the world could not tell the difference between them. at least as long as they stayed linear inside.
2) If a system is described by this ODE above, how could there be any internal information that is being revealed if putting up this system on a state space representation, compared to when using the transfer function to describe it?
if it stays linear inside and you put it in a black box, you can't tell everything about the internal structure. but if some internal state goes non-linear (perhaps because this internal state blew up because it was unstable), then you can tell something is different on the outside. if the state-variable system is completely observable and you have information about the internal structure (essentially the A, B, C, D matrices), you can determine (or "reveal") what the states must be from looking only at the output (and its derivatives). if it's completely controllable, you can manipulate the input so that the states take on whatever values you want.
this "controllable" and "observable" stuff is in sorta advanced control theory (maybe grad level, even though i first heard about it in undergrad when i learned about the state space representation). i dunno if i can give it justice in this forum. i would certainly have to dig out my textbooks and re-learn some of this again so i don't lead you astray with bu11sh1t. so far, i'm pretty sure i'm accurate about what i said, but i don't remember everything else about the topic.
Okey, I think I understand now. Correct me then if something of the following is wrong. The expression of G(s) as I was talking about above, could actually be the result of a lot of boxes so to say. That is, G(s) is a black box, and inside that black box there could be several other boxes with signals going in and out, and even feedback loops. Actually, the number of configurations inside the black box, are infinite. That corresponds to the infinite number of state space representations for G(s). But, let's now say that we know exactly what happens inside the black box. E.g., the black box is the ODE describing a bathtub with water coming in, and water going out. Let's say for simplicity, even though this is probably not physically correct, that this bathtub is described by $G(s) = \frac{1}{s^2+4s+2}$. Then, this bathtub system could just have one state space representation, since we know the system perfectly inside. Is that correct? (I here disregard the fact that you could manipulate the rows in the A-matrix, and make different matrix-operations, but it's still really the same equivalent matrix.) So in this case, I could as well solve the matrix equation, as solving the Laplace equation, it is as simple. Except that if I do it the matrix way (i.e. the state space representation way), I could have whatever initial values, it won't do it more complicated, as it would do in the Laplace way. Is that correct?
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http://mathoverflow.net/revisions/4659/list | ## Return to Answer
2 took out comment about TeX difficulty
Brian Conrad has a handout (pdf) in which he talks about tensorial maps. In it he notes that one should construct maps independently of bases, but that in order to prove the properties of such maps it made sense to choose bases or spanning sets.
I think this is generally applicable: it seems to me that picking bases should be the last part of your work on a problem, and that it mostly comes in at the level of computation. Bases provide a useful structure to a vector space that enables one to start somewhere, and proofs can be easier to do with them. But if you choose a basis too early on, you have to carry it around for the whole problem, and you might have to show how it transforms. Perhaps you could come up with ideas and proofs using bases, and then edit them to show what's really going on at the level of maps?
In class we recently constructed the determinant of a linear transformation $T:V\rightarrow V$ over and $n$-dimensional vector space V, and to do this we defined the exterior power and used the fact that $T$ became multiplication by a scalar in $\wedge^n(V)$. To be sure, we showed that given a basis $v_1, ...v_n$ of $V$, one would make the single basis element $v_1\wedge\ldots\wedge v_n$ of $\wedge^n(V)$, and used this to give the combinatorial formula for determinant. But the properties of determinant are invariant under change of basis, so we didn't prove them using a basis.
[[apologies for any sloppiness as TeX doesn't seem to be working quite right]]
1
Brian Conrad has a handout (pdf) in which he talks about tensorial maps. In it he notes that one should construct maps independently of bases, but that in order to prove the properties of such maps it made sense to choose bases or spanning sets.
I think this is generally applicable: it seems to me that picking bases should be the last part of your work on a problem, and that it mostly comes in at the level of computation. Bases provide a useful structure to a vector space that enables one to start somewhere, and proofs can be easier to do with them. But if you choose a basis too early on, you have to carry it around for the whole problem, and you might have to show how it transforms. Perhaps you could come up with ideas and proofs using bases, and then edit them to show what's really going on at the level of maps?
In class we recently constructed the determinant of a linear transformation $T:V\rightarrow V$ over and $n$-dimensional vector space V, and to do this we defined the exterior power and used the fact that $T$ became multiplication by a scalar in $\wedge^n(V)$. To be sure, we showed that given a basis $v_1, ...v_n$ of $V$, one would make the single basis element $v_1\wedge\ldots\wedge v_n$ of $\wedge^n(V)$, and used this to give the combinatorial formula for determinant. But the properties of determinant are invariant under change of basis, so we didn't prove them using a basis.
[[apologies for any sloppiness as TeX doesn't seem to be working quite right]] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9595567584037781, "perplexity_flag": "head"} |
http://mathoverflow.net/revisions/69124/list | ## Return to Answer
2 added 37 characters in body
To add to D. Savitt's comment: Given $g \in G$ ($G$ a finite group), the element $\rho(g) \in GL(V)$ is diagonalizable over the complex numbers since its minimal polynomial is separable. More specifically, it is a divisor of $x^n - 1$ for some $n$ -- once you have the minimal polynomial, you can compute the projection operators to the various eigenspaces as polynomials in $\rho(g)$, but generally this will require complex coefficients and working in a field which is not algebraically complete will only allow you to project to the kernel of irreducible factors.
Now, a diagonal matrix is determined up to conjugacy by its eigenvalues. And as was pointed out $\mbox{tr } \rho(g^k)$ gives the sum of $k$th powers of the eigenvalues. From these numbers, one can actually recover the set of eigenvalues of the matrix. You can prove this with symmetric functions, but you can also view the spectrum as a measure, and view $\mbox{tr } \rho(g^k)$ as the k'th (complex) moment / Fourier coefficient of the measure. These numbers, together with their complex conjugates give all the Fourier coefficients. They are periodic, so you do not need all of them (you could apply Stone Weierstrass, but that would be very wasteful). We already know the finitely many (say, $n$) candidate eigenvalues from the fact that they satisfy $z^n - 1 = 0$, so we can easily design a polynomial which vanishes on all but one of the eigenvalues and hence determine the multiplicity of each eigenvalue. Namely: $f(z) = \prod_{i \neq j} (z - \lambda_i)$. Then $\mbox{tr } f( \rho(g) )$ (the trace of the projection operator up to a constant) tells you the multiplicity of $\lambda_j$. (Is this exactly the proof through symmetric functions? If not, what is the relationship?)
Inspecting the above argument, it does not seem like the complex numbers play too large a role. But even for a more general unitary representation, the eigenvalues live on a circle and we can view them as a measure $\mu$. We can still obtain $\int z^k d\mu$ for all integers $k$. By Stone-Weierstrass or Fourier inversion, we get the whole spectrum in this way.
This is the extent to which I've thought about the above things so I'd be really happy to know if anyone can say more about it or give more points of view. For example, what very critical type of information do you miss out on by only considering one cyclic subgroup at a time? What about other unitary representations, possibly infinite groups or possibly infinite dimensional? (Should I start a separate thread?)
1
To add to D. Savitt's comment: Given $g \in G$ ($G$ a finite group), the element $\rho(g) \in GL(V)$ is diagonalizable over the complex numbers since its minimal polynomial is separable. More specifically, it is a divisor of $x^n - 1$ for some $n$ -- once you have the minimal polynomial, you can compute the projection operators to the various eigenspaces as polynomials in $\rho(g)$, but generally this will require complex coefficients and working in a field which is not algebraically complete will only allow you to project to the kernel of irreducible factors.
Now, a diagonal matrix is determined up to conjugacy by its eigenvalues. And as was pointed out $\mbox{tr } \rho(g^k)$ gives the sum of $k$th powers of the eigenvalues. From these numbers, one can actually recover the set of eigenvalues of the matrix. You can prove this with symmetric functions, but you can also view the spectrum as a measure, and view $\mbox{tr } \rho(g^k)$ as the k'th (complex) moment / Fourier coefficient of the measure. These numbers, together with their complex conjugates give all the Fourier coefficients. They are periodic, so you do not need all of them (you could apply Stone Weierstrass, but that would be very wasteful). We already know the finitely many (say, $n$) candidate eigenvalues from the fact that they satisfy $z^n - 1 = 0$, so we can easily design a polynomial which vanishes on all but one of the eigenvalues and hence determine the multiplicity of each eigenvalue. Namely: $f(z) = \prod_{i \neq j} (z - \lambda_i)$. Then $\mbox{tr } f( \rho(g) )$ (the trace of the projection operator up to a constant) tells you the multiplicity of $\lambda_j$. (Is this exactly the proof through symmetric functions? If not, what is the relationship?)
Inspecting the above argument, it does not seem like the complex numbers play too large a role. But even for a more general unitary representation, the eigenvalues live on a circle and we can view them as a measure $\mu$. We can still obtain $\int z^k d\mu$ for all integers $k$. By Stone-Weierstrass or Fourier inversion, we get the whole spectrum in this way.
This is the extent to which I've thought about the above things so I'd be really happy to know if anyone can say more about it or give more points of view. For example, what very critical type of information do you miss out on by only considering one cyclic subgroup at a time? What about other unitary representations, possibly infinite groups or possibly infinite dimensional? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 34, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9383366703987122, "perplexity_flag": "head"} |
http://en.m.wikipedia.org/wiki/Admissible_representation | # Admissible representation
In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra.
## Real or complex reductive Lie groups
Let G be a connected reductive (real or complex) Lie group. Let K be a maximal compact subgroup. A continuous representation (π, V) of G on a complex Hilbert space V[1] is called admissible if π restricted to K is unitary and each irreducible unitary representation of K occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of G.
An admissible representation π induces a $(\mathfrak{g},K)$-module which is easier to deal with as it is an algebraic object. Two admissible representations are said to be infinitesimally equivalent if their associated $(\mathfrak{g},K)$-modules are isomorphic. Though for general admissible representations, this notion is different than the usual equivalence, it is an important result that the two notions of equivalence agree for unitary (admissible) representations. Additionally, there is a notion of unitarity of $(\mathfrak{g},K)$-modules. This reduces the study of the equivalence classes of irreducible unitary representations of G to the study of infinitesimal equivalence classes of admissible representations and the determination of which of these classes are infinitesimally unitary. The problem of parameterizing the infinitesimal equivalence classes of admissible representations was fully solved by Robert Langlands and is called the Langlands classification.
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## Totally disconnected groups
Let G be a locally compact totally disconnected group (such as a reductive algebraic group over a local field or over the finite adeles of a global field). A representation (π, V) of G on a complex vector space V is called smooth if the subgroup of G fixing any vector of V is open. If, in addition, the space of vectors fixed by any compact open subgroup is finite dimensional then π is called admissible. Admissible representations of p-adic groups admit more algebraic description through the action of the Hecke algebra of locally constant functions on G.
Deep studies of admissible representations of p-adic reductive groups were undertaken by Casselman and by Bernstein and Zelevinsky in the 1970s. Much progress has been made more recently by Howe and Moy and Bushnell and Kutzko, who developed a theory of types and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.
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## Notes
1. I.e. a homomorphism π : G → GL(V) (where GL(V) is the group of bounded linear operators on V whose inverse is also bounded and linear) such that the associated map G × V → V is continuous.
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## References
• Bushnell, Colin J.; Henniart, Guy (2006), The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 335, Berlin, New York: Springer-Verlag, doi:10.1007/3-540-31511-X, ISBN 978-3-540-31486-8, MR 2234120
• Bushnell, Colin J.; Philip C. Kutzko (1993). The admissible dual of GL(N) via compact open subgroups. Annals of Mathematics Studies 129. Princeton University Press. ISBN 0-691-02114-7.
• Chapter VIII of Knapp, Anthony W. (2001). Representation Theory of Semisimple Groups: An Overview Based on Examples. Princeton University Press. ISBN 0-691-09089-0.
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## Free particle spectrum?
Err......no.
For simplicity, let's consider the total absorption of a photon by an electron in the lab frame, where the electron is initially at rest. For a photon of energy Eγ = pc and momentum p = Eγ/c , the final total energy of the recoiling electron would be
Ee= m0c2 + Eγ.
But the momentum of the recoiling electron would be p = Eγ/c,
leading to a total energy of
[Ee]2= [m0c2]2 + [pc]2= [m0c2]2 + [Eγ]2
These two equations are irreconcilably different, because energy and momentum cannot both be conserved..
Bob S
Is the electron a composite particle?
No, not yet at least. See the LBL Particle Data Group listing: http://pdg.lbl.gov/2002/s003.pdf Bob S
So, that's the essential difference between your and my derivation. The electron's rest mass cannot change, since it is truly an elementary particle. However, composite particles (such as atoms, molecules, nuclei and even hadrons) have a complicated internal structure described by so called "internal degrees of freedom". The rest mass of a bound system is always smaller than the sum of the rest masses of its constituents, the difference being called mass defect $\Delta m$ and being connected to a quantity called binding energy of the system $B = \Delta m c^{2}$. This is why the whole energy of the photon can be absorbed by the composite particle. In order that we compensate for the momentum of the particle (at any given energy, massless particle have the highest possible momentum, so this is the "worst case scenario"), we better give the composite particle an equal by magnitude and opposite in direction momentum and, thus even more energy is available. Now comes the paradox. The whole amount of the photon energy + kinetic energy of the particle are supposed to increase the rest energy of the particle. This is only possible if the rest mass of the particle decreases. If you remember, the rest mass was smaller than the sum of the rest masses of the constituents and the difference (by definition positive) was called mass defect. Increasing the rest mass of the particle is equivalent to decreasing the mass defect, which, in turn is directly proportional to the binding energy of the system. We can only do this until the binding energy becomes zero. Then, the composite system becomes unbounded and it disintegrates into its constituent parts.
(Bob S and Dickfore):Wow. But light does interact with an electron. It does get polarized-angular momentum gets exchanged.Why can't the electron gain mass along with the energy, or lose angular momentum, at least temporarily.
Quote by george simpso (Bob S and Dickfore):Wow. But light does interact with an electron. It does get polarized-angular momentum gets exchanged.Why can't the electron gain mass along with the energy, or lose angular momentum, at least temporarily.
Please provide evidence for a process where a free electron absorbs a photon.
Dickfore: Your request puts me back to where I was when I logged in to "free particle spectrum" For starters check out http://farside.ph.utexas.edu/teachin...s//node85.html. Next google "free electron laser" and "Absorption spectra of electrons in plasmas". There are a lot of experiments and calculation supporting absorption of a photon by an electron. For what it's worth, google "free electro absorb a photon" you'll get >200,000 responces. But, query "free electron cannot absorb a photon", and ther are ~100,000.( more data/opinions pro than con ). I think the evidence supports free electron absorption of a photon, but perhaps the collective physics wisdom does not. So, what do you come up with?
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Quote by Dickfore One can view this increase in rest mass as excitation of the intrinsic degrees of freedom of the particle by an energy $\Delta \epsilon = \Delta M \, c^{2}$. We see that this dependence is a continuous function, so the absorbtion spectrum of the photon depends only on the excitation spectrum of the particle. Notice that the translational degrees of freedom where not limited and, therefore, they are not quantized.
Not sure what your point is exactly, but these excitations of intrinsic degrees of freedom can not include excitations of translational degrees of the composite particle, since that would violate the requirement that the final particle momentum be zero in this frame.
So you are only allowed to excite the other degrees of freedom: rotational, vibrational, and interaction terms (which are all quantized).
Quote by george simpso ... "Absorption spectra of electrons in plasmas". There are a lot of experiments and calculation supporting absorption of a photon by an electron. ...
Electrons in plasmas are free? Google "collective modes in plasmas"
Quote by Gokul43201 Not sure what your point is exactly, but these excitations of intrinsic degrees of freedom can not include excitations of translational degrees of the composite particle, since that would violate the requirement that the final particle momentum be zero in this frame. So you are only allowed to excite the other degrees of freedom: rotational, vibrational, and interaction terms (which are all quantized).
Exactly what I was saying and those contribute to the rest mass of a composite object.
Recognitions: Science Advisor Free electrons can absorb EM radiation via inverse Bremsstrahlung absorption, the catch being that it can only occur in the vicinity of atoms. Claude.
Quote by Claude Bile Free electrons can absorb EM radiation via inverse Bremsstrahlung absorption, the catch being that it can only occur in the vicinity of atoms. Claude.
So, how are they free if they are in the vicinity of atoms?
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Quote by Dickfore So, how are they free if they are in the vicinity of atoms?
Free as in not bound to a nucleus. Conduction band electrons in a condensed dielectric for example.
Claude.
Quote by Claude Bile Free as in not bound to a nucleus. Conduction band electrons in a condensed dielectric for example. Claude.
What is the meaning of this 'band' you are using?
(Posted by Dickfore:)
So, how are they free if they are in the vicinity of atoms?
Quote by Claude Bile Free as in not bound to a nucleus. Conduction band electrons in a condensed dielectric for example.
The difference between the photoelectric effect, in which the photon's total energy is absorbed by the electron, and Thomson scattering on free electrons, is that a little recoil momentum is absorbed by the recoiling atom or atomic lattice. Conduction electrons are not free electrons; the work function to remove a conduction electron is >= 3 eV.
Bob S
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http://mathoverflow.net/questions/19131?sort=newest | ## Rational numbers as an extension of the field with one element?
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Greetings. I would love to have a field $\mathbb F$ which is a subfield of the field of rational numbers $\mathbb Q$, and such that the Galois group $Gal (\mathbb Q / \mathbb F)$ has preferably infinitely many elements.
While there is no such field $\mathbb F$, since $\mathbb Q$ has no proper subfields at all, I've recently heard of this field $\mathbb F_1$ with one element concept.
As far as I understand there is no definition which would be set in stone for this object, at least not yet. My question to those who know the subject: does any of the currently studied definitions of $\mathbb F_1$ allow for realization of $\mathbb F_1$ as a "subfield" of $\mathbb Q$ in some sense?
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## 1 Answer
Imo the best theory today for the field with one element is Borger's proposal to consider Lambda-rings and use their Lambda-structure as a substitute for descent from the integers (or rationals) to the field F1 with one element.
Some examples of this philosophy are contained in the nice short paper by Borger and Bart de Smit (arXiv:0801.2352) 'Galois theory and integral models of Lambda-rings'.
Lambda-rings finite etale over the rationals Q are finite discrete sets equipped with a continuous action of the monoid Gal(Qbar/Q) x N' where N' are the positive integers under multiplication. This suggest that the Galois monoid Gal(Qbar/F1) = Gal(Qbar/Q) x N'.
Likewise, Lambda-rings over Q having an integral Lambda-model correspond to finite sets with a continuous action of the monoid Zhat, that is the set of profinite integers as a topological monoid under multiplication. This suggests that the absolute Galois monoid of F1, that is Gal(F1bar/F1) = Zhat.
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Thanks for the kind words. I'd just like to emphasize (not to suggest that you meant or said otherwise) that these analogies ($\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{F}_1)$ and so on) should be taken with a grain of salt. For instance, it is good to think of the functor $\mathbf{Q}\otimes -$ from $\Lambda$-rings to $\mathbf{Q}$-algebras as the functor $\mathbf{Q}\otimes_{\mathbf{F}_1} -$, but this functor does not have the descent property, which is what Galois theory is really used for. So while it is reasonable to call $\mathbf{Q}$ an $\mathbf{F}_1$-algebra, it's a bit abusive to speak of... – James Borger Mar 24 2010 at 1:41
the relative Galois theory. This is in contrast to usual field theory, where for any extension $L/K$, the functor $L\otimes_K -$ from $K$-algebras to $L$-algebras always has the descent property; but it is similar to general ring theory where, for instance, the functor $\mathbf{Q}\otimes_{\mathbf{Z}}-$ does not have the descent property. The moral of the story is that, from the $\Lambda$-ring point of view, $\mathbf{F}_1$ fails to have lots of properties that usual fields (as opposed to rings) have. So it's a bit safer to think of this $\mathbf{F}_1$ as a generalized ring, and so... – James Borger Mar 24 2010 at 1:46
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at times field-specific concepts in descent theory do not translate so well. In retrospect, this is not so surprising, because the importance of fields, as opposed to rings, is mostly of a psychological or historical nature (IMHO). – James Borger Mar 24 2010 at 1:52 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 24, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9524713754653931, "perplexity_flag": "head"} |
http://math.stackexchange.com/questions/2093/choosing-subsets-of-a-set-with-a-specified-amount-of-maximum-overlap-between-the/2323 | # Choosing subsets of a set with a specified amount of maximum overlap between them
How can I determine the size of the largest collection of k-element subsets of an n-element set such that each pair of subsets has at most m elements in common?
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The question is slightly confusing: I believe you mean something like "what is the largest collection of k-element subsets of an n-element set such that each pair of subsets has at most m elements in common?" but I might be wrong. – Qiaochu Yuan Aug 10 '10 at 21:54
Yes, that's exactly what I meant! Thanks! – Paul Reiners Aug 10 '10 at 21:57
## 2 Answers
I think this problem is still open, but the following might be useful:
Ray-Chaudhuri-Wilson Theorem:
Let $L$ be a set of $m$ integers and $F$ be an $L$-intersecting $k$-uniform family of subsets of a set of $n$ elements, where $m \le k$, then
$|F| \le {n \choose m}$
$\bullet$
$k$-uniform family is a set of subsets, each subset being of size k.
An $L$-intersecting family is such that the intersection size of any two distinct sets in the family is in $L$.
The following result of Frankl gives us a lower bound
Frankl's Result:
For every $k \ge m \ge 1$ and $n \ge 2k^{2}$ there exists a $k$-uniform family $F$ of size $> (\frac{n}{2k})^{m}$ on $n$ points such that $|A \cap B| \le m-1$ for any two distinct sets $A,B \in F$.
$\bullet$
For an algorithm for constructing such sets (based on Frankl's result) refer: http://stackoverflow.com/questions/2955318/creating-combinations-that-have-no-more-one-intersecting-element/2955527#2955527
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http://math.stackexchange.com/questions/256906/does-factor-wise-continuity-imply-continuity | # Does factor-wise continuity imply continuity?
Let $f$ denote a map from a product space $X \times Y$ to $Z$. If for every $x\in X$, the map $f(x,-)$ is continuous, and the same holds for every $y \in Y$, then is $f$ continuous in general? If not, is there any condition to be imposed to make $f$ continuous?
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A categorical reason why you would not believe this statement holds is that you are asking of a product to satisfy a coproduct property. This is why at first glance it must feel false. The product behaves well when you project it on its components, i.e. $X \times Y \to X$ ; the coproduct behaves well when you inject its components in, i.e. $X \to X \times Y$. Notice that the maps $f(x,-)$ are nothing more than composing $\pi_Y : Y \to X \times Y$ defined by $\pi_Y(y) = (x,y)$ with $f$, i.e. $f(x,-) = f \circ \pi_Y$. – Patrick Da Silva Dec 12 '12 at 6:48
## 1 Answer
No, separate continuity does not in general imply joint continuity. The function
$$f:\Bbb R^2\to\Bbb R:\langle x,y\rangle\mapsto\begin{cases}\frac{xy}{x^2+y^2}&,\text{if }\langle x,y\rangle\ne\langle0,0\rangle\\\\0,&\text{if }\langle x,y\rangle=\langle 0,0\rangle\end{cases}$$
is continuous everywhere except at the origin. At the origin it is continuous in each variable separately, but it is not continuous.
Added: Useful keywords on which to search are separate continuity and joint continuity. Z. Piotrowski, a former colleague of mine, did quite a lot of work in this area; you’ll find many of his papers here, and in them both results and further references. Be warned, though, that the links don’t always match the text: the link to the comprehensive (if now dated) survey Separate and joint continuity is actually at number $17$, not at number $19$. As I recall, there are more results giving conditions under which the set of points of continuity contains a dense $G_\delta$-set in $X\times Y$ than there are giving conditions that guarantee that a separately continuous function is jointly continuous.
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Curiosity : Why did you use $\langle x,y \rangle$ instead of $(x,y)$ for the coordinates? This doesn't feel natural, I've never seen that before. – Patrick Da Silva Dec 12 '12 at 6:41
Well, in set theory, bivariate functions are actually functions from ordered pairs to the result set, i.e. $f(x,y) := f(\langle x,y\rangle)$. – Mario Carneiro Dec 12 '12 at 6:44
@Mario Carneiro : You are suggesting that $\langle x,y \rangle$ is a set-theoretic notation for the ordered pair? I know ordered pairs are defined something like $\langle x,y \rangle = \{x, \{x,y \} \}$ so I understand the need for a notation but just let me know if I am wrong. – Patrick Da Silva Dec 12 '12 at 6:50
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@Patrick: Yes, the notation comes from set theory. I habitually use it, partly because a lot of my background is in set theory, and partly simply because I prefer it: ordinary parentheses have enough work to do! – Brian M. Scott Dec 12 '12 at 6:51
I'm pretty sure $\langle x,y\rangle$ is the standard notation for ordered pairs in general. The definition of the ordered pair in set theory is $\langle x,y\rangle := \{\{x\},\{x,y\}\}$, which is what I think you are thinking of. – Mario Carneiro Dec 12 '12 at 6:52
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http://math.stackexchange.com/questions/162622/inconjugate-maximal-subgroups-of-a-soluble-group | # Inconjugate maximal subgroups of a soluble group
I am looking for a proof of the following proposition:
If $M_1,M_2$ are inconjugate maximal subgroups of the finite and soluble group $G$, then $M_1\cap M_2$ is maximal in at least one of $M_1$ or $M_2$.
I know there is a proof of this fact in Doerk and Hawkes' "Finite Soluble Groups", but I can't access it, at least for the time being. I was wondering if anyone knows of some other source where a proof is provided.
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## 1 Answer
I couldn't find another reference, but here are Theorem 16.6 and its proof from Doerk's book:
Theorem:
Let $L$ and $M$ be inconjugate maximal subgroups of a finite soluble group $G$. If $M^G\not\leq L^G$, then $L\cap M$ is a maximal subgroup of $L$.
Proof:
Let $R=$ Core$_G(M)$ and $S/R=$ Soc$(G/R)$. The hypothesis implies that $R\not\leq$ Core$_G(L)$ and therefore that $LR=G$ $(*)$.
Since $G/R$ is primitive, $S/R$ is a chief factor of $G$, and since $R$ centralizes $S/R$, it follows from $(*)$ that $S/R$ is $L$-irreducible.
From $(*)$ we also have that $S=S\cap LR=(S\cap L)R$, whence $S/R=(S\cap L)R/R\ \ \stackrel{\cong}{\tiny{L}}\ \ (S\cap L)/(R\cap L)$, and therefore $(S\cap L)/(R\cap L)$ is a chief factor of $L$.
Now $M$ complements $S/R$ in $G$ and $LM=G$.
Hence, $|L:L\cap M|\geq |(S\cap L)(L\cap M):L\cap M|=|S\cap L :R\cap L|=|S:R|=|G:M|=|LM:M|=|L:L\cap M|$.
Consequently $(S\cap L)(L\cap M)=L$, and $L\cap M$ complements the chief factor $(S\cap L)/(R\cap L)$ in $L$. Therefore, $L\cap M$ is a max. subgroup of $L$.$\ \ \$ QED
Since $\leq$ is a partial order, your mentioned theorem follows.
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Thanks, although I can't say I fully understand this proof. – Stefanos Jun 25 '12 at 0:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 45, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9432160258293152, "perplexity_flag": "head"} |
http://mathoverflow.net/questions/104777?sort=oldest | ## What are the algebras for the double dualization monad?
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Let $k$ be a field, and let $\mathbf{Vect}$ denote the category of vector spaces (possibly infinite-dimensional) over $k$. Taking duals gives a functor $(\ )^*\colon \mathbf{Vect}^{\mathrm{op}} \to \mathbf{Vect}$.
This contravariant functor is self-adjoint on the right, since a linear map `$X \to Y^*$` amounts to a bilinear map $X \times Y \to k$, which is essentially the same thing as a bilinear map $Y \times X \to k$, which amounts to a linear map `$Y \to X^*$`. It therefore induces a monad $(\ )^{**}$ on $\mathbf{Vect}$.
What are the algebras for this monad?
Remarks
1. I assume this is known (probably since a long time ago).
2. The first paper I came across when searching for the answer was Anders Kock, On double dualization monads, Math. Scand. 27 (1970), 151-165. I'm pretty sure it doesn't contain the answer explicitly, but it's possible that it contains some results that would help.
3. The monad isn't idempotent (that is, the multiplication part of the monad isn't an isomorphism). Indeed, take any infinite-dimensional vector space $X$. Write our monad as $(T, \eta, \mu)$. If $\mu_X$ were an isomorphism then $\eta_{TX}$ would be an isomorphism, since $\mu_X \circ \eta_{TX} = 1$. But $\eta_{TX}$ is the canonical embedding $TX \to (TX)^{**}$, and this is not surjective since $TX$ is not finite-dimensional.
4. There's another way in which the answer might be somewhat trivial, and that's if `$(\ )^*$` is monadic. But it doesn't seem obvious to me that `$(\ )^*$` even reflects isomorphisms (which it would have to if it were monadic).
5. There's a sense in which answering this question amounts to completing the analogy:
sets are to compact Hausdorff spaces as vector spaces are to ?????
Indeed, the codensity monad of the inclusion functor (finite sets) $\hookrightarrow$ (sets) is the ultrafilter monad, whose algebras are the compact Hausdorff spaces. The codensity monad of the inclusion functor (finite-dimensional vector spaces) $\hookrightarrow$ (vector spaces) is the double dualization monad, whose algebras are... what? (Maybe this will help someone to guess what the answer is.)
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My gut feeling is "the dual vector spaces" but maybe this doesn't get all the algebras... – Yemon Choi Aug 15 at 18:13
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Yemon: dunno. The point about idempotent monads is this theorem: the only algebras for an idempotent monad are the free ones, which in this case are the spaces of the form $X^{**}$ (together with the canonical map $X^{****} \to X^{**}$). So if our monad was idempotent then you'd be right. It's not idempotent, but from that I can't deduce that you're wrong, because I don't know whether the converse of the aforementioned theorem is true. – Tom Leinster Aug 15 at 18:28
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By the way, when I said dual spaces, I meant $E^*$, equipped withe canonical (co-unit?) map $E^{***} \to E^*$ - IIRC this is the answer for the Banach space version of your question – Yemon Choi Aug 16 at 0:57
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Here's a sketch of a proof that the algebras are the dual vector spaces. Given an algebra $E^{**}\to E$, topologize $E$ using the quotient topology of the weak* topology on $E^{**}$, and let $F$ be the topological dual of $E$ with respect to this topology. Now you prove that actually $E=F^*$ with the canonical algebra structure. This is plausible since if $E=F^*$, this procedure recovers $F$ from $E$ and its algebra structure. – Eric Wofsey Aug 16 at 11:29
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For more information on linearly compact vector spaces from a categorical point of view see section 24 of the Bergman and Hausknecht book "Cogroups and Co-rings in Categories of Associative Rings". – Carl Futia Aug 16 at 17:29
show 8 more comments
## 2 Answers
Tom, I believe $(-)^\ast: \mathbf{Vect}^{op} \to \mathbf{Vect}$ is monadic, essentially because all objects in $\mathbf{Vect}$, in particular $k$ as a module over $k$ as ground field, are injective.
For instance, to check that $(-)^\ast$ reflects isomorphisms, suppose $f: V \to W$ is any linear map. We have two short exact sequences
$$0 \to \ker(f) \to V \to im(f) \to 0$$
$$0 \to im(f) \to W \to coker(f) \to 0$$
Because $k$ is injective, the functor $(-)^\ast = \hom(-, k)$ preserves short exact sequences:
$$0 \to im(f)^\ast \to V^\ast \to \ker(f)^\ast \to 0$$
$$0 \to coker(f)^\ast \to W^\ast \to im(f)^\ast \to 0$$
and if $f^\ast$, the composite $W^\ast \to im(f)^\ast \to V^\ast$, is an isomorphism, then $W^\ast \to im(f)^\ast$ is injective, which forces $coker(f)^\ast = 0$ and therefore $coker(f) = 0$. By a similar argument, $\ker(f) = 0$. Therefore $f$ is an isomorphism.
The remaining hypotheses of Beck's theorem (in the form given in Theorem 2, page 179, of Mac Lane-Moerdijk) are similarly easy to check. Obviously $\mathbf{Vect}^{op}$ has coequalizers of reflexive pairs since $\mathbf{Vect}$ has equalizers. And $(-)^\ast: \mathbf{Vect}^{op} \to \mathbf{Vect}$ (which has a left adjoint, as pointed out) preserves coequalizers; this is equivalent to saying that $\hom(-, k)$, as a contravariant functor on $\mathbf{Vect}$, takes equalizers to coequalizers, or takes kernels to cokernels, but that's the same as saying that $k$ is injective, so we're done.
Oh, incidentally, double dualization is not a commutative or monoidal monad, if I recall correctly.
Edit: In a comment below, Tom asks for a more concrete description of $\mathbf{Vect}^{op}$ along the lines of topological algebra. I suspect the way to go is to see $\mathbf{Vect}$ as the Ind-completion (or Ind-cocompletion) of the category of finite-dimensional vector spaces, and therefore $\mathbf{Vect}^{op}$ as the Pro-completion of the opposite category, which is again $\mathbf{Vect}_{fd}$. I think I've seen before a result that this is equivalent to the category of topological $k$-modules which arise as projective limits of (cofiltered diagrams of) finite-dimensional spaces with the discrete topology, or something along similar lines, but I'd have to look this up to be sure. There might be pertinent material in Barr's Springer Lecture Notes on $\ast$-autonomous categories, but again I'm not sure.
Edit 2: Ah, found it. $\mathbf{Vect}^{op}$ is equivalent to the category of linearly compact vector spaces over $k$ and continuous linear maps. See Theorem 3.1 of this paper for example: arxiv.org/pdf/1202.3609. The result is credited to Lefschetz.
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Todd's answer is definitive: the category of algebras is $\mathbf{Vect}^{\mathrm{op}}$. However, I'm left wanting more, since the dual of the category of vector spaces isn't something I readily grasp. (Cf. my remark 5.) Compare the theorem that the catebgory of algebras for the double powerset monad on $\mathbf{Set}$ is $\mathbf{Set}^{\mathrm{op}}$. This is illuminated by the fact that $\mathbf{Set}^{\mathrm{op}}$ is equivalent to the category of complete atomic Boolean algebras. Is there some similar theorem for $\mathbf{Vect}^{\mathrm{op}}$? – Tom Leinster Aug 16 at 15:55
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@Tom: here is a terrible description. By Pontrjagin duality, $\text{Ab}^{op}$ is the category of compact (Hausdorff) abelian groups. A given field $k$ is a comonoid in this category (with respect to the monoidal product given by dualizing the tensor product; I am not really sure what this looks like) and $\text{Vect}^{op}$ is the category of comodules in $\text{Ab}^{op}$ over this comonoid... – Qiaochu Yuan Aug 16 at 16:22
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In case someone didn't see it, I added an edit to my answer. – Todd Trimble Aug 16 at 16:34
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And now a second edit. – Todd Trimble Aug 16 at 16:50
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This extends even further to a duality between coalgebras and pseudocompact algebras (inverse limits of finite dimensional algebras) if I remember correctly. – Benjamin Steinberg Aug 16 at 17:54
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This is not a direct answer to your question, but are you are familiar with a recent "followup" paper by Kock, Commutative Monads as a Theory of Distributions? There he considers an alternative approach to the theory of distributions starting from a general commutative monad $T$ (with a certain notion of strength), then defining double-dualization with respect to an arbitrary $T$-algebra $B$. He explains that there is a monad morphism from $T$ into any such double-dualization monad $(-\multimap B)\multimap B$, that this morphism may be factored by way of a submonad $(-\multimap B) \multimap^T B$, and states that in certain cases the map $T \Rightarrow (-\multimap B) \multimap^T B$ is an isomorphism.
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Thanks, Noam. I did see that paper, though I hadn't read the part that you have. If I don't get an answer here, I think my next step is to mail Anders Kock. – Tom Leinster Aug 16 at 13:29
PS - I very much like the idea of a "follow up" paper 42 years after the original. – Tom Leinster Aug 16 at 13:30 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 76, "mathjax_display_tex": 4, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9357696771621704, "perplexity_flag": "head"} |
http://math.stackexchange.com/questions/125871/law-of-sines-will-give-a-unique-solution-iff-a-b | # Law of Sines will give a unique solution iff a > b?
Given a triangle ABC, with known sides a=BC and b=AC, and known angle A, we wish to find angle B.
This is a typical application of the Sine Rule (Law of Sines).
In some circumstances, the sine rule gives an ambiguous result: with two possible solutions for angle B.
I am trying to find the simplest way of identifying whether or not the sine rule would give a unique solution.
Is it true to say that the sine rule will give a unique solution to this problem iff a > b?
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use the law of cosines if you want $B$: $$b^2=a^2+c^2-2ac\cos B\quad\implies$$ $$B=\arccos\frac{a^2+c^2-b^2}{2ac}$$ For each $B\ne\frac{\pi}{2}=90^\circ$, there are two angles having $\sin B$; these depend on the length of $c$. So Law of Sines only gives unique $B$ when $\sin B=1$. – bgins Mar 29 '12 at 10:23
Thanks for your revision :) sin B = k may have two solutions but often one of those solutions will not allow a valid triangle: here, I don't consider that case to be a valid solution. – Ronald Mar 29 '12 at 10:29
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This is, I think, much easier to approach by drawing diagrams of the geometry involved than by staring at the law of sines. – Chris Eagle Mar 29 '12 at 10:32
You're right. Then the question becomes - what are the necessary conditions on given sides a, b and angle A, to ensure uniqueness of the triangle to be drawn? – Ronald Mar 29 '12 at 10:36
## 3 Answers
The question isn't really (or shouldn't be) about the sine rule, but about when two sides and an angle not formed by the two sides determine a unique triangle. You found almost the right criterion; in fact if $a=b$ the triangle is also uniquely determined (unless you allow degenerate triangles). The sine rule, by contrast, always allows two different angles at $B$, since the sine is symmetric with respect to reflection at $\pi/2$. If $a\ge b$, you can exclude the greater of the two because the sum with the angle at $A$ would exceed $\pi$, whereas for $a\lt b$ both of these angles correspond to triangles.
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The triangle is also uniquely determined if $a/b=\sin A$, and impossible if $a/b<\sin A$. – Chris Eagle Mar 29 '12 at 10:38
Thanks for your insight. – Ronald Mar 29 '12 at 10:39
@Chris: Ah, you're right; thanks. – joriki Mar 29 '12 at 10:53
This is also known as the ambiguous case (or SSA), and occurs whenever $a>b\,\sin A$, i.e., whenever $B\ne90^\circ$ and $b$ is not the hypotenuse (and $a$ the leg) of a right triangle.
Angle $B$ can be acute, $B_1=\arcsin\left(\frac{b}{a}\sin A\right)$, or obtuse, $B_2=\pi-B_1=180^\circ-B_1$, corresponding to which, side $c$ will be bigger ($c_1$) or smaller ($c_2$) than $b\,\cos A$, the common length from the right triangle.
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That is correct. (Or almost correct: there is no solution at all if $a < b \sin A$.)
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http://mathoverflow.net/questions/62984/a-knot-theory-form-of-the-carpenters-ruler-question | ## A knot theory form of the carpenter’s ruler question
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Hey,
The carpenter's ruler problem is about polygons you can make with planar joints. You can formulate a similar question for knots when you introduce ball joints:
Given m ball joints connected by rods of arbitrary length, which knots can you make?
I found out that you can make the first non trivial knot with your fingers and thumbs and your thumbs are ball joints. This is probably the same question as for stick knots and since the stick number is not known for all stick knots, the answer to the question I must say is, no, the question is an open question.
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4
Yes, in general stick numbers for knots are unknown, although there is an upper bound: at most twice the crossing number. Here is table of known stick numbers: colab.sfu.ca/KnotPlot/sticknumbers. (I had earlier asked an MO question about stick numbers and crossing numbers: mathoverflow.net/questions/39870). – Joseph O'Rourke Apr 26 2011 at 0:18
The standard question for knots is :Given a knot G, what is the stick number for G (and the stick number is equal to the joint number)? My question is in the reverse sense:Given n sticks, what knot classes can one make? This comes down to: Given m crossings (m=n/2 where n is stick number), which knot classes can be created? Is this an open question? If this is not open and instead is a known theorem, then we can give all the knots which can be created with n joints... – Ben Sprott Apr 26 2011 at 19:14
1
In general this is open: "Given $n$ sticks, what knot classes can one make?" For a specific version: Given $n=c+2$ sticks, can you make a 2-braid with $c$ crossings? That is open; $c+3$ suffice, but it is unknown if $c+3$ are needed. (Here I am relying on work of Colin Adams.) – Joseph O'Rourke Apr 26 2011 at 20:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9447850584983826, "perplexity_flag": "middle"} |
http://cs.stackexchange.com/questions/9162/example-for-the-analysis-of-a-recursive-function | Example for the analysis of a recursive function
````l is a matrix of size [1...n, 1...n]
function: rec(i,j)
if (i*j == 0)
return 1
else
if (l[i,j] == 0)
l[i,j] = 1 * rec(i-1,j) + 2 * rec(i,j-1) + 3 * rec(i-1,j-1)
return l[i,j]
end_function
for i=1 to n
for j=1 to n
l[i,j] = 0
rec(n,n)
````
The nested for's are O(n2). But i have difficulties to analyse the recursive part. There is another variation of this example with l as 3d. And the essential part of 3drec function is defined as:
````if (l[i,j,k] == 0)
l[i,j,k] = 2 * rec(i-1,j,k) + 2 * rec(i,j-1,k) + 2 * rec(i,j,k-1)
````
Anyway let's think about the 2d version again. I thought something like that (that's the running time for the whole code including the nested loops):
T(n) = T(n-1, n2) + T(n, n-12) + T(n-12, n-12)
And i'm stuck here. Besides i don't know if i did right till this point.
-
$T$ is a function of one variable (or two). Make up your mind and you need a base case for your recursion to be defined properly. – saadtaame Jan 26 at 1:25
base case is not the problem, it's T(1). if i could solve the rest of the problem, i wouldn't post it here. i just need the solution till a proper recurrence. Then I could solve the rest of the recurrence equality. – moller1111 Jan 26 at 1:50
Do you want the time bounds with or without memoization? – Peter Shor Jan 26 at 4:20
@PeterShor, I understand the question as analyzing exactly the algorithm as written. Perhaps we could come up with a faster way to compute $l[i, j]$, but that would be cheating... – vonbrand Jan 26 at 5:53
3 Answers
The running time is exponential. As Yuval showed in his answer, we have
$$f(i,j) = \begin{cases} O(1), & i = 0 \text{ or } j = 0, \\ f(i-1,j) + f(i,j-1) + f(i-1,j-1) + O(1), & \text{otherwise}. \end{cases}$$
Let's look at a $g = O(f)$ defined by $g(i,0)=g(0,i)=1$ and $g(i,j)= g(i,j-1) + g(i-1,j)$.
This gives the array $$\begin{array}{ccccc} 1&1&1&1&1\cr 1&2&3&4&5\cr 1&3&6&10&15\cr 1&4&10&20&35\\1&5&15&35&70 \end{array}$$ which you should recognize as binomial coefficients. The term $g(i,i) = {2i \choose i},$ which grows as $\Theta(\frac{1}{i^{1/2}}4^i)$. This shows that the growth of $f$ is exponential.
The easiest way I know to find the exact growth formula is to compute the first few terms of the sequence and look it up on the Online Encyclopedia of Integer Sequences. Using 1 for all the $O(1)$ terms, computing them using a spreadsheet takes less than a minute, and we find that the sequence is in the OEIS. The page for the sequence tells us that the growth rate is $\Theta(\frac{1}{i^{1/2}}(3+2\sqrt{2})^i)$.
-
Here's the correct recurrence for the running time of rec: $$f(i,j) = \begin{cases} O(1), & i = 0 \text{ or } j = 0, \\ f(i-1,j) + f(i,j-1) + f(i-1,j-1) + O(1), & \text{otherwise}. \end{cases}$$ The running time of the entire program is $f(n,n) + O(n^2)$. Now it remains to solve the recurrence for $f$, which I leave to you.
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You are interested in the time `rec(i, j)` takes. If you look at the code, it doesn't depend on the contents of the `l` array, just on `i` and `j`. Just take each call to take time 1. Then, by the recursion, for the time $t_{i j}$ you have the recurrence: $$t_{i + 1, j + 1} = t_{i + 1, j} + t_{i, j + 1} + t_{i, j} \quad t_{i, 0} = t_{0, j} = 1$$ Use generating functions to solve this. Define: $$T(x, y) = \sum_{\substack{i \ge 0 \\ j \ge 0}} t_{i j} x^i y^j$$ Then $T(x, 0) = \frac{1}{1 -x}$, $T(0, y) = \frac{1}{1 - y}$. Using the properties of generating functions: $$\frac{T(x, y) - y / (1 - x) - x / (1 - y) + 1}{x y} = \frac{T(x, y) - x / (1 - y)}{x} + \frac{T(x, y) - y / (1 - x)}{y} + T(x, y)$$ This gives $T(x, y) = \frac{1 - x - y}{1 - x - y - x y}$. Luckily we aren't interested in $t_{i j}$, just in $t_{n n}$: $$\begin{align*} [x^n y^n] T(x, y) &= [x^n y^n] \left(1 + \frac{x y}{1 - x - y - x y}\right) \end{align*}$$ Let's tackle the second term: $$\begin{align*} [x^n y^n] \frac{x y}{1 - x - y - x y} &= [x^{n - 1} y^{n - 1}] \frac{1}{1 - x - y - x y} \end{align*}$$ Expanding each term of the geometric series by the multinomial theorem: $$\begin{align*} [x^{n - 1} y^{n - 1}] \frac{1}{1 - x - y - x y} &= [x^{n - 1} y^{n - 1}] \sum_{k \ge 0} \sum_{\substack{r \ge 0 \\ s \ge 0}} \binom{k}{r \, s \, k - r - s} x^r y^s (x y)^{k - r - s} \\ &= [x^{n - 1} y^{n - 1}] \sum_{k \ge 0} \sum_{\substack{r \ge 0 \\ s \ge 0}} \binom{k}{r \, s \, k - r - s} x^k y^k \\ &= \sum_{\substack{r \ge 0 \\ s \ge 0}} \binom{n - 1}{r \; s \; n - 1 - r - s} \\ &= 3^{n - 1} \end{align*}$$ This gives a complexity of $O(3^n)$.
Edits: I had messed up badly, I hope it is fixed now.
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1
Something is wrong. The run time is $O(n^2)$ with memoization (which the OP described as cheating in a comment). Without memoization, the run-time is exponential. This generating function argument should give you the answer without memoization, but clearly you've made a mistake. – Peter Shor Jan 26 at 17:05
It's not true that the coefficient of $x^ny^n$ in $T(x,y)$ is the same as the coefficient of $x^{2n}$ in $T(x,x)$: the latter is much larger. Also, you have an algebra mistake, $1-2x-x^2 \neq (1-x)^2$. The smallest (in magnitude) root of $1-2x-x^2$ is $\sqrt{2}-1$, so the growth rate of the coefficient of $x^n$ is roughly $[1/(\sqrt{2}-1)]^n = (\sqrt{2}+1)^n$. This gives us the approximation $(3+2\sqrt{2})^n$ for the coefficient of $x^{2n}$ in $T(x,x)$. This approximation is true up to polynomial factors, since $T(x,z-x)$ is maximized for $z=2x$. It also agrees with Peter Shor's answer. – Yuval Filmus Jan 26 at 18:49
You are right, I'm fixing the derivation. It turned out harder than I thought. – vonbrand Jan 26 at 19:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 34, "mathjax_display_tex": 9, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.932433009147644, "perplexity_flag": "head"} |
http://nrich.maths.org/4903/index | ### Pebbles
Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
### Adding All Nine
Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!
### Old Nuts
In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?
# Product Sudoku
### The Basic Rules of "Product Sudoku"
Like the conventional Sudoku, this Sudoku variant consists of a grid of nine rows and nine columns subdivided into nine $3 \times 3$ subgrids. Like the Sudoku Classic, it has two basic rules:
1. Each column, each row, and each box ($3 \times 3$ subgrid) must have the numbers 1 to 9.
2. No column, row or box can have two squares with the same number.
The puzzle can be solved with the help of the numbers in the top parts of certain squares. These numbers are the products of the digits in all the squares horizontally and vertically adjacent to the square.
### A Short Demonstration
The square in the top left corner of this Sudoku contains the number 20. 20 is the product of the digits in the two adjacent squares, which therefore must contain the digits 4 and 5. The 5 cannot go in the cell below the top left hand corner because 5 is not a factor of 96 (the product shown in the third cell down on the left hand side of the puzzle). Therefore 5 must be entered into the cell to the right of the cell containing 20 and 4 in the cell below.
A word document containing the problem can be found here, with an explanation for classroom use.
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9210490584373474, "perplexity_flag": "middle"} |
http://mathhelpforum.com/discrete-math/112601-induction-sequence.html | # Thread:
1. ## Induction Sequence
an = 1 + 2an-1
a1 = 1
a)Guess what an is based on inspection of this sequence
b) Prove the expression in (a) using induction
edit: ah my subscript got messed up, dunno how to post it on forums
suppose to be a subscript n = 1 + 2a subscript n-1
a subscript 1 = 1
2. Originally Posted by kentwoods
an = 1 + 2an-1
a1 = 1
a)Guess what an is based on inspection of this sequence
b) Prove the expression in (a) using induction
edit: ah my subscript got messed up, dunno how to post it on forums
suppose to be a subscript n = 1 + 2a subscript n-1
a subscript 1 = 1
$a_n = \{1,3,7,15,31,63,...\}$
Can you see the general pattern in the difference of every two successive elements?
3. yea looks like 2n + 1 ?
How would I show this by induction tho?
4. Originally Posted by kentwoods
yea looks like 2n + 1 ?
How would I show this by induction tho?
How is it 2n+1?
You have odd numbers there, yes, but the nth number is definitely not 2n+1...
Look at it this way:
$3-1 = 2$
$7-3 = 4$
$15-7 = 8$
$31-15 = 16$
What do 2,4,8,16 have in common?
Or, look at it this way:
$1= 2-1$
$3 = 4-1$
$7 = 8-1$
$15 = 16-1$
.
.
.
5. hmm
$2^n -1$
i think thats what you were trying to show me, im really bad when it comes to this
I cannot visualize it easily
6. Originally Posted by kentwoods
hmm
$2^n -1$
i think thats what you were trying to show me, im really bad when it comes to this
I cannot visualize it easily
Yes, that is correct. Now, we need to prove that $a_n = 2^n -1$ by induction.
Basis: n=1: $a_1 = 1, 2^1-1 = 1 = a_1 \Rightarrow$ correct.
Assume that it holds for n (that is, $a_n = 2^n-1$), and now we need to prove that it is correct for $n+1$, that is, we need to prove that $a_{n+1} = 2^{n+1}-1$.
$a_{n+1} = 1+2a_{n} \overbrace{=}^{\text{By the assumption for n}} 1+ 2\cdot (2^n-1) = 1 + 2 \cdot 2^n - 2 = 2 \cdot 2^n - 1 =$ $2^{n+1}-1 \Rightarrow a_{n+1} = 2^{n+1}-1$ which is what we wanted to prove, therefore the assumption holds for any $n \in \mathbb{N}$ and we are done.
7. Thanks mate, that actually makes a lot of sense when i go through it slowly
Just a quick question though, when we went over examples in class we always proved n-1 instead of n+1. Basically if one is true the one before it is true as well. I'm assuming this works the same way.
8. Originally Posted by kentwoods
Thanks mate, that actually makes a lot of sense when i go through it slowly
Just a quick question though, when we went over examples in class we always proved n-1 instead of n+1. Basically if one is true the one before it is true as well. I'm assuming this works the same way.
I don't think I follow. Did you assume that it is correct for n and prove for n-1? Or assume correct for n-1 and prove for n? If the latter, then that is equivalent to what I did. Otherwise, that is not the way to use induction (it is not correct).
9. no I meant the latter, i meant assume for n-1 and prove for n | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 19, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.957744836807251, "perplexity_flag": "middle"} |
http://mathoverflow.net/revisions/57899/list | ## Return to Answer
Post Made Community Wiki by S. Carnahan♦
3 added 144 characters in body
A bit tangential, but I'm surprised no one mentioned time in general relativity (Ken Knox discussed special relativity, but the case in general relativity is subtly different). This discusses relativity from a perspective closer to a physicist's, since it's a bit more elementary (and hence easier to understand) in my view. As discussed at the end, relativity in general is somewhat incompatible with statistical mechanics, at least under the standard approximations, so this is almost surely not what you're looking for, but it may be of use to someone.
In GR, space-time is a 4-manifold which is endowed with a Lorentzian metric $g_{ab}$, which is a rank 2 covariant tensor. The scalar product of two vectors $V$ and $W$ is then $g_{ab}V^aW^b$, from which we can compute things as in special relativity (where the metric is $\eta_{ab}$, which is $0$ if $a \ne b$, -1 if $a=b$ and $x^a$ is a spacial coordinate (i.e. x,y,z), and 1 if $a=b$ and $x^a=t$ is the time coordinate).
If a vector (field) V satisfies $g_{ab}V^aV^b >0$, we call it time-like, and similarly for trajectories based on their tangent vectors. These are the possible trajectories for particles with positive mass. Any time-like trajectory corresponds in the limit of low mass and low speed to a local frame of reference in which it is the 'time', and if the trajectory is a geodesic then the frame is inertial. Particles with 0 mass (i.e. light) have null trajectories, with $g_{ab}V^aV^b =0$. If $g_{ab}V^aV^b <0$, the vector is space-like. Particles with no forces (other than gravity) acting on them travel on geodesics. The trajectory of a massive particle is called it's world line. The sign conventions here are often reversed, so care is advised. For a time-like path, we can compute its length by $ds^2 = g_{ab} x^a x^b$, where $x$ are your coordinates.
For any observer, they observe themself as stationary, and traveling forward in time with unit speed. Time for that observer corresponds exactly to the length of their trajectory. That observer can even set up a local set of coordinates in which the metric is approximately $\eta_{ab}$, provided all masses are sufficiently far away. Locally, then, time behaves like in special relativity. The big difference between special and general relativity is that the latter has no inertial reference frames in general, so observers can only measure times in at their own location.
To summarize, time is another coordinate in space-time, just like space, but it isn't universal (it's observer dependent), and the only real restrictions on it is that it must be the part of the metric that has positive signature (in the sense above).
Since this was no-doubt useless in explaining the difference between time in special and general relativity, I recommend Hughston & Tod's An introduction to general relativity. It's fairly light reading and has an introduction to special relativity, but it's rigorous enough for mathematicians reading it. A number of other books at a higher level are available, of which Hawking & Ellis, Wald, and Misner, Thorne, & Wheeler are all good references.
However, if you're looking for a formulation in which the second law of thermodynamics is provable, or even where entropy is defined, relativity is the wrong place to look. Even in special relativity, concepts like thermal equilibrium depend on a particular reference frame, so entropy may be defined in one inertial frame but not another. There has historically been great debate over how temperature (which is the thermodynamic conjugate of entropy) should transform under Lorentz transformations, and it's still not totally resolved.
2 deleted 144 characters in body
A bit tangential, but I'm surprised no one mentioned time in general relativity (Ken Knox discussed special relativity, but the case in general relativity is subtly different). This discusses relativity from a perspective closer to a physicist's, since it's a bit more elementary (and hence easier to understand) in my view. As discussed at the end, relativity in general is somewhat incompatible with statistical mechanics, at least under the standard approximations, so this is almost surely not what you're looking for, but it may be of use to someone.
In GR, space-time is a 4-manifold which is endowed with a Riemannian Lorentzian metric $g_{ab}$, which is a rank 2 covariant tensor. The scalar product of two vectors $V$ and $W$ is then $g_{ab}V^aW^b$, from which we can compute things as in special relativity (where the metric is $\eta_{ab}$, which is $0$ if $a \ne b$, -1 if $a=b$ and $x^a$ is a spacial coordinate (i.e. x,y,z), and 1 if $a=b$ and $x^a=t$ is the time coordinate).
The length element can then be expressed by $ds^2 = g_{ab} dx^a dx^b$ where $x$ are your coordinates. Technically we should have an absolute value, since the right hand side can be negative, but it will be useful to ignore this momentarily.
If a vector (field) V satisfies $g_{ab}V^aV^b >0$, we call it time-like, and similarly for trajectories based on their tangent vectors. These are the possible trajectories for particles with positive mass. Any time-like trajectory corresponds in the limit of low mass and low speed to a local frame of reference in which it is the 'time', and if the trajectory is a geodesic then the frame is inertial. Particles with 0 mass (i.e. light) have null trajectories, with $g_{ab}V^aV^b =0$. If $g_{ab}V^aV^b <0$, the vector is space-like. Particles with no forces (other than gravity) acting on them travel on geodesics. The trajectory of a massive particle is called it's world line. The sign conventions here are often reversed, so care is advised. For a time-like path, we can compute its length by $ds^2 = g_{ab} x^a x^b$, where $x$ are your coordinates.
For any observer, they observe themself as stationary, and traveling forward in time with unit speed. Time for that observer corresponds exactly to the length of their trajectory. That observer can even set up a local set of coordinates in which the metric is approximately $\eta_{ab}$, which will be valid so long as provided all masses are sufficiently far away. Locally, then, time behaves like in special relativity. The big difference between special and general relativity is that the latter has no inertial reference frames in general, so observers can only measure times in at their own location.
To summarize, time is another coordinate in space-time, just like space, but it isn't universal (it's observer dependent), and the only real restrictions on it is that it must be the part of the metric that has positive signature (in the sense above).
Since this was no-doubt useless in explaining the difference between time in special and general relativity, I recommend Hughston & Tod's An introduction to general relativity. It's fairly light reading and has an introduction to special relativity, but it's rigorous enough for mathematicians reading it.
However, if you're looking for a formulation in which the second law of thermodynamics is provable, or even where entropy is defined, relativity is the wrong place to look. Even in special relativity, concepts like thermal equilibrium depend on a particular reference frame, so entropy may be defined in one inertial frame but not another. There has historically been great debate over how temperature (which is the thermodynamic conjugate of entropy) should transform under Lorentz transformations, and it's still not totally resolved.
1
A bit tangential, but I'm surprised no one mentioned time in general relativity (Ken Knox discussed special relativity, but the case in general relativity is subtly different). This discusses relativity from a perspective closer to a physicist's, since it's a bit more elementary (and hence easier to understand) in my view. As discussed at the end, relativity in general is somewhat incompatible with statistical mechanics, at least under the standard approximations, so this is almost surely not what you're looking for, but it may be of use to someone.
In GR, space-time is a 4-manifold which is endowed with a Riemannian metric $g_{ab}$, which is a rank 2 covariant tensor. The scalar product of two vectors $V$ and $W$ is then $g_{ab}V^aW^b$, from which we can compute things as in special relativity (where the metric is $\eta_{ab}$, which is $0$ if $a \ne b$, -1 if $a=b$ and $x^a$ is a spacial coordinate (i.e. x,y,z), and 1 if $a=b$ and $x^a=t$ is the time coordinate).
The length element can then be expressed by $ds^2 = g_{ab} dx^a dx^b$ where $x$ are your coordinates. Technically we should have an absolute value, since the right hand side can be negative, but it will be useful to ignore this momentarily.
If a vector (field) V satisfies $g_{ab}V^aV^b >0$, we call it time-like, and similarly for trajectories based on their tangent vectors. These are the possible trajectories for particles with positive mass. Any time-like trajectory corresponds in the limit of low mass and low speed to a local frame of reference in which it is the 'time', and if the trajectory is a geodesic then the frame is inertial. Particles with 0 mass (i.e. light) have null trajectories, with $g_{ab}V^aV^b =0$. If $g_{ab}V^aV^b <0$, the vector is space-like. Particles with no forces (other than gravity) acting on them travel on geodesics. The trajectory of a massive particle is called it's world line. The sign conventions here are often reversed, so care is advised.
For any observer, they observe themself as stationary, and traveling forward in time with unit speed. Time for that observer corresponds exactly to the length of their trajectory. That observer can even set up a local set of coordinates in which the metric is $\eta_{ab}$, which will be valid so long as all masses are sufficiently far away. Locally, then, time behaves like in special relativity. The big difference between special and general relativity is that the latter has no inertial reference frames in general, so observers can only measure times in at their own location.
To summarize, time is another coordinate in space-time, just like space, but it isn't universal (it's observer dependent), and the only real restrictions on it is that it must be the part of the metric that has positive signature (in the sense above).
Since this was no-doubt useless in explaining the difference between time in special and general relativity, I recommend Hughston & Tod's An introduction to general relativity. It's fairly light reading and has an introduction to special relativity, but it's rigorous enough for mathematicians reading it.
However, if you're looking for a formulation in which the second law of thermodynamics is provable, or even where entropy is defined, relativity is the wrong place to look. Even in special relativity, concepts like thermal equilibrium depend on a particular reference frame, so entropy may be defined in one inertial frame but not another. There has historically been great debate over how temperature (which is the thermodynamic conjugate of entropy) should transform under Lorentz transformations, and it's still not totally resolved. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 53, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9429616928100586, "perplexity_flag": "head"} |
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# Differential Logic and Dynamic Systems 2.0
### MyWikiBiz, Author Your Legacy — Saturday May 18, 2013
Stand and unfold yourself. Hamlet: Francsico—1.1.2
This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.
## Review and Transition
This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.
Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable $k\!$-ary scope.
• A bracketed list of propositional expressions in the form $(e_1, e_2, \ldots, e_{k-1}, e_k)$ indicates that exactly one of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ is false.
• A concatenation of propositional expressions in the form $e_1~e_2~\ldots~e_{k-1}~e_k$ indicates that all of the propositions $e_1, e_2, \ldots, e_{k-1}, e_k$ are true, in other words, that their logical conjunction is true.
All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.
This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.
While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes barred parentheses $(\!| \ldots |\!)$ may be used for logical operators.
The briefest expression for logical truth is the empty word, usually denoted by $\varepsilon\!$ or $\lambda\!$ in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression "$((~))\!$", or, especially if operating in an algebraic context, by a simple "$1\!$". Also when working in an algebraic mode, the plus sign "$+\!$" may be used for exclusive disjunction. For example, we have the following paraphrases of algebraic expressions by bracket expressions:
$\begin{matrix} x + y & = & (x, y) \end{matrix}$
$\begin{matrix} x + y + z & = & ((x, y), z) & = & (x, (y, z)) \end{matrix}$
It is important to note that the last expressions are not equivalent to the triple bracket $(x, y, z).\!$
Table 1. Syntax and Semantics of a Calculus for Propositional Logic
Expression Interpretation Other Notations
$~$ $\operatorname{True}$ $1\!$
$(~)$ $\operatorname{False}$ $0\!$
$x\!$ $x\!$ $x\!$
$(x)\!$ $\operatorname{Not}\ x$
$\begin{matrix} x' \\ \tilde{x} \\ \lnot x \\ \end{matrix}$
$x\ y\ z$ $x\ \operatorname{and}\ y\ \operatorname{and}\ z$ $x \land y \land z$
$((x)(y)(z))\!$ $x\ \operatorname{or}\ y\ \operatorname{or}\ z$ $x \lor y \lor z$
$(x\ (y))\!$
$\begin{matrix} x\ \operatorname{implies}\ y \\ \operatorname{If}\ x\ \operatorname{then}\ y \\ \end{matrix}$
$x \Rightarrow y\!$
$(x, y)\!$
$\begin{matrix} x\ \operatorname{not~equal~to}\ y \\ x\ \operatorname{exclusive~or}\ y \\ \end{matrix}$
$\begin{matrix} x \neq y \\ x + y \\ \end{matrix}$
$((x, y))\!$
$\begin{matrix} x\ \operatorname{is~equal~to}\ y \\ x\ \operatorname{if~and~only~if}\ y \\ \end{matrix}$
$\begin{matrix} x = y \\ x \Leftrightarrow y \\ \end{matrix}$
$(x, y, z)\!$
$\begin{matrix} \operatorname{Just~one~of} \\ x, y, z \\ \operatorname{is~false}. \\ \end{matrix}$
$\begin{matrix} x'y~z~ & \lor \\ x~y'z~ & \lor \\ x~y~z' & \\ \end{matrix}$
$((x),(y),(z))\!$
$\begin{matrix} \operatorname{Just~one~of} \\ x, y, z \\ \operatorname{is~true}. \\ & \\ \operatorname{Partition~all} \\ \operatorname{into}\ x, y, z. \\ \end{matrix}$
$\begin{matrix} x~y'z' & \lor \\ x'y~z' & \lor \\ x'y'z~ & \\ \end{matrix}$
$\begin{matrix} ((x, y), z) \\ & \\ (x, (y, z)) \\ \end{matrix}$
$\begin{matrix} \operatorname{Oddly~many~of} \\ x, y, z \\ \operatorname{are~true}. \\ \end{matrix}$
$x + y + z\!$
$\begin{matrix} x~y~z~ & \lor \\ x~y'z' & \lor \\ x'y~z' & \lor \\ x'y'z~ & \\ \end{matrix}$
$(w, (x),(y),(z))\!$
$\begin{matrix} \operatorname{Partition}\ w \\ \operatorname{into}\ x, y, z. \\ & \\ \operatorname{Genus}\ w\ \operatorname{comprises} \\ \operatorname{species}\ x, y, z. \\ \end{matrix}$
$\begin{matrix} w'x'y'z' & \lor \\ w~x~y'z' & \lor \\ w~x'y~z' & \lor \\ w~x'y'z~ & \\ \end{matrix}$
Note. The usage that one often sees, of a plus sign "$+\!$" to represent inclusive disjunction, and the reference to this operation as boolean addition, is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:
The expression $x + y\!$ seems indeed uninterpretable, unless it be assumed that the things represented by $x\!$ and the things represented by $y\!$ are entirely separate; that they embrace no individuals in common. (Boole, 66).
It was only later that Peirce and Jevons treated inclusive disjunction as a fundamental operation, but these authors, with a respect for the algebraic properties that were already associated with the plus sign, used a variety of other symbols for inclusive disjunction (Sty, 177, 189). It seems to have been Schröder who later reassigned the plus sign to inclusive disjunction (Sty, 208). Additional information, discussion, and references can be found in (Boole) and (Sty, 177–263). Aside from these historical points, which never really count against a current practice that has gained a life of its own, this usage does have a further disadvantage of cutting or confounding the lines of communication between algebra and logic. For this reason, it will be avoided here.
## A Functional Conception of Propositional Calculus
Out of the dimness opposite equals advance . . . . Always substance and increase, Always a knit of identity . . . . always distinction . . . . always a breed of life.
— Walt Whitman, Leaves of Grass, [Whi, 28]
In the general case, we start with a set of logical features $\{a_1, \ldots, a_n\}$ that represent properties of objects or propositions about the world. In concrete examples the features $\{a_i\!\}$ commonly appear as capital letters from an alphabet like $\{A, B, C, \ldots\}$ or as meaningful words from a linguistic vocabulary of codes. This language can be drawn from any sources, whether natural, technical, or artificial in character and interpretation. In the application to dynamic systems we tend to use the letters $\{x_1, \ldots, x_n\}$ as our coordinate propositions, and to interpret them as denoting properties of a system's state, that is, as propositions about its location in configuration space. Because I have to consider non-deterministic systems from the outset, I often use the word state in a loose sense, to denote the position or configuration component of a contemplated state vector, whether or not it ever gets a deterministic completion.
The set of logical features $\{a_1, \ldots, a_n\}$ provides a basis for generating an $n\!$-dimensional universe of discourse that I denote as $[a_1, \ldots, a_n].$ It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points $\langle a_1, \ldots, a_n \rangle$ and the set of propositions $f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}$ that are implicit with the ordinary picture of a venn diagram on $n\!$ features. Thus, we may regard the universe of discourse $[a_1, \ldots, a_n]$ as an ordered pair having the type $(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),$ and we may abbreviate this last type designation as $\mathbb{B}^n\ +\!\to \mathbb{B},$ or even more succinctly as $[\mathbb{B}^n].$ (Used this way, the angle brackets $\langle\ldots\rangle$ are referred to as generator brackets.)
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations $[n]\!$ or $\mathbf{n}$ to denote the data type of a finite set on $n\!$ elements.
Table 2. Propositional Calculus : Basic Notation
Symbol Notation Description Type
$\mathfrak{A}$ $\lbrace\!$ “$a_1\!$” $, \ldots,\!$ “$a_n\!$” $\rbrace\!$ $\operatorname{Alphabet}$ $[n] = \mathbf{n}$
$\mathcal{A}$ $\{a_1, \ldots, a_n\}$ $\operatorname{Basis}$ $[n] = \mathbf{n}$
$A_i\!$ $\{(a_i), a_i\}\!$ $\operatorname{Dimension}\ i$ $\mathbb{B}$
$A\!$
$\langle \mathcal{A} \rangle$
$\langle a_1, \ldots, a_n \rangle$
$\{(a_1, \ldots, a_n)\}$
$A_1 \times \ldots \times A_n$
$\textstyle \prod_{i=1}^n A_i$
$\operatorname{Set~of~cells},$
$\operatorname{coordinate~tuples},$
$\operatorname{points,~or~vectors}$
$\operatorname{in~the~universe}$
$\operatorname{of~discourse}$
$\mathbb{B}^n$
$A^*\!$ $(\operatorname{hom} : A \to \mathbb{B})$ $\operatorname{Linear~functions}$ $(\mathbb{B}^n)^* \cong \mathbb{B}^n$
$A^\uparrow$ $(A \to \mathbb{B})$ $\operatorname{Boolean~functions}$ $\mathbb{B}^n \to \mathbb{B}$
$A^\circ$
$[\mathcal{A}]$
$(A, A^\uparrow)$
$(A\ +\!\to \mathbb{B})$
$(A, (A \to \mathbb{B}))$
$[a_1, \ldots, a_n]$
$\operatorname{Universe~of~discourse}$
$\operatorname{based~on~the~features}$
$\{a_1, \ldots, a_n\}$
$(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))$
$(\mathbb{B}^n\ +\!\to \mathbb{B})$
$[\mathbb{B}^n]$
### Qualitative Logic and Quantitative Analogy
Logical, however, is used in a third sense, which is at once more vital and more practical; to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.
— John Dewey, How We Think, [Dew, 56]
These concepts and notations may now be explained in greater detail. In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path. On the first level of analysis we take spaces like $\mathbb{B},$ $\mathbb{B}^n,$ and $(\mathbb{B}^n \to \mathbb{B})$ at face value and treat them as the primary objects of interest. On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.
A pair of spaces, of types $\mathbb{B}^n$ and $(\mathbb{B}^n \to \mathbb{B}),$ give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, $n,\!$ counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type $\mathbb{B}^n$ correspond to what are often called propositional interpretations in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its cells, in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions $f : \mathbb{B}^n \to \mathbb{B}$ correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the models, and regions excluded represent the non-models of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations $[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}$ to stand for the pair of types $(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).$ The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or arrows) that affect the universe of discourse as an integrated whole.
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of $A, B, C,\!$ and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, $\mathcal{A} = \{a_i\}.$ Most of the time, a set such as $\mathcal{A} = \{a_i\}$ will be employed as the alphabet of a formal language. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like $(\mathbb{B}^n\ +\!\to \mathbb{B}),$ then we may use the following notations. If $\mathcal{A} = \{a_1, \ldots, a_n\}$ is an alphabet of logical features, then $A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle$ is the set of interpretations, $A^\uparrow = (A \to \mathbb{B})$ is the set of propositions, and $A^\circ = [\mathcal{A}] = [a_1, \ldots, a_n]$ is the combination of these interpretations and propositions into the universe of discourse that is based on the features $\{a_1, \ldots, a_n\}.$
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.
### Philosophy of Notation : Formal Terms and Flexible Types
Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking. Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.
— W.V. Quine, Mathematical Logic, [Qui, 7]
For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus. This rule of interpretation has exceptions, though. There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation. It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus. Just to provide a hint of what's at stake: In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about. Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information. Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation $f^{-1} \subseteq \mathbb{B} \times \mathbb{B}^n,$ or what is the same thing, $f^{-1} : \mathbb{B} \to \mathcal{P}(\mathbb{B}^n),$ and the fibers or inverse images $f^{-1}(0)\!$ and $f^{-1}(1),\!$ associated with each boolean function $f : \mathbb{B}^n \to \mathbb{B}$ that we use. In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets $f^{-1}(b),\!$ for $b \in \mathbb{B},$ is part and parcel of understanding the denotative uses of each propositional function $f.\!$
### Special Classes of Propositions
It is important to remember that the coordinate propositions $\{a_i\},\!$ besides being projection maps $a_i : \mathbb{B}^n \to \mathbb{B},$ are propositions on an equal footing with all others, even though employed as a basis in a particular moment. This set of $n\!$ propositions may sometimes be referred to as the basic propositions, the coordinate propositions, or the simple propositions that found a universe of discourse. Either one of the equivalent notations, $\{a_i : \mathbb{B}^n \to \mathbb{B}\}$ or $(\mathbb{B}^n \xrightarrow{i} \mathbb{B}),$ may be used to indicate the adoption of the propositions $a_i\!$ as a basis for describing a universe of discourse.
Among the $2^{2^n}$ propositions in $[a_1, \ldots, a_n]$ are several families of $2^n\!$ propositions each that take on special forms with respect to the basis $\{ a_1, \ldots, a_n \}.$ Three of these families are especially prominent in the present context, the linear, the positive, and the singular propositions. Each family is naturally parameterized by the coordinate $n\!$-tuples in $\mathbb{B}^n$ and falls into $n + 1\!$ ranks, with a binomial coefficient $\tbinom{n}{k}$ giving the number of propositions that have rank or weight $k.\!$
• The linear propositions, $\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),$ may be written as sums:
$\begin{matrix} \sum_{i=1}^n e_i & = & e_1 + \ldots + e_n & \operatorname{where}\ e_i = a_i\ \operatorname{or}\ e_i = 0 & \operatorname{for}\ i = 1\ \operatorname{to}\ n. \end{matrix}$
• The positive propositions, $\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),$ may be written as products:
$\begin{matrix} \prod_{i=1}^n e_i & = & e_1 \cdot \ldots \cdot e_n & \operatorname{where}\ e_i = a_i\ \operatorname{or}\ e_i = 1 & \operatorname{for}\ i = 1\ \operatorname{to}\ n. \end{matrix}$
• The singular propositions, $\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),$ may be written as products:
$\begin{matrix} \prod_{i=1}^n e_i & = & e_1 \cdot \ldots \cdot e_n & \operatorname{where}\ e_i = a_i\ \operatorname{or}\ e_i = (a_i) & \operatorname{for}\ i = 1\ \operatorname{to}\ n. \end{matrix}$
In each case the rank $k\!$ ranges from $0\!$ to $n\!$ and counts the number of positive appearances of the coordinate propositions $a_1, \ldots, a_n\!$ in the resulting expression. For example, for $n = 3,\!$ the linear proposition of rank $0\!$ is $0,\!$ the positive proposition of rank $0\!$ is $1,\!$ and the singular proposition of rank $0\!$ is $(a_1)(a_2)(a_3).\!$
The basic propositions $a_i : \mathbb{B}^n \to \mathbb{B}$ are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.
Linear propositions and positive propositions are generated by taking boolean sums and products, respectively, over selected subsets of basic propositions, so both families of propositions are parameterized by the powerset $\mathcal{P}(\mathcal{I}),$ that is, the set of all subsets $J\!$ of the basic index set $\mathcal{I} = \{1, \ldots, n\}.\!$
Let us define $\mathcal{A}_J$ as the subset of $\mathcal{A}$ that is given by $\{a_i : i \in J\}.\!$ Then we may comprehend the action of the linear and the positive propositions in the following terms:
• The linear proposition $\ell_J : \mathbb{B}^n \to \mathbb{B}$ evaluates each cell $\mathbf{x}$ of $\mathbb{B}^n$ by looking at the coefficients of $\mathbf{x}$ with respect to the features that $\ell_J$ "likes", namely those in $\mathcal{A}_J,$ and then adds them up in $\mathbb{B}.$ Thus, $\ell_J(\mathbf{x})$ computes the parity of the number of features that $\mathbf{x}$ has in $\mathcal{A}_J,$ yielding one for odd and zero for even. Expressed in this idiom, $\ell_J(\mathbf{x}) = 1$ says that $\mathbf{x}$ seems odd (or oddly true) to $\mathcal{A}_J,$ whereas $\ell_J(\mathbf{x}) = 0$ says that $\mathbf{x}$ seems even (or evenly true) to $\mathcal{A}_J,$ so long as we recall that zero times is evenly often, too.
• The positive proposition $p_J : \mathbb{B}^n \to \mathbb{B}$ evaluates each cell $\mathbf{x}$ of $\mathbb{B}^n$ by looking at the coefficients of $\mathbf{x}$ with regard to the features that $p_J\!$ "likes", namely those in $\mathcal{A}_J,$ and then takes their product in $\mathbb{B}.$ Thus, $p_J(\mathbf{x})$ assesses the unanimity of the multitude of features that $\mathbf{x}$ has in $\mathcal{A}_J,$ yielding one for all and aught for else. In these consensual or contractual terms, $p_J(\mathbf{x}) = 1$ means that $\mathbf{x}$ is AOK or congruent with all of the conditions of $\mathcal{A}_J,$ while $p_J(\mathbf{x}) = 0$ means that $\mathbf{x}$ defaults or dissents from some condition of $\mathcal{A}_J.$
### Basis Relativity and Type Ambiguity
Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.
First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis $\mathcal{A}$ will not remain singular if $\mathcal{A}$ is extended by a number of new and independent features. Even if we stick to the original set of pairwise options $\{a_i\} \cup \{(a_i)\}$ to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.
Second, the singular propositions $\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},$ picking out as they do a single cell or a coordinate tuple $\mathbf{x}$ of $\mathbb{B}^n,$ become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms $\mathbb{B}^n$ and $(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})$ and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations $\mathbf{x} : \mathbb{B}^n$ and the singular propositions $\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}$ are fully equivalent in information, and this means that every token of the type $\mathbb{B}^n$ can be reinterpreted as an appearance of the subtype $\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.$ And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.
For example, relative to the universe of discourse $[a_1, a_2, a_3]\!$ the singular proposition $a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}$ could be explicitly retyped as $a_1 a_2 a_3 : \mathbb{B}^3$ to indicate the point ‹1, 1, 1›, but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.
### The Analogy Between Real and Boolean Types
Measurement consists in correlating our subject matter with the series of real numbers; and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.
— W.V. Quine, Mathematical Logic, [Qui, 7]
There are two further reasons why it useful to spend time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms. First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern. Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the propositions as types analogy or the Curry–Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42–46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table 3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.
Table 3. Analogy Between Real and Boolean Types
$\mbox{Real Domain}\ \mathbb{R}$ $\longleftrightarrow$ $\mbox{Boolean Domain}\ \mathbb{B}$
$\mathbb{R}^n$ $\mbox{Basic Space}\!$ $\mathbb{B}^n$
$\mathbb{R}^n \to \mathbb{R}$ $\mbox{Function Space}\!$ $\mathbb{B}^n \to \mathbb{B}$
$(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}$ $\mbox{Tangent Vector}\!$ $(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}$
$\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})$ $\mbox{Vector Field}\!$ $\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})$
$(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}$ " $(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}$
$((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}$ " $((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}$
$(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})$ $\mbox{Derivation}\!$ $(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})$
$\mathbb{R}^n \to \mathbb{R}^m$
$\mbox{Basic}\!$
$\mbox{Transformation}\!$
$\mathbb{B}^n \to \mathbb{B}^m$
$(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})$
$\mbox{Function}\!$
$\mbox{Transformation}\!$
$(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})$
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space $X.\!$ Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.
It is usually expedient to take these spaces two at a time, in dual pairs of the form $X\!$ and $(X \to \mathbb{K}).$ In general, one creates pairs of type schemas by replacing any space $X\!$ with its dual $(X \to \mathbb{K}),$ for example, pairing the type $X \to Y$ with the type $(X \to \mathbb{K}) \to (Y \to \mathbb{K}),$ and $X \times Y$ with $(X \to \mathbb{K}) \times (Y \to \mathbb{K}).$ The word dual is used here in its broader sense to mean all of the functionals, not just the linear ones. Given any function $f : X \to \mathbb{K},$ the converse or inverse relation corresponding to $f\!$ is denoted $f^{-1},\!$ and the subsets of $X\!$ that are defined by $f^{-1}(k),\!$ taken over $k\!$ in $\mathbb{K},$ are called the fibers or the level sets of the function $f.\!$
### Theory of Control and Control of Theory
You will hardly know who I am or what I mean, But I shall be good health to you nevertheless, And filter and fibre your blood.
— Walt Whitman, Leaves of Grass, [Whi, 88]
In the boolean context a function $f : X \to \mathbb{B}$ is tantamount to a proposition about elements of $X,\!$ and the elements of $X\!$ constitute the interpretations of that proposition. The fiber $f^{-1}(1)\!$ comprises the set of models of $f,\!$ or examples of elements in $X\!$ satisfying the proposition $f.\!$ The fiber $f^{-1}(0)\!$ collects the complementary set of anti-models, or the exceptions to the proposition $f\!$ that exist in $X.\!$ Of course, the space of functions $(X \to \mathbb{B})$ is isomorphic to the set of all subsets of $X,\!$ called the power set of $X,\!$ and often denoted $\mathcal{P}(X)$ or $2^X.\!$
The operation of replacing $X\!$ by $(X \to \mathbb{B})$ in a type schema corresponds to a certain shift of attitude towards the space $X,\!$ in which one passes from a focus on the ostensibly individual elements of $X\!$ to a concern with the states of information and uncertainty that one possesses about objects and situations in $X.\!$ The conceptual obstacles in the path of this transition can be smoothed over by using singular functions $(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})$ as stepping stones. First of all, it's an easy step from an element $\mathbf{x}$ of type $\mathbb{B}^n$ to the equivalent information of a singular proposition $\mathbf{x} : X \xrightarrow{s} \mathbb{B},$ and then only a small jump of generalization remains to reach the type of an arbitrary proposition $f : X \to \mathbb{B},$ perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original $\mathbf{x}.$ This is frequently a useful transformation, communicating between the objective and the intentional perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.
All of the above notwithstanding, there are several points that distinguish these two tasks, namely, the theory of control and the control of theory, features that are often obscured by too much precipitation in the quickness with which we understand their similarities. In the control of uncertainty through inquiry, some of the actuators that we need to be concerned with are axiom changers and theory modifiers, operators with the power to compile and to revise the theories that generate expectations and predictions, effectors that form and edit our grammars for the languages of observational data, and agencies that rework the proposed model to fit the actual sequences of events and the realized relationships of values that are observed in the environment. Moreover, when steps must be taken to carry out an experimental action, there must be something about the particular shape of our uncertainty that guides us in choosing what directions to explore, and this impression is more than likely influenced by previous accumulations of experience. Thus it must be anticipated that much of what goes into scientific progress, or any sustainable effort toward a goal of knowledge, is necessarily predicated on long term observation and modal expectations, not only on the more local or short term prediction and correction.
### Propositions as Types and Higher Order Types
The types collected in Table 3 (repeated below) serve to illustrate the themes of higher order propositional expressions and the propositions as types (PAT) analogy.
Table 3. Analogy Between Real and Boolean Types
$\mbox{Real Domain}\ \mathbb{R}$ $\longleftrightarrow$ $\mbox{Boolean Domain}\ \mathbb{B}$
$\mathbb{R}^n$ $\mbox{Basic Space}\!$ $\mathbb{B}^n$
$\mathbb{R}^n \to \mathbb{R}$ $\mbox{Function Space}\!$ $\mathbb{B}^n \to \mathbb{B}$
$(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}$ $\mbox{Tangent Vector}\!$ $(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}$
$\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})$ $\mbox{Vector Field}\!$ $\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})$
$(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}$ " $(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}$
$((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}$ " $((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}$
$(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})$ $\mbox{Derivation}\!$ $(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})$
$\mathbb{R}^n \to \mathbb{R}^m$
$\mbox{Basic}\!$
$\mbox{Transformation}\!$
$\mathbb{B}^n \to \mathbb{B}^m$
$(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})$
$\mbox{Function}\!$
$\mbox{Transformation}\!$
$(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})$
First, observe that the type of a tangent vector at a point, also known as a directional derivative at that point, has the form $(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},$ where $\mathbb{K}$ is the chosen ground field, in the present case either $\mathbb{R}$ or $\mathbb{B}.$ At a point in a space of type $\mathbb{K}^n,$ a directional derivative operator $\vartheta\!$ takes a function on that space, an $f\!$ of type $(\mathbb{K}^n \to \mathbb{K}),$ and maps it to a ground field value of type $\mathbb{K}.$ This value is known as the derivative of $f\!$ in the direction $\vartheta\!$ [Che46, 76–77]. In the boolean case $\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}$ has the form of a proposition about propositions, in other words, a proposition of the next higher type.
Next, by way of illustrating the propositions as types idea, consider a proposition of the form $X \Rightarrow (Y \Rightarrow Z).$ One knows from propositional calculus that this is logically equivalent to a proposition of the form $(X \land Y) \Rightarrow Z.$ But this equivalence should remind us of the functional isomorphism that exists between a construction of the type $X \to (Y \to Z)$ and a construction of the type $(X \times Y) \to Z.$ The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows "$\to\!$" and products "$\times\!$" with the respective logical arrows "$\Rightarrow\!$" and products "$\land\!$". Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.
Finally, examine the middle four rows of Table 3. These display a series of isomorphic types that stretch from the categories that are labeled Vector Field to those that are labeled Derivation. A vector field, also known as an infinitesimal transformation, associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form $\xi : X \to \bigcup_{x \in X} \xi_x$ that assigns to each point $x\!$ of the space $X\!$ a tangent vector to $X\!$ at that point, namely, the tangent vector $\xi_x\!$ [Che46, 82–83]. If $X\!$ is of the type $\mathbb{K}^n,$ then $\xi\!$ is of the type $\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).$ This has the pattern $X \to (Y \to Z),$ with $X = \mathbb{K}^n,$ $Y = (\mathbb{K}^n \to \mathbb{K}),$ and $Z = \mathbb{K}.$
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table 4. Observe how the function $f : X \to \mathbb{K},$ associated with the place of $Y\!$ in the pattern, moves through its paces from the second to the first position. In this way, the vector field $\xi,\!$ initially viewed as attaching each tangent vector $\xi_x\!$ to the site $x\!$ where it acts in $X,\!$ now comes to be seen as acting on each scalar potential $f : X \to \mathbb{K}$ like a generalized species of differentiation, producing another function $\xi f : X \to \mathbb{K}$ of the same type.
Table 4. An Equivalence Based on the Propositions as Types Analogy
$\mbox{Pattern}\!$ $\mbox{Construct}\!$ $\mbox{Instance}\!$
$X \to (Y \to Z)$ $\mbox{Vector Field}\!$ $\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})$
$(X \times Y) \to Z$ $\Uparrow$ $(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}$
$(Y \times X) \to Z$ $\Downarrow$ $((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}$
$Y \to (X \to Z)$ $\mbox{Derivation}\!$ $(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})$
### Reality at the Threshold of Logic
But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.
— W.V. Quine, Mathematical Logic, [Qui, 7]
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.
| | | |
|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|
| $\mbox{Linear Space}\!$ | $\mbox{Liminal Space}\!$ | $\mbox{Logical Space}\!$ |
| $\begin{matrix} \mathcal{X} & = & \{x_1, \ldots, x_n\} \\ \end{matrix}$ | $\begin{matrix} \underline\mathcal{X} & = & \{\underline{x}_1, \ldots, \underline{x}_n\} \\ \end{matrix}$ | $\begin{matrix} \mathcal{A} & = & \{a_1, \ldots, a_n\} \\ \end{matrix}$ |
| $\begin{matrix} X_i & = & \langle x_i \rangle \\ & \cong & \mathbb{K} \\ \end{matrix}$ | $\begin{matrix} \underline{X}_i & = & \{(\underline{x}_i), \underline{x}_i \} \\ & \cong & \mathbb{B} \\ \end{matrix}$ | $\begin{matrix} A_i & = & \{(a_i), a_i \} \\ & \cong & \mathbb{B} \\ \end{matrix}$ |
| $\begin{matrix} X \\ = & \langle \mathcal{X} \rangle \\ = & \langle x_1, \ldots, x_n \rangle \\ = & X_1 \times \ldots \times X_n \\ = & \prod_{i=1}^n X_i \\ \cong & \mathbb{K}^n \\ \end{matrix}$ | $\begin{matrix} \underline{X} \\ = & \langle \underline\mathcal{X} \rangle \\ = & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\ = & \underline{X}_1 \times \ldots \times \underline{X}_n \\ = & \prod_{i=1}^n \underline{X}_i \\ \cong & \mathbb{B}^n \\ \end{matrix}$ | $\begin{matrix} A \\ = & \langle \mathcal{A} \rangle \\ = & \langle a_1, \ldots, a_n \rangle \\ = & A_1 \times \ldots \times A_n \\ = & \prod_{i=1}^n A_i \\ \cong & \mathbb{B}^n \\ \end{matrix}$ |
| $\begin{matrix} X^* & = & (\ell : X \to \mathbb{K}) \\ & \cong & \mathbb{K}^n \\ \end{matrix}$ | $\begin{matrix} \underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B}) \\ & \cong & \mathbb{B}^n \\ \end{matrix}$ | $\begin{matrix} A^* & = & (\ell : A \to \mathbb{B}) \\ & \cong & \mathbb{B}^n \\ \end{matrix}$ |
| $\begin{matrix} X^\uparrow & = & (X \to \mathbb{K}) \\ & \cong & (\mathbb{K}^n \to \mathbb{K}) \\ \end{matrix}$ | $\begin{matrix} \underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B}) \\ & \cong & (\mathbb{B}^n \to \mathbb{B}) \\ \end{matrix}$ | $\begin{matrix} A^\uparrow & = & (A \to \mathbb{B}) \\ & \cong & (\mathbb{B}^n \to \mathbb{B}) \\ \end{matrix}$ |
| $\begin{matrix} X^\circ \\ = & [\mathcal{X}] \\ = & [x_1, \ldots, x_n] \\ = & (X, X^\uparrow) \\ = & (X\ +\!\to \mathbb{K}) \\ = & (X, (X \to \mathbb{K})) \\ \cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K})) \\ = & (\mathbb{K}^n\ +\!\to \mathbb{K}) \\ = & [\mathbb{K}^n] \\ \end{matrix}$ | $\begin{matrix} \underline{X}^\circ \\ = & [\underline\mathcal{X}] \\ = & [\underline{x}_1, \ldots, \underline{x}_n] \\ = & (\underline{X}, \underline{X}^\uparrow) \\ = & (\underline{X}\ +\!\to \mathbb{B}) \\ = & (\underline{X}, (\underline{X} \to \mathbb{B})) \\ \cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\ = & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\ = & [\mathbb{B}^n] \\ \end{matrix}$ | $\begin{matrix} A^\circ \\ = & [\mathcal{A}] \\ = & [a_1, \ldots, a_n] \\ = & (A, A^\uparrow) \\ = & (A\ +\!\to \mathbb{B}) \\ = & (A, (A \to \mathbb{B})) \\ \cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\ = & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\ = & [\mathbb{B}^n] \\ \end{matrix}$ |
The left side of the Table collects mostly standard notation for an $n\!$-dimensional vector space over a field $\mathbb{K}.$ The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field $\mathbb{K},$ with a special interest in the continuous line $\mathbb{R},$ to the qualitative and discrete situations that are instanced and typified by $\mathbb{B}.$
I now proceed to explain these concepts in more detail. The most important ideas developed in Table 5 are these:
• The idea of a universe of discourse, which includes both a space of points and a space of maps on those points.
• The idea of passing from a more complex universe to a simpler universe by a process of thresholding each dimension of variation down to a single bit of information.
For the sake of concreteness, let us suppose that we start with a continuous $n\!$-dimensional vector space like $X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.$ The coordinate system $\mathcal{X} = \{x_i\}$ is a set of maps $x_i : \mathbb{R}^n \to \mathbb{R},$ also known as the coordinate projections. Given a "dataset" of points $\mathbf{x}$ in $\mathbb{R}^n,$ there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each $i\!$ we choose an $n\!$-ary relation $L_i\!$ on $\mathbb{R}^n,$ that is, a subset of $\mathbb{R}^n,$ and then we define the $i^\operatorname{th}\!$ threshold map, or limen $\underline{x}_i$ as follows:
$\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \mbox{such that:}$
$\begin{matrix} \underline{x}_i(\mathbf{x}) = 1 & \mbox{if} & \mathbf{x} \in L_i, \\ \underline{x}_i(\mathbf{x}) = 0 & \mbox{if} & \mathbf{x} \not\in L_i. \end{matrix}$
In other notations that are sometimes used, the operator $\chi (\ldots)$ or the corner brackets $\lceil\ldots\rceil$ can be used to denote a characteristic function, that is, a mapping from statements to their truth values in $\mathbb{B}.$ Finally, it is not uncommon to use the name of the relation itself as a predicate that maps $n\!$-tuples into truth values. Thus we have the following notational variants of the above definition:
$\begin{matrix} \underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}). \end{matrix}$
Notice that, as defined here, there need be no actual relation between the $n\!$-dimensional subsets $\{L_i\}\!$ and the coordinate axes corresponding to $\{x_i\},\!$ aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, $L_i\!$ is bounded by some hyperplane that intersects the $i^\operatorname{th}\!$ axis at a unique threshold value $r_i \in \mathbb{R}.$ Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set $L_i\!$ has points on the $i^\operatorname{th}\!$ axis, that is, points of the form ‹ $0, \ldots, 0, r_i, 0, \ldots, 0$ › where only the $x_i\!$ coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is real, otherwise the indexing is imaginary. For a knowledge based system $X,\!$ this should serve once again to mark the distinction between acquaintance and opinion.
States of knowledge about the location of a system or about the distribution of a population of systems in a state space $X = \mathbb{R}^n$ can now be expressed by taking the set $\underline\mathcal{X} = \{\underline{x}_i\}$ as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the $i^\operatorname{th}\!$ threshold map. This can help to remind us that the threshold operator $(\underline{~})_i$ acts on $\mathbf{x}$ by setting up a kind of a "hurdle" for it. In this interpretation the coordinate proposition $\underline{x}_i$ asserts that the representative point $\mathbf{x}$ resides above the $i^\operatorname{th}\!$ threshold.
Primitive assertions of the form $\underline{x}_i (\mathbf{x})$ may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state $\mathbf{x}$ of a contemplated system or a statistical ensemble of systems. Parentheses $(\ldots)$ may be used to indicate logical negation. Eventually one discovers the usefulness of the $k\!$-ary just one false operators of the form $(a_1, \ldots, a_k)$, as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), $\underline{X} \cong \mathbb{B}^n,$ and a space of functions (regions, propositions), $\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).$ Together these form a new universe of discourse $\underline{X}^\circ$ of the type $(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),$ which we may abbreviate as $\mathbb{B}^n\ +\!\to \mathbb{B}$ or most succinctly as $[\mathbb{B}^n].$
The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells $\underline\mathbf{x},$ the defining features $\underline{x}_i,$ and the potential shadings $f : \underline{X} \to \mathbb{B}$ all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.
Finally, let $X^*\!$ denote the space of linear functions, $(\ell : X \to \mathbb{K}),$ which has in the finite case the same dimensionality as $X,\!$ and let the same notation be extended across the Table.
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, that can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
### Tables of Propositional Forms
To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope. It provides explicit techniques for manipulating the most basic ingredients of discourse.
— W.V. Quine, Mathematical Logic, [Qui, 7–8]
To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the cactus language, the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.
Propositional forms on one variable correspond to boolean functions $f : \mathbb{B}^1 \to \mathbb{B}.$ In Table 6 these functions are listed in a variant form of truth table, one in which the axes of the usual arrangement are rotated through a right angle. Each function $f_i\!$ is indexed by the string of values that it takes on the points of the universe $X^\circ = [x] \cong \mathbb{B}^1.$ The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the $f_i,\!$ as shown in the first column of the Table. In their own right the $2^1\!$ points of the universe $X^\circ$ are coordinated as a space of type $\mathbb{B}^1,$ this in light of the universe $X^\circ$ being a functional domain where the coordinate projection $x\!$ takes on its values in $\mathbb{B}.$
| | | | | | |
|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|
| $\begin{matrix}\mathcal{L}_1 \\ \mbox{Decimal}\end{matrix}$ | $\begin{matrix}\mathcal{L}_2 \\ \mbox{Binary}\end{matrix}$ | $\begin{matrix}\mathcal{L}_3 \\ \mbox{Vector}\end{matrix}$ | $\begin{matrix}\mathcal{L}_4 \\ \mbox{Cactus}\end{matrix}$ | $\begin{matrix}\mathcal{L}_5 \\ \mbox{English}\end{matrix}$ | $\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}$ |
| $~$ | $x\colon\!$ | $1~0$ | $~$ | $~$ | $~$ |
| $f_0\!$ | $f_{00}\!$ | $0~0$ | $(~)\!$ | $\mbox{false}\!$ | $0\!$ |
| $f_1\!$ | $f_{01}\!$ | $0~1$ | $(x)\!$ | $\mbox{not}\ x$ | $\lnot x$ |
| $f_2\!$ | $f_{10}\!$ | $1~0$ | $x\!$ | $x\!$ | $x\!$ |
| $f_3\!$ | $f_{11}\!$ | $1~1$ | $((~))\!$ | $\mbox{true}\!$ | $1\!$ |
Propositional forms on two variables correspond to boolean functions $f : \mathbb{B}^2 \to \mathbb{B}.$ In Table 7 each function $f_i\!$ is indexed by the values that it takes on the points of the universe $X^\circ = [x, y] \cong \mathbb{B}^2.$ Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The $2^2\!$ points of the universe $X^\circ$ are coordinated as a space of type $\mathbb{B}^2,$ as indicated under the heading of the Table, where the coordinate projections $x\!$ and $y\!$ run through the various combinations of their values in $\mathbb{B}.$
| | | | | | |
|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|
| $\begin{matrix}\mathcal{L}_1 \\ \mbox{Decimal}\end{matrix}$ | $\begin{matrix}\mathcal{L}_2 \\ \mbox{Binary}\end{matrix}$ | $\begin{matrix}\mathcal{L}_3 \\ \mbox{Vector}\end{matrix}$ | $\begin{matrix}\mathcal{L}_4 \\ \mbox{Cactus}\end{matrix}$ | $\begin{matrix}\mathcal{L}_5 \\ \mbox{English}\end{matrix}$ | $\begin{matrix}\mathcal{L}_6 \\ \mbox{Ordinary}\end{matrix}$ |
| $~\!$ | $x\colon\!$ | $1~1~0~0\!$ | $~\!$ | $~\!$ | $~\!$ |
| $~\!$ | $y\colon\!$ | $1~0~1~0\!$ | $~\!$ | $~\!$ | $~\!$ |
| $f_{0}\!$ | $f_{0000}\!$ | $0~0~0~0\!$ | $(~)\!$ | $\mbox{false}\!$ | $0\!$ |
| $f_{1}\!$ | $f_{0001}\!$ | $0~0~0~1\!$ | $(x)(y)\!$ | $\mbox{neither}\ x\ \mbox{nor}\ y\!$ | $\lnot x \land \lnot y\!$ |
| $f_{2}\!$ | $f_{0010}\!$ | $0~0~1~0\!$ | $(x)\ y\!$ | $y\ \mbox{without}\ x\!$ | $\lnot x \land y\!$ |
| $f_{3}\!$ | $f_{0011}\!$ | $0~0~1~1\!$ | $(x)\!$ | $\mbox{not}\ x\!$ | $\lnot x\!$ |
| $f_{4}\!$ | $f_{0100}\!$ | $0~1~0~0\!$ | $x\ (y)\!$ | $x\ \mbox{without}\ y\!$ | $x \land \lnot y\!$ |
| $f_{5}\!$ | $f_{0101}\!$ | $0~1~0~1\!$ | $(y)\!$ | $\mbox{not}\ y\!$ | $\lnot y\!$ |
| $f_{6}\!$ | $f_{0110}\!$ | $0~1~1~0\!$ | $(x, y)\!$ | $x\ \mbox{not equal to}\ y\!$ | $x \ne y\!$ |
| $f_{7}\!$ | $f_{0111}\!$ | $0~1~1~1\!$ | $(x\ y)\!$ | $\mbox{not both}\ x\ \mbox{and}\ y\!$ | $\lnot x \lor \lnot y\!$ |
| $f_{8}\!$ | $f_{1000}\!$ | $1~0~0~0\!$ | $x\ y\!$ | $x\ \mbox{and}\ y\!$ | $x \land y\!$ |
| $f_{9}\!$ | $f_{1001}\!$ | $1~0~0~1\!$ | $((x, y))\!$ | $x\ \mbox{equal to}\ y\!$ | $x = y\!$ |
| $f_{10}\!$ | $f_{1010}\!$ | $1~0~1~0\!$ | $y\!$ | $y\!$ | $y\!$ |
| $f_{11}\!$ | $f_{1011}\!$ | $1~0~1~1\!$ | $(x\ (y))\!$ | $\mbox{not}\ x\ \mbox{without}\ y\!$ | $x \Rightarrow y\!$ |
| $f_{12}\!$ | $f_{1100}\!$ | $1~1~0~0\!$ | $x\!$ | $x\!$ | $x\!$ |
| $f_{13}\!$ | $f_{1101}\!$ | $1~1~0~1\!$ | $((x)\ y)\!$ | $\mbox{not}\ y\ \mbox{without}\ x\!$ | $x \Leftarrow y\!$ |
| $f_{14}\!$ | $f_{1110}\!$ | $1~1~1~0\!$ | $((x)(y))\!$ | $x\ \mbox{or}\ y\!$ | $x \lor y\!$ |
| $f_{15}\!$ | $f_{1111}\!$ | $1~1~1~1\!$ | $((~))\!$ | $\mbox{true}\!$ | $1\!$ |
## A Differential Extension of Propositional Calculus
Fire over water: The image of the condition before transition. Thus the superior man is careful In the differentiation of things, So that each finds its place.
— I Ching, Hexagram 64, [Wil, 249]
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a differential theory of qualitative equations that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.
### Differential Propositions : The Qualitative Analogues of Differential Equations
In order to define the differential extension of a universe of discourse $[\mathcal{A}],$ the initial alphabet $\mathcal{A}$ must be extended to include a collection of symbols for differential features, or basic changes that are capable of occurring in $[\mathcal{A}].$ Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.
Therefore, let us define the corresponding differential alphabet or tangent alphabet as $\operatorname{d}\mathcal{A}$ $=\!$ $\{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\},$ in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet $\mathcal{A}$ $=\!$ $\{a_1, \ldots, a_n\},$ that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in $\operatorname{d}\mathcal{A}$ is often conceived to be changeable from point to point of the underlying space $A.\!$ (Indeed, for all we know, the state space $A\!$ might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by $\mathcal{A}$ and $\operatorname{d}\mathcal{A}.$)
The tangent space to $A\!$ at one of its points $x,\!$ sometimes written $\operatorname{T}_x(A),$ takes the form $\operatorname{d}A$ $=\!$ $\langle \operatorname{d}\mathcal{A} \rangle$ $=\!$ $\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle.$ Strictly speaking, the name cotangent space is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.
Proceeding as we did with the base space $A,\!$ the tangent space $\operatorname{d}A$ at a point of $A\!$ can be analyzed as a product of distinct and independent factors:
$\operatorname{d}A\ =\ \prod_{i=1}^n \operatorname{d}A_i\ =\ \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n.$
Here, $\operatorname{d}A_i$ is a set of two differential propositions, $\operatorname{d}A_i = \{(\operatorname{d}a_i), \operatorname{d}a_i\},$ where $(\operatorname{d}a_i)$ is a proposition with the logical value of "$\mbox{not}\ \operatorname{d}a_i$". Each component $\operatorname{d}A_i$ has the type $\mathbb{B},$ operating under the ordered correspondence $\{(\operatorname{d}a_i), \operatorname{d}a_i\} \cong \{0, 1\}.$ However, clarity is often served by acknowledging this differential usage with a superficially distinct type $\mathbb{D},$ whose intension may be indicated as follows:
$\mathbb{D} = \{(\operatorname{d}a_i), \operatorname{d}a_i\} = \{\mbox{same}, \mbox{different}\} = \{\mbox{stay}, \mbox{change}\} = \{\mbox{stop}, \mbox{step}\}.$
Viewed within a coordinate representation, spaces of type $\mathbb{B}^n$ and $\mathbb{D}^n$ may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.
### An Interlude on the Path
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance.
— Michel Foucault, The Discourse on Language, [Fou, 215]
A sense of the relation between $\mathbb{B}$ and $\mathbb{D}$ may be obtained by considering the path classifier (or the equivalence class of curves) approach to tangent vectors. Consider a universe $[\mathcal{X}].$ Given the boolean value system, a path in the space $X = \langle \mathcal{X} \rangle$ is a map $q : \mathbb{B} \to X.$ In this context the set of paths $(\mathbb{B} \to X)$ is isomorphic to the cartesian square $X^2 = X \times X,$ or the set of ordered pairs chosen from $X.\!$
We may analyze $X^2 = \{ (u, v) : u, v \in X \}$ into two parts, specifically, the ordered pairs $(u, v)\!$ that lie on and off the diagonal:
$\begin{matrix} X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}. \end{matrix}$
This partition may also be expressed in the following symbolic form:
$\begin{matrix} X^2 & \cong & \operatorname{diag}(X) & + & 2 \tbinom{X}{2}. \end{matrix}$
The separate terms of this formula are defined as follows:
$\begin{matrix} \operatorname{diag}(X) & = & \{ (x, x) : x \in X \}. \end{matrix}$
$\begin{matrix} \tbinom{X}{k} & = & X\ \operatorname{choose}\ k & = & \{ k\!\mbox{-sets from}\ X \}. \end{matrix}$
Thus we have:
$\begin{matrix} \tbinom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}. \end{matrix}$
We may now use the features in $\operatorname{d}\mathcal{X} = \{ \operatorname{d}x_i \} = \{ \operatorname{d}x_1, \ldots, \operatorname{d}x_n \}$ to classify the paths of $(\mathbb{B} \to X)$ by way of the pairs in $X^2.\!$ If $X \cong \mathbb{B}^n,$ then a path $q\!$ in $X\!$ has the following form:
$\begin{matrix} q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n. \end{matrix}$
Intuitively, we want to map this $(\mathbb{B}^2)^n$ onto $\mathbb{D}^n$ by mapping each component $\mathbb{B}^2$ onto a copy of $\mathbb{D}.$ But in the presenting context "$\mathbb{D}$" is just a name associated with, or an incidental quality attributed to, coefficient values in $\mathbb{B}$ when they are attached to features in $\operatorname{d}\mathcal{X}.$
Taking these intentions into account, define $\operatorname{d}x_i : X^2 \to \mathbb{B}$ in the following manner:
$\begin{array}{lcrcl} \operatorname{d}x_i ((u, v)) & = & (\!|\ x_i (u) & , & x_i (v)\ |\!) \\ & = & x_i (u) & + & x_i (v) \\ & = & x_i (v) & - & x_i (u). \\ \end{array}$
In the above transcription, the operator bracket of the form $(\!| \ldots\ ,\ \ldots |\!)$ is a cactus lobe, in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in $\operatorname{GF}(2),$ and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.
The above definition of $\operatorname{d}x_i : X^2 \to \mathbb{B}$ is equivalent to defining $\operatorname{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}$ in the following way:
$\begin{array}{lcrcl} \operatorname{d}x_i (q) & = & (\!|\ x_i (q_0) & , & x_i (q_1)\ |\!) \\ & = & x_i (q_0) & + & x_i (q_1) \\ & = & x_i (q_1) & - & x_i (q_0). \\ \end{array}$
In this definition $q_b = q(b),\!$ for each $b\!$ in $\mathbb{B}.$ Thus, the proposition $\operatorname{d}x_i$ is true of the path $q = (u, v)\!$ exactly if the terms of $q,\!$ the endpoints $u\!$ and $v,\!$ lie on different sides of the question $x_i.\!$
The language of features in $\langle \operatorname{d}\mathcal{X} \rangle,$ indeed the whole calculus of propositions in $[\operatorname{d}\mathcal{X}],$ may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions $g : \operatorname{d}X \to \mathbb{B}.$ For example, the paths corresponding to $\operatorname{diag}(X)$ fall under the description $(\!| \operatorname{d}x_1 |\!) \cdots (\!| \operatorname{d}x_n |\!),$ which says that nothing changes against the backdrop of the coordinate frame $\{ x_1, \ldots, x_n \}.$
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space $X\!$ that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.
### The Extended Universe of Discourse
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.
— Michel Foucault, The Discourse on Language, [Fou, 215]
Next we define the extended alphabet or bundled alphabet $\operatorname{E}\mathcal{A}$ as follows:
$\begin{array}{lclcl} \operatorname{E}\mathcal{A} & = & \mathcal{A} \cup \operatorname{d}\mathcal{A} & = & \{a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n\}. \\ \end{array}$
This supplies enough material to construct the differential extension $\operatorname{E}A,$ or the tangent bundle over the initial space $A,\!$ in the following fashion:
$\begin{array}{lcl} \operatorname{E}A & = & \langle \operatorname{E}\mathcal{A} \rangle \\ & = & \langle \mathcal{A} \cup \operatorname{d}\mathcal{A} \rangle \\ & = & \langle a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle, \\ \end{array}$
and also:
$\begin{array}{lcl} \operatorname{E}A & = & A \times \operatorname{d}A \\ & = & A_1 \times \ldots \times A_n \times \operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n. \\ \end{array}$
This gives $\operatorname{E}A$ the type $\mathbb{B}^n \times \mathbb{D}^n.$
Finally, the tangent universe $\operatorname{E}A^\circ = [\operatorname{E}\mathcal{A}]$ is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features $\operatorname{E}\mathcal{A},$ and this fact is summed up in the following notation:
$\begin{array}{lclcl} \operatorname{E}A^\circ & = & [\operatorname{E}\mathcal{A}] & = & [a_1, \ldots, a_n, \operatorname{d}a_1, \ldots, \operatorname{d}a_n]. \\ \end{array}$
This gives the tangent universe $\operatorname{E}A^\circ$ the type:
$\begin{array}{lcl} (\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B}) & = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})). \\ \end{array}$
A proposition in the tangent universe $[\operatorname{E}\mathcal{A}]$ is called a differential proposition and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
With these constructions, the differential extension $\operatorname{E}A$ and the space of differential propositions $(\operatorname{E}A \to \mathbb{B}),$ we have arrived, in main outline, at one of the major subgoals of this study. Table 8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.
Table 8. Differential Extension : Basic Notation
Symbol Notation Description Type
$\operatorname{d}\mathfrak{A}$ $\lbrace\!$ “$\operatorname{d}a_1$” $, \ldots,\!$ “$\operatorname{d}a_n$” $\rbrace\!$ Alphabet of
differential
symbols
$[n] = \mathbf{n}$
$\operatorname{d}\mathcal{A}$ $\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ Basis of
differential
features
$[n] = \mathbf{n}$
$\operatorname{d}A_i$ $\{ (\operatorname{d}a_i), \operatorname{d}a_i \}$ Differential
dimension $i\!$
$\mathbb{D}$
$\operatorname{d}A$ $\langle \operatorname{d}\mathcal{A} \rangle$
$\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle$
$\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}$
$\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n$
$\textstyle \prod_i \operatorname{d}A_i$
Tangent space
at a point:
Set of changes,
motions, steps,
tangent vectors
at a point
$\mathbb{D}^n$
$\operatorname{d}A^*$ $(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})$ Linear functions
on $\operatorname{d}A$
$(\mathbb{D}^n)^* \cong \mathbb{D}^n$
$\operatorname{d}A^\uparrow$ $(\operatorname{d}A \to \mathbb{B})$ Boolean functions
on $\operatorname{d}A$
$\mathbb{D}^n \to \mathbb{B}$
$\operatorname{d}A^\circ$ $[\operatorname{d}\mathcal{A}]$
$(\operatorname{d}A, \operatorname{d}A^\uparrow)$
$(\operatorname{d}A\ +\!\to \mathbb{B})$
$(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))$
$[\operatorname{d}a_1, \ldots, \operatorname{d}a_n]$
Tangent universe
at a point of $A^\circ,$
based on the
tangent features
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$
$(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))$
$(\mathbb{D}^n\ +\!\to \mathbb{B})$
$[\mathbb{D}^n]$
The adjectives differential or tangent are systematically attached to every construct based on the differential alphabet $\operatorname{d}\mathfrak{A},$ taken by itself. Strictly speaking, we probably ought to call $\operatorname{d}\mathcal{A}$ the set of cotangent features derived from $\mathcal{A},$ but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type $(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}$ from cotangent vectors as elements of type $\mathbb{D}^n.$ In like fashion, having defined $\operatorname{E}\mathcal{A} = \mathcal{A} \cup \operatorname{d}\mathcal{A},$ we can systematically attach the adjective extended or the substantive bundle to all of the constructs associated with this full complement of $2n\!$ features.
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table 9 provides a suggestion of how these further extensions can be carried out.
$\begin{array}{lllll} \operatorname{d}^0 \mathcal{A} & = & \{a_1, \ldots, a_n\} & = & \mathcal{A} \\ \operatorname{d}^1 \mathcal{A} & = & \{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\} & = & \operatorname{d}\mathcal{A} \\ \end{array}$ $\begin{array}{lll} \operatorname{d}^k \mathcal{A} & = & \{\operatorname{d}^k a_1, \ldots, \operatorname{d}^k a_n\} \\ \operatorname{d}^* \mathcal{A} & = & \{\operatorname{d}^0 \mathcal{A}, \ldots, \operatorname{d}^k \mathcal{A}, \ldots \} \\ \end{array}$ $\begin{array}{lll} \operatorname{E}^0 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A} \\ \operatorname{E}^1 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \mathcal{A} \\ \operatorname{E}^k \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \ldots\ \cup\ \operatorname{d}^k \mathcal{A} \\ \operatorname{E}^\infty \mathcal{A} & = & \bigcup\ \operatorname{d}^* \mathcal{A} \\ \end{array}$
### Intentional Propositions
Do you guess I have some intricate purpose? Well I have . . . . for the April rain has, and the mica on the side of a rock has.
— Walt Whitman, Leaves of Grass, [Whi, 45]
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss velocities (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators $\operatorname{p}^k$ and $\operatorname{Q}^k$ are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.
$\begin{array}{lllll} \operatorname{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A} \\ \operatorname{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime \\ \operatorname{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime} \\ \cdots & & \cdots & \\ \end{array}$ $\begin{array}{lll} \operatorname{p}^k \mathcal{A} & = & \{\operatorname{p}^k a_1, \ldots, \operatorname{p}^k a_n\} \\ \end{array}$ $\begin{array}{lll} \operatorname{Q}^0 \mathcal{A} & = & \mathcal{A} \\ \operatorname{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \\ \operatorname{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}'' \\ \cdots & & \cdots \\ \operatorname{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \operatorname{p}^k \mathcal{A} \\ \end{array}$
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the intentional extension of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators $\operatorname{d}^k$ and $\operatorname{E}^k,$ and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain $X\!$ through an indefinite number of higher reaches, a particular collection of domains based on $X\!$ will be referred to as a realm of $X,\!$ and when the succession exhibits a temporal aspect, as a reign of $X.\!$
For the purposes of this discussion, an intentional proposition is defined as a proposition in the universe of discourse $\operatorname{Q}X^\circ = [\operatorname{Q}\mathcal{X}],$ in other words, a map $q : \operatorname{Q}X \to \mathbb{B}.$ The sense of this definition may be seen if we consider the following facts. First, the equivalence $\operatorname{Q}X = X \times X'$ motivates the following chain of isomorphisms between spaces:
$\begin{array}{cclcc} (\operatorname{Q}X \to \mathbb{B}) & \cong & (X & \times & X' \to \mathbb{B}) \\ & \cong & (X & \to & (X' \to \mathbb{B})) \\ & \cong & (X' & \to & (X \to \mathbb{B})). \\ \end{array}$
Viewed in this light, an intentional proposition $q\!$ may be rephrased as a map $q : X \times X' \to \mathbb{B},$ which judges the juxtaposition of states in $X\!$ from one moment to the next. Alternatively, $q\!$ may be parsed in two stages in two different ways, as $q : X \to (X' \to \mathbb{B})$ and as $q : X' \to (X \to \mathbb{B}),$ which associate to each point of $X\!$ or $X'\!$ a proposition about states in $X'\!$ or $X,\!$ respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.
In sum, the intentional proposition $q\!$ indicates a method for the systematic selection of local goals. As a general form of description, a map of the type $q : \operatorname{Q}^i X \to \mathbb{B}$ may be referred to as an "$i^\operatorname{th}$ order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.
As applied here, the word intentional is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts — aims, ends, goals, objectives, purposes, and so on — metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like conative, contingent, discretionary, experimental, kinetic, progressive, tentative, or trial would probably serve as well.
### Life on Easy Street
Failing to fetch me at first keep encouraged, Missing me one place search another, I stop some where waiting for you
— Walt Whitman, Leaves of Grass, [Whi, 88]
The finite character of the extended universe $[\operatorname{E}\mathcal{A}]$ makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition $q : \operatorname{E}A \to \mathbb{B}$ is the set of models $q^{-1}(1)\!$ in $\operatorname{E}A.$ Finding all the models of $q,\!$ the extended interpretations in $\operatorname{E}A$ that satisfy $q,\!$ can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space $[\operatorname{E}\mathcal{A}]$ with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word forging takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.
## Back to the Beginning : Exemplary Universes
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.
— Michel Foucault, The Discourse on Language, [Fou, 215]
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.
### A One-Dimensional Universe
There was never any more inception than there is now, Nor any more youth or age than there is now; And will never be any more perfection than there is now, Nor any more heaven or hell than there is now.
— Walt Whitman, Leaves of Grass, [Whi, 28]
Let $\mathcal{X} = \{ x_1 \} = \{ A \}$ be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "$A\!$" in a more usual informal way, to name a feature and not a space, in departure from my formerly stated formal conventions. At any rate, the basis element $A = x_1\!$ may be interpreted as a simple proposition or a coordinate projection $A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.$ The space $X = \langle A \rangle = \{ (A), A \}$ of points (cells, vectors, interpretations) has cardinality $2^n = 2^1 = 2\!$ and is isomorphic to $\mathbb{B} = \{ 0, 1 \}.$ Moreover, $X\!$ may be identified with the set of singular propositions $\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.$ The space of linear propositions $X^* = \{ \operatorname{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}$ is algebraically dual to $X\!$ and also has cardinality $2.\!$ Here, "$0\!$" is interpreted as denoting the constant function $0 : \mathbb{B} \to \mathbb{B},$ amounting to the linear proposition of rank $0,\!$ while $A\!$ is the linear proposition of rank $1.\!$ Last but not least we have the positive propositions $\{ \operatorname{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},$ of rank $1\!$ and $0,\!$ respectively, where "$1\!$" is understood as denoting the constant function $1 : \mathbb{B} \to \mathbb{B}.$ In sum, there are $2^{2^n} = 2^{2^1} = 4$ propositions altogether in the universe of discourse, comprising the set $X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, (A), A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).$
The first order differential extension of $\mathcal{X}$ is $\operatorname{E}\mathcal{X} = \{ x_1, \operatorname{d}x_1 \} = \{ A, \operatorname{d}A \}.$ If the feature $A\!$ is understood as applying to some object or state, then the feature $\operatorname{d}A$ may be interpreted as an attribute of the same object or state that says that it is changing significantly with respect to the property $A,\!$ or that it has an escape velocity with respect to the state $A.\!$. In practice, differential features acquire their logical meaning through a class of temporal inference rules.
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that $A\!$ and $\operatorname{d}A$ are true at a given moment one may infer that $(A)\!$ will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:
| | | | | | | |
|------|------------------------------------------------------------------------------|-----|------------------------------------------------------------------------------|-------|------------------------------------------------------------------------------|-------|
| From | $(A)\!$ | and | $(\operatorname{d}A)\!$ | infer | $(A)\!$ | next. |
| From | $(A)\!$ | and | $\operatorname{d}A\!$ | infer | $A\!$ | next. |
| From | $A\!$ | and | $(\operatorname{d}A)\!$ | infer | $A\!$ | next. |
| From | $A\!$ | and | $\operatorname{d}A\!$ | infer | $(A)\!$ | next. |
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a clock — a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.
The clock indicates the moment . . . . but what does eternity indicate?
— Walt Whitman, Leaves of Grass, [Whi, 79]
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta $\{ (\operatorname{d}A), \operatorname{d}A \}$ are changed or unchanged in the next instance. In order to know this, one would have to determine $\operatorname{d}^2 A,$ and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that $\operatorname{d}^k A = 0$ for all $k\!$ greater than some fixed value $M.\!$ Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.
### Example 1. A Square Rigging
Urge and urge and urge, Always the procreant urge of the world.
— Walt Whitman, Leaves of Grass, [Whi, 28]
By way of example, suppose that we are given the initial condition $A = \operatorname{d}A$ and the law $\operatorname{d}^2 A = (A).$ Since the equation $A = \operatorname{d}A$ is logically equivalent to the disjunction $A\ \operatorname{d}A\ \operatorname{or}\ (A)(\operatorname{d}A),$ we may infer two possible trajectories, as displayed in Table 11. In either case the state $A\ (\operatorname{d}A)(\operatorname{d}^2 A)$ is a stable attractor or a terminal condition for both starting points.
Table 11. A Pair of Commodious Trajectories
$\operatorname{Time}$ $\operatorname{Trajectory}\ 1$ $\operatorname{Trajectory}\ 2$
0
1
2
3
4
| | | |
|-----|------|-------|
| A | dA | (d2A) |
| (A) | dA | d2A |
| A | (dA) | (d2A) |
| A | (dA) | (d2A) |
| " | " | " |
| | | |
|-----|------|-------|
| (A) | (dA) | d2A |
| (A) | dA | d2A |
| A | (dA) | (d2A) |
| A | (dA) | (d2A) |
| " | " | " |
Because the initial space X = 〈A〉 is one-dimensional, we can easily fit the second order extension E2X = 〈A, dA, d2A〉 within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.
Figure 12. The Anchor
If we eliminate from view the regions of E2X that are ruled out by the dynamic law d2A = (A), then what remains is the quotient structure that is shown in Figure 13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties A and d2A. As it happens, this fact might have been expressed "right off the bat" by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as (A, d2A).
Figure 13. The Tiller
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an n-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a n-cube without necessarily being forced to actualize all of its points.
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires "the infinite use of finite means". This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.
This consequence of dealing with extensions that are "practically infinite" becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.
### Back to the Feature
I guess it must be the flag of my disposition, out of hopeful green stuff woven.
— Walt Whitman, Leaves of Grass, [Whi, 31]
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that I may continue with outlining the structure of the differential extension [EX] = [A, dA]. Over the extended alphabet EX = {x1, dx1} = {A, dA}, of cardinality 2n = 2, we generate the set of points, EX, of cardinality 22n = 4, that bears the following chain of equivalent descriptions:
| | | |
|----|----|----------------------------------|
| EX | = | 〈A, dA〉 |
| | = | {(A), A} × {(dA), dA} |
| | = | {(A)(dA), (A) dA, A (dA), A dA}. |
The space EX may be assigned the mnemonic type B × D, which is really no different than B × B = B2. An individual element of EX may be regarded as a disposition at a point or a situated direction, in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.
To complete the construction of the extended universe of discourse EX • = [x1, dx1] = [A, dA], one must add the set of differential propositions EX^ = {g : EX → B} $\cong$ (B × D → B) to the set of dispositions in EX. There are $2^{2^{2n}}$ = 16 propositions in EX^, as detailed in Table 14.
Table 14. Differential Propositions
A : 1 1 0 0
dA : 1 0 1 0
f0 g0 0 0 0 0 ( ) False 0
g1 g2 g4 g8
0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
(A)(dA) (A) dA A (dA) A dA
Neither A nor dA Not A but dA A but not dA A and dA
¬A ∧ ¬dA ¬A ∧ dA A ∧ ¬dA A ∧ dA
f1 f2
g3 g12
0 0 1 1 1 1 0 0
(A) A
Not A A
¬A A
g6 g9
0 1 1 0 1 0 0 1
(A, dA) ((A, dA))
A not equal to dA A equal to dA
A ≠ dA A = dA
g5 g10
0 1 0 1 1 0 1 0
(dA) dA
Not dA dA
¬dA dA
g7 g11 g13 g14
0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0
(A dA) (A (dA)) ((A) dA) ((A)(dA))
Not both A and dA Not A without dA Not dA without A A or dA
¬A ∨ ¬dA A → dA A ← dA A ∨ dA
f3 g15 1 1 1 1 (( )) True 1
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for X^. Thus the first set of propositions {fi} is automatically embedded in the present set {gj}, and the corresponding inclusions are indicated at the far left margin of the table.
### Tacit Extensions
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.
— Michel Foucault, The Discourse on Language, [Fou, 215]
Strictly speaking, however, there is a subtle distinction in type between the function $f_i : X \to \mathbb{B}$ and the corresponding function $g_j : \operatorname{E}X \to \mathbb{B},$ even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet $\mathcal{X}$ is a subset of another alphabet $\mathcal{Y},$ then we say that any proposition $f : \langle \mathcal{X} \rangle \to \mathbb{B}$ has a tacit extension to a proposition $\epsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},$ and that the space $(\langle \mathcal{X} \rangle \to \mathbb{B})$ has an automatic embedding within the space $(\langle \mathcal{Y} \rangle \to \mathbb{B}).$ The extension is defined in such a way that $\epsilon f\!$ puts the same constraint on the variables of $\mathcal{X}$ that are contained in $\mathcal{Y}$ as the proposition $f\!$ initially did, while it puts no constraint on the variables of $\mathcal{Y}$ outside of $\mathcal{X},$ in effect, conjoining the two constraints.
If the variables in question are indexed as $\mathcal{X} = \{ x_1, \ldots, x_n \}$ and $\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},$ then the definition of the tacit extension from $\mathcal{X}$ to $\mathcal{Y}$ may be expressed in the form of an equation:
$\epsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) = f(x_1, \ldots, x_n).$
On formal occasions, such as the present context of definition, the tacit extension from $\mathcal{X}$ to $\mathcal{Y}$ is explicitly symbolized by the operator $\epsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),$ where the appropriate alphabets $\mathcal{X}$ and $\mathcal{Y}$ are understood from context, but normally one may leave the "$\epsilon\!$" silent.
Let's explore what this means for the present Example. Here, $\mathcal{X} = \{ A \}$ and $\mathcal{Y} = \operatorname{E}\mathcal{X} = \{ A, \operatorname{d}A \}.$ For each of the propositions $f_i\!$ over $X\!,$ specifically, those whose expression $e_i\!$ lies in the collection $\{ 0, (A), A, 1 \},\!$ the tacit extension $\epsilon f\!$ of $f\!$ to $\operatorname{E}X$ can be phrased as a logical conjunction of two factors, $f_i = e_i \cdot \tau\ ,$ where $\tau\!$ is a logical tautology that uses all the variables of $\mathcal{Y} - \mathcal{X}.$ Working in these terms, the tacit extensions $\epsilon f\!$ of $f\!$ to $\operatorname{E}X$ may be explicated as shown in Table 15.
Table 15. Tacit Extension of $[A]\!$ to $[A, \operatorname{d}A]$
| | | | | | | |
|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|------------------------------------------------------------------------------|
| $0\!$ | $=\!$ | $0\!$ | $\cdot\!$ | $((\operatorname{d}A),\ \operatorname{d}A)\!$ | $=\!$ | $0\!$ |
| $(A)\!$ | $=\!$ | $(A)\!$ | $\cdot\!$ | $((\operatorname{d}A),\ \operatorname{d}A)\!$ | $=\!$ | $(A)(\operatorname{d}A)\ +\ (A)\ \operatorname{d}A\!$ |
| $A\!$ | $=\!$ | $A\!$ | $\cdot\!$ | $((\operatorname{d}A),\ \operatorname{d}A)\!$ | $=\!$ | $A\ (\operatorname{d}A)\ +\ A\ \operatorname{d}A\!$ |
| $1\!$ | $=\!$ | $1\!$ | $\cdot\!$ | $((\operatorname{d}A),\ \operatorname{d}A)\!$ | $=\!$ | $1\!$ |
In its effect on the singular propositions over $X,\!$ this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like $A\!$ or $(A),\!$ to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.
### Example 2. Drives and Their Vicissitudes
I open my scuttle at night and see the far-sprinkled systems, And all I see, multiplied as high as I can cipher, edge but the rim of the farther systems.
— Walt Whitman, Leaves of Grass, [Whi, 81]
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.
Again, let X = {x1} = {A}. In the discussion that follows I will consider a class of trajectories having the property that dkA = 0 for all k greater than some fixed m, and I indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference dmA exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature dmA the drive at that point. Curves of constant drive dmA are then referred to as "mth gear curves".
• Scholium. The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, § 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].
Given this language, the particular Example that I take up here can be described as the family of 4th gear curves through E4X = 〈A, dA, d2A, d3A, d4A〉. These are the trajectories generated subject to the dynamic law d4A = 1, where it is understood in such a statement that all higher order differences are equal to 0. Since d4A and all higher dkA are fixed, the temporal or transitional conditions (initial, mediate, terminal — transient or stable states) vary only with respect to their projections as points of E3X = 〈A, dA, d2A, d3A〉. Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E3X. It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.
Figure 16. A Couple of Fourth Gear Orbits
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states q in EmX with the dyadic rationals (or the binary fractions) in the half-open interval [0, 2). Formally and canonically, a state qr is indexed by a fraction r = s/t whose denominator is the power of two t = 2m and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next. The differential coefficients of the state q are just the values dkA(q), for k = 0 to m, where d0A is defined as being identical to A. To form the binary index d0.d1…dm of the state q the coefficient dkA(q) is read off as the binary digit dk associated with the place value 2–k. Expressed by way of algebraic formulas, the rational index r of the state q can be given by the following equivalent formulations:
| | | | | |
|------------------------------------------------------------------------------|----|------------------------------------------------------------------------------|----|------------------------------------------------------------------------------|
| $r(q)\!$ | = | $\sum_k d_k \cdot 2^{-k}$ | = | $\sum_k \mbox{d}^k A(q) \cdot 2^{-k}$ |
| = | | | | |
| $\frac{s(q)}{t}$ | = | $\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}$ | = | $\frac{\sum_k \mbox{d}^k A(q) \cdot 2^{(m-k)}}{2^m}$ |
Applied to the example of fourth gear curves, this scheme results in the data of Tables 17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs ‹pi, qj›, where pi may be read as a temporal parameter that indicates the present time of the state, and where j is the decimal equivalent of the binary numeral s. Informally and more casually, the Tables exhibit the states qs as subscripted with the numerators of their rational indices, taking for granted the constant denominators of 2m = 24 = 16. Within this set-up, the temporal successions of states can be reckoned as given by a kind of parallel round-up rule. That is, if ‹dk, dk+1› is any pair of adjacent digits in the state index r, then the value of dk in the next state is dk′ = dk + dk+1.
Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1
Time State A dA
pi qj d0A d1A d2A d3A d4A
p0
p1
p2
p3
p4
p5
p6
p7
q01
q03
q05
q15
q17
q19
q21
q31
| | | | | |
|----|----|----|----|----|
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 1 |
| 0 | 1 | 1 | 1 | 1 |
| 1 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 1 | 1 | 1 |
Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2
Time State A dA
pi qj d0A d1A d2A d3A d4A
p0
p1
p2
p3
p4
p5
p6
p7
q25
q11
q29
q07
q09
q27
q13
q23
| | | | | |
|----|----|----|----|----|
| 1 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 | 1 |
| 1 | 1 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 1 | 1 |
## Transformations of Discourse
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the fons et origo of an unfathomable transformation.
— Robert Musil, The Man Without Qualities, [Mus, 39]
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.
As a first step I discuss the kinds of transformations that we already know as extensions and projections, and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.
### Foreshadowing Transformations : Extensions and Projections of Discourse
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.
— Gaston Leroux, The Phantom of the Opera, [Ler, 126]
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type [X] → [Y] is implied any time that we make use of one alphabet X that happens to be included in another alphabet Y. When we are discussing differential issues we usually have in mind that the extended alphabet Y has a special construction or a specific lexical relation with respect to the initial alphabet X, one that is marked by characteristic types of accents, indices, or inflected forms.
#### Extension from 1 to 2 Dimensions
Figure 18-a lays out the angular form of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type B1 → B2 and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an areal view of each universe of discourse.
Figure 18-a. Extension from 1 to 2 Dimensions: Areal
Figure 18-b shows the differential extension from X • = [x] to EX • = [x, dx] in a bundle of boxes form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a proposition at a point, in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
Figure 18-c shows the same extension in a compact style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or digraph form of representation. (Notice that my definition of a digraph allows for loops or slings at individual points, in addition to arcs or arrows between the points.)
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
#### Extension from 2 to 4 Dimensions
Figure 19-a lays out the areal view or the angular form of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type B2 → B4. In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
Figure 19-b shows the differential extension from U • = [u, v] to EU • = [u, v, du, dv] in the bundle of boxes form of venn diagram.
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.
Figure 19-c illustrates the extension from 2 to 4 dimensions in the compact style of venn diagram. Here, just the changes with respect to the center cell are shown.
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
Figure 19-d gives the digraph form of representation for the differential extension U • → EU •, where the 4 nodes marked "@" are the cells uv, u(v), (u)v, (u)(v), respectively, and where a 2-headed arc counts as two arcs of the differential digraph.
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
### Thematization of Functions : And a Declaration of Independence for Variables
And as imagination bodies forth The forms of things unknown, the poet's pen Turns them to shapes, and gives to airy nothing A local habitation and a name. A Midsummer Night's Dream, 5.1.18
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.
#### Thematization : Venn Diagrams
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.
— Walt Whitman, Leaves of Grass, [Whi, 11–12]
Figure 20-i traces the first couple of steps in this order of thematic progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when one considers the proposition u·v in [u, v].
The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition u·v a distinctive functional name "J ". Second, one has come to think explicitly about the target domain that contains the functional values of J, as when one writes J : 〈u, v〉 → B.
Figure 20-i. Thematization of Conjunction (Stage 1)
In Figure 20-ii the proposition J is viewed explicitly as a transformation from one universe of discourse to another.
Figure 20-ii. Thematization of Conjunction (Stage 2)
```o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ / \ /
\ / \ /
\ / \ J /
\ / \ /
\ / \ /
o----------\---------/----------o o----------\---------/----------o
| \ / | | \ / |
| \ / | | \ / |
| o-----@-----o | | o-----@-----o |
| /`````````````\ | | /`````````````\ |
| /```````````````\ | | /```````````````\ |
| /`````````````````\ | | /`````````````````\ |
| o```````````````````o | | o```````````````````o |
| |```````````````````| | | |```````````````````| |
| |```````` J ````````| | | |```````` x ````````| |
| |```````````````````| | | |```````````````````| |
| o```````````````````o | | o```````````````````o |
| \`````````````````/ | | \`````````````````/ |
| \```````````````/ | | \```````````````/ |
| \`````````````/ | | \`````````````/ |
| o-----------o | | o-----------o |
| | | |
| | | |
o-------------------------------o o-------------------------------o
J = u v x = J<u, v>
Figure 20-ii. Thematization of Conjunction (Stage 2)
```
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function J : 〈u, v〉 → B to serve as the name of its dependent variable J : B does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when one writes J : 〈u, v〉 → 〈x〉 and thereby assigns a concrete type 〈x〉 to the abstract codomain B. To make this induction of variables more formal one can append subscripts, as in xJ, to indicate the origin or the derivation of these parvenu characters. However, it is not always convenient to keep inventing new variable names in this way. For use at these times, I introduce a lexical operator "¢", read cents or obelus, that converts a function name into a variable name. For example, one may think of x = xJ = ¢(J) = J ¢ = J ¢ as "the cache variable of J ", "J circumscript", "J made circumstantial", or "J considered as a contingent variable".
In Figure 20-iii we arrive at a stage where the functional equations, J = u·v and x = u·v, are regarded as propositions in their own right, reigning in and ruling over 3-feature universes of discourse, [u, v, J] and [u, v, x], respectively. Subject to the cautions already noted, the function name "J " can be reinterpreted as the name of a feature J ¢, and the equation J = u·v can be read as the logical equivalence ((J, u v)). To give it a generic name let us call this newly expressed, collateral proposition the thematization or the thematic extension of the original proposition J.
Figure 20-iii. Thematization of Conjunction (Stage 3)
The first venn diagram represents the thematization of the conjunction J with shading in the appropriate regions of the universe [u, v, J]. Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name "J " are resolved by introducing a new variable name "x " to take the place of J ¢, and the region that represents this fresh featured x is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name "J " to the proposition u·v, we now give the name "ι" to its thematization ((x, u v)). Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function ι : 〈u, v, x〉 → B.
From now on, the terms thematic extension and thematization will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from J to ι, I introduce a class of operators symbolized by the Greek letter θ, writing ι = θJ in the present instance. The operator θ, in the present situation bearing the type θ : [u, v] →> [u, v, x], provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.
Figure 21 shows how the thematic extension operator θ acts on two further examples, the disjunction ((u)(v)) and the equality ((u, v)). Referring to the disjunction as f‹u, v› and the equality as g‹u, v›, I write the thematic extensions as φ = θf and γ = θg.
Figure 21. Thematization of Disjunction and Equality
#### Thematization : Truth Tables
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.
— Walt Whitman, Leaves of Grass, [Whi, 19]
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.
A preliminary step, as illustrated in Table 22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions f‹u, v› = ((u)(v)) and g‹u, v› = ((u, v)).
Table 22. Disjunction f and Equality g
u v
f g
0 0
0 1
1 0
1 1
0 1
1 0
1 0
1 1
Next, each propositional form is individually represented in the fashion shown in Tables 23-i and 23-ii, using "f " and "g " as function names and creating new variables x and y to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the 2-dimensional universes of f and g to the 3-dimensional universes of θf and θg. The top halves of the Tables replicate the truth table patterns for f and g in the form f : [u, v] → [x] and g : [u, v] → [y]. The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for (f) and (g) under the copies for f and g. At this stage, the columns for θf and θg are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions f and g.
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)
Table 23-i. Disjunction f
u v f
x φ
| | | |
|----|----|----|
| 0 | 0 | → |
| 0 | 1 | → |
| 1 | 0 | → |
| 1 | 1 | → |
0 1
1 1
1 1
1 1
0 0
0 1
1 0
1 1
1 0
0 0
0 0
0 0
Table 23-ii. Equality g
u v g
y γ
| | | |
|----|----|----|
| 0 | 0 | → |
| 0 | 1 | → |
| 1 | 0 | → |
| 1 | 1 | → |
1 1
0 1
0 1
1 1
0 0
0 1
1 0
1 1
0 0
1 0
1 0
0 0
All the data are now in place to give the truth tables for θf and θg. In the remaining steps all we do is to permute the rows and change the roles of x and y from dependent to independent variables. In Tables 24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples ‹u, v, x› and ‹u, v, y› in binary numerical order, suitable for viewing as the arguments of the maps θf = φ : [u, v, x] → B and θg = γ : [u, v, y] → B. Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions f and g to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables x := f ¢ and y := g ¢ are now to be regarded as independent variables.
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)
Table 24-i. Disjunction f
u v f x
φ
| | | | |
|----|----|----|----|
| 0 | 0 | → | 0 |
| 0 | 0 | | 1 |
| 0 | 1 | | 0 |
| 0 | 1 | → | 1 |
1
0
0
1
| | | | |
|----|----|----|----|
| 1 | 0 | | 0 |
| 1 | 0 | → | 1 |
| 1 | 1 | | 0 |
| 1 | 1 | → | 1 |
0
1
0
1
Table 24-ii. Equality g
u v g y
γ
| | | | |
|----|----|----|----|
| 0 | 0 | | 0 |
| 0 | 0 | → | 1 |
| 0 | 1 | → | 0 |
| 0 | 1 | | 1 |
0
1
1
0
| | | | |
|----|----|----|----|
| 1 | 0 | → | 0 |
| 1 | 0 | | 1 |
| 1 | 1 | | 0 |
| 1 | 1 | → | 1 |
1
0
0
1
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables 25-i and 25-ii sort the rows in a different order, in effect treating x and y as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form φ : [x, u, v] → B and γ : [y, u, v] → B makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable F ¢ is true then θF exhibits the pattern of the original F, and when F ¢ is false then θF exhibits the pattern of its negation (F).
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)
Table 25-i. Disjunction f
u v f x
φ
| | | | |
|----|----|----|----|
| 0 | 0 | → | 0 |
| 0 | 1 | | 0 |
| 1 | 0 | | 0 |
| 1 | 1 | | 0 |
1
0
0
0
| | | | |
|----|----|----|----|
| 0 | 0 | | 1 |
| 0 | 1 | → | 1 |
| 1 | 0 | → | 1 |
| 1 | 1 | → | 1 |
0
1
1
1
Table 25-ii. Equality g
u v g y
γ
| | | | |
|----|----|----|----|
| 0 | 0 | | 0 |
| 0 | 1 | → | 0 |
| 1 | 0 | → | 0 |
| 1 | 1 | | 0 |
0
1
1
0
| | | | |
|----|----|----|----|
| 0 | 0 | → | 1 |
| 0 | 1 | | 1 |
| 1 | 0 | | 1 |
| 1 | 1 | → | 1 |
1
0
0
1
Finally, Tables 26-i and 26-ii compare the tacit extensions ε : [u, v] → [u, v, x] and ε : [u, v] → [u, v, y] with the thematic extensions of the same types, as applied to the propositions f and g, respectively.
Tables 26-i and 26-ii. Tacit Extension and Thematization
Table 26-i. Disjunction f
u v x
εf θf
| | | |
|----|----|----|
| 0 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 1 | 1 |
0 1
0 0
1 0
1 1
| | | |
|----|----|----|
| 1 | 0 | 0 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| 1 | 1 | 1 |
1 0
1 1
1 0
1 1
Table 26-ii. Equality g
u v y
εg θg
| | | |
|----|----|----|
| 0 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 0 | 1 | 1 |
1 0
1 1
0 1
0 0
| | | |
|----|----|----|
| 1 | 0 | 0 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| 1 | 1 | 1 |
0 1
0 0
1 0
1 1
Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form (( f ¢ , f ¢‹u, v› )) and Column 5 simplifies these equations into the form of algebraic expressions. (As always, "+" refers to exclusive disjunction, and "f " should be read as "fi¢" in the body of the Table.)
Table 27. Thematization of Bivariate Propositions
u :
v :
1100
1010
f
θf
θf
f0
f1
f2
f3
f4
f5
f6
f7
0000
0001
0010
0011
0100
0101
0110
0111
()
(u)(v)
(u) v
(u)
u (v)
(v)
(u, v)
(u v)
(( f , () ))
(( f , (u)(v) ))
(( f , (u) v ))
(( f , (u) ))
(( f , u (v) ))
(( f , (v) ))
(( f , (u, v) ))
(( f , (u v) ))
f + 1
f + u + v + uv
f + v + uv + 1
f + u
f + u + uv + 1
f + v
f + u + v + 1
f + uv
f8
f9
f10
f11
f12
f13
f14
f15
1000
1001
1010
1011
1100
1101
1110
1111
u v
((u, v))
v
(u (v))
u
((u) v)
((u)(v))
(())
(( f , u v ))
(( f , ((u, v)) ))
(( f , v ))
(( f , (u (v)) ))
(( f , u ))
(( f , ((u) v) ))
(( f , ((u)(v)) ))
(( f , (()) ))
f + uv + 1
f + u + v
f + v + 1
f + u + uv
f + u + 1
f + v + uv
f + u + v + uv + 1
f
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables 28 and 29 present ordinary truth tables for the functions fi : B2 → B and for the corresponding thematizations θfi = φi : B3 → B.
Table 28. Propositions on Two Variables
| | | | | | | | | | | | | | | | | | |
|----|----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| u | v | f00 | f01 | f02 | f03 | f04 | f05 | f06 | f07 | f08 | f09 | f10 | f11 | f12 | f13 | f14 | f15 |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Table 29. Thematic Extensions of Bivariate Propositions
| | | | | | | | | | | | | | | | | | | |
|----|----|----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| u | v | f¢ | φ00 | φ01 | φ02 | φ03 | φ04 | φ05 | φ06 | φ07 | φ08 | φ09 | φ10 | φ11 | φ12 | φ13 | φ14 | φ15 |
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
### Propositional Transformations
If only the word ‘artificial’ were associated with the idea of art, or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that logical refers to artificial thought.
— John Dewey, How We Think, [Dew, 56–57]
In this Subdivision I develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general context the source and the target universes of a transformation are allowed to be distinct, but may also be one and the same. When these concepts are applied to dynamic systems one focuses on the important special cases of transformations that map a universe into itself, and transformations of this shape may be interpreted as the state transitions of a discrete dynamical process, as these take place among the myriad ways that a universe of discourse might change, and by that change turn into itself.
#### Alias and Alibi Transformations
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:
1. A perspectival or alias transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.
2. A transitional or alibi transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.
#### Transformations of General Type
Es ist passiert, "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
— Robert Musil, The Man Without Qualities, [Mus, 34]
Consider the situation illustrated in Figure 30, where the alphabets U = {u, v} and X = {x, y, z} are used to label basic features in two different logical universes, U • = [u, v] and X • = [x, y, z].
``` o-------------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------o---------------------------o
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
o-------------------------o o-------------------------o o-------------------------o
| U | | U | | U |
| o---o o---o | | o---o o---o | | o---o o---o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / / \ \ | | / / \ \ | | / / \ \ |
| o o o o | | o o o o | | o o o o |
| | u | | v | | | | u | | v | | | | u | | v | |
| o o o o | | o o o o | | o o o o |
| \ \ / / | | \ \ / / | | \ \ / / |
| \ o / | | \ o / | | \ o / |
| \ / \ / | | \ / \ / | | \ / \ / |
| o---o o---o | | o---o o---o | | o---o o---o |
| | | | | |
o-------------------------o o-------------------------o o-------------------------o
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o x o | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o
Figure 30. Generic Frame of a Logical Transformation
```
Enter the picture, as we usually do, in the middle of things, with features like x, y, z that present themselves to be simple enough in their own right and that form a satisfactory, if a temporary, foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps p, q : X → B. Then we discover that the simple features {x, y, z} are really more complex than we thought at first, and it becomes useful to regard them as functions {f, g, h} of other features {u, v}, that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse U • = [u, v]. It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.
A particular transformation F : [u, v] → [x, y, z] may be expressed by a system of equations, as shown below. Here, F is defined by its component maps F = ‹F1, F2, F3› = ‹f, g, h›, where each component map in {f, g, h} is a proposition of type Bn → B1.
| | | |
|----|----|---------|
| x | = | f‹u, v› |
| y | = | g‹u, v› |
| z | = | h‹u, v› |
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions {f, g, h} in one universe of discourse and the special collection of simple propositions {x, y, z} on which are founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.
### Analytic Expansions : Operators and Functors
Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object.
— C.S. Peirce, "The Maxim of Pragmatism", CP 5.438
Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.
#### Operators on Propositions and Transformations
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to "get the drift" of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators that I will explicitly consider here are of this kind. Figure 31 illustrates the typical situation.
```o---------------------------------------o
| |
| |
| U% F X% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| !W! | | !W! |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| !W!U% !W!F !W!X% |
| |
| |
o---------------------------------------o
Figure 31. Operator Diagram (1)
```
In this Figure "W" serves as a generic name for an operator, in this case one that takes a logical transformation F of type (U • → X •) into a logical transformation WF of type (WU • → WX •). Thus, the operator W must be viewed as making assignments for both families of objects that we have previously considered, both for universes of discourse like U • and X • and for logical transformations like F.
NB. Strictly speaking, an operator like W works between two whole categories of universes and transformations, which we call the source and the target categories of W. Given this setting, W specifies for each universe U • in its source category a definite universe WU • in its target category, and to each transformation F in its source category it assigns a unique transformation WF in its target category. Naturally, this only works if W takes the source U • and the target X • of the map F over to the source WU • and the target WX • of the map WF. With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation F, and thus we can take it for granted that the assignment of universes under W is defined appropriately at the source and the target ends of F. It is not always the case, though, that we need to use the particular names (like "WU •" and "WX •") that W assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names, and it is only necessary that we can tell from the information associated with an operator W what universes they are.
In Figure 31 the maps F and WF are displayed horizontally, the way that one normally orients functional arrows in a written text, and W rolls the map F downward into the images that are associated with WF. In Figure 32 the same information is redrawn so that the maps F and WF flow down the page, and W unfurls the map F rightward into domains that are the eminent purview of WF.
```o---------------------------------------o
| |
| |
| U% !W! !W!U% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| F | | !W!F |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| X% !W! !W!X% |
| |
| |
o---------------------------------------o
Figure 32. Operator Diagram (2)
```
The latter arrangement, as it appears in Figure 32, is more congruent with the thinking about operators that we shall be doing in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30.
#### Differential Analysis of Propositions and Transformations
The resultant metaphysical problem now is this: Does the man go round the squirrel or not?
— William James, Pragmatism, [Jam, 43]
The approach to the differential analysis of logical propositions and transformations of discourse that will be pursued here is carried out in terms of particular operators W that act on propositions F or on transformations F to yield the corresponding operator maps WF. The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.
• Remark on Strategy. At this point I run into a set of conceptual difficulties that force me to make a strategic choice in how I proceed. Part of the problem can be remedied by extending my discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead me to try two different types of solution. The approach that I develop first makes use of a variant type of extension operator, the trope extension, to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of contingency spaces. These are an even more generous type of extended universe than the kind I currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces me to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, I call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well my first approach deals with them.
I now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form W : (U • → X •) → (EU • → EX •). If we assume that the source universe U • and the target universe X • have finite dimensions n and k, respectively, then each operator W is encompassed by the same abstract type:
W : ( [ Bn ] → [ Bk ] ) → ( [ Bn × Dn ] → [ Bk × Dk ] ) .
Since the range features of the operator result WF : [Bn × Dn] → [Bk × Dk] can be sorted out by their ordinary versus their differential qualities and the component maps can be examined independently, the complete operator W can be separated accordingly into two components, in the form W = ‹ε, W›. Given a fixed context of source and target universes of discourse, ε is always the same type of operator, a multiple component elaboration of the tacit extension operators that were articulated earlier. In this context ε has the shape:
| | | | | | | | | | | | | | |
|---------------|----|----|----|------|----|------|----|----|----|-----------|----|------|----|
| Concrete type | ε | : | ( | U • | → | X • | ) | → | ( | EU • | → | X • | ) |
| Abstract type | ε | : | ( | [Bn] | → | [Bk] | ) | → | ( | [Bn × Dn] | → | [Bk] | ) |
On the other hand, the operator W is specific to each W. In this context W always has the form:
| | | | | | | | | | | | | | |
|---------------|----|----|----|------|----|------|----|----|----|-----------|----|------|----|
| Concrete type | W | : | ( | U • | → | X • | ) | → | ( | EU • | → | dX • | ) |
| Abstract type | W | : | ( | [Bn] | → | [Bk] | ) | → | ( | [Bn × Dn] | → | [Dk] | ) |
In the types just assigned to ε and W, and implicitly to their results εF and WF, I have listed the most restrictive ranges defined for them, rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:
| | | | | | | | | | | | | | | | | |
|----|----|----|------|----|------|----|------|----|------------------------------------------------------------------------------|----|-----------|----|------|----|-----------|----|
| εF | : | ( | EU • | → | X • | ⊆ | EX • | ) | $\cong$ | ( | [Bn × Dn] | → | [Bk] | ⊆ | [Bk × Dk] | ) |
| WF | : | ( | EU • | → | dX • | ⊆ | EX • | ) | $\cong$ | ( | [Bn × Dn] | → | [Dk] | ⊆ | [Bk × Dk] | ) |
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.
In giving names to these operators I am attempting to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the operators W and their components W, which forces me to find two distinct but parallel sets of terminology. Here is the plan that I have settled on. First, the component operators W are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators W = ‹ε, W› are assigned their titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition I am still working toward, comes out fit with its customary name. Finally, the operator results WF and WF can be fixed in this frame of reference by tethering the operative adjective for W or W to the anchoring epithet map, in conformity with an already standard practice.
##### The Secant Operator : E
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.
— William James, Pragmatism, [Jam, 46]
Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted "E", which receives the principal investment of analytic attention, and on the constituent parts of E, which derive their shares of significance as developed by the analysis. In the sequel, I refer to E as the secant operator, taking it for granted that a context has been chosen that defines its type. The secant operator has the component description E = ‹ε, E›, and its active ingredient E is known as the enlargement operator. (Here, I have named E after the literal ancestor of the shift operator in the calculus of finite differences, defined so that Ef(x) = f(x+1) for any suitable function f, though of course the logical analogue that we take up here must have a rather different definition.)
```U% $E$ $E$U% $E$U% $E$U%
o------------------>o============o============o
| | | |
| | | |
| | | |
| | | |
F | | $E$F = | $d$^0.F + | $r$^0.F
| | | |
| | | |
| | | |
v v v v
o------------------>o============o============o
X% $E$ $E$X% $E$X% $E$X%
Figure 33-i. Analytic Diagram (1)
```
```U% $E$ $E$U% $E$U% $E$U% $E$U%
o------------------>o============o============o============o
| | | | |
| | | | |
| | | | |
| | | | |
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F
| | | | |
| | | | |
| | | | |
v v v v v
o------------------>o============o============o============o
X% $E$ $E$X% $E$X% $E$X% $E$X%
Figure 33-ii. Analytic Diagram (2)
```
In its action on universes E yields the same result as E, a fact that can be expressed in equational form by writing EU • = EU • for any universe U •. Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of EF are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure 30.
Acting on a transformation F from universe U • to universe X •, the operator E determines a transformation EF from EU • to EX •. The map EF forms the main body of evidence to be investigated in performing a differential analysis of F. Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the "big picture", it is critically important to emphasize that the map EF is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation F until we can lay out the full "parts diagram" of EF along the lines of the generic frame in Figure 30.
If one is working within the confines of propositional calculus, it is possible to give an elementary definition of EF by means of a system of propositional equations, as will now be described.
Given a transformation:
F = ‹F1, …, Fk› : Bn → Bk
of concrete type:
F : [u1, …, un] → [x1, …, xk]
the transformation:
EF = ‹F1, …, Fk, EF1, …, EFk› : Bn × Dn → Bk × Dk
of concrete type:
EF : [u1, …, un, du1, …, dun] → [x1, …, xk, dx1, …, dxk]
is defined by means of the following system of logical equations:
| | | | | |
|-----|----|-----------------------------|----|---------------|
| x1 | = | εF1‹u1, …, un, du1, …, dun› | = | F1‹u1, …, un› |
| ... | | | | |
| xk | = | εFk‹u1, …, un, du1, …, dun› | = | Fk‹u1, …, un› |
| | | | | |
|-----|----|-----------------------------|----|---------------------------|
| dx1 | = | EF1‹u1, …, un, du1, …, dun› | = | F1‹u1 + du1, …, un + dun› |
| ... | | | | |
| dxk | = | EFk‹u1, …, un, du1, …, dun› | = | Fk‹u1 + du1, …, un + dun› |
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse that is generated by all of the named variables. Specifically, this is the universe of discourse over 2(n+k) variables that is denoted by:
E[U ∪ X] = [u1, …, un, x1, …, xk, du1, …, dun, dx1, …, dxk].
In this light, it should be clear that the system of equations defining EF embodies, in a higher rank and in a differentially extended version, an analogy with the process of thematization that was treated earlier for propositions of the type F : Bn → B.
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing EF = ‹εF, EF›, for any map F. This is tantamount to regarding E as a complex operator, E = ‹ε, E›, with a form of application that distributes each component of the operator to work on each component of the operand:
EF = ‹ε, E›F = ‹εF, EF› = ‹εF1, …, εFk, EF1, …, EFk›.
Quite a lot of "thematic infrastructure" or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the angle brackets, which were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves, but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the angle bracket notation ‹ , › can be regarded as a kind of thematic frame, an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of EF.
The generic notations d0F, d1F, …, dmF in Figure 33 refer to the increasing orders of differentials that are extracted in the course of analyzing F. When the analysis is halted at a partial stage of development, notations like r0F, r1F, …, rmF may be used to summarize the contributions to EF that remain to be analyzed. The Figure illustrates a convention that renders the remainder term rmF, in effect, the sum of all differentials of order strictly greater than m.
I next discuss the set of operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number I will introduce along the way.
##### The Radius Operator : e
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.
— William James, Pragmatism, [Jam, 46]
The operator identified as d0 in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for F in the appropriately extended context. Construed in terms of its broadest components, d0 is equivalent to the doubly tacit extension operator ‹ε, ε›, in recognition of which let us redub it as "e". Pursuing a geometric analogy, we may refer to e = ‹ε, ε› = d0 as the radius operator. The operation that is intended by all of these forms is defined by the equation:
| | | |
|----|----|------------------------------|
| eF | = | ‹ε, ε›F |
| | = | ‹εF, εF› |
| | = | ‹εF1, …, εFk, εF1, …, εFk› , |
which is tantamount to the system of equations given below.
| | | | | |
|-----|----|-----------------------------|----|---------------|
| x1 | = | εF1‹u1, …, un, du1, …, dun› | = | F1‹u1, …, un› |
| ... | | | | |
| xk | = | εFk‹u1, …, un, du1, …, dun› | = | Fk‹u1, …, un› |
| | | | | |
|-----|----|-----------------------------|----|---------------|
| dx1 | = | εF1‹u1, …, un, du1, …, dun› | = | F1‹u1, …, un› |
| ... | | | | |
| dxk | = | εFk‹u1, …, un, du1, …, dun› | = | Fk‹u1, …, un› |
##### The Phantom of the Operators : η
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, which was playing the most perfect music!
— Gaston Leroux, The Phantom of the Opera, [Ler, 81]
I now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost me some painstaking trouble to detect. In the end I shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.
Given a transformation F : [u1, …, un] → [x1, …, xk], we often need to make a separate treatment of a related family of transformations of the form F* : [u1, …, un, du1, …, dun] → [dx1, …, dxk]. The operator η is introduced to deal with the simplest one of these maps:
ηF : [u1, …, un, du1, …, dun] → [dx1, …, dxk]
which is defined by the equations:
| | | | | |
|-----|----|-----------------------------|----|---------------|
| dx1 | = | εF1‹u1, …, un, du1, …, dun› | = | F1‹u1, …, un› |
| ... | | | | |
| dxk | = | εFk‹u1, …, un, du1, …, dun› | = | Fk‹u1, …, un› |
In effect, the operator η is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator e. Operating independently, η achieves precisely the same results that the second ε in ‹ε, ε› accomplishes by working within the context of its adjuvant thematic frame, "‹ , ›". From this point on, because the use of ε and η in this setting combines the aims of both the tacit and the thematic extensions, and because η reflects in regard to ε little more than the application of a differential twist, a mere turn of phrase, I refer to η as the trope extension operator.
##### The Chord Operator : D
What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.
— William James, Pragmatism, [Jam, 45]
Next I discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.
This is the operator that is referred to as r0 in the initial stage of analysis (Figure 33-i), and that is expanded as d1 + r1 in the subsequent step (Figure 33-ii). In congruence, but not quite harmony, with my allusions of analogy that are not quite geometry, I call this the chord operator and denote it D. In the more casual terms that are here introduced, D is defined as the remainder of E and e, and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise E and the bar of exigency e.
The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we may write D = ‹ε, D›, calling D the difference operator and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord D is not one that need be lost at any stage of development. At the mth stage of play it can always be reconstituted in the following form:
| | | |
|----|----|----------------------|
| D | = | E – e |
| | = | r0 |
| | = | d1 + r1 |
| | = | d1 + … + dm + rm |
| | = | ∑(i = 1 … m) di + rm |
##### The Tangent Operator : T
They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.
— William James, Pragmatism, [Jam, 300]
The operator tagged as d1 in the analytic diagram (Figure 33) is called the tangent operator, and is usually denoted in this text as d or T. Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composure among transformations, it also earns the title of a tangent functor. According to the custom adopted here, we dissect it as T = d = ‹ε, d›, where d is the operator that yields the first order differential dF when applied to a transformation F, and whose name is legion.
Figure 34 illustrates a stage of analysis where we ignore everything but the tangent functor T, and attend to it chiefly as it bears on the first order differential dF in the analytic expansion of F. In this situation, we often refer to the extended universes EU • and EX • under the equivalent designations TU • and TX •, respectively. The purpose of the tangent functor T is to extract the tangent map TF at each point of U •, and the tangent map TF = ‹ε, d›F tells us not only what the transformation F is doing at each point of the universe U • but also what F is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.
```U% $T$ $T$U% $T$U%
o------------------>o============o
| | |
| | |
| | |
| | |
F | | $T$F = | <!e!, d> F
| | |
| | |
| | |
v v v
o------------------>o============o
X% $T$ $T$X% $T$X%
Figure 34. Tangent Functor Diagram
```
• NB. There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators W in {η, E, D, d, r} so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps WF have equally good reasons for placing their values in differential stocks. The only explanation I can devise at present is that, without doing this, I cannot justify the comparison and combination of their values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now, the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.
### Transformations of Type B2 → B1
To study the effects of these analytic operators in the simplest possible situation, let us revert to a still more primitive case. Consider the singular proposition J‹u, v› = uv, regarded either as the functional product of the maps u and v or as the logical conjunction of the features u and v, a map whose fiber of truth J–1(1) picks out the single cell of that logical description in the universe of discourse U •. Thus J, or uv, may be treated as a pseudonym for the point whose coordinates are ‹1, 1› in U •.
#### Analytic Expansion of Conjunction
In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a soul. What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.
— Robert Musil, The Man Without Qualities, [Mus, 118]
Figure 35 pictures the form of conjunction J : B2 → B as a transformation from the 2-dimensional universe [u, v] to the 1-dimensional universe [x]. This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition J : 〈u, v〉 → B is being recast into the thematized role of a transformation J : [u, v] → [x], where the new variable x takes the part of a thematic variable ¢(J).
Figure 35. Conjunction as Transformation
##### Tacit Extension of Conjunction
I teach straying from me, yet who can stray from me? I follow you whoever you are from the present hour; My words itch at your ears till you understand them.
— Walt Whitman, Leaves of Grass, [Whi, 83]
Earlier I defined the tacit extension operators ε : X • → Y • as maps embedding each proposition of a given universe X • in a more generously given universe Y • containing X •. Of immediate interest are the tacit extensions ε : U • → EU •, that locate each proposition of U • in the enlarged context of EU •. In its application to the propositional conjunction J = u v in [u, v], the tacit extension operator ε produces the proposition εJ in EU • = [u, v, du, dv]. The extended proposition εJ may be computed according to the scheme in Table 36, in effect, doing nothing more than conjoining a tautology of [du, dv] to J in U •.
Table 36. Computation of εJ
| | | | | | | | | |
|----|----|--------------|----|-------------|----|-------------|----|-----------|
| εJ | = | J‹u, v› | | | | | | |
| | = | u v | | | | | | |
| | = | u v (du)(dv) | + | u v (du) dv | + | u v du (dv) | + | u v du dv |
| | | | |
|----|----|--------------|----|
| εJ | = | u v (du)(dv) | + |
| | | u v (du) dv | + |
| | | u v du (dv) | + |
| | | u v du dv | |
The lower portion of the Table contains the dispositional features of εJ arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function F that is being employed in a differential context is equivalent to εF, for a suitable ε.
Figures 37-a through 37-d present several pictures of the proposition J and its tacit extension εJ. Notice in these Figures how εJ in EU • visibly extends J in U •, by annexing to the indicated cells of J all of the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all of the dispositions that spring from them, in other words, it attributes to these cells all of the conceivable changes that are their issue.
Figure 37-a. Tacit Extension of J (Areal)
Figure 37-b. Tacit Extension of J (Bundle)
Figure 37-c. Tacit Extension of J (Compact)
Figure 37-d. Tacit Extension of J (Digraph)
The computational scheme that was shown in Table 36 treated J as a proposition in U • and formed εJ as a proposition in EU •. When J is regarded as a mapping J : U • → X • then εJ must be obtained as a mapping εJ : EU • → X •. By default, the tacit extension of the map J : [u, v] → [x] is naturally taken to be a particular map, of the following form:
εJ : [u, v, du, dv] → [x] ⊆ [x, dx]
This is the map that looks like J when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that J already employs.
But the choice of a particular thematic variable, for example x for ¢(J), is a shade more arbitrary than the initial choice of variable names {u, v}. This means that the map I am calling the trope extension, specifically:
ηJ : [u, v, du, dv] → [dx] ⊆ [x, dx]
since it looks just the same as εJ in the way that its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.
These considerations have the practical consequence that all of our computations and illustrations of εJ perform the double duty of capturing an image of ηJ as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension ηJ, because the exercise would be identical to the work already done for εJ. Since the computations given for εJ are expressed solely in terms of the variables {u, v, du, dv}, these variables work equally well for finding ηJ. Furthermore, since each of the above Figures shows only how the level sets of εJ partition the extended source universe EU • = [u, v, du, dv], all of them serve equally well as portraits of ηJ.
##### Enlargement Map of Conjunction
No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.
— Robert Musil, The Man Without Qualities, [Mus, 62]
The enlargement map EJ is computed from the proposition J by making a particular class of formal substitutions for its variables, in this case u + du for u and v + dv for v, and subsequently expanding the result in whatever way happens to be convenient for the end in view.
Table 38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing EJ over the cells of [u, v]. The critical step of this procedure uses the facts that (0, x) = 0 + x = x and (1, x) = 1 + x = (x) for any boolean variable x.
Table 38. Computation of EJ (Method 1)
| | | | |
|----|----|--------------------------|----|
| EJ | = | J‹u + du, v + dv› | |
| | | | |
| | = | (u, du)(v, dv) | |
| | | | |
| | = | u v J‹1 + du, 1 + dv› | + |
| | | u (v) J‹1 + du, 0 + dv› | + |
| | | (u) v J‹0 + du, 1 + dv› | + |
| | | (u)(v) J‹0 + du, 0 + dv› | |
| | | | |
| | = | u v J‹(du), (dv)› | + |
| | | u (v) J‹(du), dv › | + |
| | | (u) v J‹ du , (dv)› | + |
| | | (u)(v) J‹ du , dv › | |
| | | | | |
|----|----------------|-----------------|-----------------|----------------|
| EJ | = u v (du)(dv) | | | |
| | | + u (v) (du) dv | | |
| | | | + (u) v du (dv) | |
| | | | | + (u)(v) du dv |
Table 39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.
Table 39. Computation of EJ (Method 2)
| | | | | |
|----|--------------------------------------------------------------------------------------------------|-----------------|-----------------|----------------|
| EJ | = ‹u + du› $\cdot$ ‹v + dv› | | | |
| | | | | |
| | = u v + u dv + v du + du dv | | | |
| | | | | |
| EJ | = u v (du)(dv) | + u (v) (du) dv | + (u) v du (dv) | + (u)(v) du dv |
Figures 40-a through 40-d present several views of the enlarged proposition EJ.
Figure 40-a. Enlargement of J (Areal)
Figure 40-b. Enlargement of J (Bundle)
Figure 40-c. Enlargement of J (Compact)
Figure 40-d. Enlargement of J (Digraph)
An intuitive reading of the proposition EJ becomes available at this point, and may be useful. Recall that propositions in the extended universe EU • express the dispositions of system and the constraints that are placed on them. In other words, a differential proposition in EU • can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand EJ as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the truth of J, that is, the region of the universe where J is true. This interpretation is visibly clear in the Figures above, and appeals to the imagination in a satisfying way, but it has the added benefit of giving fresh meaning to the original name of the shift operator E. Namely, EJ can be read as a proposition that enlarges on the meaning of J, in the sense of explaining its practical bearings and clarifying what it means in terms of the available options for differential action and the consequential effects that result from each choice.
Treated this way, the enlargement EJ has strong ties to the normal use of J, no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of J, in effect, pointing to the interpretive elements in its fiber of truth J–1(1). It is this kind of use that is often compared with the mention of a proposition, and thereby hangs a tale.
##### Digression : Reflection on Use and Mention
Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.
— John Dewey, How We Think, [Dew, 57]
The contrast drawn in logic between the use and the mention of a proposition corresponds to the difference that we observe in functional terms between using "J " to indicate the region J–1(1) and using "J " to indicate the function J. You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name "J " is used as a sign of the function J, and if the function J has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "J " by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise we have an inference like the following: If a buffalo is white, and white is a color, then a buffalo is a color. But a buffalo is not, only buff is.
The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.
The well-known capacity that thoughts have — as doctors have discovered — for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.
— Robert Musil, The Man Without Qualities, [Mus, 130]
##### Difference Map of Conjunction
"It doesn't matter what one does", the Man Without Qualities said to himself, shrugging his shoulders. "In a tangle of forces like this it doesn't make a scrap of difference." He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.
— Robert Musil, The Man Without Qualities, [Mus, 8]
With the tacit extension map εJ and the enlargement map EJ well in place, the difference map DJ can be computed along the lines displayed in Table 41, ending up, in this instance, with an expansion of DJ over the cells of [u, v].
Table 41. Computation of DJ (Method 1)
| | | | | |
|----|----|-------------------|----|---------|
| DJ | = | EJ | + | εJ |
| | = | J‹u + du, v + dv› | + | J‹u, v› |
| | = | (u, du)(v, dv) | + | u v |
| | | | | | |
|----|----|--------------|----------------|-----------------|----------------|
| DJ | = | 0 | | | |
| | + | u v (du) dv | + u (v)(du) dv | | |
| | + | u v du (dv) | | + (u) v du (dv) | |
| | + | u v du dv | | | + (u)(v) du dv |
DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv
Alternatively, the difference map DJ can be expanded over the cells of [du, dv] to arrive at the formulation shown in Table 42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns of the middle portion of the Table.
Table 42. Computation of DJ (Method 2)
| | | | | | | | | |
|----|----|---------|----|-------------------|----|-----------|----|----------------|
| DJ | = | εJ | + | EJ | | | | |
| | = | J‹u, v› | + | J‹u + du, v + dv› | | | | |
| | = | u v | + | (u, du)(v, dv) | | | | |
| | = | 0 | + | u dv | + | v du | + | du dv |
| DJ | = | 0 | + | u (du) dv | + | v du (dv) | + | ((u, v)) du dv |
Even more simply, the same result is reached by matching up the propositional coefficients of εJ and EJ along the cells of [du, dv] and adding the pairs under boolean sums (that is, "mod 2", where 1 + 1 = 0), as shown in Table 43.
Table 43. Computation of DJ (Method 3)
DJ = εJ + EJ
| | | | | |
|----|----------------|----------------|-----------------|----------------|
| εJ | = u v (du)(dv) | + u v (du) dv | + u v du (dv) | + u v du dv |
| EJ | = u v (du)(dv) | + u (v)(du) dv | + (u) v du (dv) | + (u)(v) du dv |
DJ = 0 $\cdot$ (du)(dv) + u $\cdot$ (du) dv + v $\cdot$ du (dv) + ((u, v)) du dv
The difference map DJ can also be given a dispositional interpretation. First, recall that εJ exhibits the dispositions to change from anywhere in J to anywhere at all, and EJ enumerates the dispositions to change from anywhere at all to anywhere in J. Next, observe that each of these classes of dispositions may be divided in accordance with the case of J versus (J) that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to εJ and EJ have in common the dispositions to preserve J, their symmetric difference (εJ, EJ) is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of J in one direction or the other. In other words, we may conclude that DJ expresses the collective disposition to make a definite change with respect to J, no matter what value it holds in the current state of affairs.
| | | |
|----|---------------------------------|---------------------------------|
| εJ | = {Dispositions from J to J } | + {Dispositions from J to (J) } |
| | | |
| EJ | = {Dispositions from J to J } | + {Dispositions from (J) to J } |
| | | |
| DJ | = (εJ, EJ) | |
| | | |
| DJ | = {Dispositions from J to (J) } | + {Dispositions from (J) to J } |
Figures 44-a through 44-d illustrate the difference proposition DJ.
Figure 44-a. Difference Map of J (Areal)
Figure 44-b. Difference Map of J (Bundle)
Figure 44-c. Difference Map of J (Compact)
Figure 44-d. Difference Map of J (Digraph)
##### Differential of Conjunction
By deploying discourse throughout a calendar, and by giving a date to each of its elements, one does not obtain a definitive hierarchy of precessions and originalities; this hierarchy is never more than relative to the systems of discourse that it sets out to evaluate.
— Michel Foucault, The Archaeology of Knowledge, [Fou, 143]
Finally, at long last, the differential proposition dJ can be gleaned from the difference proposition DJ by ranging over the cells of [u, v] and picking out the linear proposition of [du, dv] that is "closest" to the portion of DJ that touches on each point. The idea of distance that would give this definition unequivocal sense has been referred to in cautionary quotes, the kind we use to distance ourselves from taking a final position. There are obvious notions of approximation that suggest themselves, but finding one that can be justified as ultimately correct is not as straightforward as it seems.
He had drifted into the very heart of the world. From him to the distant beloved was as far as to the next tree.
— Robert Musil, The Man Without Qualities, [Mus, 144]
Let us venture a guess about where these developments might be heading. From the present vantage point, it appears that the ultimate answer to the quandary of distances and the question of a fitting measure may be that, rather than having the constitution of an analytic series depend on our familiar notions of approach, proximity, and approximation, it will be found preferable, and perhaps unavoidable, to turn the tables and let the orders of approximation be defined in terms of our favored and operative notions of formal analysis. Only the aftermath of this conversion, if it does converge, could be hoped to prove whether this hortatory form of analysis and the cohort idea of an analytic form — the limitary concept of a self-corrective process and the coefficient concept of a completable product — are truly (in practical reality) the more inceptive and persistent of principles and really (for all practical purposes) the more effective and regulative of ideas.
Awaiting that determination, I proceed with what seems like the obvious course, and compute dJ according to the pattern in Table 45.
Table 45. Computation of dJ
| | | | | |
|----|------------------|----------------|-----------------|-----------------------------------------------------------------------------------------|
| DJ | = u v ((du)(dv)) | + u (v)(du) dv | + (u) v du (dv) | + (u)(v) du dv |
| ⇒ | | | | |
| dJ | = u v (du, dv) | + u (v) dv | + (u) v du | + (u)(v) $\cdot$ 0 |
Figures 46-a through 46-d illustrate the proposition dJ, rounded out in our usual array of prospects. This proposition of EU • is what we refer to as the (first order) differential of J, and normally regard as the differential proposition corresponding to J.
Figure 46-a. Differential of J (Areal)
Figure 46-b. Differential of J (Bundle)
Figure 46-c. Differential of J (Compact)
Figure 46-d. Differential of J (Digraph)
##### Remainder of Conjunction
I bequeath myself to the dirt to grow from the grass I love, If you want me again look for me under your bootsoles. You will hardly know who I am or what I mean, But I shall be good health to you nevertheless, And filter and fibre your blood. Failing to fetch me at first keep encouraged, Missing me one place search another, I stop some where waiting for you
— Walt Whitman, Leaves of Grass, [Whi, 88]
Let us now recapitulate the story so far. In effect, we have been carrying out a decomposition of the enlarged proposition EJ in a series of stages. First, we considered the equation EJ = εJ + DJ, which was involved in the definition of DJ as the difference EJ – εJ. Next, we contemplated the equation DJ = dJ + rJ, which expresses DJ in terms of two components, the differential dJ that was just extracted and the residual component rJ = DJ – dJ. This remaining proposition rJ can be computed as shown in Table 47.
Table 47. Computation of rJ
rJ = DJ + dJ
| | | | | |
|----|------------------|----------------|-----------------|-----------------------------------------------------------------------------------------|
| DJ | = u v ((du)(dv)) | + u (v)(du) dv | + (u) v du (dv) | + (u)(v) du dv |
| dJ | = u v (du, dv) | + u (v) dv | + (u) v du | + (u)(v) $\cdot$ 0 |
rJ = u v du dv + u (v) du dv + (u) v du dv + (u)(v) du dv
As it happens, the remainder rJ falls under the description of a second order differential rJ = d2J. This means that the expansion of EJ in the form:
| | | | | | | |
|----|----|-----|----|-----|----|-----|
| EJ | = | εJ | + | DJ | | |
| | = | εJ | + | dJ | + | rJ |
| | = | d0J | + | d1J | + | d2J |
which is nothing other than the propositional analogue of a Taylor series, is a decomposition that terminates in a finite number of steps.
Figures 48-a through 48-d illustrate the proposition rJ = d2J, which forms the remainder map of J and also, in this instance, the second order differential of J.
Figure 48-a. Remainder of J (Areal)
Figure 48-b. Remainder of J (Bundle)
Figure 48-c. Remainder of J (Compact)
Figure 48-d. Remainder of J (Digraph)
##### Summary of Conjunction
To establish a convenient reference point for further discussion, Table 49 summarizes the operator actions that have been computed for the form of conjunction, as exemplified by the proposition J.
Table 49. Computation Summary for J
| | | | | | | | | | | | | | | | | |
|----|----|----|-------------------------------------------------------------------------------|------------|----|------|-------------------------------------------------------------------------------|--------|----|------|-------------------------------------------------------------------------------|--------|----|--------|-------------------------------------------------------------------------------|-------|
| εJ | = | uv | $\cdot$ | 1 | + | u(v) | $\cdot$ | 0 | + | (u)v | $\cdot$ | 0 | + | (u)(v) | $\cdot$ | 0 |
| EJ | = | uv | $\cdot$ | (du)(dv) | + | u(v) | $\cdot$ | (du)dv | + | (u)v | $\cdot$ | du(dv) | + | (u)(v) | $\cdot$ | du dv |
| DJ | = | uv | $\cdot$ | ((du)(dv)) | + | u(v) | $\cdot$ | (du)dv | + | (u)v | $\cdot$ | du(dv) | + | (u)(v) | $\cdot$ | du dv |
| dJ | = | uv | $\cdot$ | (du, dv) | + | u(v) | $\cdot$ | dv | + | (u)v | $\cdot$ | du | + | (u)(v) | $\cdot$ | 0 |
| rJ | = | uv | $\cdot$ | du dv | + | u(v) | $\cdot$ | du dv | + | (u)v | $\cdot$ | du dv | + | (u)(v) | $\cdot$ | du dv |
#### Analytic Series : Coordinate Method
And if he is told that something is the way it is, then he thinks: Well, it could probably just as easily be some other way. So the sense of possibility might be defined outright as the capacity to think how everything could "just as easily" be, and to attach no more importance to what is than to what is not.
— Robert Musil, The Man Without Qualities, [Mus, 12]
Table 50 exhibits a truth table method for computing the analytic series (or the differential expansion) of a proposition in terms of coordinates.
Table 50. Computation of an Analytic Series in Terms of Coordinates
u v
du dv
u’ v’
0 0
0 0
0 1
1 0
1 1
0 0
0 1
1 0
1 1
0 1
0 0
0 1
1 0
1 1
0 1
0 0
1 1
1 0
1 0
0 0
0 1
1 0
1 1
1 0
1 1
0 0
0 1
1 1
1 1
1 0
0 1
0 0
0 0
0 1
1 0
1 1
εJ EJ
DJ
dJ d2J
0 0
0
0
1
0
0
0
1
0 0
0 0
0 0
0 1
0 0
0
1
0
0
0
1
0
0 0
0 0
1 0
1 1
0 0
1
0
0
0
1
0
0
0 0
1 0
0 0
1 1
0 1
0
0
0
0
1
1
1
0 0
1 0
1 0
0 1
The first six columns of the Table, taken as a whole, represent the variables of a construct that I describe as the contingent universe [u, v, du, dv, u′, v′ ], or the bundle of contingency spaces [du, dv, u′, v′ ] over the universe [u, v]. Their placement to the left of the double bar indicates that all of them amount to independent variables, but there is a co-dependency among them, as described by the following equations:
| | | | | |
|----|----|--------|----|---------|
| u’ | = | u + du | = | (u, du) |
| v’ | = | v + du | = | (v, dv) |
These relations correspond to the formal substitutions that are made in defining EJ and DJ. For now, the whole rigamarole of contingency spaces can be regarded as a technical device for achieving the effect of these substitutions, adapted to a setting where functional compositions and other symbolic manipulations are difficult to contemplate and execute.
The five columns to the right of the double bar in Table 50 contain the values of the dependent variables {εJ, EJ, DJ, dJ, d2J}. These are normally interpreted as values of functions WJ : EU → B or as values of propositions in the extended universe [u, v, du, dv], but the dependencies prevailing in the contingent universe make it possible to regard these same final values as arising via functions on alternative lists of arguments, say, ‹u, v, u′, v′›.
The column for εJ is computed as J‹u, v› = uv. This, along with the columns for u and v, illustrates the Table's structure-sharing scheme, listing only the initial entries of each constant block.
The column for EJ is computed by means of the following chain of identities, where the contingent variables u′ and v′ are defined as u′ = u + du and v′ = v + dv.
EJ‹u, v, du, dv› = J‹u + du, v + dv› = J‹u’, v’›
This makes it easy to determine EJ by inspection, computing the conjunction J‹u′, v′› = u′ v′ from the columns headed u′ and v′. Since all of these forms express the same proposition EJ in EU •, the dependence on du and dv is still present but merely left implicit in the final variant J‹u′, v′›.
• NB. On occasion, it is tempting to use the further notation J′‹u, v› = J‹u′, v′›, especially to suggest a transformation that acts on whole propositions, for example, taking the proposition J into the proposition J′ = EJ. The prime [′] then signifies an action that is mediated by a field of choices, namely, the values that are picked out for the contingent variables in sweeping through the initial universe. But this heaps an unwieldy lot of construed intentions on a rather slight character, and puts too high a premium on the constant correctness of its interpretation. In practice, therefore, it is best to avoid this usage.
Given the values of εJ and EJ, the columns for the remaining functions can be filled in quickly. The difference map is computed according to the relation DJ = εJ + EJ. The first order differential dJ is found by looking in each block of constant ‹u, v› and choosing the linear function of ‹du, dv› that best approximates DJ in that block. Finally, the remainder is computed as rJ = DJ + dJ, in this case yielding the second order differential d2J.
#### Analytic Series : Recap
Let us now summarize the results of Table 50 by writing down for each column, and for each block of constant ‹u, v›, a reasonably canonical symbolic expression for the function of ‹du, dv› that appears there. The synopsis formed in this way is presented in Table 51. As one has a right to expect, it confirms the results that were obtained previously by operating solely in terms of the formal calculus.
Table 51. Computation of an Analytic Series in Symbolic Terms
u v
J
EJ
DJ
dJ
d2J
0 0
0 1
1 0
1 1
0
0
0
1
du dv
du (dv)
(du) dv
(du)(dv)
du dv
du (dv)
(du) dv
((du)(dv))
()
du
dv
(du, dv)
du dv
du dv
du dv
du dv
Figures 52 and 53 provide a quick overview of the analysis performed so far, giving the successive decompositions of EJ = J + DJ and DJ = dJ + rJ in two different styles of diagram.
Figure 52. Decomposition of EJ
Figure 53. Decomposition of DJ
#### Terminological Interlude
Lastly, my attention was especially attracted, not so much to the scene, as to the mirrors that produced it. These mirrors were broken in parts. Yes, they were marked and scratched; they had been "starred", in spite of their solidity …
— Gaston Leroux, The Phantom of the Opera, [Ler, 230]
At this point several issues of terminology have accrued enough substance to intrude on our discussion. The remarks of this Section are intended to accomplish two goals. First, I call attention to important aspects of the previous series of Figures, translating into literal terms what they depict in iconic forms, and I restress the most important structural elements that they indicate. Next, I prepare the way for taking on more complex examples of transformations, whose target universes have more than a single dimension.
In talking about the actions of operators it is important to keep in mind the distinctions between the operators per se, their operands, and their results. Furthermore, in working with composite forms of operators W = ‹W1, …, Wn› , transformations F = ‹F1, …, Fn› , and target domains X • = [x1, …, xn], we need to preserve a clear distinction between the compound entity of each given type and any one of its separate components. It is curious, given the usefulness of the concepts operator and operand, that we seem to lack a generic term, formed on the same root, for the corresponding result of an operation. Following the obvious paradigm would lead on to words like opus, opera, and operant, but these words are too affected with clang associations to work well at present, though they might be adapted in time. One current usage gets around this problem by using the substantive map as a systematic epithet to express the result of each operator's action. I am following this practice as far as possible, for example, using the phrase tangent map to denote the end product of the tangent functor acting on its operand map.
• Scholium. See [JGH, 6-9] for a good account of tangent functors and tangent maps in ordinary analysis, and for examples of their use in mechanics. This work as a whole is a model of clarity in applying functorial principles to problems in physical dynamics.
Whenever we focus on isolated propositions, on single components of composite operators, or on the portions of transformations that have 1-dimensional ranges, we are free to shift between the native form of a proposition J : U → B and the thematized form of a mapping J : U • → [x] without much trouble. In these cases we are able to tolerate a higher degree of ambiguity about the precise nature of the input and output domains of an operator than we otherwise might. For example, in the preceding treatment of the example J, and for each operator W in the set {ε, η, E, D, d, r}, both the operand J and the result WJ could be viewed in either one of two ways. On the one hand, we could regard them as propositions J : U → B and WJ : EU → B, ignoring the qualitative distinction between the range [x] $\cong$ B of εJ and the range [dx] $\cong$ D of the other types of WJ. This is what we usually do when we content ourselves simply with coloring in regions of venn diagrams. On the other hand, we could view these entities as maps J : U • → [x] = X • and εJ : EU • → [x] ⊆ EX • or WJ : EU • → [dx] ⊆ EX •, in which case the qualitative characters of the output features are not allowed to go without saying, nor thus at the risk of being forgotten.
At the beginning of this Division I recast the natural form of a proposition J : U → B into the thematic role of a transformation J : U • → [x], where x was a variable recruited to express the newly independent ¢(J). However, in my computations and representations of operator actions I immediately lapsed back to viewing the results as native elements of the extended universe EU •, in other words, as propositions WJ : EU → B, where W ranged over the set {ε, E, D, d, r}. That is as it should be. In fact, I have worked hard to devise a language that gives us all of these competing advantages, the flexibility to exchange terms and types that bear equal information value, and the capacity to reflect as quickly and as wittingly as a controlled reflex on the fibers of our propositions, independently of whether they express amusements, beliefs, or conjectures.
As we take on target spaces of increasing dimension, however, these types of confusions (and confusions of types) become less and less permissible. For this reason, Tables 54 and 55 present a rather detailed summary of the notation and the terminology that I am using here, as applied to the case of J = uv. The rationale of these Tables is not so much to train more elephant guns on this poor drosophila of an example, but to establish the general paradigm with enough solidity to bear the weight of abstraction that is coming on down the road.
Table 54 provides basic notation and descriptive information for the objects and operators that are used used in this Example, giving the generic type (or broadest defined type) for each entity. Here, the operators W in {e, E, D, d, r} and their components W in {ε, η, E, D, d, r} both have the same broad type W, W : (U • → X •) → (EU • → EX •), as would be expected of operators that map transformations J : U • → X • to extended transformations WJ, WJ : EU • → EX •.
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators
Item Notation Description Type
U • = [u, v] Source Universe [B2]
X • = [x] Target Universe [B1]
EU • = [u, v, du, dv] Extended Source Universe [B2 × D2]
EX • = [x, dx] Extended Target Universe [B1 × D1]
J J : U → B Proposition (B2 → B) ∈ [B2]
J J : U • → X • Transformation, or Mapping [B2] → [B1]
W
W :
U • → EU • ,
X • → EX • ,
(U • → X •)
→
(EU • → EX •) ,
for each W in the set:
{ε, η, E, D, d}
Operator
[B2] → [B2 × D2] ,
[B1] → [B1 × D1] ,
([B2] → [B1])
→
([B2 × D2] → [B1 × D1])
ε
η
E
D
d
Tacit Extension Operator ε
Trope Extension Operator η
Enlargement Operator E
Difference Operator D
Differential Operator d
W
W :
U • → TU • = EU • ,
X • → TX • = EX • ,
(U • → X •)
→
(TU • → TX •) ,
for each W in the set:
{e, E, D, T}
Operator
[B2] → [B2 × D2] ,
[B1] → [B1 × D1] ,
([B2] → [B1])
→
([B2 × D2] → [B1 × D1])
e
E
D
T
Radius Operator e = ‹ε, η›
Secant Operator E = ‹ε, E›
Chord Operator D = ‹ε, D›
Tangent Functor T = ‹ε, d›
Table 55 supplies a more detailed outline of terminology for operators and their results. Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol. Accordingly, each of the component operator maps WJ, since their ranges are 1-dimensional (of type B1 or D1), can be regarded either as propositions WJ : EU → B or as logical transformations WJ : EU • → X •. As a rule, the plan of the Table allows us to name each entry by detaching the adjective at the left of its row and prefixing it to the generic noun at the top of its column. In one case, however, it is customary to depart from this scheme. Because the phrase differential proposition, applied to the result dJ : EU → D, does not distinguish it from the general run of differential propositions G : EU → B, it is usual to single out dJ as the tangent proposition of J.
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes
Operator Proposition Map
Tacit
Extension
ε :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → X •)
εJ :
〈u, v, du, dv〉 → B
B2 × D2 → B
εJ :
[u, v, du, dv] → [x]
[B2 × D2] → [B1]
Trope
Extension
η :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
ηJ :
〈u, v, du, dv〉 → D
B2 × D2 → D
ηJ :
[u, v, du, dv] → [dx]
[B2 × D2] → [D1]
Enlargement
Operator
E :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
EJ :
〈u, v, du, dv〉 → D
B2 × D2 → D
EJ :
[u, v, du, dv] → [dx]
[B2 × D2] → [D1]
Difference
Operator
D :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
DJ :
〈u, v, du, dv〉 → D
B2 × D2 → D
DJ :
[u, v, du, dv] → [dx]
[B2 × D2] → [D1]
Differential
Operator
d :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
dJ :
〈u, v, du, dv〉 → D
B2 × D2 → D
dJ :
[u, v, du, dv] → [dx]
[B2 × D2] → [D1]
Remainder
Operator
r :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
rJ :
〈u, v, du, dv〉 → D
B2 × D2 → D
rJ :
[u, v, du, dv] → [dx]
[B2 × D2] → [D1]
Radius
Operator
e = ‹ε, η› :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → EX •)
eJ :
[u, v, du, dv] → [x, dx]
[B2 × D2] → [B × D]
Secant
Operator
E = ‹ε, E› :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → EX •)
EJ :
[u, v, du, dv] → [x, dx]
[B2 × D2] → [B × D]
Chord
Operator
D = ‹ε, D› :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → EX •)
DJ :
[u, v, du, dv] → [x, dx]
[B2 × D2] → [B × D]
Tangent
Functor
T = ‹ε, d› :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → EX •)
dJ :
〈u, v, du, dv〉 → D
B2 × D2 → D
TJ :
[u, v, du, dv] → [x, dx]
[B2 × D2] → [B × D]
#### End of Perfunctory Chatter : Time to Roll the Clip!
Two steps remain to finish the analysis of J that I began so long ago. First, I need to paste the accumulated heap of flat pictures into the frames of transformations, filling out the shapes of the operator maps WJ : EU • → EX •. This scheme is executed in two styles, using the areal views in Figures 56-a and the box views in Figures 56-b. Finally, in Figures 57-1 to 57-4 I put all the pieces together to construct the full operator diagrams for W : J → WJ. There is a large amount of redundancy in these three series of figures. At this early stage of exposition I thought that it would be better not to tax the reader's imagination, and to guarantee that the author, at least, has worked through the relevant exercises. I hope the reader will excuse the flagrant use of space and try to view these snapshots as successive frames in the animation of logic that they are meant to become.
##### Operator Maps : Areal Views
Figure 56-a1. Radius Map of the Conjunction J = uv
Figure 56-a2. Secant Map of the Conjunction J = uv
Figure 56-a3. Chord Map of the Conjunction J = uv
Figure 56-a4. Tangent Map of the Conjunction J = uv
##### Operator Maps : Box Views
Figure 56-b1. Radius Map of the Conjunction J = uv
Figure 56-b2. Secant Map of the Conjunction J = uv
Figure 56-b3. Chord Map of the Conjunction J = uv
Figure 56-b4. Tangent Map of the Conjunction J = uv
##### Operator Diagrams for the Conjunction J = uv
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
### Taking Aim at Higher Dimensional Targets
The past and present wilt . . . . I have filled them and emptied them, And proceed to fill my next fold of the future.
— Walt Whitman, Leaves of Grass, [Whi, 87]
In the next Subdivision I consider a logical transformation F that has the concrete type F : [u, v] → [x, y] and the abstract type F : [B2] → [B2]. From the standpoint of propositional calculus, the task of understanding such a transformation is naturally approached by parsing it into component maps with 1-dimensional ranges, as follows:
| | | | | | | | | |
|-------|----|--------|----|----------|----|--------|----|--------|
| F | = | ‹f, g› | = | ‹F1, F2› | : | [u, v] | → | [x, y] |
| where | | f | = | F1 | : | [u, v] | → | [x] |
| and | | g | = | F2 | : | [u, v] | → | [y] |
Then one tackles the separate components, now viewed as propositions Fi : U → B, one at a time. At the completion of this analytic phase, one returns to the task of synthesizing all of these partial and transient impressions into an agile form of integrity, a solidly coordinated and deeply integrated comprehension of the ongoing transformation. (Very often, of course, in tangling with refractory cases, one never gets as far as the beginning again.)
Let us now refer to the dimension of the target space or codomain as the toll (or tole) of a transformation, as distinguished from the dimension of the range or image that is customarily called the rank. When we keep to transformations with a toll of 1, as J : [u, v] → [x], we tend to get lazy about distinguishing a logical transformation from its component propositions. However, if we deal with transformations of a higher toll, this form of indolence can no longer be tolerated.
Well, perhaps we can carry it a little further. After all, the operator result WJ : EU • → EX • is a map of toll 2, and cannot be unfolded in one piece as a proposition. But when a map has rank 1, like εJ : EU → X ⊆ EX or dJ : EU → dX ⊆ EX, we naturally choose to concentrate on the 1-dimensional range of the operator result WJ, ignoring the final difference in quality between the spaces X and dX, and view WJ as a proposition about EU.
In this way, an initial ambivalence about the role of the operand J conveys a double duty to the result WJ. The pivot that is formed by our focus of attention is essential to the linkage that transfers this double moment, as the whole process takes its bearing and wheels around the precise measure of a narrow bead that we can draw on the range of WJ. This is the escapement that it takes to get away with what may otherwise seem to be a simple duplicity, and this is the tolerance that is needed to counterbalance a certain arrogance of equivocation, by all of which machinations we make ourselves free to indicate the operator results WJ as propositions or as transformations, indifferently.
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map F : [B2] → [B2], and begin to pave the way, to some extent, for discussing any transformation of the form F : [Bn] → [Bk].
Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
Item Notation Description Type
U • = [u, v] Source Universe [Bn]
X •
= [x, y]
= [f, g]
Target Universe [Bk]
EU • = [u, v, du, dv] Extended Source Universe [Bn × Dn]
EX •
= [x, y, dx, dy]
= [f, g, df, dg]
Extended Target Universe [Bk × Dk]
F F = ‹f, g› : U • → X • Transformation, or Mapping [Bn] → [Bk]
f
g
f, g : U → B
f : U → [x] ⊆ X •
g : U → [y] ⊆ X •
Proposition
Bn → B
∈ (Bn, Bn → B)
= (Bn +→ B) = [Bn]
W
W :
U • → EU • ,
X • → EX • ,
(U • → X •)
→
(EU • → EX •) ,
for each W in the set:
{ε, η, E, D, d}
Operator
[Bn] → [Bn × Dn] ,
[Bk] → [Bk × Dk] ,
([Bn] → [Bk])
→
([Bn × Dn] → [Bk × Dk])
ε
η
E
D
d
Tacit Extension Operator ε
Trope Extension Operator η
Enlargement Operator E
Difference Operator D
Differential Operator d
W
W :
U • → TU • = EU • ,
X • → TX • = EX • ,
(U • → X •)
→
(TU • → TX •) ,
for each W in the set:
{e, E, D, T}
Operator
[Bn] → [Bn × Dn] ,
[Bk] → [Bk × Dk] ,
([Bn] → [Bk])
→
([Bn × Dn] → [Bk × Dk])
e
E
D
T
Radius Operator e = ‹ε, η›
Secant Operator E = ‹ε, E›
Chord Operator D = ‹ε, D›
Tangent Functor T = ‹ε, d›
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
Operator
or
Operand
Proposition
or
Component
Transformation
or
Mapping
Operand
F = ‹F1, F2›
F = ‹f, g› : U → X
Fi : 〈u, v〉 → B
Fi : Bn → B
F : [u, v] → [x, y]
F : Bn → Bk
Tacit
Extension
ε :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → X •)
εFi :
〈u, v, du, dv〉 → B
Bn × Dn → B
εF :
[u, v, du, dv] → [x, y]
[Bn × Dn] → [Bk]
Trope
Extension
η :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
ηFi :
〈u, v, du, dv〉 → D
Bn × Dn → D
ηF :
[u, v, du, dv] → [dx, dy]
[Bn × Dn] → [Dk]
Enlargement
Operator
E :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
EFi :
〈u, v, du, dv〉 → D
Bn × Dn → D
EF :
[u, v, du, dv] → [dx, dy]
[Bn × Dn] → [Dk]
Difference
Operator
D :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
DFi :
〈u, v, du, dv〉 → D
Bn × Dn → D
DF :
[u, v, du, dv] → [dx, dy]
[Bn × Dn] → [Dk]
Differential
Operator
d :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
dFi :
〈u, v, du, dv〉 → D
Bn × Dn → D
dF :
[u, v, du, dv] → [dx, dy]
[Bn × Dn] → [Dk]
Remainder
Operator
r :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → dX •)
rFi :
〈u, v, du, dv〉 → D
Bn × Dn → D
rF :
[u, v, du, dv] → [dx, dy]
[Bn × Dn] → [Dk]
Radius
Operator
e = ‹ε, η› :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → EX •)
eF :
[u, v, du, dv] → [x, y, dx, dy]
[Bn × Dn] → [Bk × Dk]
Secant
Operator
E = ‹ε, E› :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → EX •)
EF :
[u, v, du, dv] → [x, y, dx, dy]
[Bn × Dn] → [Bk × Dk]
Chord
Operator
D = ‹ε, D› :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → EX •)
DF :
[u, v, du, dv] → [x, y, dx, dy]
[Bn × Dn] → [Bk × Dk]
Tangent
Functor
T = ‹ε, d› :
U • → EU • , X • → EX • ,
(U • → X •) → (EU • → EX •)
dFi :
〈u, v, du, dv〉 → D
Bn × Dn → D
TF :
[u, v, du, dv] → [x, y, dx, dy]
[Bn × Dn] → [Bk × Dk]
### Transformations of Type B2 → B2
To take up a slightly more complex example, but one that remains simple enough to pursue through a complete series of developments, consider the transformation from U • = [u, v] to X • = [x, y] that is defined by the following system of equations:
| | | | | |
|----|----|---------|----|----------|
| x | = | f‹u, v› | = | ((u)(v)) |
| y | = | g‹u, v› | = | ((u, v)) |
The component notation F = ‹F1, F2› = ‹f, g› : U • → X • allows us to give a name and a type to this transformation, and permits us to define it by means of the compact description that follows:
‹x, y› = F‹u, v› = ‹((u)(v)), ((u, v))›
The information that defines the logical transformation F can be represented in the form of a truth table, as in Table 60. To cut down on subscripts in this example I continue to use plain letter equivalents for all components of spaces and maps.
Table 60. Propositional Transformation
u v f g
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
1
((u)(v)) ((u, v))
Figure 61 shows how one might paint a picture of the logical transformation F on the canvass that was earlier primed for this purpose (way back in Figure 30).
Figure 61. Propositional Transformation
Figure 62 extracts the gist of Figure 61, exemplifying a style of diagram that is adequate for most purposes.
Figure 62. Propositional Transformation (Short Form)
Figure 63 give a more complete picture of the transformation F, showing how the points of U • are transformed into points of X •. The lines that cross from one universe to the other trace the action that F induces on points, in other words, they depict the aspect of the transformation that acts as a mapping from points to points, and chart its effects on the elements that are variously called cells, points, positions, or singular propositions.
Figure 63. Transformation of Positions
Table 64 shows how the action of the transformation F on cells or points is computed in terms of coordinates.
Table 64. Transformation of Positions
u v x y x y x (y) (x) y (x)(y) X • = [x, y ]
0 0
0 1
1 0
1 1
0
1
1
1
1
0
0
1
0
0
0
1
0
1
1
0
1
0
0
0
0
0
0
0
↑
F
‹f, g ›
↑
((u)(v)) ((u, v)) u v (u, v) (u)(v) ( ) U • = [u, v ]
Table 65 extends this scheme from single cells to arbitrary regions of the source and target universes, and illustrates a form of computation that can be used to determine how a logical transformation acts on all of the propositions in the universe. The way that a transformation of positions affects the propositions, or any other structure that can be built on top of the positions, is normally called the induced action of the given transformation on the system of structures in question.
Table 65. Induced Transformation on Propositions
X •
← F = ‹f , g› ←
U •
fi‹x, y›
u =
v =
1 1 0 0
1 0 1 0
= u
= v
fj‹u, v›
x =
y =
1 1 1 0
1 0 0 1
= f‹u, v›
= g‹u, v›
f0
f1
f2
f3
f4
f5
f6
f7
()
(x)(y)
(x) y
(x)
x (y)
(y)
(x, y)
(x y)
0 0 0 0
0 0 0 1
0 0 1 0
0 0 1 1
0 1 0 0
0 1 0 1
0 1 1 0
0 1 1 1
()
()
(u)(v)
(u)(v)
(u, v)
(u, v)
(u v)
(u v)
f0
f0
f1
f1
f6
f6
f7
f7
f8
f9
f10
f11
f12
f13
f14
f15
x y
((x, y))
y
(x (y))
x
((x) y)
((x)(y))
(())
1 0 0 0
1 0 0 1
1 0 1 0
1 0 1 1
1 1 0 0
1 1 0 1
1 1 1 0
1 1 1 1
u v
u v
((u, v))
((u, v))
((u)(v))
((u)(v))
(())
(())
f8
f8
f9
f9
f14
f14
f15
f15
Given the alphabets U = {u, v} and X = {x, y}, along with the corresponding universes of discourse U • and X • $\cong$ [B2], how many logical transformations of the general form G = ‹G1, G2› : U • → X • are there?
Since G1 and G2 can be any propositions of the type B2 → B, there are 24 = 16 choices for each of the maps G1 and G2, and thus there are 24$\cdot$24 = 28 = 256 different mappings altogether of the form G : U • → X •. The set of all functions of a given type is customarily denoted by placing its type indicator in parentheses, in the present instance writing (U • → X •) = {G : U • → X •}, and so the cardinality of this function space can be most conveniently summed up by writing |(U • → X •)| = |(B2 → B2)| = 44 = 256.
Given any transformation G = ‹G1, G2› : U • → X • of this type, one can define a couple of further transformations, related to G, that operate between the extended universes, EU • and EX •, of its source and target domains.
First, the enlargement map (or the secant transformation) EG = ‹EG1, EG2› : EU • → EX • is defined by the following set of component equations:
EGi = Gi‹u + du, v + dv›
Second, the difference map (or the chordal transformation) DG = ‹DG1, DG2› : EU • → EX • is defined in component-wise fashion as the boolean sum of the initial proposition Gi and the enlarged proposition EGi, for i = 1, 2, according to the following set of equations:
| | | | | |
|-----|----|----------|----|--------------------|
| DGi | = | Gi‹u, v› | + | EGi‹u, v, du, dv› |
| | = | Gi‹u, v› | + | Gi‹u + du, v + dv› |
Maintaining a strict analogy with ordinary difference calculus would perhaps have us write DGi = EGi – Gi, but the sum and difference operations are the same thing in boolean arithmetic. It is more often natural in the logical context to consider an initial proposition q, then to compute the enlargement Eq, and finally to determine the difference Dq = q + Eq, so we let the variant order of terms reflect this sequence of considerations.
Viewed in this light the difference operator D is imagined to be a function of very wide scope and polymorphic application, one that is able to realize the association between each transformation G and its difference map DG, for instance, taking the function space (U • → X •) into (EU • → EX •). Given the interpretive flexibility of contexts in which we are allowing a proposition to appear, it should be clear that an operator of this scope is not at all a trivial matter to define properly, and may take some trouble to work out. For the moment, let's content ourselves with returning to particular cases.
In their application to the present example, namely, the logical transformation F = ‹f, g› = ‹((u)(v)), ((u, v))›, the operators E and D respectively produce the enlarged map EF = ‹Ef, Eg› and the difference map DF = ‹Df, Dg›, whose components can be given as follows, if the reader, in lieu of a special font for the logical parentheses, can forgive a syntactically bilingual formulation:
| | | |
|----|----|--------------------|
| Ef | = | ((u + du)(v + dv)) |
| Eg | = | ((u + du, v + dv)) |
| | | | | |
|----|----|----------|----|--------------------|
| Df | = | ((u)(v)) | + | ((u + du)(v + dv)) |
| Dg | = | ((u, v)) | + | ((u + du, v + dv)) |
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components f and g that we earlier used on J. This work is recorded in Appendix 1 and a summary of the results is presented in Tables 66-i and 66-ii.
Table 66-i. Computation Summary for f‹u, v› = ((u)(v))
| | | | | | | | | | | | | | | | | |
|----|----|----|-------------------------------------------------------------------------------|---------|----|------|-------------------------------------------------------------------------------|-----------|----|------|-------------------------------------------------------------------------------|-----------|----|--------|-------------------------------------------------------------------------------|------------|
| εf | = | uv | $\cdot$ | 1 | + | u(v) | $\cdot$ | 1 | + | (u)v | $\cdot$ | 1 | + | (u)(v) | $\cdot$ | 0 |
| Ef | = | uv | $\cdot$ | (du dv) | + | u(v) | $\cdot$ | (du (dv)) | + | (u)v | $\cdot$ | ((du) dv) | + | (u)(v) | $\cdot$ | ((du)(dv)) |
| Df | = | uv | $\cdot$ | du dv | + | u(v) | $\cdot$ | du (dv) | + | (u)v | $\cdot$ | (du) dv | + | (u)(v) | $\cdot$ | ((du)(dv)) |
| df | = | uv | $\cdot$ | 0 | + | u(v) | $\cdot$ | du | + | (u)v | $\cdot$ | dv | + | (u)(v) | $\cdot$ | (du, dv) |
| rf | = | uv | $\cdot$ | du dv | + | u(v) | $\cdot$ | du dv | + | (u)v | $\cdot$ | du dv | + | (u)(v) | $\cdot$ | du dv |
Table 66-ii. Computation Summary for g‹u, v› = ((u, v))
| | | | | | | | | | | | | | | | | |
|----|----|----|-------------------------------------------------------------------------------|------------|----|------|-------------------------------------------------------------------------------|----------|----|------|-------------------------------------------------------------------------------|----------|----|--------|-------------------------------------------------------------------------------|------------|
| εg | = | uv | $\cdot$ | 1 | + | u(v) | $\cdot$ | 0 | + | (u)v | $\cdot$ | 0 | + | (u)(v) | $\cdot$ | 1 |
| Eg | = | uv | $\cdot$ | ((du, dv)) | + | u(v) | $\cdot$ | (du, dv) | + | (u)v | $\cdot$ | (du, dv) | + | (u)(v) | $\cdot$ | ((du, dv)) |
| Dg | = | uv | $\cdot$ | (du, dv) | + | u(v) | $\cdot$ | (du, dv) | + | (u)v | $\cdot$ | (du, dv) | + | (u)(v) | $\cdot$ | (du, dv) |
| dg | = | uv | $\cdot$ | (du, dv) | + | u(v) | $\cdot$ | (du, dv) | + | (u)v | $\cdot$ | (du, dv) | + | (u)(v) | $\cdot$ | (du, dv) |
| rg | = | uv | $\cdot$ | 0 | + | u(v) | $\cdot$ | 0 | + | (u)v | $\cdot$ | 0 | + | (u)(v) | $\cdot$ | 0 |
Table 67 shows how to compute the analytic series for F = ‹f, g› = ‹((u)(v)), ((u, v))› in terms of coordinates, and Table 68 recaps these results in symbolic terms, agreeing with earlier derivations.
Table 67. Computation of an Analytic Series in Terms of Coordinates
u v
du dv
u’ v’
0 0
0 0
0 1
1 0
1 1
0 0
0 1
1 0
1 1
0 1
0 0
0 1
1 0
1 1
0 1
0 0
1 1
1 0
1 0
0 0
0 1
1 0
1 1
1 0
1 1
0 0
0 1
1 1
1 1
1 0
0 1
0 0
0 0
0 1
1 0
1 1
εf εg
Ef Eg
Df Dg
df dg
d2f d2g
0 1
0 1
1 0
1 0
1 1
0 0
1 1
1 1
1 0
0 0
1 1
1 1
0 0
0 0
0 0
0 0
1 0
1 0
1 0
0 1
1 1
1 0
0 0
1 1
0 1
0 0
0 0
1 1
0 1
1 0
0 0
0 0
0 0
1 0
1 0
1 0
1 1
0 1
1 0
0 0
0 1
1 1
0 0
0 0
0 1
1 1
1 0
0 0
0 0
0 0
1 0
1 1
1 1
1 0
1 0
0 1
0 0
0 1
0 1
1 0
0 0
0 1
0 1
0 0
0 0
0 0
0 0
1 0
Table 68. Computation of an Analytic Series in Symbolic Terms
u v f g Df Dg df dg d2f d2g
0 0
0 1
1 0
1 1
0 1
1 0
1 0
1 1
((du)(dv))
(du) dv
du (dv)
du dv
(du, dv)
(du, dv)
(du, dv)
(du, dv)
(du, dv)
dv
du
( )
(du, dv)
(du, dv)
(du, dv)
(du, dv)
du dv
du dv
du dv
du dv
( )
( )
( )
( )
Figure 69 gives a graphical picture of the difference map DF = ‹Df, Dg› for the transformation F = ‹f, g› = ‹((u)(v)), ((u, v))›. This depicts the same information about Df and Dg that was given in the corresponding rows of the computation summary in Tables 66-i and 66-ii, excerpted here:
| | | | | | | | | | | | | | | | | |
|----|----|----|-------------------------------------------------------------------------------|----------|----|------|-------------------------------------------------------------------------------|----------|----|------|-------------------------------------------------------------------------------|----------|----|--------|-------------------------------------------------------------------------------|------------|
| | | | | | | | | | | | | | | | | |
| Df | = | uv | $\cdot$ | du dv | + | u(v) | $\cdot$ | du (dv) | + | (u)v | $\cdot$ | (du) dv | + | (u)(v) | $\cdot$ | ((du)(dv)) |
| | | | | | | | | | | | | | | | | |
| Dg | = | uv | $\cdot$ | (du, dv) | + | u(v) | $\cdot$ | (du, dv) | + | (u)v | $\cdot$ | (du, dv) | + | (u)(v) | $\cdot$ | (du, dv) |
| | | | | | | | | | | | | | | | | |
Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))›
Figure 70-a shows a graphical way of picturing the tangent functor map dF = ‹df, dg› for the transformation F = ‹f, g› = ›((u)(v)), ((u, v))›. This amounts to the same information about df and dg that was given in the computation summary of Tables 66-i and 66-ii, the relevant rows of which are repeated here:
| | | | | | | | | | | | | | | | | |
|----|----|----|-------------------------------------------------------------------------------|----------|----|------|-------------------------------------------------------------------------------|----------|----|------|-------------------------------------------------------------------------------|----------|----|--------|-------------------------------------------------------------------------------|----------|
| | | | | | | | | | | | | | | | | |
| df | = | uv | $\cdot$ | 0 | + | u(v) | $\cdot$ | du | + | (u)v | $\cdot$ | dv | + | (u)(v) | $\cdot$ | (du, dv) |
| | | | | | | | | | | | | | | | | |
| dg | = | uv | $\cdot$ | (du, dv) | + | u(v) | $\cdot$ | (du, dv) | + | (u)v | $\cdot$ | (du, dv) | + | (u)(v) | $\cdot$ | (du, dv) |
| | | | | | | | | | | | | | | | | |
Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›
Figure 70-b shows another way to picture the action of the tangent functor on the logical transformation F‹u, v› = ‹((u)(v)), ((u, v))›, roughly in the style of the bundle of universes type of diagram.
Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›
• Nota Bene. The original Figure 70-b lost some of its labeling in a succession of platform metamorphoses over the years, so I have included an Ascii version below to indicate where the missing labels go.
```o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u) v o-----------------------o dv' @ (u) v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u (v) o-----------------------o dv' @ u (v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u v o-----------------------o dv' @ u v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o o-----------------------o o-----------------------o
= u' o-----------------------o v' =
= | U' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
```
## Epilogue, Enchoiry, Exodus
It is time to explain myself . . . . let us stand up.
— Walt Whitman, Leaves of Grass, [Whi, 79]
## Appendices
### Appendix A
| | | | | | |
|---------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------|
| $\mathcal{L}_1$ Decimal | $\mathcal{L}_2$ Binary | $\mathcal{L}_3$ Vector | $\mathcal{L}_4$ Cactus | $\mathcal{L}_5$ English | $\mathcal{L}_6$ Ordinary |
| | $x\colon\!$ | $1~1~0~0\!$ | | | |
| | $y\colon\!$ | $1~0~1~0\!$ | | | |
| $\begin{matrix} f_0 \\[4pt] f_1 \\[4pt] f_2 \\[4pt] f_3 \\[4pt] f_4 \\[4pt] f_5 \\[4pt] f_6 \\[4pt] f_7 \end{matrix}$ | $\begin{matrix} f_{0000} \\[4pt] f_{0001} \\[4pt] f_{0010} \\[4pt] f_{0011} \\[4pt] f_{0100} \\[4pt] f_{0101} \\[4pt] f_{0110} \\[4pt] f_{0111} \end{matrix}$ | $\begin{matrix} 0~0~0~0 \\[4pt] 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~0~1~1 \\[4pt] 0~1~0~0 \\[4pt] 0~1~0~1 \\[4pt] 0~1~1~0 \\[4pt] 0~1~1~1 \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (x)(y) \\[4pt] (x)~y~ \\[4pt] (x)~~~ \\[4pt] ~x~(y) \\[4pt] ~~~(y) \\[4pt] (x,~y) \\[4pt] (x~~y) \end{matrix}$ | $\begin{matrix} \text{false} \\[4pt] \text{neither}~ x ~\text{nor}~ y \\[4pt] y ~\text{without}~ x \\[4pt] \text{not}~ x \\[4pt] x ~\text{without}~ y \\[4pt] \text{not}~ y \\[4pt] x ~\text{not equal to}~ y \\[4pt] \text{not both}~ x ~\text{and}~ y \end{matrix}$ | $\begin{matrix} 0 \\[4pt] \lnot x \land \lnot y \\[4pt] \lnot x \land y \\[4pt] \lnot x \\[4pt] x \land \lnot y \\[4pt] \lnot y \\[4pt] x \ne y \\[4pt] \lnot x \lor \lnot y \end{matrix}$ |
| $\begin{matrix} f_8 \\[4pt] f_9 \\[4pt] f_{10} \\[4pt] f_{11} \\[4pt] f_{12} \\[4pt] f_{13} \\[4pt] f_{14} \\[4pt] f_{15} \end{matrix}$ | $\begin{matrix} f_{1000} \\[4pt] f_{1001} \\[4pt] f_{1010} \\[4pt] f_{1011} \\[4pt] f_{1100} \\[4pt] f_{1101} \\[4pt] f_{1110} \\[4pt] f_{1111} \end{matrix}$ | $\begin{matrix} 1~0~0~0 \\[4pt] 1~0~0~1 \\[4pt] 1~0~1~0 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~0 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \\[4pt] 1~1~1~1 \end{matrix}$ | $\begin{matrix} ~~x~~y~~ \\[4pt] ((x,~y)) \\[4pt] ~~~~~y~~ \\[4pt] ~(x~(y)) \\[4pt] ~~x~~~~~ \\[4pt] ((x)~y)~ \\[4pt] ((x)(y)) \\[4pt] ((~)) \end{matrix}$ | $\begin{matrix} x ~\text{and}~ y \\[4pt] x ~\text{equal to}~ y \\[4pt] y \\[4pt] \text{not}~ x ~\text{without}~ y \\[4pt] x \\[4pt] \text{not}~ y ~\text{without}~ x \\[4pt] x ~\text{or}~ y \\[4pt] \text{true} \end{matrix}$ | $\begin{matrix} x \land y \\[4pt] x = y \\[4pt] y \\[4pt] x \Rightarrow y \\[4pt] x \\[4pt] x \Leftarrow y \\[4pt] x \lor y \\[4pt] 1 \end{matrix}$ |
| | | | | | |
|---------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------|---------------------------------------------------------------------------------------|----------------------------------------------------------------------------------------|
| $\mathcal{L}_1$ Decimal | $\mathcal{L}_2$ Binary | $\mathcal{L}_3$ Vector | $\mathcal{L}_4$ Cactus | $\mathcal{L}_5$ English | $\mathcal{L}_6$ Ordinary |
| | $x\colon\!$ | $1~1~0~0\!$ | | | |
| | $y\colon\!$ | $1~0~1~0\!$ | | | |
| $f_0\!$ | $f_{0000}\!$ | $0~0~0~0$ | $(~)$ | $\text{false}\!$ | $0\!$ |
| $\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}$ | $\begin{matrix} f_{0001} \\[4pt] f_{0010} \\[4pt] f_{0100} \\[4pt] f_{1000} \end{matrix}$ | $\begin{matrix} 0~0~0~1 \\[4pt] 0~0~1~0 \\[4pt] 0~1~0~0 \\[4pt] 1~0~0~0 \end{matrix}$ | $\begin{matrix} (x)(y) \\[4pt] (x)~y~ \\[4pt] ~x~(y) \\[4pt] ~x~~y~ \end{matrix}$ | $\begin{matrix} \text{neither}~ x ~\text{nor}~ y \\[4pt] y ~\text{without}~ x \\[4pt] x ~\text{without}~ y \\[4pt] x ~\text{and}~ y \end{matrix}$ | $\begin{matrix} \lnot x \land \lnot y \\[4pt] \lnot x \land y \\[4pt] x \land \lnot y \\[4pt] x \land y \end{matrix}$ |
| $\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}$ | $\begin{matrix} f_{0011} \\[4pt] f_{1100} \end{matrix}$ | $\begin{matrix} 0~0~1~1 \\[4pt] 1~1~0~0 \end{matrix}$ | $\begin{matrix} (x) \\[4pt] ~x~ \end{matrix}$ | $\begin{matrix} \text{not}~ x \\[4pt] x \end{matrix}$ | $\begin{matrix} \lnot x \\[4pt] x \end{matrix}$ |
| $\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}$ | $\begin{matrix} f_{0110} \\[4pt] f_{1001} \end{matrix}$ | $\begin{matrix} 0~1~1~0 \\[4pt] 1~0~0~1 \end{matrix}$ | $\begin{matrix} ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ | $\begin{matrix} x ~\text{not equal to}~ y \\[4pt] x ~\text{equal to}~ y \end{matrix}$ | $\begin{matrix} x \ne y \\[4pt] x = y \end{matrix}$ |
| $\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}$ | $\begin{matrix} f_{0101} \\[4pt] f_{1010} \end{matrix}$ | $\begin{matrix} 0~1~0~1 \\[4pt] 1~0~1~0 \end{matrix}$ | $\begin{matrix} (y) \\[4pt] ~y~ \end{matrix}$ | $\begin{matrix} \text{not}~ y \\[4pt] y \end{matrix}$ | $\begin{matrix} \lnot y \\[4pt] y \end{matrix}$ |
| $\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}$ | $\begin{matrix} f_{0111} \\[4pt] f_{1011} \\[4pt] f_{1101} \\[4pt] f_{1110} \end{matrix}$ | $\begin{matrix} 0~1~1~1 \\[4pt] 1~0~1~1 \\[4pt] 1~1~0~1 \\[4pt] 1~1~1~0 \end{matrix}$ | $\begin{matrix} ~(x~~y)~ \\[4pt] ~(x~(y)) \\[4pt] ((x)~y)~ \\[4pt] ((x)(y)) \end{matrix}$ | $\begin{matrix} \text{not both}~ x ~\text{and}~ y \\[4pt] \text{not}~ x ~\text{without}~ y \\[4pt] \text{not}~ y ~\text{without}~ x \\[4pt] x ~\text{or}~ y \end{matrix}$ | $\begin{matrix} \lnot x \lor \lnot y \\[4pt] x \Rightarrow y \\[4pt] x \Leftarrow y \\[4pt] x \lor y \end{matrix}$ |
| $f_{15}\!$ | $f_{1111}\!$ | $1~1~1~1$ | $((~))$ | $\text{true}\!$ | $1\!$ |
| | | | | | |
|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------------------------------------------------------------------------|
| | $f\!$ | $\operatorname{T}_{11} f$ $\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}$ | $\operatorname{T}_{10} f$ $\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}$ | $\operatorname{T}_{01} f$ $\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}$ | $\operatorname{T}_{00} f$ $\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ |
| $f_0\!$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ |
| $\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}$ | $\begin{matrix} (x)(y) \\[4pt] (x)~y~ \\[4pt] ~x~(y) \\[4pt] ~x~~y~ \end{matrix}$ | $\begin{matrix} ~x~~y~ \\[4pt] ~x~(y) \\[4pt] (x)~y~ \\[4pt] (x)(y) \end{matrix}$ | $\begin{matrix} ~x~(y) \\[4pt] ~x~~y~ \\[4pt] (x)(y) \\[4pt] (x)~y~ \end{matrix}$ | $\begin{matrix} (x)~y~ \\[4pt] (x)(y) \\[4pt] ~x~~y~ \\[4pt] ~x~(y) \end{matrix}$ | $\begin{matrix} (x)(y) \\[4pt] (x)~y~ \\[4pt] ~x~(y) \\[4pt] ~x~~y~ \end{matrix}$ |
| $\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}$ | $\begin{matrix} (x) \\[4pt] ~x~ \end{matrix}$ | $\begin{matrix} ~x~ \\[4pt] (x) \end{matrix}$ | $\begin{matrix} ~x~ \\[4pt] (x) \end{matrix}$ | $\begin{matrix} (x) \\[4pt] ~x~ \end{matrix}$ | $\begin{matrix} (x) \\[4pt] ~x~ \end{matrix}$ |
| $\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}$ | $\begin{matrix} ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ | $\begin{matrix} ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ | $\begin{matrix} ((x,~y)) \\[4pt] ~(x,~y)~ \end{matrix}$ | $\begin{matrix} ((x,~y)) \\[4pt] ~(x,~y)~ \end{matrix}$ | $\begin{matrix} ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ |
| $\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}$ | $\begin{matrix} (y) \\[4pt] ~y~ \end{matrix}$ | $\begin{matrix} ~y~ \\[4pt] (y) \end{matrix}$ | $\begin{matrix} (y) \\[4pt] ~y~ \end{matrix}$ | $\begin{matrix} ~y~ \\[4pt] (y) \end{matrix}$ | $\begin{matrix} (y) \\[4pt] ~y~ \end{matrix}$ |
| $\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}$ | $\begin{matrix} (~x~~y~) \\[4pt] (~x~(y)) \\[4pt] ((x)~y~) \\[4pt] ((x)(y)) \end{matrix}$ | $\begin{matrix} ((x)(y)) \\[4pt] ((x)~y~) \\[4pt] (~x~(y)) \\[4pt] (~x~~y~) \end{matrix}$ | $\begin{matrix} ((x)~y~) \\[4pt] ((x)(y)) \\[4pt] (~x~~y~) \\[4pt] (~x~(y)) \end{matrix}$ | $\begin{matrix} (~x~(y)) \\[4pt] (~x~~y~) \\[4pt] ((x)(y)) \\[4pt] ((x)~y~) \end{matrix}$ | $\begin{matrix} (~x~~y~) \\[4pt] (~x~(y)) \\[4pt] ((x)~y~) \\[4pt] ((x)(y)) \end{matrix}$ |
| $f_{15}\!$ | $((~))$ | $((~))$ | $((~))$ | $((~))$ | $((~))$ |
| $\text{Fixed Point Total}\!$ | | $4\!$ | $4\!$ | $4\!$ | $16\!$ |
| | | | | | |
|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|
| | $f\!$ | $\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}$ | $\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}$ | $\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}$ | $\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}$ |
| $f_0\!$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ |
| $\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}$ | $\begin{matrix} (x)(y) \\[4pt] (x)~y~ \\[4pt] ~x~(y) \\[4pt] ~x~~y~ \end{matrix}$ | $\begin{matrix} ((x,~y)) \\[4pt] ~(x,~y)~ \\[4pt] ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ | $\begin{matrix} (y) \\[4pt] ~y~ \\[4pt] (y) \\[4pt] ~y~ \end{matrix}$ | $\begin{matrix} (x) \\[4pt] (x) \\[4pt] ~x~ \\[4pt] ~x~ \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (~) \\[4pt] (~) \\[4pt] (~) \end{matrix}$ |
| $\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}$ | $\begin{matrix} (x) \\[4pt] ~x~ \end{matrix}$ | $\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}$ | $\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (~) \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (~) \end{matrix}$ |
| $\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}$ | $\begin{matrix} ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (~) \end{matrix}$ | $\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}$ | $\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (~) \end{matrix}$ |
| $\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}$ | $\begin{matrix} (y) \\[4pt] ~y~ \end{matrix}$ | $\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (~) \end{matrix}$ | $\begin{matrix} ((~)) \\[4pt] ((~)) \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (~) \end{matrix}$ |
| $\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}$ | $\begin{matrix} ~(x~~y)~ \\[4pt] ~(x~(y)) \\[4pt] ((x)~y)~ \\[4pt] ((x)(y)) \end{matrix}$ | $\begin{matrix} ((x,~y)) \\[4pt] ~(x,~y)~ \\[4pt] ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ | $\begin{matrix} ~y~ \\[4pt] (y) \\[4pt] ~y~ \\[4pt] (y) \end{matrix}$ | $\begin{matrix} ~x~ \\[4pt] ~x~ \\[4pt] (x) \\[4pt] (x) \end{matrix}$ | $\begin{matrix} (~) \\[4pt] (~) \\[4pt] (~) \\[4pt] (~) \end{matrix}$ |
| $f_{15}\!$ | $((~))$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ |
| | | | | | |
|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|
| | $f\!$ | $\operatorname{E}f|_{xy}$ | $\operatorname{E}f|_{x(y)}$ | $\operatorname{E}f|_{(x)y}$ | $\operatorname{E}f|_{(x)(y)}$ |
| $f_0\!$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ |
| $\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}$ | $\begin{matrix} (x)(y) \\[4pt] (x)~y~ \\[4pt] ~x~(y) \\[4pt] ~x~~y~ \end{matrix}$ | $\begin{matrix} ~\operatorname{d}x~~\operatorname{d}y~ \\[4pt] ~\operatorname{d}x~(\operatorname{d}y) \\[4pt] (\operatorname{d}x)~\operatorname{d}y~ \\[4pt] (\operatorname{d}x)(\operatorname{d}y) \end{matrix}$ | $\begin{matrix} ~\operatorname{d}x~(\operatorname{d}y) \\[4pt] ~\operatorname{d}x~~\operatorname{d}y~ \\[4pt] (\operatorname{d}x)(\operatorname{d}y) \\[4pt] (\operatorname{d}x)~\operatorname{d}y~ \end{matrix}$ | $\begin{matrix} (\operatorname{d}x)~\operatorname{d}y~ \\[4pt] (\operatorname{d}x)(\operatorname{d}y) \\[4pt] ~\operatorname{d}x~~\operatorname{d}y~ \\[4pt] ~\operatorname{d}x~(\operatorname{d}y) \end{matrix}$ | $\begin{matrix} (\operatorname{d}x)(\operatorname{d}y) \\[4pt] (\operatorname{d}x)~\operatorname{d}y~ \\[4pt] ~\operatorname{d}x~(\operatorname{d}y) \\[4pt] ~\operatorname{d}x~~\operatorname{d}y~ \end{matrix}$ |
| $\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}$ | $\begin{matrix} (x) \\[4pt] ~x~ \end{matrix}$ | $\begin{matrix} ~\operatorname{d}x~ \\[4pt] (\operatorname{d}x) \end{matrix}$ | $\begin{matrix} ~\operatorname{d}x~ \\[4pt] (\operatorname{d}x) \end{matrix}$ | $\begin{matrix} (\operatorname{d}x) \\[4pt] ~\operatorname{d}x~ \end{matrix}$ | $\begin{matrix} (\operatorname{d}x) \\[4pt] ~\operatorname{d}x~ \end{matrix}$ |
| $\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}$ | $\begin{matrix} ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ | $\begin{matrix} ~(\operatorname{d}x,~\operatorname{d}y)~ \\[4pt] ((\operatorname{d}x,~\operatorname{d}y)) \end{matrix}$ | $\begin{matrix} ((\operatorname{d}x,~\operatorname{d}y)) \\[4pt] ~(\operatorname{d}x,~\operatorname{d}y)~ \end{matrix}$ | $\begin{matrix} ((\operatorname{d}x,~\operatorname{d}y)) \\[4pt] ~(\operatorname{d}x,~\operatorname{d}y)~ \end{matrix}$ | $\begin{matrix} ~(\operatorname{d}x,~\operatorname{d}y)~ \\[4pt] ((\operatorname{d}x,~\operatorname{d}y)) \end{matrix}$ |
| $\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}$ | $\begin{matrix} (y) \\[4pt] ~y~ \end{matrix}$ | $\begin{matrix} ~\operatorname{d}y~ \\[4pt] (\operatorname{d}y) \end{matrix}$ | $\begin{matrix} (\operatorname{d}y) \\[4pt] ~\operatorname{d}y~ \end{matrix}$ | $\begin{matrix} ~\operatorname{d}y~ \\[4pt] (\operatorname{d}y) \end{matrix}$ | $\begin{matrix} (\operatorname{d}y) \\[4pt] ~\operatorname{d}y~ \end{matrix}$ |
| $\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}$ | $\begin{matrix} (~x~~y~) \\[4pt] (~x~(y)) \\[4pt] ((x)~y~) \\[4pt] ((x)(y)) \end{matrix}$ | $\begin{matrix} ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] ((\operatorname{d}x)~\operatorname{d}y~) \\[4pt] (~\operatorname{d}x~(\operatorname{d}y)) \\[4pt] (~\operatorname{d}x~~\operatorname{d}y~) \end{matrix}$ | $\begin{matrix} ((\operatorname{d}x)~\operatorname{d}y~) \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] (~\operatorname{d}x~~\operatorname{d}y~) \\[4pt] (~\operatorname{d}x~(\operatorname{d}y)) \end{matrix}$ | $\begin{matrix} (~\operatorname{d}x~(\operatorname{d}y)) \\[4pt] (~\operatorname{d}x~~\operatorname{d}y~) \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] ((\operatorname{d}x)~\operatorname{d}y~) \end{matrix}$ | $\begin{matrix} (~\operatorname{d}x~~\operatorname{d}y~) \\[4pt] (~\operatorname{d}x~(\operatorname{d}y)) \\[4pt] ((\operatorname{d}x)~\operatorname{d}y~) \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \end{matrix}$ |
| $f_{15}\!$ | $((~))$ | $((~))$ | $((~))$ | $((~))$ | $((~))$ |
| | | | | | |
|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|-------------------------------------------------------------------------------|
| | $f\!$ | $\operatorname{D}f|_{xy}$ | $\operatorname{D}f|_{x(y)}$ | $\operatorname{D}f|_{(x)y}$ | $\operatorname{D}f|_{(x)(y)}$ |
| $f_0\!$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ | $(~)$ |
| $\begin{matrix} f_1 \\[4pt] f_2 \\[4pt] f_4 \\[4pt] f_8 \end{matrix}$ | $\begin{matrix} (x)(y) \\[4pt] (x)~y~ \\[4pt] ~x~(y) \\[4pt] ~x~~y~ \end{matrix}$ | $\begin{matrix} ~~\operatorname{d}x~~\operatorname{d}y~~ \\[4pt] ~~\operatorname{d}x~(\operatorname{d}y)~ \\[4pt] ~(\operatorname{d}x)~\operatorname{d}y~~ \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \end{matrix}$ | $\begin{matrix} ~~\operatorname{d}x~(\operatorname{d}y)~ \\[4pt] ~~\operatorname{d}x~~\operatorname{d}y~~ \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] ~(\operatorname{d}x)~\operatorname{d}y~~ \end{matrix}$ | $\begin{matrix} ~(\operatorname{d}x)~\operatorname{d}y~~ \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] ~~\operatorname{d}x~~\operatorname{d}y~~ \\[4pt] ~~\operatorname{d}x~(\operatorname{d}y)~ \end{matrix}$ | $\begin{matrix} ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] ~(\operatorname{d}x)~\operatorname{d}y~~ \\[4pt] ~~\operatorname{d}x~(\operatorname{d}y)~ \\[4pt] ~~\operatorname{d}x~~\operatorname{d}y~~ \end{matrix}$ |
| $\begin{matrix} f_3 \\[4pt] f_{12} \end{matrix}$ | $\begin{matrix} (x) \\[4pt] ~x~ \end{matrix}$ | $\begin{matrix} \operatorname{d}x \\[4pt] \operatorname{d}x \end{matrix}$ | $\begin{matrix} \operatorname{d}x \\[4pt] \operatorname{d}x \end{matrix}$ | $\begin{matrix} \operatorname{d}x \\[4pt] \operatorname{d}x \end{matrix}$ | $\begin{matrix} \operatorname{d}x \\[4pt] \operatorname{d}x \end{matrix}$ |
| $\begin{matrix} f_6 \\[4pt] f_9 \end{matrix}$ | $\begin{matrix} ~(x,~y)~ \\[4pt] ((x,~y)) \end{matrix}$ | $\begin{matrix} (\operatorname{d}x,~\operatorname{d}y) \\[4pt] (\operatorname{d}x,~\operatorname{d}y) \end{matrix}$ | $\begin{matrix} (\operatorname{d}x,~\operatorname{d}y) \\[4pt] (\operatorname{d}x,~\operatorname{d}y) \end{matrix}$ | $\begin{matrix} (\operatorname{d}x,~\operatorname{d}y) \\[4pt] (\operatorname{d}x,~\operatorname{d}y) \end{matrix}$ | $\begin{matrix} (\operatorname{d}x,~\operatorname{d}y) \\[4pt] (\operatorname{d}x,~\operatorname{d}y) \end{matrix}$ |
| $\begin{matrix} f_5 \\[4pt] f_{10} \end{matrix}$ | $\begin{matrix} (y) \\[4pt] ~y~ \end{matrix}$ | $\begin{matrix} \operatorname{d}y \\[4pt] \operatorname{d}y \end{matrix}$ | $\begin{matrix} \operatorname{d}y \\[4pt] \operatorname{d}y \end{matrix}$ | $\begin{matrix} \operatorname{d}y \\[4pt] \operatorname{d}y \end{matrix}$ | $\begin{matrix} \operatorname{d}y \\[4pt] \operatorname{d}y \end{matrix}$ |
| $\begin{matrix} f_7 \\[4pt] f_{11} \\[4pt] f_{13} \\[4pt] f_{14} \end{matrix}$ | $\begin{matrix} (~x~~y~) \\[4pt] (~x~(y)) \\[4pt] ((x)~y~) \\[4pt] ((x)(y)) \end{matrix}$ | $\begin{matrix} ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] ~(\operatorname{d}x)~\operatorname{d}y~~ \\[4pt] ~~\operatorname{d}x~(\operatorname{d}y)~ \\[4pt] ~~\operatorname{d}x~~\operatorname{d}y~~ \end{matrix}$ | $\begin{matrix} ~(\operatorname{d}x)~\operatorname{d}y~~ \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] ~~\operatorname{d}x~~\operatorname{d}y~~ \\[4pt] ~~\operatorname{d}x~(\operatorname{d}y)~ \end{matrix}$ | $\begin{matrix} ~~\operatorname{d}x~(\operatorname{d}y)~ \\[4pt] ~~\operatorname{d}x~~\operatorname{d}y~~ \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \\[4pt] ~(\operatorname{d}x)~\operatorname{d}y~~ \end{matrix}$ | $\begin{matrix} ~~\operatorname{d}x~~\operatorname{d}y~~ \\[4pt] ~~\operatorname{d}x~(\operatorname{d}y)~ \\[4pt] ~(\operatorname{d}x)~\operatorname{d}y~~ \\[4pt] ((\operatorname{d}x)(\operatorname{d}y)) \end{matrix}$ |
| $f_{15}\!$ | $((~))$ | $((~))$ | $((~))$ | $((~))$ | $((~))$ |
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## Document History
```Author: Jon Awbrey
Created: 16 Dec 1993
Relayed: 31 Oct 1994
Revised: 03 Jun 2003
Recoded: 03 Jun 2007
```
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http://physics.stackexchange.com/questions/55252/is-angular-momentum-always-conserved-in-the-absence-of-an-external-torque?answertab=oldest | # Is angular momentum always conserved in the absence of an external torque?
Consider either the angular momentum of the earth around the sun or equivalently swinging a ball horizontally on a string.
I know that with respect to the point of rotation of the swinging ball, angular momentum is constant. Consider now an origin outside of the 'orbit'. It seems to me that the angular momentum is different as the orbit progresses but there is no external torque (the physical scenario is unaltered!). The easiest way to see this is to consider when the ball is at opposite ends of the circle - the radius changes but the velocity only flips sign.
In addition to this, how about we consider an event where we cut the string. Now place our reference point on the line of motion when the ball flies away in a straight line. By definition the ball will have zero angular momentum with reference to any point on this line ($m|r||v|sin\theta=0$) but certainly before the event it had some angular momentum.
The main question is in the title, but essentially what am I missing conceptually about this problem?
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## 1 Answer
Yes. For any system of particles, the following statement is true:
If the net torque on a system of particles is zero, then the total angular momentum of the system is conserved.
The proof in the context of classical mechanics is below.
For the ball on the string example, if you are only considering the ball, then there is an external torque on the ball: that of the string. One subtlety is that if you pick the origin of your coordinates to be the center of the circle about which it rotates, then in that case there is no torque and the angular momentum of the ball is, in fact, conserved. However, if you pick a different point as your origin, then it's not the case that the position vector is always along the line of the tension vector, and therefore there will be a nonzero torque. Remember that when you calculate the angular momentum and the torque, you need to use the same origin for both to be consistent.
For the orbiting example, you need to consider the system consisting of both planets, then there is no external torque on this system and the total angular momentum is conserved.
proof.
Let $m_i$ denote the mass of particle $i$ and let $\mathbf x_i$ denote the position of particle $i$, then the total angular momentum of the system is defined as $$\mathbf L = \sum_i \mathbf x_i\times\mathbf (m_i \dot{\mathbf x}_i)$$ Taking a time-derivative gives $$\dot{\mathbf L} = \sum_i\Big(m_i\dot{\mathbf x}_i\times\dot{\mathbf x_i} + \mathbf x_i\times(m_i\ddot{\mathbf{x}}_i)\Big) = \sum_i\mathbf x_i\times\mathbf F_i$$ where $\mathbf F_i$ is the net force on each particle. Now split the force on each particle into the net external force $\mathbf F_i^e$ and the net force due to all of the other particles in the system $$\mathbf F_i = \mathbf F_i^e + \sum_j \mathbf f_{ij}$$ where $\mathbf f_{ij}$ denotes the force on particle $i$ due to particle $j$. Then we have $$\dot{\mathbf L} = \sum_i\mathbf x_i\times\mathbf F_i^e + \sum_{ij}\mathbf x_i\times\mathbf f_{ij}$$ by Newton's third law, we have $\mathbf f_{ij} = -\mathbf f_{ji}$ which causes the last sum to vanish, we are left with $$\dot{\mathbf L} = \sum_i\mathbf x_i\times\mathbf F_i^e$$ where the expression on the right is precisely the net external torque on the system.
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But the tension in the string acts in the same direction as the radius vector thus the cross product is zero. Ideally, this is a torque-less system no? (edit: I agree with the proof - the bold question/title was to illustrate the point of my examples) – Fire Feb 27 at 5:04
Well if you pick the origin of your coordinates to be the center of the circle about which it rotates, then in that case there is no torque and the angular momentum of the ball is, in fact, conserved. However, if you pick a different point as your origin, then it's not the case that the position vector is always along the line of the tension vector, and therefore there will be a nonzero torque. Remember that when you calculate the angular momentum and the torque, then you need to use the same origin for both to be consistent. – joshphysics Feb 27 at 5:18
I realized not long after I typed my comment my fatal error - torque depends on the position vector too! – Fire Feb 27 at 5:23
Yesssssss. Very well done! I know the feeling. That usually happens when I say things out loud to a friend and realize the "fatal error" as you call it. I'm going to edit the post to include the content of the comment also in order to clarify. – joshphysics Feb 27 at 5:24
It should also be pointed out that angular momentum is only conserved when the system's Lagrangian is invariant under rotations. – elfmotat Feb 27 at 5:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 11, "mathjax_display_tex": 5, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9239278435707092, "perplexity_flag": "head"} |
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A charge +Q is located at the origin and a second charge, +4Q is at a distance d on the x-axis. where should a third charge, q, be placed, and what should be its sign and magnitude, so that all three ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9076465964317322, "perplexity_flag": "middle"} |
http://math.stackexchange.com/questions/35381/is-every-encryption-a-bijective-function?answertab=votes | # Is every encryption a bijective function?
Is there any encryption algorithm that is not bijective function ?
Should an encryption always give the same result given same key ?
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## 4 Answers
Formally a (symmetric) encryption function is a function $E$ from $\mathcal{M} \times \mathcal{K}$ to $\mathcal{M}$, where $\mathcal{M}$ is a message space (here we use, for convenience, the same for cipher and plaintext) and $\mathcal{K}$ a keyspace. For a fixed $k \in \mathcal{K}$ we want $E(.,k)$ to be a bijection from $\mathcal{M}$ to $\mathcal{M}$: this just means every message has a unique encryption under the key $k$ and every message can be decrypted too. (There are also notions of randomized encryption, etc., but this is the simplest model).
For algorithms like RSA, or cipher modes like CBC, it is true that one message, under one key, has multiple encryptions, but the decryption should always be doable in a unique way (we can filter out, so to say, the randomness we have added as an extra ingredient in encryption). It's necessary in assymetric cryptography (like RSA) in order to have a secure system, and most likely in symmetric systems too. It avoids some attacks that would exist otherwise.
To give an example, for RSA we use a large key (say a 1024 bit modulus $n$, with some encryption exponent $e$ ), so we can encrypt stuff that has size around 128 bytes (a bit less, because we consider numbers $x < n$ as messages, that can be encrypted as $x^e \mod n$). Now, if we have a message (like, say "Hello", or some 16 byte key we want to use for later communications, as RSA is often used, but say we use "Hello", in its ASCII encoding (hex) `48 65 6c 6c 6f` and we want to send that as a message, we could use the string, as a big number `0x48656c6c6f` which is less than $n$ (I won't give an example, but recall we chose it of size 128 bytes), so we could just encrypt it, and decrypt it uniquely. However, this leaves an attacker (who also knows $n$ and $e$, so could do the same encryption) with the option of trying out the encryption (under this key) of all words in a dictionary and so find the message, without really breaking RSA. So in practice (this is a commonly used standard; other exist but are harder to explain), we generate a number $x$ from the message by forming a 128 byte number as follows: first a byte `0x00`, then a byte `0x01`, then 120 random non-zero bytes, then again a byte `0x00` and then the message `0x48 65 6c 6c 6f` (5 bytes), in total a number of 128 bytes, that starts with `0x00` to ensure it's less than any $n$ of that size 128 bytes and this is the $x$ we use to compute $x^e \mod n$. After decryption (uniquely, by doing another exponentiation) we strip off the random bytes (all non-zero, so we know the original message starts after the first `0x00` after the initial one. This way, the attacker cannot precompute all possible ciphertexts for a given plaintext. This is called "padding the message", and the standard requires at least non-zero random bytes.
So in a way, decryption is still unique, but we made sure that the possible ciphertexts (as number less than $n$) are not all possible such numbers, only the ones we get after a valid padding. The "actual", intended message is shorter. So RSA as a function is a bijection, but as a "real message" to "ciphertext" mapping it is not a real function. It will be if we add the randomization as a factor: given a message, a random number (for padding) and a key, the result is still uniquely decipherable, and we can, given a key, separate out the message and the randomness. But note that that we changed the encryption function a bit: message space and ciphertext space are different now, and this has added security value. The disadvantage is that we cannot directly encrypt any string anymore. But in practice, only small values are encrypted under RSA, which are then used as a new key for a faster algorithm.
For an example of this in symmetric crypto, look at cipher modes like CBC wikipedia link that also expand the ciphertext by an extra IV block to get more than one possible ciphertext for a single plaintext.
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So the answer is YES given the same key ? – iKid Apr 27 '11 at 9:39
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for the simplest systems, yes, for real life systems most often no. – Henno Brandsma Apr 27 '11 at 9:41
could you give me an example with "NO" ? if an encryption can give two different results given the same key or 1 cipher text can come from 2 different encryption, how could it be decrypted ? – iKid Apr 27 '11 at 9:43
Given the key, the decryption should be unique. I'll edit to give an example. – Henno Brandsma Apr 27 '11 at 10:21
I think his point is that modern encryption isn't even a true function, but a "set-valued"/multi-valued function. – Nicholas Dixon Jun 14 '11 at 12:34
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An encryption algorithm which is not guaranteed to give the same result for the same inputs (i.e. encryption key and message) is certainly possible. In fact, the initialisation vector for chained block ciphers is usually random, but we usually consider it as one of the inputs because for a fixed encryption key and initialisation vector, the algorithm will be deterministic.
As such, an encryption algorithm is not necessarily a well-defined function, in the mathematical sense. Nonetheless, it is essential that under no circumstance the algorithm map two different inputs to the same output — otherwise decryption is impossible. So, if the encryption algorithm is deterministic, it is at least an injective map. Moreover, if the decryption algorithm can accept any string as a ciphertext, then the encryption algorithm must also be surjective, and in this situation it is bijective.
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so If I add in that the plain text and cipher text domain is any string and the encryption function including any other inputs ( key, IV ..) except plain text. The answer should be "YES" ? How could an encryption algorithm is non-deterministic algorithm? Is there any example ? – iKid Apr 27 '11 at 9:55
Yes, given the rest of the encryption scheme data, the mapping from plain to cipher and back is a bijection, between appropriate sets, see the RSA example. – Henno Brandsma Apr 27 '11 at 13:21
Encryption algorithms will not be surjective if you want to have a checksum feature. (A checksum can be detect submission errors. For example you can add a digit that makes the number divisible by 9. If you receive a number that is not divisible by 9, you know that there has been a transmission error.)
I am not sure about your second question. Of course, the encryption can vary by time, but maybe this is not your question. (If the encryption varies by circumstances this can also be viewed as a key change, but I note that there are very simple "encryption" systems that are not injective. For example, the word "lead" encodes very different meanings that have to be reconstructed from context. This problem is also very present for alphabets without vowels.)
Clearly, if you construct modern encryption algorithms you will want to have an injective function, but there are situation (like some hash functions) where people are satisfied that the function will be almost always injective in practice.
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could you explain more ? what is checksum feature ? If encryption with same key can give 2 different results, how can it be decrypted ? – iKid Apr 27 '11 at 9:32
"Should an encryption always give the same result given same key ?"
No. If the ciphertext message is ever the same, then the system is vulnerable to replay attacks.
You probably already know several encryption algorithms that, if you cipher the same plaintext with the same key, always deterministically produce identically the same ciphertext.
There are many encryption algorithms (some listed below) that, if you cipher the same plaintext with the same key a hundred times, will (almost certainly) produce a hundred different ciphertexts.
If you decrypt those 100 ciphertexts with the right key and the appropriate decryption algorithm, they will all produce a perfect copy of the original plaintext.
These cryptosystems generally have the transmitter pull numbers from a random number generator, and somehow transmit those random numbers to the receiver inside the ciphertext message. Given exactly the same inputs -- i.e., the random number generator is broken and produces exactly the same bytes again, a bug in the software lets you keep using the same key even after the message counter rolls over to zero, and we use the same key and plaintext message -- these cryptosystems would produce the same ciphertext message.
symmetric ciphers with an IV
Every cipher that uses an initialization vector will encrypt the same plaintext in many different ways. (They are not bijective).
Typically the ciphertext is composed of a random initialization vector (sent unencrypted) concatenated with the encrypted data (generally the same length as the plaintext, or rounded up to a full block). The initialization vector is freshly generated for every message. (With TKIP and CCMP, the IV is a simple counter -- the TSC; with some other encryption algorithms, the IV is freshly pulled from a random number generator). The initialization vector is 10 bytes for CipherSaber, 3 bytes for Wired Equivalent Privacy, 6 bytes for TKIP, 6 bytes for CCMP, etc.
public-key ciphers
Hybrid cryptosystems encrypt the same plaintext in many different ways. (They are not bijective).
PGP, GPG, SSL, TLS, SSH, etc. use hybrid cryptosystems.
The ciphertext message is composed of an encrypted temporary key concatenated with the encrypted data (generally the same length as the plaintext, or rounded up to a full block).
The temporary key is freshly pulled from a random number generator for every message. So the same plaintext message, sent at another time, will almost certainly be encrypted with a different temporary key and so almost certainly will generate a different ciphertext message.
The plaintext message is encrypted with a symmetric cipher using the temporary key. The temporary key is encrypted with a public-key cipher using the public key of the recipient.
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http://math.stackexchange.com/questions/279886/why-do-the-reals-need-to-be-constructed-do-they-somehow-span-the-rationals-t/279897 | # Why do the reals need to be constructed? Do they somehow “span” the rationals, the roots, and the transcendentals like e and pi?
Here is my question. Why do the reals need to be "constructed" by this bizarre "Dedekind cut" or "equivalence class of Cauchy sequences" argument? Why can't they simply be "observed" as consisting of all numbers that "span" some known sets of numbers?
I am thinking here, in part, by analogy with linear algebra and with the complex numbers, where $i$, the square root of $-1$, is really all you need, in addition to the reals, to get all complex numbers as a spanning set of $1$ and $i$ over $\mathbb R$. (Every complex number may be expressed as $a\cdot 1 + b\cdot i$ where $a, b\in\mathbb R$.)
We have a couple of known transcendental numbers, $e$ and $\pi$. We have all the rationals. We have all the square roots most of which are irrational. We have all the fractional roots of $e$ and $\pi$. We have the $e$th roots of all the numbers that exist, and the $\pi$th roots. Maybe we also have some other sets of transcendental numbers out there that we can use?
What I am trying to ask is, are we using these "Dedekind cuts" and "equivalence classes of Cauchy sequences" just because we don't "know enough real numbers yet", because their characterization hasn't occurred to us yet, or do we already have enough real numbers in our arsenal, like e and pi, to make a "spanning set" without using equivalence classes of infinite sequences and the like, or, is it the case that we really have to use these kinds of constructions of the reals, for some deep mathematical reason?
It just doesn't seem right. Because you have to admit, by identifying "sets of numbers" like Dedekind cuts and equivalent classes of Cauchy sequences which are both sets of numbers, with actual numbers, mathematicians create (at least in my mind) some cause for doubt about what they are doing here with the reals. A "set" seems like a strangely undefined term, which I understand, but not well, is subject to various kinds of paradoxes and levels of analysis problems. (This last paragraph may be more of a separate question, about the validity of using sets of numbers as numbers, from the first question, which is more about why aren't there simpler ways to define or understand the real numbers in terms of numbers and operations we already understand.)
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There are way more reals ($\beth_1$, continuity) than constructible reals (mere $\beth_0$, enumerable). – Jan Dvorak Jan 16 at 7:21
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In math, we don't "observe", we prove. As a matter of fact, already the natural numbes are usually constructed set-theoretically in such a way that each number is in fact a set of numbers. – Hagen von Eitzen Jan 16 at 7:22
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The point of integers is to close $\mathbb{N}$ under subtraction. The point of the rationals is to close $\mathbb{Z}$ under division (as best as possible). The point of the reals is to close $\mathbb{Q}$ under limits. Limits are not an algebraic sort of operation, and are much more closely tied to converging sequences and cutting a line than they are to any algebraic operation. – Robert Mastragostino Jan 16 at 7:33
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Also set are perfectly well-defined, though your teacher may not have been so thorough about it because the subject can get hairy quickly. And the integers and rationals are often constructed by equivalence classes from the naturals, so I'm not sure what should be so bizarre about doing this one more time. Sets are better defined than numbers are. If you think using sets to define numbers is odd, I think the biggest issue with resolving that issue would be deciding on what a number exactly is. – Robert Mastragostino Jan 16 at 7:35
@Robert Mastragostino, I wish you would expand your comment about $\mathbb{R}$ being the closure of $\mathbb{Q}$ under limits into an answer. It gets exactly at the OP's original question as to whether or not we already have "enough" numbers without the reals. To do analysis, we don't. You should get the credit for raising this point. – trb456 Jan 16 at 21:02
show 3 more comments
## 7 Answers
If you like, and some people do, you can forget about any construction of the reals from the rationals (or anything else) and instead define them axiomatically. One such axiomatization is Tarski's.
This approach will avoid any weird feeling you might have about a real number being an equivalence class of whatnot.
Usually, the reason to provide an explicit construction of something from a simpler things is that it proves that that something exists (mathematically). Moreover, it allows you to study properties of that something in terms of the simpler things that you presumably know better.
Nobody things of real numbers as equivalence classes of anything. Once the construction is done you can just forget about it if you like. Having a construction just means that the model of the real numbers that you fantasize about is at least as consistent as a model you might have of the simpler things. To some people it gives reassurance, to others a headache.
As for your attempt to define the read as something spanned by those things we have names for, together with some operations on there. The problem is that there are only countably many such things while there are uncountably many real numbers (at least if you believe that every real numbers admits at most two decimal representations). So this can't work. It might be strange to think about there being more reals then potential names or ways to approximate reals but it's a real fact (pardon the pun).
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Many thanks for the suggestion to look at Tarski. I think for me, if I understand correctly what this is, replacing one set of axioms (axioms defining a well-ordered field) with another, smaller set of axioms (Tarski), doesn't do much. – user58450 Jan 16 at 8:51
Someone please add an answer that refers to "least upper bound", thanks! – barrycarter Jan 16 at 18:35
Something is still bugging me about all this and that is, aren't e and Pi counterexamples! Pi is not an equivalence class of Cauchy sequences. It is not a Dedekind cut. It is a real number. Obviously, it is intrinsic to the relationship between the length of the radius and the area (and circumference) of the circle. So: can real numbers be fully classified by their geometric relationships or implications, and if so, do those geometric relationships or implications form a basis for describing the reals that is better than using the mere property "every real number gets converged to"? – user58450 Jan 18 at 23:22
@user58450 in a construction of the real numbers using equivalence class of Cauchy sequences the numbers pi and e will both be represented as equivalence classes. In a construction using Dedekind cuts they will be represented by Dedekind cuts. "... can real numbers be fully classified by their geometric relationships and implications..."? No! – Ittay Weiss Jan 18 at 23:40
Many thanks for your answer. Obviously still thinking this through and all these responses really help. – user58450 Jan 18 at 23:48
Constructing the reals is important if you want to do analysis. If you want to talk meaningfully about sequences or continuity, you need to fill in the "holes" in your space. You're coming from the perspective that we built the reals because we need "more stuff", but that's not the case. The reals are designed to fit together a certain way, and it just so happens that you need a lot of stuff to do that. If all the interesting analysis we wanted to do could be done with a smaller, countably infinite structure, it's possible that's what we'd call "the real line". In fact, I think some people do try and do analysis with the computable numbers.
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– user58450 Jan 16 at 8:44
– kahen Jan 16 at 21:38
The vector space of $\mathbb R$ over the field $\mathbb Q$ is an infinite dimensional vector space. The reason is that $\mathbb Q$ is a countable set therefore $\mathbb Q^n$ is also countable, but $\mathbb R$ is not countable. So, we will need an uncountable basis to constuct real numbers from rational numbers.
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– Samuel Jan 16 at 7:37
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Yes, I need to think through and read about this issue of how big R is, a bit more. It seems that all the commenters, though each may suggest a slightly different way of looking at the question, are comfortable with the work that has been done to date to construct or describe R. In other words, very educated commenters seem satisfied that this formalization for R is both quite solid and, perhaps, as useful, or comprehensible, as it needs to be to describe the subject. At some level, if it's not going to impact any practical problems in science and technology, then maybe it doesn't matter. – user58450 Jan 16 at 9:10
– AakashM Jan 16 at 12:19
@user58450 I think 'comfortable with the work to describe $\mathbb{R}$' actually depends a lot on who you're talking to. To analysts (those who care about the elements of $\mathbb{R}$ as numbers that can be added, multiplied, etc.) it's a comfortable, familiar place; but for set theorists and some topologists (who tend to think of it as 'subsets of $\mathbb{N}$' or 'functions from $\mathbb{N} \to \mathbb{N}$'), $\mathbb{R}$ is a scary, wild place where almost anything can happen. – Steven Stadnicki Jan 16 at 22:06
You can get away as follows: You demand axiomatically that there exists a complete ordered field. It can be shown that any two such fields are canonically isomorphic and thus whatever someone assumes to be his personal idea or mental representation of $\mathbb R$, it is essentially the same as other people's idea as long as they agree to talk about a complete ordered field. (You than rather obtain $\mathbb Q$, $\mathbb Z$, $\mathbb N$ as subsets instead of constructing the other way round)
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You can read more about constructing the naturals, integers and rationals here. Also you can read Kahen's magnificent answer about showing that $\mathbb R$ is the only complete field here. – JSchlather Jan 16 at 7:41
Many thanks for your link. It will take some time to digest but I really appreciate that someone went to such lengths on this site to explain something!! – user58450 Jan 16 at 8:53
But there are many complete ordered fields not isomorphic to $\Bbb R.$ In order to characterize $\Bbb R$ one needs to further require the Archimedean property (or some equivalent). – Math Gems Jan 16 at 17:49
@MathGems Ah, it depends on what one understands as complete. Yeah, one better say complete Archimidean ordered field. – Hagen von Eitzen Jan 16 at 19:58
Thank you for the kind words, @JacobSchlather – kahen Jan 16 at 21:16
You can actually do math without explicitly constructing the real numbers, although you end up constructing them implicitly.
If you accept that the number "1" exists, as well as the basic operations plus, minus, multiply, and divide, you can construct the rationals.
Even though you can't picture fractions like 3559/3571 in your head, you can certainly see how they could be constructed.
Sadly, there are several problems you can't solve with rational numbers:
• x = x + 7
• x*0 = 5
• x^2 = 2
• x^2 = -1
Why does not having a solution to "x^2=2" bother us more than not having a solution to the other problems above?
Answer: you can find rational numbers p and q such that p^2 < 2 and q^2 > 2 AND make q-p < epsilon, for any rational value of epsilon, no matter how small.
In other words, you can "squeeze" rational number squares as close to 2 as you want, without actually touching it. This offends our intuition, although Zeno claims it's quite normal (we try to punch him, but can get only arbitrarily close).
How do we solve this problem? Several possibilities:
• Create a new number "s" and declare that s^2=2. Of course, this doesn't help with problems like "x^2=3" or "x^3=2".
• Declare that every polynomial with rational coefficients is now also a number, namely the number that solves the polynomial itself.[1]
• This seems to work fairly well, until someone points out the ratio of a circle's circumference to its diameter is not part of your number system. Again, you can arbitrarily close to that ratio, but never quite hit it.
• So, how do you solve this new problem? You declare every set of rational numbers to be a number. Notice that you still haven't explicitly constructed the real numbers: your number system consists only of rational numbers and sets of rational numbers (we throw out the solutions to polynomial equations with rational coefficients since it's redundant).
• With a little cleverness, you can define rules for adding, subtracting, multiplying, and dividing these new numbers you've created, both with each other, and with the rational numbers themselves.
• How do these new numbers (ie, arbitrary sets of rationals) solve "x^2=2" and similar problems? You declare that a set S of rational numbers is a solution to f(x) = y, provided that:
• For all r in S, f(r) <= y
• For any rational epsilon, there exists r in S such that |f(r)-y| < epsilon
• You have now implicitly constructed the reals, simply using rational numbers and sets. No real numbers anywhere.
• Of course, there are a few problems with declaring any set of rationals to be a number. For example "x^2=2" now has an infinite number of solutions.
• At this point, you might want to declare two sets to be equivalent under certain conditions (eg, the "least upper bound" condition), but this isn't really necessary: if you're OK with having infinite solutions to problems like "x^2=2", you can stop here.
There you have it: mathematics without explicitly constructing the real numbers!
DISCLAIMER: I realize this probably has some errors (eg, removing unbounded rational sets), and I intend it solely as a general guideline.
[1] In traditional mathematics, polynomials have multiple solutions, so declaring a polynomial to be a single number is admittedly a bit odd. However, I'm using this as a throwaway example.
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Lots of people have given good answers, but they seem to me not to be hitting your main question. You asked why the reals have to be constructed, and is the reason perhaps that there are "not enough" numbers. The answer is, if you are doing analysis, then no, there are not enough numbers, and the constructions you currently find bizarre are actually the way the need for the reals arise.
Say you just want to use the rational numbers. They're a nice, ordered field, and they are the space where we in essence take all "real" measurements, like in science. But in analysis, and its more well-known application calculus, the main new tool one uses is limit-taking: approaching points infinitesimally. Suppose you say you'll only approach points "nicely", with sequences that never diverge off to infinity, don't oscillate wildly, etc. A great class of "nice" sequences are the Cauchy sequences, where the points get "arbitrarily" close together the further out in the sequence you get.
Unfortunately, this idea won't work. Look at this sequence:
$3, 3.1, 3.14, 3.141, 3.1415,...$
It's monotone, bounded above and below, and so Cauchy, and it consists only of rationals, but it is obviously meant to converge to $\pi$. So the logical thing to ask is: what is the smallest number of additional points I need to add to the rationals to make it so all rational Cauchy sequences converge? But that's the whole real line.
This is why things like the computable numbers seem appealing, and may be needed in explicitly constructive applications. But nothing short of the reals guarantees rational Cauchy convergence, and this is such a basic need in analysis that anything less is simply a huge headache; i.e. you'd have to condition any theorem involving limits on whether or not the limit exists in your chosen space that is not the whole real line.
I think most of the other answers here are better than this one, and give more interesting details, but I did not want this main point to slip.
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But what if I only demand that computable Cauchy sequences converge? Even with the Dedekind cut construction, there is a subtlety about what kinds of subsets "exist"... – Zhen Lin Jan 17 at 23:43
I just checked the Wikipedia article on computable numbers. Note that all rationals are computable, so your idea won't work. But if you take computable sequences with a computable modulus of convergence (i.e. the rate of convergence is also computable) in addition, then you get computable analysis, which seems similar to constructive real analysis. But this is a very different animal, quite different from classical analysis (e.g. equality is not computable; all computable real functions are continuous). – trb456 Jan 17 at 23:53
Each rational number is computable, but it is not at all obvious to me that a Cauchy sequence of rational numbers is computable. – Zhen Lin Jan 18 at 0:01
It's not: the set of all limits of rational Cauchy sequences is the reals. But if you take a sequence of computable numbers, and add the requirement that the modulus of convergence also be computable, you get computable limits. So this means that there are fewer such sequences: i.e. there must exist sequences of computable numbers which converge "normally" (real modulus of convergence) but don't have a computable modulus of convergence. – trb456 Jan 18 at 0:12
A very good question, in part because it will be hard to say whether an answer is the right answer. There are many good answers here already, many correct answers. But the question has interesting ambiguities, whether intended or not. Here I offer a few things, from my own perspective, which I think will contribute something to the set of answers.
First, "need." In an extreme sense, we don't even need all the integers. All the calculation we need to make things can be done on a finite machine. A slightly less extreme position is that the computations in making things have always been done with finite precision. I am not being facetious, because at some point the need comes down to what one is willing to assume just works and what one is unwilling to accept without justification. Many people are willing to assume that there are numbers that work just fine, and they go out and design and build bridges and financial systems and so forth. This sort of choice happens in mathematics as well. One may choose to work from a set of assumptions while another may choose to investigate how those assumptions may be justified. While everyone recognizes the importance of sound footing, they also realize there are important problems someone should think about and not wait on the highly unlikely outcome that there are fundamental problems with our assumptions.
Second, there is the question of whether such a manner of construction of the reals is necessary. The suggestion that the real numbers are "all the numbers that span a certain something" presupposes that some numbers have been constructed or otherwise exist; and among those numbers, some have a property that would distinguish them as real. The supposition raises the question of how these other numbers came to exist, were they constructed or are they assumed to exist. It would be simpler to construct the reals directly via cuts or sequences or, as has been suggested, to assume they exist with the necessary properties.
Third, to do analysis, it is not strictly necessary to construct the reals, for one could construct the surreal numbers instead. They "work" (that is, could be used to do analysis), since they contain a field isomorphic to the reals. But the reals tend to be more convenient.
Fourth, there is the question of the purpose or a "deep mathematical reason" in constructing the reals. Someone summarized one reason very nicely in a comment: for the sake of limits. There is an older reason. The basic idea in Dedekind's development of irrationals may be found in Euclid (Bk. V, Def. 5.). It is attributed to Eudoxus. The purpose was to develop a rigorous theory of similar figures that could handle irrational proportions. Since the concept of a real (or even a rational) number was absent, a definition in terms of integral multiples was needed. With it, Euclid proved the gem of Bk. VI, a generalization of the Pythagorean theorem, that if similar figures be constructed on the sides of a right triangle, the area on the hypotenuse equals the sum of areas on the sides.
Finally, another purpose was to develop calculus without appealing to any intuitive notions of what a geometrical line is like and or what an inifinitesimal is. Sometimes one might "see" something a certain way but it turns out not to be that way. Something like if $f'(0)=10$ then $f$ is increasing near $0$. It turns out not to be true always, even though one sometimes (loosely) talks of the graph having an upward slope. Such loose thinking might produce errors in analysis, and some thought it was important to show that the analysis of real-valued functions of real variables can be founded on a theory that depends only on numbers and not on geometric properties.
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Michael Ex on your comment about the nonstandard reals -- I did take a look at these briefly. I don't think they have actually constructed the "nonstandard reals" at all. Instead, what they appear to have done is to tack on their theory of infinitesimals, to the existing reals which they assume to exist without question and without any detailed definition or construction. Thus, these guys get away with neatly bypassing almost all the issues in my question about R. [There is a character limit to my comment so I am going to finish this response with additional comment.] – user58450 Jan 18 at 22:13
If they were really "constructing the nonstandard reals" then they would say something like "assume the rationals" and add to them, numbers infinitesimally close to the rationals. They do NOT do this! Instead, they say "assume the reals" (!) for example citing the free text Elementary Calculus: an Infinitesimal Approach by Keisler online at math.wisc.edu/~keisler/calc.html page 27: "The real numbers form a subset of the hyperreal numbers..." They use the reals to define the hyperreal extension of R. Perhaps an awesome way to do calculus but so far not helpful to me re R. – user58450 Jan 18 at 22:21
These nonstandard analysis guys I think actually explain why somebody can't do a "nonstandard analysis" version of the reals that assumes only the rationals and augments them with numbers "infinitely close" to the rationals. Their reason is that if you pick any real number, it is demonstrably not "infinitely close" to any particular rational number whatsoever. In other words, that idea doesn't work. But this whole thing with R still really bugs me; I just really dislike this "infinite sequences (or infinite sets in the case of Dedekind) of numbers" definition of R. – user58450 Jan 18 at 22:31
I will post back if/when after further thinking, I feel I really honestly "get" why this Cauchy/Dedekind thing is such an absolutely necessity. I do see how there is this need for the "continuum" to be proven to "exist" so that there aren't any "holes", and I sort of see how it is helpful to actually be able to say "please refer to exhibit A, your Honor: equivalence classes of Cauchy sequences! These are the same thing as this ordered continuum that we are trying to prove exists, so that means that this ordered continuum thing exists." All these comments have been helpful. – user58450 Jan 19 at 0:40
@user58450 Thank you for your comment(s). My point was not, I think, of terrible significance: if B suffices instead of A, then A is not, strictly, logically necessary (B = nonstandard $R$, A = standard $R$). Nonetheless, standard $R$ is the traditional foundation of analysis and the question of its construction retains importance. Who are "they" you refer to? You seem to have a notion of "nonstandard $R$" that is anterior to a definition of it. Surely, Robinson is free to define or construct a set ("nonstandard" $R$) as he pleases. That he did so successfully is generally acknowledged. – Michael E2 Jan 19 at 1:50
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http://www.physicsforums.com/showthread.php?t=228415 | Physics Forums
## Q factor (Quality Factor)
1. The problem statement, all variables and given/known data
The amplitude of a driven harmonic oscillator reaches a value of 20.0 F_0 /m at a resonant frequency of 390 Hz.
What is the Q value of this system?
Since our Professor accidentally gave us the problem set from a different book instead of the one we have, and the one we have makes no mention of Q, the only introduction I have to Q value is a few sentences near the end of class, and it is just confusing me to no end.
2. Relevant equations
Okay, so as far as I know, Q value is equal to the resonant frequency (w_0) over the width of the resonance. I have several equations relating q to nu (v) and the resonant frequency:
Q = (resonant frequency)/(width of resonance)
Q = (w_0/ v) where v is nu
and
width of resonance at K = Kmax/sqrt(2) = W
3. The attempt at a solution
The main trouble I am having with this problem is trying to figure out how to use the two units given to convert and insert them into the equations I was given. I have assumed that the resonant frequency of 390 Hz was equal to w_0. After this, I have tried several methods to get the final answer:
At first I attempted to insert 390 into w_0 and 20 F_0/m into v and find Q that way.
This resulted in Q = (w_0/v) = 19.5, which was incorrect, and the units are probably off (since Q is unitless)
After this, I tried to use the equation where width = Kmax/sqrt(2). I substituted 20 F_0/m into Kmax and got a width of 14.142 F_0/m. I then substituted this value into the Q equation where: Q = (390 Hz / 14.142 F_0/m) = 27.577, which is incorrect, and again I am pretty sure that my units are off.
After this, I had only two more chances to answer the question so I have just been searching for how to convert the Amplitude given into a width of resonance with no success. Can anybody help me go through the process of converting the two values given into values that I can use in the Q equation?
Thank you to anyone who replies
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Actually I just figured out another part of the problem. I now think that the units $$F_0$$ / m are actually values themselves represented by variables. F0 = maximum driving force and m is the mass of the object. This just means that now I think I have to find a way to take this amplitude and plug it into something that gives me the range of resonance frequency values to divide the resonant frequency by. Anyone have any idea how I would go about doing that?
anybody have any ideas at all?
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http://mathoverflow.net/questions/17424/looking-for-reference-talking-about-torsion-theory-on-coherent-sheaves-on-project/17438 | ## Looking for reference talking about torsion theory on coherent sheaves on projective space
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I am looking for reference talking about how torsion theory play roles in algebraic geometry. I will be really happy to see some concrete examples. Say, talking about torsion theory in $Coh(P^{1})$.
Thanks in advance
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## 1 Answer
Depending upon how strict you are with your definition of torsion theory a good source of examples is the theory of semi-orthogonal decompositions. A really nice example of this is the appearance of such decompositions which parallel operations in the minimal model program. A good introduction to this is Kawamata's survey. Of course there are other interesting things one can do with such decompositions in algebraic geometry (e.g. the work of Bondal, Orlov, Kapranov, and many others).
For torsion theories on abelian categories a good example is stability conditions. Here what is interesting is the interplay between torsion theory on hearts and t-structures. The original paper is by Bridgeland and in the case of $\mathbb{P}^1$ the stability manifold has been computed by Okada (I suggest looking at the journal version, I recall that there were at one point some typos in the arxiv version which were fixed in the published one).
As far as torsion theories on $\mathrm{Coh}(\mathbb{P}^1)$ goes it is a reasonable exercise to actually classify them (I did this at one point but never wrote it up properly). The closest place to this being written down that I know of is in the paper of Gorodentsev, Kuleshov, and Rudakov "t-stabilities and t-structures on triangulated categories" where they classify the minimal t-stabilities on the derived category of coherent sheaves on $\mathbb{P}^1$.
An example of something similar but that is not quite what you asked for is the application of cotorsion theories to relative homological algebra.
Definition: Suppose that $\mathcal{A}$ is an abelian category and that $(\mathcal{F},\mathcal{C})$ is a pair of full subcategories. Then we say that $(\mathcal{F},\mathcal{C})$ is a cotorsion theory if
$\mathcal{F} = \{F \in \mathcal{A} \; \vert \; \mathrm{Ext}^1(F,\mathcal{C}) = 0\}$ and $\mathcal{C} = \{C \in \mathcal{A} \; \vert \; \mathrm{Ext}^1(\mathcal{F},C) = 0\}$
where the subcategories appearing in the Ext's just signifies that it is true for every object of that subcategory.
There is a notion of a cotorsion theory having enough injectives and projectives and this guarantees for sufficiently good $\mathcal{A}$, say $R$-modules, (by a theorem of Eklof and Trlifaj) that $\mathcal{F}$-covers and $\mathcal{C}$-envelopes exist. In particular this can be used to show that flat covers exist. A good reference for this is Chapter 7 of Relative Homological Algebra by Enochs and Jenda.
The application to algebraic geometry/commutative algebra is using this formalism to build Gorenstein injective/projective/flat covers and envelopes.
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Thank you very much! – Shizhuo Zhang Mar 8 2010 at 1:39
1
No problem :) Hopefully some other people will contribute suggestions as well. – Greg Stevenson Mar 8 2010 at 2:40
The paper t-stabilities and t-structures on triangulated categories is really nice, it is the very thing I am looking for. It seems that the t-stabilities described t-structures on derived category. This is good! In fact, in our lecture course, Rosenberg took the Keller's observations on t-structure and built the machinery based on certain spectrum on triangulated category to describe t-structures. According to him, there is a decomposition for this spectrum, different t-structures sitting in different component. He is going to lecture this soon, now he is talking about cohomology induction – Shizhuo Zhang Mar 10 2010 at 2:30
in the language of spectrum of triangulated category and how it goes to abelian induction(in the language of spectrum of abelian category)via t-structures to get representations – Shizhuo Zhang Mar 10 2010 at 2:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 13, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9357896447181702, "perplexity_flag": "middle"} |
http://cms.math.ca/10.4153/CMB-2011-184-7 | Canadian Mathematical Society
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# A Cohomological Property of $\pi$-invariant Elements
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[PDF: 165KB]
Published:2012-02-03
• M. Filali,
Department of Mathematical Sciences, University of Oulu, Oulu 90014, Finland
• M. Sangani Monfared,
Department of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4
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## Abstract
Let $A$ be a Banach algebra and $\pi \colon A \longrightarrow \mathscr L(H)$ be a continuous representation of $A$ on a separable Hilbert space $H$ with $\dim H =\frak m$. Let $\pi_{ij}$ be the coordinate functions of $\pi$ with respect to an orthonormal basis and suppose that for each $1\le j \le \frak m$, $C_j=\sum_{i=1}^{\frak m} \|\pi_{ij}\|_{A^*}\lt \infty$ and $\sup_j C_j\lt \infty$. Under these conditions, we call an element $\overline\Phi \in l^\infty (\frak m , A^{**})$ left $\pi$-invariant if $a\cdot \overline\Phi ={}^t\pi (a) \overline\Phi$ for all $a\in A$. In this paper we prove a link between the existence of left $\pi$-invariant elements and the vanishing of certain Hochschild cohomology groups of $A$. Our results extend an earlier result by Lau on $F$-algebras and recent results of Kaniuth-Lau-Pym and the second named author in the special case that $\pi \colon A \longrightarrow \mathbf C$ is a non-zero character on $A$.
Keywords: Banach algebras, $\pi$-invariance, derivations, representations
MSC Classifications: 46H15 - Representations of topological algebras 46H25 - Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 13N15 - Derivations | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 21, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.7000617980957031, "perplexity_flag": "middle"} |
http://mathoverflow.net/questions/22567/diameter-of-a-circle-in-an-embedded-riemannian-manifold/22589 | Diameter of a circle in an embedded Riemannian manifold
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This question was inspired by an answer to the "Magic trick based on deep mathematics" question. I wanted to post it as a comment, but I ran out of characters! I'm sure there must be a collection of standard results related to this question, but I don't know where to start looking.
First, a quick definition. The diameter of a set $S \in \mathbb{R}^n$ is $\sup\{d(x, y) \mid x, y \in S\}$.
A sheet of paper is a good physical example of a Riemannian 2-manifold with boundary, and a table is a good physical model of (a subset of) $\mathbb{R}^2$. Embed the paper isometrically in $\mathbb{R}^2$ by laying flat on a table.
Draw the outline of a circular cup on the paper. It seems obvious that no matter how you embed the paper in $\mathbb{R}^2$, the outline of the cup will always be a metric circle, and it will always have the same diameter $D$.
Now, lift the paper into the air, embedding it isometrically in $\mathbb{R}^3$. If you let the paper flop around, the outline of the cup might not be a metric circle anymore... but will it still have diameter $D$?
Finally, cut along the outline of the cup, removing an open disk from the sheet of paper. The paper now has a second boundary component, and it's no longer simply connected. The paper has also gained a surprising property: you can bend it around in midair (that is, embed it isometrically in $\mathbb{R}^3$) so that the outline of the cup has diameter greater than $D$! What's the important property of the paper that we changed to make this possible?
Comments
I don't think you need to cut along the outline of the cup to make this work... you could probably just cut out any disk contained within the outline of the cup. So maybe simply-connectedness is the important property?
My gut tells me that if you draw two dots on the sheet of paper, the distance between the dots is maximized when the paper is flat on the table. When you bend the paper around in midair, the dots can get closer together, but they can never get farther apart. I think this is equivalent to the statement that if $\delta$ is the natural distance function on the paper, $d$ is the distance function in $\mathbb{R}^3$, and $F$ is an isometric embedding of the paper in $\mathbb{R}^3$, $d(Fx, Fy) \le \delta(x, y)$ for all points $x$ and $y$ on the paper.
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1
Of course, the most immediate change you made was to cut out something, so that the intrinsic diameter of your radius-$1$ circle is actually $\pi$, since that's how far someone would have to travel to get from one end to the other moving inside the paper. – Theo Johnson-Freyd Apr 26 2010 at 16:12
2 Answers
For the second question:
As commented above, if you remove an open disc from the inside of the region bounded by a circle, you can bend the paper so as to increase the diameter of the circle in $\mathbb{R}^3$. This doesn't work if you remove a point rather than an open disc - but with a point removed you can instead bend the paper so as to decrease the diameter of the circle by making the removed point a cone point of a wavy cone. This gives us a good way of thinking about what's going on here: embedding $\mathbb{R}^2$ into $\mathbb{R}^3$ preserves diameter of circles, but when you ignore part of $\mathbb{R}^2$, (such as when your circle is on a small piece of paper, or when you've removed a point inside the circle) you can introduce singularities there. Perhaps what's important is probably not the fundamental group, but the type and location of singularities that would arise if the embedding was extended to all of $\mathbb{R}^2$. Any thoughts?
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The answer to the first is NO.
As mentioned by another answer you can use Do Carmo to prove a local result of the type: at points where the normal curvature is non-zero the 0-directions defines a foliation of straight lines.
With this one can prove that embedding $\mathbb{R}²$ into $\mathbb{R}^3$ preserves the diameter of circles, but one can construct a counter example in the case of a piece of paper:
Take a equilateral triangle on your table with points barely out side of the piece of paper. Then since these sides do not meet inside the piece you may fold (using very sharp foldings) the piece of paper up at the sides of the triangle such that out side of a nighborhood of the triangle the paper lies in planes which are perpendicular to the table. Note that this can not be extended to all of $\mathbb{R}^2$ because the folding lines interset! Then shrinking the circumscribed sphere slightly of the triangle to lie in the paper - this is folded up so that the diameter becomes strictly less.
For the second question I really dont see $\pi_1$ any where. What I see is the following: for me it is natural to defined the distance on a Riemmanian manifold $X$ as:
$d(x,y) = \inf_{\gamma} \textrm{len}(\gamma)$
where the infimum is over curves $\gamma$ from $x$ to $y$, and len($\gamma$) is the length of the curve. This is intrinsic and gives the same diameters for the piece of paper, but when you remove an open set it becomes different, because the curves has to avoid this set.
Since isometric embeddings preserve lengths of curves you get with this definition
$d(Fx,Fy) \leq d(x,y)$
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The embedding is not a product in general. A piece of a cone is a simples counter-example. – Sergei Ivanov Apr 26 2010 at 11:57
Your right, I will change that. – Thomas Kragh Apr 26 2010 at 11:59
1
Nice example, but I disagree with your last paragraph: the diameter is for the distance in $\mathbb{R}^3$, (otherwise your counter example would not be one). – Benoît Kloeckner Apr 26 2010 at 12:03
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Another comment: if the sheet of paper is large enough compared to the circle (more precisely, as soon as all triangles having the given circle as inscribed circle have one of their vertex in the sheet), then your construction does not work. In this case, is the answer positive? – Benoît Kloeckner Apr 26 2010 at 12:10
2
@Benoît: yes, if the sheet contains a circle, say, of radius 3 times larger than the cup (with the same center), you will have to leave one of the diameters intact. – Sergei Ivanov Apr 26 2010 at 12:21
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http://mathoverflow.net/questions/24836?sort=votes | ## Centralizing four red vectors in six green sectors
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Four red vectors are given, one per quadrant, $[0,90^\circ)$, $[90^\circ,180^\circ)$, etc. A rigid star of six green vectors separated by $60^\circ$ can be positioned at $(\theta, \theta+60^\circ, \theta+120^\circ, \theta+180^\circ, \theta+240^\circ, \theta+300^\circ)$. The goal is to spin the green star so that the red vectors are centralized in green sectors as much as possible. Define the deviation $\delta(r)$ of a red vector $r$ as the larger of the (absolute value of the) two angles from the red vector to the boundaries of the green sector in which it lies. I want to minimize the largest red deviation.
For example, let the red vectors be at $(0^\circ, 90^\circ, 180^\circ, 270^\circ)$. Then choosing $\theta=15^\circ$ yields a deviation of $45^\circ$ for all red vectors. For example, `$\delta(0^\circ) = \max \{ 15^\circ, 45^\circ \}$`.
My question is: What is the largest deviation of any four red vectors? I thought it might approach $60^\circ$, but it seems that perhaps $52.5^\circ$ is the worst ($52.5^\circ = 7 \pi / 24$).
The problem generalizes to $k$ red vectors and $m > k$ green sectors. Likely the logic to establish the answer for $(k,m)=(4,6)$ will work for any $(k,m)$.
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Thanks to both Sergei Ivanov and David Eppstein for their clear answers! – Joseph O'Rourke May 16 2010 at 10:06
## 2 Answers
Replace each of your red vectors by its value modulo 60. You are then seeking to find a choice for $\theta$ that is as far as possible (mod 60) from any of the red vectors. The best choice for theta is to put it into the largest gap (mod 60) between red vectors, so the worst choice for the red vectors is for them to be equally spaced 15 degrees apart. With this choice, you get theta at distance 7.5 degrees from its nearest red vector or (as you already calculated) $\delta = 52.5 = 60 - 7.5$.
More generally, if you have k and m, the worst case is when the k vectors are equally spaced modulo $2\pi/m$, in which case the gaps between them have size $2\pi/km$, the distance from $\theta$ to the nearest red vector when it's placed in the middle of a gap will be $2\pi/2km$, and $\delta=\frac{2\pi}{m}-\frac{2\pi}{2km}=(1-\frac{1}{2k})\frac{2\pi}{m}$.
I don't understand why you want to restrict $m>k$ since the answer doesn't depend on that.
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Assuming $k\le m$, the answer is $\frac{2\pi}{m}-\frac{\pi}{mk}$. I prefer to speak about red and green points on the circle rather than vectors. The angle between vectors is the (intrinsic) distance between points on the circle.
First, it does not matter that the red points are in different sectors: moving a red point the distance $\frac{2\pi}m$ along the circle does not change anything and point can be moved to any sector (this is where I use the assumption that $k\le m$). Second, it is easier to deal with the minimum distance between red and green, and maximize it (the answer is $2\pi/m$ minus the original one).
Fold the circle $m$ times along a circle of length $2\pi/m$. The red points are mapped to $k$ red points on the small circle, these are given. The green points are all mapped to one green point on the small circle, this one is variable. Now the goal is to place the green point as far as possible from the red ones. Obviously the best place is the midpoint of the longest arc between the red points, and the worst case is when the red points divide the small circle into equal arcs (of length $2\pi/mk$). Hence the answer is $\pi/mk$.
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http://mathematica.stackexchange.com/questions/11522/getting-the-557th-element-of-a-table?answertab=votes | # Getting the 557th element of a table
Make a table with elements $\sin(\sqrt{8 n}/4) + r\,$for $n$ from $1$ to $1000$, where $r$ is a random real number between $-0.1$ and $0.1$ that is different for each value of $n$. Also make a plot of all the points of this table. What is the $557^{th}$ element of this table?
Everything is working well, but I can't get the 557th element of the table. I've tried to use the `Select` command.
````Select[TableSin[Sqrt[(a*n)/4]] + Random[Real, {-0.1, 0.1}], {n, 1, 1000}] ,557]
````
but it's not working. I can't figure out how to fix it.
Can anyone give me some hints?
-
Look at `Take` in the documentation – rm -rf♦ Oct 3 '12 at 21:59
Sounds more like `Part` is what he is looking for. – Mr Alpha Oct 3 '12 at 22:01
My professor gave the hint: use "Select" – user43523 Oct 3 '12 at 22:04
3
And we gave you two more hints... `Select` would be a really terrible way to do it and I would not advise it. You can probably teach your professor something new today :) – rm -rf♦ Oct 3 '12 at 22:11
1
The problem specifies "different" random reals to be used. Even for such a small list length as 1000, so far as I'm aware there's no guarantee that Mathematica will provide distinct ones. To be extra certain (if it's important), you can take additional measures, e.g.: generate a considerably longer list of random reals; use `Union` to extract just the distinct entries; and then, if necessary, shorten that result to the 1000 you need. However, the actual intention of the problem statement may have been merely not to use the same randomly chosen number for each item. – murray Oct 4 '12 at 0:25
show 9 more comments
## 1 Answer
`Select` is wrong in this case, you can't use it to get the n-th element of a list without using additional helper functions.
You'll get the most out of this exercise by looking at `Part`, which is a very flexible function for extracting elements out of a list based on their position, for example
````x = {1,1,2,3,5,8,13,21,34,55}
(* Explicit syntax. Extracts the 3rd element out of a list
(first element has index 1). *)
Part[x, 3]
(* ==> 2 *)
(* Syntactic sugar: using [[ ]] for Part *)
x[[3]]
(* ==> 2 *)
````
`Part` can do much more, e.g. extracting ranges, submatrices and so on. While not necessary to solve this problem, I highly recommend reading a few paragraphs in the Mathematica help - you will need it again soon. :-)
For the fun of it, here's the easiest way of doing the same task with `Select` instead of `Part` I could come up with:
````x = {1, 1, 2, 3, 5, 8, 13, 21, 34, 55}
Select[
MapIndexed[{#1, First@#2} &, x],
Last[#] == 3 &
] // First // First
(* ==> 2 *)
````
(Manually index the list wasting a ton of memory, then pick the element and discard the indexing again.)
-
That `Select` method is just silly... Go Syntactic sugar! – kale Oct 4 '12 at 0:45
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http://sbseminar.wordpress.com/2007/11/03/a-probability-puzzle/ | ## A probability puzzle November 3, 2007
Posted by David Speyer in fun problems, Uncategorized.
trackback
I learned this puzzle from Henry Cohn. Henry tells me it is probably due to Peter Glynn and Phillip Heidelberger (JSTOR).
I’ve just come back from trick-or-treating, and in my bag are two types of candy — peppermints and toffee. I’d like to know what fraction of the bag is peppermints. Not wanting to dump out that whole big bag, I’ll just draw a few random samples and make an estimate. (The bag is big enough that it doesn’t matter whether or not I return candies to the bag after sampling.) The responsible way to do this would be to decide ahead of time how many samples I will draw — say ten pieces — and stick to that plan. Instead, though, I decide to just sample for five seconds and count however much candy I can grab in that time. This would be fine too, if it took me the same amount of time to draw a toffee or to draw a peppermint. But it doesn’t. I can draw a peppermint from the bag in one second, while the toffee sticks to my fingers and takes three seconds to unwrap. This makes my sampling process flawed. Suppose that the probability of drawing a peppermint is $p$, and the probability of drawing a toffee is $q$ (with $p+q=1$). So, for example, I might draw a toffee, a peppermint and then another toffee (which I would finish unwrapping after time expired). The probability of this sequence of draws is $q*p*q$ and, if I encountered that series of draws, I would esitmate that my bag was 1/3 peppermint. In the table below, I have listed all possible sequences of draws, the odds of obtaining that sample, and the estimate that sample produces
| | | | | | | | | | |
|-----------|-------|-------|-------|-------|-------|-------|-------|-------|-----|
| Draws: | PPPPP | PPPPT | PPPT | PPT | PTP | PTT | TPP | TPT | TT |
| Odds: | p^5 | p^4 q | p^3 q | p^2 q | p^2 q | p q^2 | p^2 q | p q^2 | q^2 |
| Estimate: | 1 | 4/5 | 3/4 | 2/3 | 2/3 | 1/3 | 2/3 | 1/3 | 0 |
So the expected value of our estimate is
$p^5+2 p^2 q+(3/4) p^3 q+(4/5) p^4 q+(2/3) p q^2 = \\ \hphantom{p^5+} (2/3) p +(2/3) p^2-(7/12) p^3+(1/20) p^4+(1/5) p^5$.
Note that this is not equal to the true value, $p$. It is a slight underestimate because those sticky toffee get counted more than their fair share. In statistical jargon, this is not a consistent estimator. Here is a graph of our estimate as a function of the true value of $p$.
Incidently, this has a “practical” application. Suppose a pollster comes to your door and asks what candidate you plan on voting for. If you suspect that he will poll for eight hours and then stop, you should stall as long as possible, as that will lead to a higher reported total for your candidate.
Ok, that was nifty, but it wasn’t the puzzle. Here is the puzzle. Suppose that I adjust my sampling procedure in just the smallest way: if I am still unwrapping a piece of toffee when time runs out, I don’t count it to my sample. So that changes the bottom row of our table as follows:
| | | | | | | | | | |
|---------------|-------|-------|-------|-------|-------|-------|-------|-------|-----|
| Draws: | PPPPP | PPPPT | PPPT | PPT | PTP | PTT | TPP | TPT | TT |
| Odds: | p^5 | p^4 q | p^3 q | p^2 q | p^2 q | p q^2 | p^2 q | p q^2 | q^2 |
| Old Estimate: | 1 | 4/5 | 3/4 | 2/3 | 2/3 | 1/3 | 2/3 | 1/3 | 0 |
| New Estimate: | 1 | 1 | 1 | 2/3 | 2/3 | 1/2 | 2/3 | 1/2 | 0 |
The new expected value of our estimate is $p^5+2 p^2 q+p^3 q+p^4 q+p q^2=p$. That’s right, this slight change made our procedure accurate again! So the puzzle is, why? I have two proofs, but they both involve some messy computations. There must be an argument which makes this just jump out at us. (Of course, there is nothing important about the numbers 3 and 5, or about having two types of candy. This happens with any number of types of candy, any unwrapping times and any sampling time, as long as the sampling time is longer than the longest unwrapping time.)
By the way, I was going to suggest that pollsters should use this modified sampling to defeat the attack mentioned above. But the modified method is vulnerable to a slightly more sophisticated attack: you should now answer slowly near the beginning of the day but rapidly at the end of the day. The modified method only works when the time to draw a particular candy is independent of past history, including being independent of how much history has passed.
Update: Terence Tao gives a very nice solution in the comments.
## Comments»
1. John Armstrong - November 3, 2007
My first guess is that if you allow overspills, you’re effectively shortening the toffee time by a factor of $\frac{n+1}{n}$ where $n$ is the a expected number of toffees, while still calculating as if the original value of the time is correct. But somehow this seems a mite simplistic.
The one who I think will know the right answer is Isabel.
2. carlbrannen - November 3, 2007
This reminds me of the calculation of how many people are on a bus (with infinite capacity, of course) if they arrive by Poisson process. Say you get N as the average. That is the average seen by the bus drivers.
What is the average seen by the passengers? It is N+1.
The reasoning is also similar to this old canard. Suppose that half of children are male and half female. Suppose that everybody stops having children as soon as they get their first girl. Will this increase or decrease the ratio of boy to girl babies born?
3. Dean Pangelinan - November 4, 2007
With the a-priori knowledge that determining that a candy is a peppermint takes only one second, and unwrapping a toffe takes three seconds, why wouldn’t you simply state that you take one second to unwrap each candy. If it is fully unwrapped in one second, it is a peppermint, and if it is not unwrapped fully, then it is a toffee.
Nothing in the original statement said that you had to EAT the candy, only that you didn’t want to dump out the whole bag, and you were presuming that each piece was wrapped!
4. Terence Tao - November 4, 2007
In order to avoid division by zero issues, one should adopt the convention that if no candy has been fully unwrapped when the time runs out, then one uses the partially unwrapped candy as the sample (thus the fraction would be 1 if it was a peppermint and 0 if it was a toffee).
One can see why the algorithm works by the following “martingale” argument. For each n=0,1,2,…, let X_n be the random variable defined as the fraction of peppermints one sees when one has unwrapped n candies or when the time runs out, whichever occurs sooner. For instance, in the TPP case mentioned above, one has X_0=X_1=0, X_2=1/2, X_3=2/3, and X_n=2/3 for all n > 3. The objective is to show that X_n has expectation p for all sufficiently large n.
On the other hand, it is clear (with our above conventions) that X_0 has expectation p. So it suffices to show that X_n – X_{n-1} has expectation 0 for every n; informally, this means that each new candy one gets to unwrap does not affect the expected fraction.
We can condition on the event that we actually get to unwrap n candies, since otherwise X_n-X_{n-1} vanishes. We can then condition on the distribution of candies one sees at this stage, e.g. m peppermints and n-m toffees (so $X_n = m/n$).
Now for the key observation: once one performs this conditioning, the _order_ in which the m peppermints and n-m toffees appears is completely random (each of the $\binom{n}{m}$ possibilities is equally likely).
We can now proceed in one of two ways. One is direct computation: the last candy was equal to a peppermint with probability $m/n$, and a toffee with probability $(n-m)/n$. In the former case $X_{n-1} = (m-1)/(n-1)$, and in the latter $X_{n-1} = m/(n-1)$. Since $\frac{m}{n} \frac{m-1}{n-1} + \frac{n-m}{n} \frac{m}{n-1} = \frac{m}{n}$, the claim follows. (The case n=1 has to be treated separately, of course.)
The other way is to view $X_n$ as the probability that a randomly selected candy among the n candies already unwrapped is a peppermint. $X_{n-1}$, on the other hand, is the probability that after randomly deleting one of those candies, a randomly selected candy among the remaining candies is a peppermint. It is then intuitively obvious that these probabilities are the same (they can be joined by the joint distribution of selecting two distinct candies at random among the n candies, deleting one, and inspecting the type of the other).
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http://mathoverflow.net/questions/69099/shortest-closed-curve-to-inspect-a-sphere | ## Shortest closed curve to inspect a sphere
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of $C$, in the sense that the segment $xy$ intersects $S$ at precisely the one point $x$. I am interested in the shortest $C$ with this property. In computational geometry, such paths are called watchman tours, and there are many results concerning polygons in the plane finding such tours.
This question arose at a conference I'm attending, and I was pointed to a paper by V. A. Zalgaller:
"Shortest Inspection Curves for the Sphere" (Journal of Mathematical Sciences, Volume 131, Number 1, 5307-5320; Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 87–108.)
I cannot access the paper from the conference, but from the abstract it appears he focused on open rather than closed curves.
Has anyone heard of this natural question? Can you point me to relevant literature? Thanks!
Addendum. Here is the $4\pi$ saddle / baseball-stitches curve suggested by Gjergji Zaimi:
-
10
My guess is $4\pi R$, which I can achieve gluing 4 semicircles of radius $R$ (saddle shape). My attempts at finding a clever way to prove the lower bound using a Crofton formula have failed though... – Gjergji Zaimi Jun 29 2011 at 9:51
2
It is also the single boundary curve remaining on this figure: take the favorite cylinder of Archimedes, having height and diameter $2R.$ Take a saw and cut it in half on a plane that contains the axis of rotation. Now you have two shapes, each a cylinder over a semicircle, and each having a flat square as one face. Rotate one piece 90 degrees with respect to the other, so that the squares match up, and glue back together. – Will Jagy Jun 29 2011 at 17:57
4
(of course if we allow unions of closed curves, the infimum of length is 0, e.g. C = 6 small circles, each passing through a vertex of the cube $[−1,+1]^3$, or even C = the vertices themselves, as a degenerate case). – Pietro Majer Jun 30 2011 at 8:35
6
It is not difficult to see that if the curve is at constant distance from the sphere, then its length is at least $4\pi$. Indeed suppose that it is a constant distance $r$ from the center. The area of the set of points newly seen from a short segment of length $ds$ is $a(r)ds$, with $a(r)=2\sqrt{r^2-1}/r^2$. This is maximal for $r=\sqrt{2}$, then $a=1$. Since each point of the sphere is "newly seen" from at least one point of the curve, the result follows. The curve proposed by Gjergji Zaimi is at constant distance from the center, so it is optimal at least in this restricted sense. – Jean-Marc Schlenker Jul 2 2011 at 6:24
3
This problem appears in the last lines of a math popularization article by Jean-Baptiste Hiriart-Urruty, Du calcul différentiel au calcul variationnel, in Quadrature 70(2008):8-18, see www.math.univ-toulouse.fr/~jbhu/Fermat_Quadrature.pdf There the problem is presented as open and attributed to Alain Grigis. – Jean-Marc Schlenker Jul 5 2011 at 6:15
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http://mathhelpforum.com/statistics/64519-probability-consecutive-wins-print.html | # Probability of consecutive wins
Printable View
• December 11th 2008, 09:35 AM
ZPlayer
Probability of consecutive wins
Hello all,
I have played binomial game and recorded wins and losses. I played 179 rounds and then analyzed the resulting sequence. I recorded the number of times when victories appear in runs. For example, out of 179 rounds, there is one time when there are six wins in a row. Five wins in a row shows up two times. There are three times when there are four wins in a row. Three wins in a row occur five times. There are 11 times when there are two wins in a row. Finally, there are 25 times when there is a one win alone.
I would like to know what statistical law or distribution explains this observation.
Thanks to all.
• December 11th 2008, 03:25 PM
meymathis
Warning: This comment is intuition (plus a few matlab simulations to verify).
For large p (the probability of 1 success), I'm not sure that this is an easy distribution to describe. At least, I don't see it.
As p get's smaller, I think that it may start to converge to a negative binomial. Suppose p is the probability of 1 success. For large N, the chance of a run of 2 is p times less likely than a run of 1. A run of 3 is p times less likely than a run of 2. Etc. BTW, a run of length 0 is a single failure.
p needs to be small so that you have lots of "trials" (chances for runs) compared to the length of the run lengths.
I should also point out that although I am waving my hands, I did run some simulations using matlab. For p = 0.5 and smaller, my experimental results matched the negative binomial distribution pretty well. I didn't actually try to run anything larger. So 'small' may not actually have to be too small.
• December 12th 2008, 02:12 PM
awkward
Quote:
Originally Posted by ZPlayer
Hello all,
I have played binomial game and recorded wins and losses. I played 179 rounds and then analyzed the resulting sequence. I recorded the number of times when victories appear in runs. For example, out of 179 rounds, there is one time when there are six wins in a row. Five wins in a row shows up two times. There are three times when there are four wins in a row. Three wins in a row occur five times. There are 11 times when there are two wins in a row. Finally, there are 25 times when there is a one win alone.
I would like to know what statistical law or distribution explains this observation.
Thanks to all.
Hi ZPlayer,
Just to be sure of the problem statement, let's say we flip a fair coin N times and count the number of runs of heads of length r, where a run of length r is defined as a sequence of exactly r heads not preceded or followed by a head. It's fairly easy to find the expected number of runs of length r. The actual probability distribution, on the other hand, seems to be a much harder problem and I don't know a formula for it.
So let's work on the expected value problem:
Define
$X_i = 1$ if a run of r heads ends on the ith coin flip, for $r \leq i \leq N$,
$= 0$ otherwise.
A run of r heads ends on flip r if we have r heads followed by a tail, so
$P(X_r = 1) = 2^{-r-1}$
A run of r heads ends on flip N if it is preceded by a tail, so
$P(X_N = 1) = 2^{-r-1}$.
For other values of i, i.e., $r < i < N$, a run of r heads must be preceded and followed by a tail, so
$P(X_i = 1) = 2^{-r-2} \text{ for } r < i < N$.
The total number of runs of length r is $X_r + X_{r+1} + \dots + X_N$, and
$E(X_r + X_{r+1} + \dots + X_N)$
$= E(X_r) + E(X_{r+1}) + \dots + E(X_N)$
$=2^{-r-1} + (N-r-1) 2^{-r-2} + 2^{-r-1}$
$=(3 + N - r) 2^{-r-2}$
Here we have used the theorem that E(X+Y) = E(X) + E(Y). It's important to know that this theorem holds even if X and Y are not independent. That's good for us here, because the $X_i$'s are not independent. (There are probably some people who think this is the only probability theorem I know, because I seem to apply it on every problem. Maybe they're right. (Nerd))
I think if you work out the numbers you will find they agree fairly well with your data.
• December 19th 2008, 11:01 AM
meymathis
Quote:
Originally Posted by awkward
(There are probably some people who think this is the only probability theorem I know, because I seem to apply it on every problem. Maybe they're right. (Nerd))
Perhaps, awkward, but you do use it to great effect. (Clapping) I thought about that technique but couldn't quite see how to use it.
I thought about this some more and came to the following conclusion. The negative binomial will be an excellent approximation. In order for it to be a good approximation, I think that the requirement would be something like $(1-p)N>100$ (to get absolute errors of both PDF and CDF around 1e-3 or smaller - based on simulations).
The experiment you are performing (almost) consists of some number of negative binomial experiments (NBEs). The number of NBEs depends on the outcomes of the NBEs themselves. This is where the difficulty lies. The number of runs of length 5 is not independent from the number of runs of length 9. For example if N=10, and you know there was 1 run of length 9, then you know there was NOT a run of length 5.
However, if $(1-p)N$ is big, this says that the expected number of failures is quite large, hence the number of NBEs is quite large. When this happens, then the number of runs of length 5 becomes more independent then the number of runs of length 9. Or another way of thinking about it is that the number of NBEs relative to N converges to $(1-p)$.
So in essence you can think of the problem as being $(1-p)N$ independent negative binomial experiments. And thus you are doing a finite sampling of a negative binomial distribution.
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http://mathoverflow.net/questions/61058/what-is-the-size-of-the-smallest-rigid-extension-field-of-the-complex-numbers | ## What is the size of the smallest rigid extension field of the complex numbers?
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
Suppose we consider a rigid extension field $F$, i.e., $\text{Aut}(F) = 1$ over the complex numbers $\mathbb C$. What is the minimal cardinality of $F$? In particular it should hold that in this case $|F| > |\mathbb C|$.
Moreover, if we replace $\mathbb C$ with any other algebraically closed field, what can one say in this case?
Any comment, reference, or pointer is highly appreciated.
All the best, Sebastian
-
## 2 Answers
Firstly, it may be worth linking to this related question on MO.
Pröhle proved that all fields of characteristic 0 can be embedded in a rigid field - see "Does a given subfield of characteristic zero imply any restriction to the endomorphism monoids of fields?" for his particular construction.
Later, Dugas and Göbel showed in "All infinite groups are Galois groups over any field" that for ${\mathbb C}$ it can be done in a field of cardinality the successor cardinal of $2^{\aleph_0}$ which is as good as you could expect.
In follow-up papers "Automorphism groups of fields I" and "Automorphism groups of fields II", they show that for any group $G$ and any field $K$, there exists an extension with automorphism group isomorphic to $G$ and cardinality $\aleph_0|K||G|$.
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thanks a lot. The $\aleph_0 |K| |G|$ construction does not apply here as it only holds for fields that are not algebraically closed. In fact, the Dugas-Göbel construction, as well as the older ones all fail for algebraically closed fields. In Pröhle's work there is an early remark that the construction fails in that case and that for algebraically closed fields it is not possible to maintain the size - that is what the lemma states with $G = \{1\}$. – sebastian Apr 11 2011 at 15:15
Ah okay, I don't have access to the last of the D-G papers and the MathSciNet review unfortunately doesn't detail the construction or give the restrictions on the field, though of course, as you say, the cardinality must increase. However, they do state in the earlier papers that they obtain rigid extensions of ${\mathbb C}$, with the necessary jump in cardinality. – dke Apr 11 2011 at 17:10
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I deleted what I had written, as it was irrelevant.
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@Angelo: I think sebastian means rigid, not rigid over C. – Martin Brandenburg Apr 8 2011 at 13:58
Martin, thanks for pointing this out. – Angelo Apr 8 2011 at 14:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9489890336990356, "perplexity_flag": "head"} |
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The Hamiltonian, like the Lagrangian, is a function that summarizes equations of motion. It has the additional interpretation of giving the total energy of a system. Though originally stated for classical mechanics, it is also an important part of quantum mechanics.
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Start from the Lagrangian and define a canonical momentum $p_a(t)$ for each canonical coordinate $q_a(t)$: $p_a = \frac{\partial L}{\partial \dot q_a}$ The Hamiltonian is given by $\left(\sum_a p_a \dot q_a \right) - L$ Hamilton's equations of motion are $\dot q_a = \frac{\partial H}{\partial p_a}$ $\dot p_a = - \frac{\partial H}{\partial q_a}$ The Hamiltonian has the interesting property that $\dot H = \frac{\partial H}{\partial t}$ meaning that if the Hamiltonian has no explicit time dependence, it is a constant of the motion.
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To illustrate the derivation of the Hamiltonian, let us start with the Lagrangian for a particle with Newtonian kinetic energy and potential energy V(q): $L = T - V$ where $T = \frac12 m \left( \frac{dq}{dt} \right)^2$ For canonical coordinate q, we find canonical momentum p: $p = m \frac{dq}{dt}$ and from that, we find the Hamiltonian: $H = T + V$ where the kinetic energy is now given by $T = \frac{p^2}{2m}$
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## Initial Velocity of a falling object
Hello everyone
I know that $$\sqrt { \frac {2d} {g} }$$ but what about including initial velocity?
Thanks in advanced
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Quote by eNathan I know that $$\sqrt { \frac {2d} {g} }$$ but what about including initial velocity?
that _what_ is $$\sqrt { \frac {2d} {g} }$$?
It's $$t= \sqrt { \frac {2d} {g} }$$.
Where did that equation come from?
Mentor Blog Entries: 1 $$y = y_0 + v_0 t - \frac {g t^2}{2}$$ This is the kinematic equation describing a falling body (ignoring air resistance). Here "up" is positive and "down" is negative; "y" is the position after t seconds; $y_0$ is the initial height; $v_0$ is the initial speed.
## Initial Velocity of a falling object
robphy, we all know what I meant.
Thanks Doc_Al!
Quote by Doc Al $$y = y_0 + v_0 t - \frac {g t^2}{2}$$ This is the kinematic equation describing a falling body (ignoring air resistance). Here "up" is positive and "down" is negative; "y" is the position after t seconds; $y_0$ is the initial height; $v_0$ is the initial speed.
I am still not clear why the time of the initial velocity (going up) is equal to the t in 1/2gt^2. Will someone enlighten me?
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Quote by RENATO I am still not clear why the time of the initial velocity (going up) is equal to the t in 1/2gt^2. Will someone enlighten me?
I'm unclear what you are asking. What do you mean by "time of the initial velocity"? Please rephrase your question. What problem are you trying to solve?
If a person is standing on top of the cliff, throws the ball upwards at 15 ft/sec from an initial height of 50 ft. How high is the rock after 2 seconds. Also, what is the total time when it hits the ground at 50 ft below. I know I can solve the problem by following the formula y=1/2 at^2 +Vot +yo, yo being the initial height. What is not clear to me is why is t (time) the same throughout the equation.
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Quote by RENATO What is not clear to me is why is t (time) the same throughout the equation.
The time (t) in that equation is a parameter that continually changes. t = 0 is the moment when the ball is first thrown. That equation tells you how the position of the ball changes as a function of time, where time is measured from the moment the ball was thrown.
Thank you so much. I do understand now.
That equation still will not give you the correct answer. That is for if the initial velocity is orthaginol to gravity. You are not accounting for the distance traveled in the positive y direction. You need velocity to start out positive at a decreasing rate, and end up negative at an increasing rate. Either work it piecewise find a path function.
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Quote by cstoos That equation still will not give you the correct answer. That is for if the initial velocity is orthaginol to gravity. You are not accounting for the distance traveled in the positive y direction.
You have it backwards. That equation only deals with vertical motion (in the y direction). The v0 in that equation is just the y-component of the initial velocity.
[QUOTE=RENATO;2723823]If a person is standing on top of the cliff, throws the ball upwards at 15 ft/sec from an initial height of 50 ft. How high is the rock after 2 seconds. Also, what is the total time when it hits the ground at 50 ft below. I know the formula y=1/2 at^2 +Vot +yo, yo being the initial height. What is not clear to me is why is t (time) the same throughout the equation. Will someone solve for the height after 2 seconds and the total time so I would know if my answers are correct.
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Quote by RENATO Will someone solve for the height after 2 seconds and the total time so I would know if my answers are correct.
Why don't you show what you did and we'll take a look at your work? (As I thought was explained before, time is just a parameter. That formula gives the position as a function of time.)
In one case, the time is given and you'll calculate the position. In the other case you know the final position and you have to solve for the time. You use the same equation for both parts.
I can only solve for the height after 2 seconds which is 15.6 ft, but I do not know how to solve for the total time.
Is the following correct: -50 = Vot -32.2/2(t^2) -50 = 15t -16.1t^2 t = 2.28
or is it 3 seconds, now I am guessing.
Another way of solving, which I am not sure is: d =Vot + (0.5)at^2 50= (-15)t + 16.1 t^2 t= 2.28 seconds
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http://mathoverflow.net/questions/92748?sort=votes | ## Can convolution on $R_+$ be discontinuous everywhere ?
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Let $L^+$ be a set of all real valued functions defined on a real line which are Lebesgue integrable on each $[0,c]$, where $c>0$, and are zero for $x<0$. Let for $a>0$, $f_a(t)=(t-a)^{-3/4}$ for $t>a$ and $0$ for $t \leq a$. Then, for $a,b>0$, convolution $(f_a*f_b)(x):=\int_0^x f_a(x-y)f_b(y)dy=\beta (x-a-b)^{-\frac{1}{2}}$ for $x> a+b$ and $0$ for $x \leq a+b$, where $\beta=B(\frac{1}{4}, \frac{1}{4})$. It shows that convolution of two functions from $L^+$ need not be continuous. Is it possible, maybe by condensation of singularities and above example to show existence of two functions from $L^+$ which convolution is discontinuous everywhere
on $[0, \infty)$?
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You say $\int_0^x f_a(x-y)f_b(y)dy=f_{a+b}(x)$. My calculations disagree. But the convolution $f_a * f_b$ is, indeed, unbounded at $a+b$. – Gerald Edgar Mar 31 2012 at 17:12
Sorry, I did mistake. I just have edited. – arc Mar 31 2012 at 18:21
## 2 Answers
How about this. If $g \in L^+$, then almost every point is a Lebesgue point for $g$. And if $h \in L^+$, then almost every point is a Lebesgue point for $h$. We should then be able to show that the convolution $g * h$ is differentiable almost everywhere. Or something like that.
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Certainly $f*g$ is locally $L^1$, so almost all points are also Lebesgue points for $f*g$. For every such point $x$ and every $\varepsilon>0$ there exists a $\delta>0$ with almost all points $y\in B(x,\delta)$ satisfying $|f*g(x)-f*g(y)|<\varepsilon$. (Why? Just assume not...) So there exists a version of $f*g$ continuous at $x$. With a little thought you can now devise a scheme to modify $f*g$ so it really is continuous except at countably many points. – Ollie Margetts Apr 1 2012 at 0:59
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You can't have a convolution of two functions in $L_1(0,c)$ that is not in $L_1(0,c)$, else you violate Young's inequality, or the fact that $L_1(0,c)$ (i.e. the space you refer to as $L^{+}$) is a Banach algebra under convolution, so the answer is no. I assume you are taking $a < c$ and $b < c$ (otherwise you can't work in $L_1(\mathbb{R})$ because $f_a$ and $f_b$ don't belong there).
EDIT: Excuse me, Arc. I do apologize. I did not make the relevant distinction between an everywhere discontinuous function and a function defined only on a set of measure zero. I now understand the definition, so my answer is largely unhelpful.
However, if you are trying to construct a function that is discontinuous on a dense set, I do get the feeling that your construction wont do because the output of your convolution is (as you would expect) better behaved than the inputs, it is locally in $L_{2-\epsilon}$ and still has only one point of discontinuity. Perhaps I am missing the point here, though. Can you explain a bit more about what you want to achieve and your approach?
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An $L^1$ function can be discontinuous everywhere. – Gerald Edgar Mar 31 2012 at 23:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 49, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9498714208602905, "perplexity_flag": "head"} |
http://mathoverflow.net/questions/9000/intermediate-value-theorem-on-computable-reals/9116 | ## Intermediate value theorem on computable reals
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Wikipedia says that the intermediate value theorem “depends on (and is actually equivalent to) the completeness of the real numbers.” It then offers a simple counterexample to the analogous proposition on ℚ and a proof of the theorem in terms of the completeness property.
Does an analogous result hold for the computable reals (perhaps assuming that the function in question is computable)? If not, is there a nice counterexample?
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Have you read en.wikipedia.org/wiki/… already? – Qiaochu Yuan Dec 15 2009 at 16:48
I'm afraid I don't understand your question. The IVT says that if a continuous function has range (y1,y2) over some interval (x1,x2) then for any yc between y1 and y2, there is some xc between x1 and x2 such that the function at f(xc) = yc. I don't understand how this is meant to be extended to the computable reals... – Gabriel Benamy Dec 15 2009 at 16:56
Qiaochu Yuan: Thanks for the link. I'm not interested in adopting intuitionist logic; an existence proof would satisfy me. – Jason Orendorff Dec 15 2009 at 17:37
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Gabriel Benamy: Like this-- "If a continuous computable function f maps the computable reals in the interval [x1, x2] to computable reals, with f(x1)=y1 and f(x2)=y2, then for any computable number yc between y1 and y2, there is some computable number xc between x1 and x2 such that the function at f(xc)=yc." (Continuity is defined the same for the computable reals as for the reals, in case that's the difficulty.) – Jason Orendorff Dec 15 2009 at 17:45
I can see that there's an existential quantifier in the IVT, and if it is to be proven, then that has to come from somewhere. If you write out the proof for the reals, that quantifier clearly comes from invoking the least upper bound property. If it's obvious that one just can't get an existential quantifier on the computable reals, then that's the piece I'm missing (and, in that case, sorry for the dumb question!) – Jason Orendorff Dec 15 2009 at 17:51
## 5 Answers
Let me assume that you are speaking about computable reals and functions in the sense of computable analysis, which is one of the most successful approaches to the topic. (One must be careful, since there are several incompatible notions of computability on the reals.)
In computable analysis, the computable real numbers are those that can be computed to within any desired precision by a finite, terminating algorithm (e.g. using Turing machines). One should imagine receiving rational approximations to the given real. In this subject, functions on the reals are said to be computable, if there is an algorithm that can compute, for any desired degree of accuracy, the value of the function, for any algorithm that produces approximations to the input with sufficient accuracy. That is, if we want to know f(x) to within epsilon, then the algorithm is allowed to ask for x to within any delta it cares to.
The Computable Intermediate Value Theorem would be the assertion that if f is a computable continuous function and f(a)< c<f(b) for computable reals a, b, c, then there is a computable real d with f(d)=c.
The book Computable analysis: an introduction by Klaus Weihrauch discusses exactly this question in Example 6.3.6.
The basic situation is as follows. The answer is Yes. If f happens to be increasing, then the usual bisection proof of existence turns out to be effective. For other f, however, one can use a trisection proof. Theorem 6.3.8 says that if f is computable and f(x)*f(z)<0, then f has a computable zero. This implies the Computable Intermediate Value theorem above.
In contrast, the same theorem also says that there is a non-negative computable continuous function f on [0,1], such that the sets of zeros of f has Lebesgue measure greater than 1/2, but f has no computable zero.
In summary, if the function crosses the line, you can compute a crossing point, but if it stays on one side, then you might not be able to compute a kissing point, even if it is kissing on a large measure set.
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What a great answer. Thanks. – Jason Orendorff Dec 17 2009 at 6:53
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Vasco Brattka, the most prominent of Weihrauch's followers, commented <a href="cs.nyu.edu/pipermail/fom/2009-May/…; on the FOM list<a> that the constructive critique of the IVT may be interpreted classically as (the failure of) a <i>uniform</i> assignment of a zero, ie continuously wrt the function (or parameters that describe it). This is the only observation about the IVT that I have heard from that school since I presented my paper on the IVT at <a href="cca-net.de/cca2005/">their conference</a>. – Paul Taylor Jan 1 2010 at 16:35
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Thanks first to Andrej for drawing attention to my paper on the IVT, and indeed for his contributions to the work itself. This paper is the introduction to Abstract Stone Duality (my theory of computable general topology) for the general mathematician, but Sections 1 and 2 discuss the IVT in traditional language first. The following are hints at the ideas that you will find there and at the end of Section 14.
I think it's worth starting with a warning about the computable situation in ${\bf R}^2$, where it is customary to talk about fixed points instead of zeroes.
Gunter Baigger described a computable endofunction of the square. The classical Brouwer thereom says that it has a fixed point, but no such fixed point can be defined by a program. This is in contrast to the classical response to the constructive IVT, that either there is a computable zero, or the function hovers at zero over an interval. (I have not yet managed to incorporate Baigger's counterexample into my thinking.)
Returning to ${\bf R}^1$, we have a lamentable failure of classical and contructive mathematicians to engage in a meaningful debate. The former claim that the result in full generality is "obvious", and argue by quoting random fragments of what their opponents have said in order to make them look stupid. On the other hand, to say that "constructively, the intermediate value theorem fails" by showing that it implies excluded middle is equally unconstructive.
Even amongst mainstream mathematicians several arguments are conflated, so I would like to sort them out on the basis of the generality of the functions to which they apply.
On the cone hand we have the classical IVT, and the approximate construtive one that Neel mentions. These apply to any continuous function with $f(0) < 0 < f(1)$.
There are several other results that impose other pre-conditions:
• the exact constructive IVT, for non-hovering functions, described by Reid;
• using Newton's algorithm, for continuously differentiable functions such that $f(x)$ and $f'(x)$ are never simultaneously zero; and
• the Brouwer degree, with an analogous condition in higher dimensions.
These conditions are all weaker forms of saying that the function is an open map.
Any continuous function $f:X\to Y$ between compact Hausdorff spaces is proper: the inverse image $Z=f^{-1}(0)\subset X$ of $0\in Y$ is compact (albeit possibly empty).
If $f:X\to Y$ is also an open map then $Z$ is overt too. I'll come back to that word in a moment.
When $f$ is an open map between compact Hausdorff spaces and $Z$ is nonempty, there is a compact subspace $K\subset X$ and an open one $V\subset Y$ with $0\in V$ and $V\subset f(K)$.
So for real manifolds we might think of $K$ is a (filled-in) ball and $f(K)\setminus V$ as the non-zero values that $f$ takes on the enclosing sphere.
Could I have forgotten that the original question was about computability?
No, that's exactly what I'm getting at.
In ${\bf R}^1$ an enclosing sphere is a straddling interval, $[d,u]$ such that $f(d) < 0 < f(u)$ or $f(d) > 0 > f(u)$.
The interval-halving (or, I suspect, any computational) algorithm generates a convergent sequence of straddling intervals.
More abstractly, write $\lozenge U$ if the open subset $U$ contains a straddling interval. The interval-halving algorithm (known historically as the Bolzano--Weierstrass theorem or lion hunting) depends exactly on the property that $\lozenge$ takes unions to disjunctions, and in particular $$\lozenge(U\cup V) \Longrightarrow \lozenge U \lor \lozenge V.$$ (Compare this with the Brouwer degree, which takes disjoint unions to sums of integers.)
I claim, therefore, that the formulation of the constructive IVT should be the identification of suitable conditions (more than continuity but less than openness) on $f$ in order to prove the above property of $\lozenge$.
Alternatively, instead of restricting the function $f$, we could restrict the open subsets $U$ and $V$. This is what the argument at the end of Section 14 of my paper does. This gives a factorisation $f=g\cdot p$ of any continuous function $f:{\bf R}\to{\bf R}$ into a proper surjection $p$ with compact connected fibres and a non-hovering map $g$.
To a classical mathematician, $p$ is obviously surjective in the pointwise sense, whereas this is precisely the situation that a constructivist finds unacceptable. Meanwhile, they agree on finding zeroes of $g$.
In fact, this process finds interval-valued zeroes of any continuous function that takes opposite signs, which was the common sense answer to the question in the first place.
The operator $\lozenge$ defines an overt subspace, but I'll leave you to read the paper to find out what that means.
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I am afraid Joel has missed an important detail there, which is worth pointing out. Suppose $f$ is continuous and computable on $[a,b]$ and $f(a) \cdot f(b) < 0$. We must be careful to distinguish between
1. there exists a computable $x$ in $[a,b]$ such that $f(x) = 0$, and
2. there is an algorithm which accepts as input $f$, $a$, and $b$ and outputs $x$ in $[a,b]$ such that $f(x) = 0$.
In the first case we have a classical existence of a computable entity $x$, while in case (b) we have a computable existence of a computable entity.
I am pretty sure Weihrauch only proves 1., and it is impossible to prove 2., even if we further assume that $f$ is not only computable but computably continuous, or even Lipshitz with a known computable constant. The basic reason why 2. does not hold is that the $x$ cannot be chosen continuously with respect to the input data: essentially, a very small perturbation of $f$ can cause $x$ to jump around. Because all computable maps are continuos, we cannot have an algorithm computing $x$ (this is not a proof, just the idea, you have to work a bit harder to get all the details right).
However, you can impose fairly mild conditions on $f$ that are typically satisfied in practice. For example, if $f$ is locally non-constant, by which I mean that for every $y$ in $[a,b]$ we can compute nearby points $z$ and $w$ such that $f(z) \neq f(w)$, then IVT holds computably in the sense of 2. To see this, just perform bisection, but always avoid hitting a zero by going to a nearby non-zero point (because either $f(z)$ or $f(w)$ is non-zero, and we can compute which one). This condition is satisfied by non-trivial polynomials, for example, as well as for any differentiable function wose derivative only has isolated zeroes.
Let me also say a bit about the use of completeness of reals in IVT. Neel's remark translats from constructive mathematics to computability as follows: we can compute arbitrarily good approximations to the IVT. The trouble is that the approximations need not converge to anything, at least not computably. Classically they have an accumulation point, but we can't compute any information from it.
A second point is that IVT holds not because $\mathbb{R}$ is complete, but because it is connected. A very thorough analysis of this was made by Paul Taylor in his paper "A lambda-calculus for real analysis", see http://www.paultaylor.eu/ASD/lamcra/ . It's not easy reading, but it is very educational.
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Indeed, the Computable IVT as I stated it is equivalent to (a), and this is what Weirich proves (and what the questioner seems to have asked). A major goal of computable analysis is to undertand the computable reals as a mathematical structure, using classical logic, and it is statement (a) that implies, for example, that the computable reals are a real-closed field. (See Weirich Corollary 6.3.10.) Statement (b), in contrast, requires a uniform algorithm, which is a stronger notion. Such uniformity issues arise throughout compubility theory, both on the natural numbers and on the reals. – Joel David Hamkins Dec 22 2009 at 1:48
This question and its answers confused me for a while, but I think I get it now, so I'll describe my experience in the hopes that it will help other non-experts, and the experts can tell me if I've still got anything wrong.
My initial reaction was: given a computable function f : [a, b] → ℝ with f(a) < 0 and f(b) > 0, "of course" we can compute an element z of [a, b] with f(z) = 0, as follows: choose c to be the midpoint (a+b)/2, check whether f(c) is positive or negative, report that z is in [a, c] or [c, b] respectively, and recurse on that interval. Ironically, the problem is that f(c) might actually be exactly 0. If so, we can ask for its value to arbitrarily large precision, but will never learn anything about whether it is positive, negative, or zero, so we cannot output c as the answer, nor guarantee that either subinterval [a, c] or [c, b] contains a root of f.
Under the locally non-constant hypothesis, however, we can choose c' near c (so that both intervals [a, c'] and [c', b] are less than k times as wide as [a, b] for some fixed k < 1) so that f(c) and f(c') are distinct. Then we can compute f(c) and f(c') in parallel, stopping when we know that one of them is either positive or negative, as must eventually happen since they cannot both be zero, and proceed as before.
It's also clear now why classically there is a computable root of f without any hypotheses besides computability of f: either some number of the form a + (s/2r)(b-a) is a root of f—these numbers are all computable—or the original algorithm will succeed forever and thus compute a root of f.
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Tjhat is an accurate description of what is going on, yes. – Andrej Bauer Dec 22 2009 at 16:36
Yes, I agree... – Joel David Hamkins Dec 24 2009 at 20:36
This is the "interval-halving" proof of the "exact" constructive IVT for functions that "don't hover" or are "locally not constantly zero". – Paul Taylor Jan 1 2010 at 19:18
Constructively, the intermediate value theorem fails, so there is no computable procedure to calculate an intermediate value on computable reals.
However, the following theorem does hold, remembering that constructively all real-valued functions are computable, as are the reals themselves. For every continuous real-valued function f on an interval [a,b] with a < b, for every c between f(a) and f(b), the following holds:
$\forall n.\; \exists x \in [a,b].\; |f(x) − c| < 2^{−n}$
Classically, I think you can use this to derive the intermediate value theorem, since you can use it to cook up a Cauchy sequence. But then you'll pass out of the set of computable reals, of course.
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This is known as the "approximate" intermediate value theorem. – Paul Taylor Jan 1 2010 at 18:42
Thanks! I think I didn't really understand what continuity did and didn't do until I understood why the approximate version didn't imply the full version, constructively. It's a little embarrassing to admit that I didn't get high-school mathematics, but only a little: high-school mathematics is some of the greatest mathematics (in Whitehead's sense) ever created. – Neel Krishnaswami Jan 2 2010 at 11:09 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 67, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9396081566810608, "perplexity_flag": "head"} |
http://www.physicsforums.com/showthread.php?t=572470 | Physics Forums
## Solving Poisson's equation with the help of Greens function
Hey all,
some weeks ago in a tutorial our TA solved Poissons equation with Greens functions..would be very short, but he also derived the Greens function using a Fourier transform. Two points I really don't get and he could also not explain it. Maybe you can help me? There might be even a short way..but if you could help out here and tell me what went wrong I would be really helpful :)!
Function to solve:
$\nabla^2$ $\Phi$ = -$\frac{\rho}{\epsilon}$ = f(x) = $\nabla^2$ u(x)
Having the Green's function:
$\nabla^2$ G(r,r') = $\delta$(r-r')
$\nabla^2$ G(x) = $\delta$(x)
taking the fourier transform:
∫$\nabla^2$ G(x)exp(-ikx)dV = 1
and now the big mess is starting: using Greens first identity, namely:
∫$\nabla*\nabla$G exp(-ikx)dV = ∫$\nabla$ G exp(-ikx) dS - ∫$\nabla$G $\nabla$ exp(-ikx)dV
then he states that the surface term is equal to zero. I understand that using the fourier transform we integrate from minus to plus infinity and that we have the condition that our Greens function is usually prone to go to zero at infinity, but how is he doing that step? A integration by parts?
From that he is then coming to the point that
∫G Δexp(-ikx) dV = k^2 ∫ G(r) exp(-ikx) dV
After a substitution he then concludes that G(k) = 1/k^2
To be honest...after applying Greens identity i am somewhat lost. And don't see the steps anymore ;(.Maybe I copied something wrong from the blackboard..it is just driving me nuts!
In the end when transforming back he actually integrates from zero to inf
∫sinc(u) du = pi/2
I always thought this is not possible...that the integral of sinc cannot be given analytically?
As you might have guessed..it is a big constrution site for me...would be very happy if someone could help me out!!
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Quote by spookyfw Hey all, some weeks ago in a tutorial our TA solved Poissons equation with Greens functions..would be very short, but he also derived the Greens function using a Fourier transform. Two points I really don't get and he could also not explain it. Maybe you can help me? There might be even a short way..but if you could help out here and tell me what went wrong I would be really helpful :)! Function to solve: $\nabla^2$ $\Phi$ = -$\frac{\rho}{\epsilon}$ = f(x) = $\nabla^2$ u(x) Having the Green's function: $\nabla^2$ G(r,r') = $\delta$(r-r') $\nabla^2$ G(x) = $\delta$(x) taking the fourier transform: ∫$\nabla^2$ G(x)exp(-ikx)dV = 1 and now the big mess is starting: using Greens first identity, namely: ∫$\nabla*\nabla$G exp(-ikx)dV = ∫$\nabla$ G exp(-ikx) dS - ∫$\nabla$G $\nabla$ exp(-ikx)dV then he states that the surface term is equal to zero. I understand that using the fourier transform we integrate from minus to plus infinity and that we have the condition that our Greens function is usually prone to go to zero at infinity, but how is he doing that step? A integration by parts? From that he is then coming to the point that ∫G Δexp(-ikx) dV = k^2 ∫ G(r) exp(-ikx) dV After a substitution he then concludes that G(k) = 1/k^2 To be honest...after applying Greens identity i am somewhat lost. And don't see the steps anymore ;(.Maybe I copied something wrong from the blackboard..it is just driving me nuts! In the end when transforming back he actually integrates from zero to inf ∫sinc(u) du = pi/2 I always thought this is not possible...that the integral of sinc cannot be given analytically? As you might have guessed..it is a big constrution site for me...would be very happy if someone could help me out!!
$$\int ( \nabla^2 G ) e^{-ikx}dV = \int (\nabla G) e^{-ikx} dS - \int \nabla G \nabla e^{-ikx}V$$
This is a simple invocation of the chain rule (which you can rearrange to make something like integration by parts) and using the divergence theorem on the first term. Then he does it again to remove the gradient on the Green's function in the remaining term on the right hand side. So from there it's a simple step to see that the k-space Green's function is something like 1/k^2. However, the following,
∫G Δexp(-ikx) dV = k^2 ∫ G(r) exp(-ikx) dV
is incorrect. The gradient of the exponential is simply -ik, you are confusing this step. Work out the integration by parts step twice to see that
$$1 = k^2 \int d\mathbf{r} G(\mathbf{r})e^{-i\mathbf{k}\cdot\mathbf{r}}$$
where we see that the integration on the right hand side is simply the Fourier transform of the spatial Green's function. Then you apply the inverse Fourier transform and convert it to spherical coordinates in the k-space (above I made the wavevector to be the 3D cartesian coordinate system). So when you do that you then arrive at the integral over the sinc function. Something like,
$$G(\mathbf{r}) = \frac{1}{2\pi^2} \int_0^\infty d k \ \rm{sinc} (k r)$$
You can then just do a change of variables to integrate over the variable k*r, which is why you pull out a factor of r in the denominator and then this integration over the sinc corrects the constants. In the end, you should get a Green's function that is something like,
$$G(\mathbf{r},\mathbf{r'}) = \frac{1}{\left| \mathbf{r}-\mathbf{r'} \right|}$$
There may be a factor of 4\pi in the denominator, looks like it will be for your system of units.
Thank you very much Born2bwire! Now I also get: ∫(∇2G)e−ikxdV=∫(∇G)e−ikxdS−∫∇G∇e−ikxV. That makes sense now, but when I apply the 'integration-by-parts' again on the last term, I have a slight problem because the divergence theorem..well is working with divergences, but here we have a grad, no? I don't see how u can simply apply it again and get grid of the grad. U would just blow up the terms and not get rid of them... what is my mistake here?
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## Solving Poisson's equation with the help of Greens function
The first is easiest by just looking at using the chain rule. You take,
$$\nabla \cdot \left[ \left(\nabla G \right) e^{-i\mathbf{k}\cdot\mathbf{r}} \right] = \left( \nabla^2 G \right) e^{-i\mathbf{k}\cdot\mathbf{r}} + \left(\nabla G \right) \nabla e^{-i\mathbf{k}\cdot\mathbf{r}} = \left( \nabla^2 G \right) e^{-i\mathbf{k}\cdot\mathbf{r}} - ik \left(\nabla G \right) e^{-i\mathbf{k}\cdot\mathbf{r}}$$
You can then rearrange it so that the Laplacian is on the LHS and the rest are on the RHS. Obviously the volume integral over the divergence because a surface integral because of the divergence theorem. Since the volume here is infinite, we know that this surface integral is zero because the gradient of the Green's function is zero at infinity (likewise the Green's function at infinity is also zero). This is deduced from physical insight of our problem where we know that we consider the potential of a point source at infinity to be zero.
Thus, we are now interested in the volume integral,
$$-ik \int d\mathbf{r} \left(\nabla G \right) e^{-i\mathbf{k}\cdot\mathbf{r}}$$
Once again we use the chain rule,
$$\nabla \left( G e^{-i\mathbf{k}\cdot\mathbf{r}} \right) = \left( \nabla G \right) e^{-i\mathbf{k}\cdot\mathbf{r}} + G \nabla e^{-i\mathbf{k}\cdot\mathbf{r}} = \left( \nabla G \right) e^{-i\mathbf{k}\cdot\mathbf{r}} - ik G e^{-i\mathbf{k}\cdot\mathbf{r}}$$
Thus, to evaluate the gradient of the LHS, we can use the fundamental rule of calculus and it just because the evaluation of the Green's function and the exponential function at the limits of integration (infinity which again evaluates to zero). Consider for example, the problem in Cartesian coordinates. In one-dimension we have,
$$\int_{-\infty}^{\infty} dx \frac{\partial}{\partial x} \left( G e^{-ik_xx} \right) = \left. G e^{-ik_xx} \right|_{-\infty}^\infty$$
A simple example. Then in two-dimensions,
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} dx dy \left[ \frac{\partial}{\partial x} \left( G e^{-i(k_xx+k_yy)} \right) \hat{x} + \frac{\partial}{\partial y} \left( G e^{-i(k_xx+k_yy)} \right) \hat{y} \right] = \int_{-\infty}^{\infty} dy \left. G e^{-i(k_xx+k_yy)} \right|_{x=-\infty}^{x=\infty} \hat{x} + \int_{-\infty}^{\infty} dx \left. G e^{-i(k_xx+k_yy)} \right|_{y=-\infty}^{y=\infty} \hat{y}$$
The extension to three-dimensions is not hard to imagine. So we are just left with my second equation given in my previous post.
EDIT: Regarding my last post,
However, the following, ∫G Δexp(-ikx) dV = k^2 ∫ G(r) exp(-ikx) dV is incorrect.
This is correct, I was misinterpreting your symbol for the Laplacian to meaning gradient.
Also, I have been a bit sloppy in the above, obviously we should be getting scalars but I seem to have accidentally omitted the dot products with the volume elements in the above when we have gradients.
Excellent. Now I a understand. Thank you very much and thanks for your time :)!!!
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http://mathoverflow.net/questions/119416?sort=votes | ## Probability of all combinations of k numbers among n being coprime
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A simple argument shows that if we choose $n$ positive integers at random, the probability of their greatest common divisor being 1 is $1/ \zeta (n)$ (in the sense that if we choose the numbers among $\lbrace 1, 2, 3, \dots , N \rbrace$ uniformly and independently, the above probability tends to some number $p(n)$ as $N \rightarrow \infty$).
In the above sense, what is the probability $p(n,k)$ that among n integers $x_1, x_2, \dots, x_n$ chosen at random we have $\gcd (x_{i_1}, x_{i_2}, \dots, x_{i_k})=1$ for all possible combinations of the $x_{i_j}$ (with $s \neq t$ if for $u \neq v$, $x_{i_u}=x_s$ and $x_{i_v}=x_t$) ?
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How are the $x_1,\dots,x_n$ chosen? If they are all multiples of two, then the gcd condition never happens. – Eric Naslund Jan 20 at 18:28
I thought it was clear that they are chosen at random. Just fixed the possible ambiguity. – Alex G Jan 20 at 18:35
You will need fewer than k of them even for that to be possible (which I think is Eric Naslund's point), which will likely be the most prominent condition determining the probability. Even if you study the relaxed condition that no k of the n numbers have a small prime factor in common, I think you will get good bounds on the probability you seek. Gerhard "Ask Me About System Design" Paseman, 2013.01.20 – Gerhard Paseman Jan 20 at 20:12
## 1 Answer
For any prime $q$, define $$\ell(q,n,k) = q^{-n} \sum_{j=0}^{n-k} \binom nj (q-1)^j,$$ so that $\ell(q,n,k)$ is the probability, if a biased coin that comes up heads only $1/q$ of the time is tossed $n$ times, that at least $k$ heads are obtained. Equivalently, $\ell(q,n,k)$ is the probability, if $n$ numbers are chosen uniformly and independently from the set ${0,1,\dots,q-1}$, that at least $k$ of the numbers equal $0$.
Then the same argument giving the $1/\zeta(n)$ result shows that the probability that the greatest common divisor of every $k$-subset of $n$ "randomly chosen" integers is $1$ equals $$\prod_q \big( 1 - \ell(q,n,k) \big),$$ where the product is over all primes $q$. This product is convergent as long as $k\ge2$; it diverges to $0$ if $k=1$ (appropriately).
For example, if three integers are chosen at random, the probability that they are pairwise coprime is $$\prod_q \big( 1 - \ell(q,3,2) \big) = \prod_q \bigg( 1-\frac1q \bigg)^2 \bigg( 1+\frac2q \bigg) \approx 0.286747.$$ (A million trials with random integers between $1$ and $10^{60}$ yielded $286912$ pairwise coprime triples, so this limiting probability seems accurate.) I don't believe this product has a closed form in terms of the zeta function.
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1
This .286747... is also the probability that two positive integers are "strongly carefree," that is, coprime and both squarefree, see Finch, Mathematical Constants, p. 110. Next page, Finch notes the probability $k$ integers are pairwise coprime is $\prod(1-p^{-1})^{k-1}(1+(k-1)p^{-1})$ for $k=2,3$, unproved for $k\gt3$. Some of Finch's references are to unpublished work of Moree. A published reference is Schroeder's book, Number Theory in Science and Communication, pp 25, 48-51, and 54. – Gerry Myerson Jan 20 at 22:13
Nice. Even though as you might have guessed, I was hoping for an answer involving the zeta function. – Alex G Jan 21 at 2:11
The formula in the answer certainly reduces, when $k=2$ (pairwise coprime), to $\prod_q (1-q^{-1})^{n-1}(1+(n-1)q^{-1})$, validating Finch's statement. – Greg Martin Jan 22 at 1:28
Indeed, Toth seems to have proved this statement (about $n$ randomly chosen integers being pairwise coprime) in a 2002 Fibonacci Quarterly paper. ams.org/mathscinet-getitem?mr=1885265 – Greg Martin Jan 25 at 1:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 42, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9341465830802917, "perplexity_flag": "head"} |
http://crypto.stackexchange.com/questions/2115/is-sha1-secure-with-such-many-inputs-z-that-z-constant-secret-x-variable-pub/2122 | # Is SHA1 secure with such many inputs Z that Z = constant secret X + variable public Y?
Let me ask whether SHA1 is designed to be secure for the following case. You compute each SHA1 of many strings,for example 1 million, where each string is a concatenation of X+Y , where X is secret and constant and Y is public and variable.
Thank you in advance.
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You should explain what you want to protect against. i.e. what can an attacker do, and what do you want to prevent him from doing. You can't simply say some scheme is "secure", you need to specify clearly what your requirements are. – CodesInChaos Mar 18 '12 at 9:28
## 2 Answers
Answering the question as worded in its body: NO, $\mathrm{SHA1}$ is not designed so that the proposed construction is secure under the stated conditions. The design objective of the $\mathrm{SHA1}$ and $\mathrm{SHA2}$ hashes, as explained by NIST, is that
it is computationally infeasible to find a message that corresponds to a given message digest, or to find two different messages that produce the same message digest.
and this does not lead to the conclusion that the proposed construction is secure, under any reasonable definition.
In order to answer the (different) question as worded in its title, we need to define "secure". The question is about a construction $F: (X,Y) \to F(X,Y) = \mathrm{SHA1}(X||Y)$. One possible definition of "secure" could be that $Y \to F(X,Y)$ is computationally indistinguishable from a random function for an attacker not knowing $X$ (which implies that $X$ does not leak from examples).
With this wide definition, NO, $\mathrm{SHA1}$ is not secure. It is vulnerable to a length-extension attack, a classical weakness of the Merkle–Damgård construction used in $\mathrm{SHA1}$. An adversary knowing $\mathrm{SHA1}(X||Y)$ can compute $\mathrm{SHA1}(X||Y')$ for some $Y'\ne Y$, on the conditions that $Y'=Y||P||E$; and he knows $E$ and the length of $X||Y$; and $P$ is a specific 65-to-576-bit string depending only on that length. Having many examples of $(Y,\mathrm{SHA1}(X||Y))$ helps the adversary, in that it widens the choices for $Y'$.
To perform this length-extension attack, the adversary can choose $P$ as the 65-to-576-bit string constituting the padding for $X||Y$ when computing $\mathrm{SHA1}(X||Y)$. Then, $\mathrm{SHA1}(X||Y||P||E)$ is computed by starting the normal computation where the block with $E$ starts, with the 160-bit hash state $\mathrm{SHA1}(X||Y)$ rather than the constant specified in $\mathrm{SHA1}$. The attack has very low cost, lower than if the adversary had computed $\mathrm{SHA1}(X||Y')$ directly with knowledge of $X||Y'$.
When the conditions for the length-extension attack are not met (including if the length of $Y$ is fixed, exactly or with a small leeway of +64 bits), or when the objective of the adversary is finding $X$, the construction proposed has considerable cryptanalytic resistance. Up to some limit on the size of $X$, we know no attack much better than brute-forcing $X$. A security argument could be given based on the assumption that the cipher used in the $\mathrm{SHA1}$ round function is secure; but we know that this assumption does not hold at all, if only because of some near-practical attacks on the collision-resistance of $\mathrm{SHA1}$. Therefore I can't quantify the level of resistance better than "AFAIK unbroken for practical purposes", which is not strong enough for a recommendation.
Fortunately, there is a recommendable construct based on $\mathrm{SHA1}$ that is designed to be secure (in the wide definition we considered); this is HMAC with $\mathrm{SHA1}$ as the hash function, $X$ as the key, $Y$ as the message. This construction has a security argument that is not directly harmed by a collision-resistance attack on $\mathrm{SHA1}$. Although the absolute confidence we can have in this argument is the subject of debate and some controversy, it is still very reassuring to me about the security of HMAC, and I am not alone.
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I'd recommend HMAC. It's designed to mix a key and a message in a secure way.
````HMAC(K,m) = H((Key ⊕ opad) ∥ H((Key ⊕ ipad) ∥ Message)).
````
Where `H` is a normal hash function such as SHA1, `opad` and `ipad` are constants. `∥` stands for concatenation.
The main disadvantage of HMAC is that it requires two hash function calls, which makes significantly more expensive for short messages, whereas for large messages the overhead is negligible. But as always, only optimize if benchmarks indicate that something is a bottleneck. Typically IO is much slower than crypto.
Your construction is vulnerable to length extension attacks. An attacker who knows `SHA1(Key ∥ Message)` can compute `SHA1(Key ∥ Message ∥ EvilMessage)`
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The attacker can always calculate SHA1(Key||Message||Padding||Evil message). The Davies-Meyer step is supposed to prevent an attacker from rolling back the final hash to an (unknown) intermediary step, even if the message part, including padding, corresponding to that operation is known. – Henrick Hellström Mar 18 '12 at 10:18
@CodeInChaos: I disagree with the stated condition for the length extension attack to work. The right condition is that the attacker knows `EvilMessage` and that starts by the appropriate 65 to 576 bits. The imposed prefix of `EvilMessage` depends, uniquely, on the length of `Key ∥ Message`, and is the padding specified by `SHA-1` when hashing a message of that length. Nothing special happens when that length is a multiple of 512. Knowing `Message` does not help. – fgrieu Mar 18 '12 at 11:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 38, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9362896680831909, "perplexity_flag": "middle"} |
http://mathhelpforum.com/trigonometry/177670-calculating-trig-ratios-w-out-calculator.html | # Thread:
1. ## calculating trig. ratios w/ out a calculator
Hello,
So i know how to construct a table for 0, 30, 45, 60, and 90 degree angles (and the respective radians) that shows the trig. ratios for these degree/radian values. If i am asked to determine the trig. ratio for 30 degrees without a calculator i'm cool w/ that. What I cannot always do is determine trig ratios for csc 120 (or another degree value other than the values i construct the table for). I figured I could just take the inverse of 2 times the sin of 60 degrees, but this didn't result in the correct answer as determined w/ the use of a calculator. I would think this would always work (that is, adding up the trig ratios for an angle to determine a greater angle that is a multiple of the smaller angles). Sometimes it works, but sometimes it doesn't.
Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).
2. Originally Posted by jonnygill
Hello,
So i know how to construct a table for 0, 30, 45, 60, and 90 degree angles (and the respective radians) that shows the trig. ratios for these degree/radian values. If i am asked to determine the trig. ratio for 30 degrees without a calculator i'm cool w/ that. What I cannot always do is determine trig ratios for csc 120 (or another degree value other than the values i construct the table for). I figured I could just take the inverse of 2 times the sin of 60 degrees, but this didn't result in the correct answer as determined w/ the use of a calculator. I would think this would always work (that is, adding up the trig ratios for an angle to determine a greater angle that is a multiple of the smaller angles). Sometimes it works, but sometimes it doesn't.
Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).
Yes. Lookup "unit circle".
The format of the points around the circle is (cosx, sinx). So find the point corresponding to 120 degrees, find the sin value (√3/2) and flip it.
3. thank you.
but what if such a chart is not available?
4. Originally Posted by jonnygill
Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).
$\displaystyle csc(120) = \frac{1}{sin(120)} = \frac{1}{sin(2 \cdot 60 )} = \frac{1}{2~sin(60)~cos(60)}$
$\displaystyle = \frac{1}{2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}$
$\displaystyle = \frac{2}{\sqrt{3}}$
-Dan
5. You have to commit to memory the special triangles for 30,45 and 60 and then know the sign of each function corresponding to each quadrant. Have you seen CAST?
6. Originally Posted by topsquark
$\displaystyle csc(120) = \frac{1}{sin(120)} = \frac{1}{sin(2 \cdot 60 )} = \frac{1}{2~sin(60)~cos(60)}$
$\displaystyle = \frac{1}{2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}$
$\displaystyle = \frac{2}{\sqrt{3}}$
-Dan
Where did cos come from? how does one know when to use cos instead of sin in problems like this?
7. Originally Posted by jonnygill
Where did cos come from? how does one know when to use cos instead of sin in problems like this?
I used the identity $sin(2 \theta) = 2~sin( \theta )~cos( \theta )$. It's one of the double angle formulas.
-Dan
8. Better yet, note that $\displaystyle \sin{\left(120^{\circ}\right)} = \sin{\left(180^{\circ} - 60^{\circ}\right)}$
$\displaystyle = \sin{\left(60^{\circ}\right)}$
$\displaystyle = \frac{\sqrt{3}}{2}$.
9. Originally Posted by jonnygill
thank you.
but what if such a chart is not available?
live it ... learn it ... luv it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8958474397659302, "perplexity_flag": "middle"} |
http://crypto.stackexchange.com/questions/tagged/cryptanalysis+algorithm-design | # Tagged Questions
6answers
300 views
### Where can I begin to study the math behind modern cryptography?
I've been studying (more like breathing and eating) crypto for almost a year now, implementing algorithms, reading books, studying code, etc. The deeper I go in, the more I realize there is; I feel ...
2answers
157 views
### How hard is to find the operators of an addition knowing the sum of them?
I want to learn whether or no there is a cryptographic primitive,scheme assumption that is based on the following hard problem if it is hard . By hard we mean that we have a polynomial adversary: The ...
2answers
688 views
### Why do block ciphers need a non-linear component (like an S-box)?
Why is there a requirement of "Non-Linear functions" as a component of many popular block ciphers (e.g. the S-box in DES or 3DES)? How does it make the cipher more secure? The only intuition I have ...
1answer
211 views
### Is XOR in a CBC-like mode secure?
Assuming that $K_{n}$, $P_{n}$, and $C_{n}$ are individual bytes of the key, plaintext, and ciphertext respectively. The first byte of ciphertext is computed like this: $C_{1} = K_{1} \oplus P_{1}$ ...
3answers
446 views
### What is the best way to put a backdoor in an encryption system?
How can you put a backdoor into an encryption algorithm? Are there any techniques that can be used to reduce the time it takes to break a key? I am looking for practical examples encryption schemes ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9207504391670227, "perplexity_flag": "middle"} |
http://en.wikipedia.org/wiki/Rayleigh_distribution | # Rayleigh distribution
Not to be confused with Rayleigh mixture distribution.
Parameters
Probability density function
Cumulative distribution function
$\sigma>0\,$
$x\in [0;\infty)$
$\frac{x}{\sigma^2} e^{-x^2/2\sigma^2}$
$1 - e^{-x^2/2\sigma^2}$
$\sigma \sqrt{\frac{\pi}{2}}$
$\sigma\sqrt{\ln(4)}\,$
$\sigma\,$
$\frac{4 - \pi}{2} \sigma^2$
$\frac{2\sqrt{\pi}(\pi - 3)}{(4-\pi)^{3/2}}$
$-\frac{6\pi^2 - 24\pi +16}{(4-\pi)^2}$
$1+\ln\left(\frac{\sigma}{\sqrt{2}}\right)+\frac{\gamma}{2}$
$1+\sigma t\,e^{\sigma^2t^2/2}\sqrt{\frac{\pi}{2}} \left(\textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right)\!+\!1\right)$
$1\!-\!\sigma te^{-\sigma^2t^2/2}\sqrt{\frac{\pi}{2}}\!\left(\textrm{erfi}\!\left(\frac{\sigma t}{\sqrt{2}}\right)\!-\!i\right)$
In probability theory and statistics, the Rayleigh distribution (pron.: ) is a continuous probability distribution for positive-valued random variables.
A Rayleigh distribution is often observed when the overall magnitude of a vector is related to its directional components. One example where the Rayleigh distribution naturally arises is when wind velocity is analyzed into its orthogonal 2-dimensional vector components. Assuming that the magnitudes of each component are uncorrelated, normally distributed with equal variance, and zero mean, then the overall wind speed (vector magnitude) will be characterized by a Rayleigh distribution. A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are i.i.d. (independently and identically distributed) Gaussian with equal variance and zero mean. In that case, the absolute value of the complex number is Rayleigh-distributed.
The distribution is named after Lord Rayleigh.[citation needed]
## Definition
The probability density function of the Rayleigh distribution is[1]
$f(x;\sigma) = \frac{x}{\sigma^2} e^{-x^2/2\sigma^2}, \quad x \geq 0,$
where $\sigma >0,$ is the scale parameter of the distribution. The cumulative distribution function is[1]
$F(x) = 1 - e^{-x^2/2\sigma^2}$
for $x \in [0,\infty).$
## Properties
The raw moments are given by:
$\mu_k = \sigma^k2^\frac{k}{2}\,\Gamma\left(1 + \frac{k}{2}\right)$
where $\Gamma(z)$ is the Gamma function.
The mean and variance of a Rayleigh random variable may be expressed as:
$\mu(X) = \sigma \sqrt{\frac{\pi}{2}}\ \approx 1.253 \sigma$
and
$\textrm{var}(X) = \frac{4 - \pi}{2} \sigma^2 \approx 0.429 \sigma^2$
The mode is $\sigma$ and the maximum pdf is
$f_\text{max} = f(\sigma;\sigma) = \frac{1}{\sigma} e^{-\frac{1}{2}} \approx \frac{1}{\sigma} 0.606$
The skewness is given by:
$\gamma_1 = \frac{2\sqrt{\pi}(\pi - 3)}{(4 - \pi)^\frac{3}{2}} \approx 0.631$
The excess kurtosis is given by:
$\gamma_2 = -\frac{6\pi^2 - 24\pi + 16}{(4 - \pi)^2} \approx 0.245$
The characteristic function is given by:
$\varphi(t) = 1 - \sigma te^{-\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}} \left[\textrm{erfi} \left(\frac{\sigma t}{\sqrt{2}}\right) - i\right]$
where $\operatorname{erfi}(z)$ is the imaginary error function. The moment generating function is given by
$M(t) = 1 + \sigma t\,e^{\frac{1}{2}\sigma^2t^2}\sqrt{\frac{\pi}{2}} \left[\textrm{erf}\left(\frac{\sigma t}{\sqrt{2}}\right) + 1\right]$
where $\operatorname{erf}(z)$ is the error function.
### Information entropy
The information entropy is given by[citation needed]
$H = 1 + \ln\left(\frac{\sigma}{\sqrt{2}}\right) + \frac{\gamma}{2}$
where $\gamma$ is the Euler–Mascheroni constant.
## Parameter estimation
Given N independent and identically distributed Rayleigh random variables with parameter $\sigma$, the maximum likelihood estimate of $\sigma$ is
$\hat{\sigma}\approx \!\,\sqrt{\frac{1}{2N}\sum_{i=1}^N x_i^2}.$
An application of the estimation of $\sigma$ can be found in magnetic resonance imaging (MRI). As MRI images are recorded as complex images but most often viewed as magnitude images, the background data is Rayleigh distributed. Hence, the above formula can be used to estimate the noise variance in an MRI image from background data.[2]
## Generating random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
$X=\sigma\sqrt{-2 \ln(U)}\,$
has a Rayleigh distribution with parameter $\sigma$. This is obtained by applying the inverse transform sampling-method.
## Related distributions
• $R \sim \mathrm{Rayleigh}(\sigma)$ is Rayleigh distributed if $R = \sqrt{X^2 + Y^2}$, where $X \sim N(0, \sigma^2)$ and $Y \sim N(0, \sigma^2)$ are independent normal random variables. (This gives motivation to the use of the symbol "sigma" in the above parameterization of the Rayleigh density.)
• If $R \sim \mathrm{Rayleigh} (1)$, then $R^2$ has a chi-squared distribution with parameter $N$, degrees of freedom, equal to two (N=2) : $[Q=R^2] \sim \chi^2(N)\ .$
• If $R \sim \mathrm{Rayleigh}(\sigma)$, then $\sum_{i=1}^N R_i^2$ has a gamma distribution with parameters $N$ and $2\sigma^2$: $[Y=\sum_{i=1}^N R_i^2] \sim \Gamma(N,2\sigma^2) .$
• The Chi distribution with v=2 is equivalent to Rayleigh Distribution with sigma=1
• The Rice distribution is a generalization of the Rayleigh distribution.
• The Weibull distribution is a generalization of the Rayleigh distribution. In this instance, parameter $\sigma$ is related to the Weibull scale parameter $\lambda$: $\lambda = \sigma \sqrt{2} .$
• The Maxwell–Boltzmann distribution describes the magnitude of a normal vector in three dimensions.
• If $X$ has an exponential distribution $X \sim \mathrm{Exponential}(\lambda)$, then $Y=\sqrt{2X\sigma^2\lambda} \sim \mathrm{Rayleigh}(\sigma) .$
## References
1. ^ a b Papoulis, Athanasios; Pillai, S. (2001) Probability, Random Variables and Stochastic Processe. ISBN 0073660116, ISBN 9780073660110[]
2. Sijbers J., den Dekker A. J., Raman E. and Van Dyck D. (1999) "Parameter estimation from magnitude MR images", International Journal of Imaging Systems and Technology, 10(2), 109–114 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 56, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.757896900177002, "perplexity_flag": "head"} |
http://mathhelpforum.com/pre-calculus/80142-proving-sin-cos-identities.html | # Thread:
1. ## proving sin and cos identities
i'm not sure im doing this right and im having trouble understanding this lesson
what i have is:
$\cos\theta+\sin\theta \tan\theta = \displaystyle{\frac{1}{\cos\theta}}$
$\cos\theta+\sin\theta\displaystyle{\frac{\sin\thet a}{\cos\theta}}$
\cos\theta+\displaystyle{\frac{\sin^2\theta\}{\cos \theta}}
$\displaystyle{\frac{\cos^2\theta+\sin^2\theta}{\co s\theta}}$
which would give me the correct answer but im worried i've done something incorrectly with the fractions
also i'm kind of new to this latex thing, but the third line was giving me a syntax error and i don't know what i did wrong exactly so i took the tags off
2. Originally Posted by allywallyrus
i'm not sure im doing this right and im having trouble understanding this lesson
what i have is:
$\cos\theta+\sin\theta \tan\theta = \displaystyle{\frac{1}{\cos\theta}}$
$\cos\theta+\sin\theta\displaystyle{\frac{\sin\thet a}{\cos\theta}}$
$cos(\theta)+ \frac{sin^2(\theta)}{cos(\theta)}$
$\displaystyle{\frac{\cos^2\theta+\sin^2\theta}{\co s\theta}}$
which would give me the correct answer but im worried i've done something incorrectly with the fractions
also i'm kind of new to this latex thing, but the third line was giving me a syntax error and i don't know what i did wrong exactly so i took the tags off
Nope your maths is good, you just need to use the Pythagorean identity ( $sin^2(x) + cos^2(x) = 1$) to get to the final answer
3. oh good, thats what i was hoping to hear
and i dont suppose there is any rule against bumping this thread with new questions a little later on? i will probably have a couple more | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9380452036857605, "perplexity_flag": "middle"} |
http://mathoverflow.net/questions/82365/is-every-finitely-generated-idempotent-ring-singly-generated-as-a-two-sided-ideal/82380 | ## Is every finitely generated idempotent ring singly generated as a two-sided ideal?
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
In this post, a ring is understood to be what one usually calls a ring, not assuming that it has a unit. Some people call such objects rng.
Question: Let R be a finitely generated (non-unital and associative) ring, such that $R=R^2$, i.e. the multiplication map $R \otimes R \to R$ is surjective (every element is a sum of products of other elements). Is it possible that every element of $R$ is contained in a proper two-sided ideal of $R$? Or, must it be the case that $R$ is singly generated as a two-sided ideal in itself?
Note, if $Z \subset R$, then the ideal generated by $Z$ is the span of $Z \cup RZ \cup ZR \cup RZR$, which in the case of idempotent rings is equal to the span of $RZR$.
Question: For a fixed natural number $k$, can it happen that every set of $k$ elements of $R$ generates a proper ideal of $R$?
So far, I do not know of any example where the ring $R$ is not generated by a single element as a two-sided ideal in itself. I first thought that it must be easy to find counterexamples, but I learned from Narutaka Ozawa that the free non-unital ring on a finite number of idempotents is singly generated as a two-sided ideal in itself. He also showed that no finite ring can give an interesting example. The commutative case is also well-known; Kaplansky showed that every finitely generated commutative idempotent ring must have a unit.
Update: Some partial results about this question and a relation to the Wiegold problem in group theory can be found in http://arxiv.org/abs/1112.1802
-
1
@Andreas: Never mind. The problem is nice and well written. Normal people still call a ring a ring. – Mark Sapir Dec 1 2011 at 16:21
1
Todd, I intended to make clear that I will not assume that a unit is included. en.wikipedia.org/wiki/… Anyhow, I think the interesting part of the question starts with the third sentence. – Andreas Thom Dec 1 2011 at 16:34
1
Is it possible that the algebra of a finitely generated semigroup $S$ satisfying $S^2=S$ could work (or can you prove that such an example never works? I can prove that inverse semigroups don't work. – Benjamin Steinberg Dec 1 2011 at 19:51
1
@Ben: I tried. It is not easy, and may be not possible. In fact, take one representative $a_i$ of each maximal $J$-class of $S$. Then $\sum a_i$ seem to generate $KS$ for every field $K$. – Mark Sapir Dec 1 2011 at 20:10
1
@Mark, this was the problem my first few attempts ran into. But I couldn't prove it in general. – Benjamin Steinberg Dec 1 2011 at 20:29
show 9 more comments
## 2 Answers
Consider the ring generated by $a,b,c,d,e$ subject to the relation $a=bc+de$ and all its cyclic shifts: $b=cd+ea$, and 3 more. It is "idempotent" obviously. Can it be killed by one relation? I will ask Agata Smoktunowicz. She should be able to figure it out quickly.
Update Agata responded saying that the problem, while interesting, is too difficult. She did try using Groebner-Shirshov bases but without success. She did manage to prove the statement for semigroup algebras using an argument similar to Ozawa's (as Ben Steinberg asked here). If $S$ is a semigroup, $S^2=S$, then $KS$ is generated as an ideal by one element.
-
Thanks. I asked Agata already, and she could not come up with an example. – Andreas Thom Dec 1 2011 at 13:45
Anyway, your example looks very interesting. – Andreas Thom Dec 1 2011 at 13:52
3
If Agata cannot do it, nobody can. I have sent her the example. – Mark Sapir Dec 1 2011 at 15:29
Mark, thanks. Does K have to be a field or does it work over the integral semigroup ring? I assume the latter. – Benjamin Steinberg Dec 2 2011 at 18:59
1
George Bergman has shown me a proof which works over every unital commutative ring; even commutativity is not necessary I think. – Andreas Thom Dec 7 2011 at 12:43
show 1 more comment
### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
What about the ring generated by $a,b,c$ subject to the relations $a^2=a$, $b^2=b$ and $c^2=c$? If this ring is generated as an ideal by a single element $r$ one can show that $r= \pm a \pm b \pm c + h$ where $h$ is of order at least $2$ (in the ideal generated by $ab, ba, ac, \dots$). I can not inagine how this can happen.
If this works then one can add extra generators which is likely to answer also the second question.
-
@Martin: Andreas wrote in his question that for rings generated by idempotents the statement is true. By the way, do you know how to deal with my example? – Mark Sapir Dec 1 2011 at 19:17
@Mark: Andreas has a very cute proof that this will not work. Since for finite rings everything is OK, one need to build some elaborate obstruction -- the only thing which comes to my mind is something build out of ideal class groups, but I do not see how to encode this into a ring.... Your example "lacks" any structure and I have no idea how even to start. – kassabov Dec 1 2011 at 21:56
@Martin: I do not think structure can help here at all. Some manipulation with words. That is why I asked Agata. – Mark Sapir Dec 1 2011 at 22:05
1
To elaborate on my last comment: to study a f.g. ring $R$ one usually takes its Jacobson radical $J$ and study the semi-simple part $R/J$ where the density theorem gives structure. But for semi-simple rings, I think, the statement is true, so the most interesting case is when $R=J$. I think, in particular, that a nil-ring can be an example. If $R=J$, there is no structure theory as far as I know, and the only way to treat such rings is by studying generators and relations (Groebner bases, etc.). – Mark Sapir Dec 2 2011 at 1:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 42, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9445332884788513, "perplexity_flag": "head"} |
http://www.citizendia.org/Frequency_spectrum | Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. Frequency is a measure of the number of occurrences of a repeating event per unit Time.
Electromagnetic emission spectrum of Iron in the visible region. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of Iron (ˈаɪɚn is a Chemical element with the symbol Fe (ferrum and Atomic number 26
A source of light can have many colors mixed together and in different amounts (intensities). A rainbow, or prism, sends the different frequencies in different directions, making them individually visible at different angles. A graph of the intensity plotted against the frequency (showing the amount of each color) is the frequency spectrum of the light. When all the visible frequencies are present in equal amounts, the effect is the "color" white, and the spectrum is a flat line. Therefore, flat-line spectrums in general are often referred to as white, whether they represent light or something else.
Similarly, a source of sound can have many different frequencies mixed together. Each frequency stimulates a different length receptor in our ears. When only one length is predominantly stimulated, we hear a note. A steady hissing sound or a sudden crash stimulates all the receptors, so we say that it contains some amounts of all frequencies in our audible range. Things in our environment that we refer to as noise often comprise many different frequencies. Therefore, when the sound spectrum is flat, it is called white noise. White noise is a random signal (or process with a flat Power spectral density. This term carries over into other types of spectrums than sound.
Each broadcast radio and TV station transmits a wave on an assigned frequency (aka channel). A radio antenna adds them all together into a single function of amplitude (voltage) vs. time. The radio tuner picks out one channel at a time (like each of the receptors in our ears). Some channels are stronger than others. If we made a graph of the strength of each channel vs. the frequency of the tuner, it would be the frequency spectrum of the antenna signal.
## Spectrum analysis
Example of voice waveform and its frequency spectrum
A triangle wave pictured in the time domain (top) and frequency domain (bottom). A triangle wave is a Non-sinusoidal Waveform named for its triangular shape The fundamental frequency component is at 220 Hz (A2).
Analysis means decomposing something complex into simpler, more basic parts. As we have seen, there is a physical basis for modeling light, sound, and radio waves as being made up of various amounts of all different frequencies. Any process that quantifies the various amounts vs. frequency can be called spectrum analysis. It can be done on many short segments of time, or less often on longer segments, or just once for a deterministic function (such as $\begin{matrix} \frac{\sin (t) }{t} \end{matrix}\,$).
The Fourier transform of a function produces a spectrum from which the original function can be reconstructed (aka synthesized) by an inverse transform, making it reversible. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In order to do that, it preserves not only the magnitude of each frequency component, but also its phase. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 This information can be represented as a 2-dimensional vector or a complex number, or as magnitude and phase (polar coordinates). In graphical representations, often only the magnitude (or squared magnitude) component is shown. This is also referred to as a power spectrum. In Statistical signal processing and Physics, the spectral density, power spectral density ( PSD) or energy spectral density (
Because of reversibility, the Fourier transform is called a representation of the function, in terms of frequency instead of time, thus, it is a frequency domain representation. Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency Linear operations that could be performed in the time domain have counterparts that can often be performed more easily in the frequency domain. It is also helpful just for understanding and interpreting the effects of various time-domain operations, both linear and non-linear. For instance, only non-linear operations can create new frequencies in the spectrum.
The Fourier transform of a random (aka stochastic) waveform (aka noise) is also random. Stochastic (from the Greek "Στόχος" for "aim" or "guess" means Random. is a one volume manga created by Tsutomu Nihei as a prequel to his ten-volume work Blame!. Some kind of averaging is required in order to create a clear picture of the underlying frequency content (aka frequency distribution). In Statistics, a frequency distribution is a list of the values that a variable takes in a sample. Typically, the data is divided into time-segments of a chosen duration, and transforms are performed on each one. Then the magnitude or (usually) squared-magnitude components of the transforms are summed into an average transform. This is a very common operation performed on digitized (aka sampled) time-data, using the discrete Fourier transform (see Welch method). In Mathematics, the discrete Fourier transform (DFT is one of the specific forms of Fourier analysis. In Physics, Engineering, and applied Mathematics, Welch's method, named after P When the result is flat, as we have said, it is commonly referred to as white noise. White noise is a random signal (or process with a flat Power spectral density.
## Physical acoustics of music
Main article: Musical Acoustics
Sound spectrum is one of the determinants of the timbre or quality of a sound or note. Musical acoustics or music acoustics is the branch of Acoustics concerned with researching and describing the Physics of Music — how sounds In Music, timbre (ˈtæm-bər' like timber, or, from Fr timbre tɛ̃bʁ is the quality of a Musical note or sound that distinguishes different Sound' is Vibration transmitted through a Solid, Liquid, or Gas; particularly sound means those vibrations composed of Frequencies In Music, the term note has two primary meanings 1 a sign used in Musical notation to represent the relative duration and pitch of a Sound; It is the relative strength of pitches called harmonics and partials (collectively overtones) at various frequencies usually above the fundamental frequency, which is the actual note named (eg. In Acoustics and Telecommunication, the harmonic of a Wave is a component Frequency of the signal that is an Integer An overtone is a natural resonance or vibration frequency of a system an A).
## See also
The electromagnetic (EM spectrum is the range of all possible Electromagnetic radiation frequencies Musical acoustics or music acoustics is the branch of Acoustics concerned with researching and describing the Physics of Music — how sounds This article is about the technique in signal processing The term "frequency estimation" can also refer to Probability estimation. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9224271774291992, "perplexity_flag": "middle"} |
http://mathhelpforum.com/number-theory/77173-congruency.html | # Thread:
1. ## Congruency
I'm having trouble solving this problem.
Solve the congruence 2x is equivalent to 7 (mod 17)
Can someone work it out for me in step by step form with some explinations
2. Add the modulus to $7$ :
$\begin{array}{rcll} 2x & \equiv & 7 & \left(\text{mod } 17\right) \\ 2x & \equiv & 7 {\color{red}\ + \ 17} & \left(\text{mod } 17\right) \\ 2x & \equiv & 24 & \left(\text{mod } 17\right) \\ 2x & \equiv & 2 \cdot 12 & \left(\text{mod } 17\right) \end{array}$
Since $\gcd (2,17) = 1$, we can cancel both sides by 2 without worrying about the modulus.
3. Originally Posted by o_O
$\begin{array}{rcll}2x & \equiv & 24 & \left(\text{mod } 17\right) \\ 2x & \equiv & 2 \cdot 17 & \left(\text{mod } 17\right) \end{array}$
how does this follow
4. Sorry that was a typo. Fixed it now.
5. Congruence ax = b (mod c) for some a, b, c in Z can be written as a linear diophantine equation ax - cy = b. And that is easy to solve using the box method.
For this problem, 2x = 7 (mod 17) can be written as 2x - 17y = 7.
Using the box method, we found the Bezout relation: -56.2 + 7.17 = 7
Therefore, x = -56 = 12 (mod 17). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9246892333030701, "perplexity_flag": "middle"} |
http://www.physicsforums.com/showthread.php?s=446eafbcf5a291880d956d50737efc2e&p=4252353 | Physics Forums
## Relationship between current and current density for a volume conductor
(I'd like to preface this with the warning that the following question may be a very dumb one.)
My understanding is that current density (or flux) $\vec{J} = \vec{J}(x, y, z)$ is the rate of flow of charge (or the current) per unit area. (Units of $\frac{\text{C/s}}{\text{cm}^2}$.)
Say we know that in an irregularly shaped volume conductor (e.g. a nerve fiber, where the irregularity is due to some sort of biological obstruction), the current density $\vec{J}$ has the form given in Figure 1 (see attachments).
The current $I$ is --- I think --- given from Ohm's law: specifically, if the potential difference across the ends of the fiber and the total resistance of the fiber are known, then
[tex]
\begin{equation*}
I = \frac{V}{R},
\end{equation*}
[/tex]
which is a scalar quantity. I believe this value is constant for each point $(x, y, z)$ in the fiber, assuming no build-up of current anywhere (by the law of conservation of charge).
Now I know that the surface integral of a flux gives a flow rate, so the surface integral of $\vec{J}$ should give a current. But is the current equal to $I$? I mean, does
[tex]
\begin{equation*}
\iint_S \vec{J} \cdot \vec{n} dS = I
\end{equation*}
[/tex]
for every $S$, or only for $S$ = cross-sectional area of the conductor? The reason I ask is because, if I draw three example surfaces $S_1, S_2, S_3$ (Figure 2), the surface integral of $\vec{J}$ over $S_1$ is obviously not equal to that over $S_2$.
So I guess what I am asking is: is
[tex]
\begin{equation*}
I = \iint_{S_1} \vec{J} \cdot \vec{n} dS_1 = \iint_{S_3} \vec{J} \cdot \vec{n} dS_3 \neq \iint_{S_2} \vec{J} \cdot \vec{n} dS_2
\end{equation*}
[/tex]
true, where $I$ is current as found from Ohm's law? And if so, does this mean that any current $I$ through a conductor is not simply the rate of flow of charge, but a rate of flow of charge implicitly with respect to the cross-sectional area?
Thanks
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Quote by Cole A. (I'd like to preface this with the warning that the following question may be a very dumb one.) My understanding is that current density (or flux) $\vec{J} = \vec{J}(x, y, z)$ is the rate of flow of charge (or the current) per unit area. (Units of $\frac{\text{C/s}}{\text{cm}^2}$.) Say we know that in an irregularly shaped volume conductor (e.g. a nerve fiber, where the irregularity is due to some sort of biological obstruction), the current density $\vec{J}$ has the form given in Figure 1 (see attachments). The current $I$ is --- I think --- given from Ohm's law: specifically, if the potential difference across the ends of the fiber and the total resistance of the fiber are known, then $$\begin{equation*} I = \frac{V}{R}, \end{equation*}$$ which is a scalar quantity.
Fine up to here.
Quote by Cole A. I believe this value is constant for each point $(x, y, z)$ in the fiber, assuming no build-up of current anywhere (by the law of conservation of charge).
No, this is wrong. Current I is a total amount, that is, it's the integral of J across a surface as you've written below. It doesn't make sense to talk about I at an infinitesimal point (x,y,z).
Quote by Cole A. Now I know that the surface integral of a flux gives a flow rate, so the surface integral of $\vec{J}$ should give a current. But is the current equal to $I$? I mean, does $$\begin{equation*} \iint_S \vec{J} \cdot \vec{n} dS = I \end{equation*}$$ for every $S$, or only for $S$ = cross-sectional area of the conductor?
In your example, it is the second option.
Quote by Cole A. The reason I ask is because, if I draw three example surfaces $S_1, S_2, S_3$ (Figure 2), the surface integral of $\vec{J}$ over $S_1$ is obviously not equal to that over $S_2$. So I guess what I am asking is: is $$\begin{equation*} I = \iint_{S_1} \vec{J} \cdot \vec{n} dS_1 = \iint_{S_3} \vec{J} \cdot \vec{n} dS_3 \neq \iint_{S_2} \vec{J} \cdot \vec{n} dS_2 \end{equation*}$$ true, where $I$ is current as found from Ohm's law?
Yes.
Quote by Cole A. And if so, does this mean that any current $I$ through a conductor is not simply the rate of flow of charge, but a rate of flow of charge implicitly with respect to the cross-sectional area? Thanks
Yes. Usually it's obvious what that surface is (current in a wire refers to the area of the wire), but if the surface isn't obvious then you need to specify it.
Thank you for clearing things up.
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http://en.wikipedia.org/wiki/Scalable | # Scalability
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In electronics (including hardware, communication and software), scalability is the ability of a system, network, or process to handle a growing amount of work in a capable manner or its ability to be enlarged to accommodate that growth.[1] For example, it can refer to the capability of a system to increase total throughput under an increased load when resources (typically hardware) are added. An analogous meaning is implied when the word is used in an economic context, where scalability of a company implies that the underlying business model offers the potential for economic growth within the company.
Scalability, as a property of systems, is generally difficult to define[2] and in any particular case it is necessary to define the specific requirements for scalability on those dimensions that are deemed important. It is a highly significant issue in electronics systems, databases, routers, and networking. A system whose performance improves after adding hardware, proportionally to the capacity added, is said to be a scalable system.
An algorithm, design, networking protocol, program, or other system is said to scale if it is suitably efficient and practical when applied to large situations (e.g. a large input data set, a large number of outputs or users, or a large number of participating nodes in the case of a distributed system). If the design or system fails when a quantity increases, it does not scale. In practice, if there are a large number of things n that affect scaling, then n must grow less than n2. An example is a search engine, that must scale not only for the number of users, but for the number of objects it indexes. Scalability refers to the ability of a site to increase in size as demand warrants.[3]
The concept of scalability is desirable in technology as well as business settings. The base concept is consistent – the ability for a business or technology to accept increased volume without impacting the contribution margin (= revenue − variable costs). For example, a given piece of equipment may have capacity from 1–1000 users, and beyond 1000 users, additional equipment is needed or performance will decline (variable costs will increase and reduce contribution margin).
## Measures
Scalability can be measured in various dimensions, such as:
• Administrative scalability: The ability for an increasing number of organizations or users to easily share a single distributed system.
• Functional scalability: The ability to enhance the system by adding new functionality at minimal effort.
• Geographic scalability: The ability to maintain performance, usefulness, or usability regardless of expansion from concentration in a local area to a more distributed geographic pattern.
• Load scalability: The ability for a distributed system to easily expand and contract its resource pool to accommodate heavier or lighter loads or number of inputs. Alternatively, the ease with which a system or component can be modified, added, or removed, to accommodate changing load.
## Examples
• A routing protocol is considered scalable with respect to network size, if the size of the necessary routing table on each node grows as O(log N), where N is the number of nodes in the network.
• A scalable online transaction processing system or database management system is one that can be upgraded to process more transactions by adding new processors, devices and storage, and which can be upgraded easily and transparently without shutting it down.
• Some early peer-to-peer (P2P) implementations of Gnutella had scaling issues. Each node query flooded its requests to all peers. The demand on each peer would increase in proportion to the total number of peers, quickly overrunning the peers' limited capacity. Other P2P systems like BitTorrent scale well because the demand on each peer is independent of the total number of peers. There is no centralized bottleneck, so the system may expand indefinitely without the addition of supporting resources (other than the peers themselves).
• The distributed nature of the Domain Name System allows it to work efficiently even when all hosts on the worldwide Internet are served, so it is said to "scale well".
## Horizontal and vertical scaling
Methods of adding more resources for a particular application fall into two broad categories: horizontal and vertical scaling.[4]
To scale horizontally (or scale out) means to add more nodes to a system, such as adding a new computer to a distributed software application. An example might be scaling out from one Web server system to three. As computer prices drop and performance continues to increase, low cost "commodity" systems can be used for high performance computing applications such as seismic analysis and biotechnology workloads that could in the past only be handled by supercomputers. Hundreds of small computers may be configured in a cluster to obtain aggregate computing power that often exceeds that of single traditional RISC processor based scientific computers. This model has further been fueled by the availability of high performance interconnects such as Myrinet and InfiniBand technologies. It has also led to demand for features such as remote maintenance and batch processing management previously not available for "commodity" systems. The scale-out model has created an increased demand for shared data storage with very high I/O performance, especially where processing of large amounts of data is required, such as in seismic analysis. This has fueled the development of new storage technologies such as object storage devices. Scale out solutions for database servers generally seek to move toward a shared nothing architecture going down the path blazed by Google of sharding.
To scale vertically (or scale up) means to add resources to a single node in a system, typically involving the addition of CPUs or memory to a single computer. Such vertical scaling of existing systems also enables them to use virtualization technology more effectively, as it provides more resources for the hosted set of operating system and application modules to share. Taking advantage of such resources can also be called "scaling up", such as expanding the number of Apache daemon processes currently running.
There are tradeoffs between the two models. Larger numbers of computers means increased management complexity, as well as a more complex programming model and issues such as throughput and latency between nodes; also, some applications do not lend themselves to a distributed computing model. In the past, the price difference between the two models has favored "scale up" computing for those applications that fit its paradigm, but recent advances in virtualization technology have blurred that advantage, since deploying a new virtual system over a hypervisor (where possible) is almost always less expensive than actually buying and installing a real one.[dubious ] Configuring an existing idle system has always been less expensive than buying, installing, and configuring a new one, regardless of the model.
## Database scalability
A number of different approaches enable databases to grow to very large size while supporting an ever-increasing rate of transactions per second. Not to be discounted, of course, is the rapid pace of hardware advances in both the speed and capacity of mass storage devices, as well as similar advances in CPU and networking speed. Beyond that, a variety of architectures are employed in the implementation of very large-scale databases.
One technique supported by most of the major database management system (DBMS) products is the partitioning of large tables, based on ranges of values in a key field. In this manner, the database can be scaled out across a cluster of separate database servers. Also, with the advent of 64-bit microprocessors, multi-core CPUs, and large SMP multiprocessors, DBMS vendors have been at the forefront of supporting multi-threaded implementations that substantially scale up transaction processing capacity.
Network-attached storage (NAS) and Storage area networks (SANs) coupled with fast local area networks and Fibre Channel technology enable still larger, more loosely coupled configurations of databases and distributed computing power. The widely supported X/Open XA standard employs a global transaction monitor to coordinate distributed transactions among semi-autonomous XA-compliant database resources. Oracle RAC uses a different model to achieve scalability, based on a "shared-everything" architecture that relies upon high-speed connections between servers.
While DBMS vendors debate the relative merits of their favored designs, some companies and researchers question the inherent limitations of relational database management systems. GigaSpaces, for example, contends that an entirely different model of distributed data access and transaction processing, Space based architecture, is required to achieve the highest performance and scalability. On the other hand, Base One makes the case for extreme scalability without departing from mainstream relational database technology.[5] For specialized applications, NoSQL architectures such as Google's BigTable can further enhance scalability. Google's massively distributed Spanner technology, positioned as a successor to BigTable, supports general-purpose database transactions and provides a more conventional SQL-based query language.[6] In any case, whether or not adhering to traditional relational concepts, there appears to be no limit in sight to database scalability.[citation needed]
## Design for scalability
It is often advised to focus system design on hardware scalability rather than on capacity. It is typically cheaper to add a new node to a system in order to achieve improved performance than to partake in performance tuning to improve the capacity that each node can handle. But this approach can have diminishing returns (as discussed in performance engineering). For example: suppose 70% of a program can be sped up if parallelized and run on multiple CPUs instead of one. If $\alpha$ is the fraction of a calculation that is sequential, and $1-\alpha$ is the fraction that can be parallelized, the maximum speedup that can be achieved by using P processors is given according to Amdahl's Law: $\frac{1}{\alpha+\frac{1-\alpha}{P}}$. Substituting the value for this example, using 4 processors we get $\frac{1}{0.3+\frac{1-0.3}{4}} = 2.105$. If we double the compute power to 8 processors we get $\frac{1}{0.3+\frac{1-0.3}{8}} = 2.581$. Doubling the processing power has only improved the speedup by roughly one-fifth. If the whole problem was parallelizable, we would, of course, expect the speed up to double also. Therefore, throwing in more hardware is not necessarily the optimal approach.
## Weak versus strong scaling
In the context of high performance computing there are two common notions of scalability.
• The first is strong scaling, which is defined as how the solution time varies with the number of processors for a fixed total problem size.[7]
• The second is weak scaling, which is defined as how the solution time varies with the number of processors for a fixed problem size per processor.
## References
1. André B. Bondi, 'Characteristics of scalability and their impact on performance', Proceedings of the 2nd international workshop on Software and performance, Ottawa, Ontario, Canada, 2000, ISBN 1-58113-195-X, pages 195–203
2. See for instance, Mark D. Hill, 'What is scalability?' in ACM SIGARCH Computer Architecture News, December 1990, Volume 18 Issue 4, pages 18–21, (ISSN 0163-5964) and Leticia Duboc, David S. Rosenblum, Tony Wicks, 'Doctoral symposium: presentations: A framework for modelling and analysis of software systems scalability' in Proceeding of the 28th international conference on Software engineering ICSE '06, May 2006. ISBN 1-59593-375-1, pages 949–952
3. Laudon & Traver, 2008.
4. Michael, M.; J.E. Moreira, D. Shiloach, R.W. Wisniewski (March 26, 2007). "Scale-up x Scale-out: A Case Study using Nutch/Lucene". Parallel and Distributed Processing Symposium, 2007. IPDPS 2007. IEEE International. Retrieved 2008-01-10. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 5, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9184965491294861, "perplexity_flag": "middle"} |
http://math.stackexchange.com/questions/238946/the-minimal-number-of-triangles-or-edges-whose-union-is-a-graph-g?answertab=oldest | # The minimal number of triangles or edges whose union is a graph $G$.
Let $G$ be a simple graph of order $n$. If $G$ is triangle-free, then we know that there is a bipartite graph of the same order and the same size. So $G$ has size less than $n^2/4$.
Now if it has triangles, it might have more than $n^2/4$ edges. But then if we think of the graph as a smallest union of triangles and edges, then I guess that this union involves less than $n^2/4$ triangles or edges. I tried with several examples and my guess holds for them.
Is this a known result, or is it just false?
-
What is order and size? What if $n$ is odd? $C_5$ has no bipartite graph with the same number of vertices. – draks ... Nov 28 '12 at 22:02
If I understand correctly, the question basically asks if every $n$-vertex graph can be decomposed into at most $n^2/4$ triangles and edges. Turán's theorem implies every triangle-free graph has at most $n^2/4$ vertices, so any counter-example must have a triangle. – Douglas S. Stones Jan 6 at 0:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 13, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9420974850654602, "perplexity_flag": "head"} |
http://mathoverflow.net/questions/66323/for-an-l-adic-sheaf-f-n-why-is-the-complex-f-n-of-finite-tor-dimension | For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?
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Let $X$ be a variety and let `$\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$` be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of sheaves of $\mathbf{Z}/\ell^{n+1}$-modules defined by $\mathcal{F}_n$, i.e., the complex with $\mathcal{F}_n$ concentrated in degree zero and zeroes everywhere else.
In the proof of the generalized Trace formula (see Deligne's Rapport sur la formule des traces or de Jong's Stacks Project: Etale cohomology) the following fact is used.
Fact. We have that $K_n$ is of finite Tor dimension.
Equivalently:
Fact. We have that $K_n$ is isomorphic in $D^-(X,\mathbf{Z}/\ell^{n+1})$ to a bounded complex of constructible sheaves of flat $\mathbf{Z}/\ell^{n+1}$-modules.
Unfortunately, I haven't been able to find a proof of this fact in the literature.
Question. How does one prove the above Fact?
[Edit: Explanation of Torsten Ekedahl's answer] By an $\ell$-adic sheaf, one usually means a `$\mathbf{Q}_\ell$`-sheaf. My mistake was that I considered an arbitrary `$\mathbf{Z}_\ell$`-sheaf $\mathcal{F}$. In fact, given a `$\mathbf{Q}_\ell$`-sheaf $\mathcal{G}$, we have that `$\mathcal{G}= \mathcal{F}\otimes \mathbf{Q}_\ell$` for some torsion-free `$\mathbf{Z}_\ell$-sheaf $\mathcal{F}$`. But then it's clear that the complex defined by $\mathcal{F}_n$ is of finite Tor-dimension.
This Fact is very strange to me for the following reason:
Example. Consider the complex of $\mathbf{Z}/\ell^{n+1}$-modules $$\ldots \stackrel{\cdot l^n}{\longrightarrow} \mathbf{Z}/\ell^{n+1} \stackrel{\cdot l}{\longrightarrow} \mathbf{Z}/\ell^{n+1} \stackrel{\cdot l^n}{\longrightarrow} \mathbf{Z}/\ell^{n+1} \stackrel{\cdot l}{\longrightarrow}\mathbf{Z}/\ell^{n+1} \longrightarrow \mathbf{Z}/\ell \rightarrow 0 .$$ From this it follows that Tor_i$(\mathbf{Z}/\ell, \mathbf{Z}/\ell) = \mathbf{Z}/\ell \neq 0$ if $i>0$. In particular, the $\mathbf{Z}/\ell^{n+1}$-module $\mathbf{Z}/\ell$ is NOT of finite Tor dimension. Thus, we see that $K_n = \mathbf{Z}/\ell$ does not give an $\ell$-adic sheaf. (This would contradict the Fact.)
This Example suggests that the proof of the above Fact relies on the compatibility between the K_n.
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1 Answer
I think you have (slightly) misread your sources. If you take Rapport for instance (which is the one I am familiar with) Deligne never make this claim (as far as I can see). As a typical example consider 4.11 (proof of 2.30) where he assumes that $(K_n)$ is a torsion free $\mathbb Z_\ell$-complex in which case $K_n$ is indeed of finite Tor-dimension. Alternatively, he could have considerd (even when $K$ is just a sheaf) the derived reduction modulo $\ell^n$, `$K\bigotimes^{\mathbb L}_{\mathbb Z_\ell}\mathbb Z/\ell^n$` which always will have finite Tor-dimension. Your example is a proof that not taking the derived reduction but just the reduction will not necessarily give something of finite Tor-dimension.
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Thank you very much! – Ariyan Javanpeykar May 29 2011 at 7:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 28, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9184149503707886, "perplexity_flag": "head"} |
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http://cms.math.ca/10.4153/CMB-2007-050-x | Canadian Mathematical Society
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# Asymptotic Existence of Resolvable Graph Designs
Read article
[PDF: 175KB]
http://dx.doi.org/10.4153/CMB-2007-050-x
Canad. Math. Bull. 50(2007), 504-518
Published:2007-12-01
Printed: Dec 2007
• Peter Dukes
• Alan C. H. Ling
Features coming soon:
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Format: HTML LaTeX MathJax PDF PostScript
## Abstract
Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A \emph{block design} $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$-subsets of a $v$-set $X$ in which every unordered pair of elements from $X$ is contained in exactly $\lambda$ elements of $\cA$. More generally, for a fixed simple graph $G$, a \emph{graph design} $\GD(v,G,\lambda)$ is a collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such that every unordered pair of elements from $X$ is an edge of exactly $\lambda$ elements of $\cA$. A famous result of Wilson says that for a fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all sufficiently large $v$ satisfying certain necessary conditions. A block (graph) design as above is \emph{resolvable} if $\cA$ can be partitioned into partitions of (graphs whose vertex sets partition) $X$. Lu has shown asymptotic existence in $v$ of resolvable $\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.
Keywords: graph decomposition, resolvable designs
MSC Classifications: 05B05 - Block designs [See also 51E05, 62K10] 05C70 - Factorization, matching, partitioning, covering and packing 05B10 - Difference sets (number-theoretic, group-theoretic, etc.) [See also 11B13] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 27, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8382383584976196, "perplexity_flag": "middle"} |
http://mathoverflow.net/questions/62843?sort=votes | ## Path connectedness of varieties
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
Let $X$ be a variety. Then, is $X$ path connected? And by path connected, I mean any two closed points $P, Q$ on the variety can be connected by the image of a finite number of non-singular curves.
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3
If $X$ is quasi-projective and of dim $\ge 2$, you can use Bertini's theorem on a sufficiently general hyperplane section through P and Q. – J.C. Ottem Apr 24 2011 at 14:36
The version in Hartshorne requires $X$ has at most a finite number of singular points and that $X$ projective (or equivalently, projective with a finite number of points removed). Do you have a more general form in mind? Also, your answer leads to another question (probably a dumb one that I cannot think of): curves are parametrizable, i.e. any segment on a curve is an image of a non-singular curve? – Brian Apr 24 2011 at 14:46
4
Brian, J.C. Ottem is right. You can just use Bertini. To your question of whether every curve is the image of a non-singular one, the answer is yes, just take the normalization of the curve (see the section on curves in the first chapter of Hartshorne). I don't know what you mean by segment on a curve though. – Karl Schwede Apr 24 2011 at 15:24
Dear Karl Schwede: Thanks a lot for your answer. My question about the curve is indeed a very dumb one. – Brian Apr 24 2011 at 15:29
1
We have some more than adequate answers given in the comments. Would one of the commenters be willing to step up and actually answer the question in the formal MO sense? – Pete L. Clark Apr 24 2011 at 22:29
show 1 more comment
## 1 Answer
Given any two points on a projective variety, blow them up and re embed the blownup variety in P^N. Then by Bertini, any general linear section of the right codimension will meet the variety in an irreducible curve which also meets both exceptional divisors. Then blowing back down gives an irreducible curve connecting the original two points. Normalizing that curve gives a map from just one smooth connected curve that connects your two points. (I learned this trick from David Mumford.) – roy smith 11 hours ago
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Thanks, Roy. – Pete L. Clark Apr 25 2011 at 4:10
This is very nice! Thanks. – QcH Apr 26 2011 at 0:06
1
The above proof is attributed to C. P. Ramanujam. You can find it (and a few other such gems) in the article by Ramanan in the volume `CPR-A tribute' published by TIFR. – Mohan Apr 26 2011 at 17:21
thanks for the attribution Mohan. – roy smith Apr 26 2011 at 23:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 8, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9181855320930481, "perplexity_flag": "middle"} |
http://mathhelpforum.com/algebra/10960-solve-equation-w-rational-exponent.html | # Thread:
1. ## Solve Equation w/ Rational Exponent.
I am lost. The Algebra II book I use doesn't explain concepts very well. Does anyone know how to solve this problem as well as clearly explain their chosen steps?
$x^{\frac{1}{2}} -5x^{\frac{1}{4}} +6=0$
Thanks in advance.
2. Hello, Mulya66!
I hope there's a typo in the problem.
. . Otherwise, I can't solve it.
. . . . . . . . . . . . . . . . . . . . .
↓
Is it supposed to be: . $x^{\frac{2}{3}} -5x^{\frac{1}{3}} + 6 \:=\:0$
Then it factors: . $\left(x^{\frac{1}{3}} - 2\right)\left(x^{\frac{1}{3}} - 3\right) \:=\:0$
And we have: . $\begin{Bmatrix}x^{\frac{1}{3}} - 2 \:=\:0\quad\Rightarrow\quad x^{\frac{1}{3}}\:=\:2\quad\Rightarrow\quad x\,=\,8 \\<br /> x^{\frac{1}{3}} - 3\:=\:0\quad\Rightarrow\quad x^{\frac{1}{3}}\:=\:3\quad\Rightarrow\quad x\,=\,27 \end{Bmatrix}$
3. Oops! It's all a typo! It's really $x^{\frac{1}{2}} -5x^{\frac{1}{4}} + 6 \:=\:0$
(Well, that just shows how tired I am...)
Originally Posted by Soroban
Hello, Mulya66!
I hope there's a typo in the problem.
. . Otherwise, I can't solve it.
. . . . . . . . . . . . . . . . . . . . .
↓
Is it supposed to be: . $x^{\frac{2}{3}} -5x^{\frac{1}{4}} + 6 \:=\:0$
Then it factors: . $\left(x^{\frac{1}{3}} - 2\right)\left(x^{\frac{1}{3}} - 3\right) \:=\:0$
And we have: . $\begin{Bmatrix}x^{\frac{1}{3}} - 2 \:=\:0\quad\Rightarrow\quad x^{\frac{1}{3}}\:=\:2\quad\Rightarrow\quad x\,=\,8 \\<br /> x^{\frac{1}{3}} - 3\:=\:0\quad\Rightarrow\quad x^{\frac{1}{3}}\:=\:3\quad\Rightarrow\quad x\,=\,27 \end{Bmatrix}$
4. Hello, Mulya66!
$x^{\frac{1}{2}} -5x^{\frac{1}{4}} + 6 \:=\:0$
Use the same technique . . .
It factors: . $\left(x^{\frac{1}{4}} - 2\right)\left(x^{\frac{1}{4}} - 3\right) \:=\:0$
Can you finish it now?
5. Originally Posted by Soroban
$\left(x^{\frac{1}{4}} - 2\right)\left(x^{\frac{1}{4}} - 3\right) \:=\:0$
$\begin{Bmatrix}x^{\frac{1}{4}} - 2 \:=\:0\quad\Rightarrow\quad x^{\frac{1}{4}}\:=\:2\quad\Rightarrow\quad x\,=\,16 \\<br /> x^{\frac{1}{4}} - 3\:=\:0\quad\Rightarrow\quad x^{\frac{1}{4}}\:=\:3\quad\Rightarrow\quad x\,=\,81 \end{Bmatrix}$
Thanks! It's very easy now. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9126717448234558, "perplexity_flag": "middle"} |
http://physics.stackexchange.com/questions/39602/why-do-we-use-hermitian-operators-in-qm?answertab=votes | # Why do we use Hermitian operators in QM?
Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') operators. The eigenvalues of the operator are the allowed values of the observable. Since Hermitian operators have a real spectrum, all is well.
However, there are non-Hermitian operators with real eigenvalues, too. Consider the real triangular matrix:
$$\left( \begin{array}{ccc} 1 & 0 & 0 \\ 8 & 4 & 0 \\ 5 & 9 & 3 \end{array} \right)$$
Obviously this matrix isn't Hermitian, but it does have real eigenvalues, as can be easily verified.
Why can't this matrix represent an observable in QM? What other properties do Hermitian matrices have, which (for example) triangular matrices lack, that makes them desirable for this purpose?
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## 3 Answers
One problem with the given $3\times 3$ matrix example is that the eigenspaces are not orthogonal.
Thus it doesn't make sense to say that one has with 100% certainty measured the system to be in some eigenspace but not in the others, because there may be a non-zero overlap to a different eigenspace.
One may prove$^{1}$ that an operator is Hermitian if and only if it is diagonalizable in an orthonormal basis with real eigenvalues. See also this Phys.SE post.
$^{1}$We will ignore subtleties with unbounded operators, domains, selfadjoint extensions, etc., in this answer.
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Thanks for the answer @Qmechanic. Please could you elaborate a little on how a non-orthogonal eigenbasis differs from a classical uncertainty (ie a mixed state with a density operator)? After all, we're never 100% sure that a system is in some eigenspace but not the others. – poorsod Oct 11 '12 at 21:47
Also, if I encounter an operator with both real eigenvalues and an orthogonal eigenbasis, is that enough for me to conclude that it is Hermitian? – poorsod Oct 11 '12 at 21:49
1. Here I'm considering an ideal quantum system. 2. Yes, cf. my update. – Qmechanic♦ Oct 13 '12 at 12:44
If you want to see something different, there are actually a few articles by Carl Bender developing quantum mechanics formulated with parity-time symmetric operators. He shows that some Hamiltonians are not Hermitian, yet they have real eigenvalues and seem to represent valid physical systems. If you think about it, the requirement that your operator is parity-time symmetric makes more sense physically than hermiticity. In a later article, his quantum mechanics approach was proven to be equivalent to the standard one where operators are hermitian.
If you are interested, you can read http://arxiv.org/abs/quant-ph/0501052
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It should be mentioned that PT-symmetric operators are Hermitian operators wrt. a non-standard sesqui-linear form (inner product), so in that sense one may argue, that they do not constitute a departure from standard quantum mechanical axioms. – Qmechanic♦ Oct 11 '12 at 23:01
To give an answer that is a little more general than what you're asking I can think of three reasons for having hermitian operators in quantum theory:
1) Quantum theory relies on unitary transforms, for symmetries, basis changes or time evolution. Unitary transforms are generated by hermitian operators as in $U=\exp(iH)$. And unitary Lie group representations come with a lie algebra of hermitian operators.
2) Outcomes of measurements are taken from a set of orthogonal states with real measurement values. This structure if efficiently represented by a hermitian operator that comes with an eigenstructure that matches these requirements precisely.
3) State representations of subsystems and ensembles lead to hermitian operators. For ensembles this can be seen from the construction as a convex sum of projectors, which are necessarily hermitian. For subsystem states it comes out of tracing a projector over tensor factor spaces. This is related to point 2) because processes like decoherence connect measurement outcomes with density operators.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9212732911109924, "perplexity_flag": "middle"} |
http://physics.stackexchange.com/questions/22146/what-meaning-does-the-slope-of-the-efficiency-path-on-a-mollier-diagram-have-in/22210 | # What meaning does the slope of the efficiency path on a Mollier diagram have in terms of temperature?
Let's say I have a steam turbine that I have modified to increase its isentropic efficiency. As a specific case, consider the modification outlined in the Mollier diagram below. The arrows represent the before and after modification paths of the turbine on the Mollier diagram.
The slope of the arrows represent how efficient the turbine is. A horizontal slope is a throttling process whereas a vertical slope is 100% isentropic efficiency. Slopes in between obviously are something less than 100%.
The slope is $\Delta{H} \over \Delta{S}$ which has units of temperature (degR).
Does the fact that the units of the slope are temperature hold any special meaning? Why would the value of a temperature inherently represent a performance level of isentropic efficiency? Does the fact that the values are always negative mean anything? Is the reason 100% efficiency is impossible related to the fact that the value of the temperature is $-\infty$?
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## 1 Answer
Yes, the fact that the slope is infinite means 100% efficiency is not possible. This is because $\Delta S$ is zero while $\Delta H$ is non-zero, which is not possible. Entropy will be created when heat is released.
In other situations, for example when considering Mollier diagrams of pure substances, the slope of the isobaric line is the saturation temperature. In this case, it will likely be the absolute temperature of the system because:
$Tds = dh$
at constant pressure from the second law of thermodynamics. So, the slope on the diagram is the temperature of the system.
In your figure, I don't understand why the slope is negative because the curves clearly have positive slopes.
-
I don't fully follow your comments. Ok on infinite slope. My lines aren't isobaric since I'm reducing the pressure to internally generate the kinetic energy to turn the turbine blades. As you say this will always be done w/ some irreversibility and so net P will have to decrease. Just from the math the slope has to be negative since $\Delta H$ is negative whereas &\Delta S\$ is positive. – Jason Waldrop Mar 15 '12 at 16:26
I'm just looking at your plot you put in the post. As the lines move right, they also move up which is a positive slope. Also, isn't it isobaric -- the lines in the figure are labeled with a single pressure? Typically the diagrams consist of constant pressure, constant temperature, and constant volume lines. What is your plot showing? – tpg2114 Mar 15 '12 at 19:46
Sorry I was talking about the straight lines which represent the paths taken by the two turbines. I understand your original comments on the isobaric and isothermal lines now. I was talking about the path lines slope. – Jason Waldrop Mar 15 '12 at 20:53 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.949657678604126, "perplexity_flag": "middle"} |
http://math.stackexchange.com/questions/32077/what-is-the-dimension-of-a-representation | What is the dimension of a representation
What is meant by dimension of a representation in the following excersize: "Prove that any irreducible representation of an abelian group has the dimension of 1"? I looked at the solution, and it proves that any irreducible representation of an abelian group is scalar. I understand the proof, but I still can't figure out what is meant by dimension.
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1 Answer
A representation of a group $G$ is a vector space $V$ together with an action of $G$ on $V$ by linear transformations. The dimension of the representation is just the dimension of $V$.
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So should I interpret the exceresize as "Prove that an abelian group can only have irreducible representation on a vector space with dimension of 1"? – Alexei Averchenko Apr 10 '11 at 13:08
Well, yes, that's a tautology now. – wildildildlife Apr 10 '11 at 14:38
1
@wild: why is that a tautology now? – Rasmus Apr 10 '11 at 14:47
I mean, given that Chris has just defined the dimension of a representation to be the dimension of the corresponding vector space, the exercise and your interpretation of it are tautologically the same. Anyway, the answer to your question is 'yes' :) – wildildildlife Apr 10 '11 at 18:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9346666932106018, "perplexity_flag": "head"} |
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When the light hits on a surface, it reflects with the "same" angle as the one that hits the surface. I was wondering if this choice of angle can be explained by a minimality condition?
0answers
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### Trigonometry in the plane mirror [closed]
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### How does this trick with mirrors work?
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When light reaches a boundary between materials below the critical angle, some of it refracts and some of it reflects. For example, glass acts as a partial mirror with a dark background. Assuming ...
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Why the mirror changes the left with the right side but not the top with the bottom? Go to the mirror and check this out.
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Okay, I am not really good in physics (rather terrible), but nonetheless. So, I was just wondering if you can see a mirage, is there something special in our eyes that we can see it or what? I mean, ...
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In perspective of a given example, if a man was to stand $2\ m$ away from a mirror which was $0.9\ m$ in height and was able to see his full reflection, what would the height of the mirror have to be ...
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It is supposedly possible to trap a beam of light bouncing back and fourth between two mirrors in a stable configuration. As I understand it, this means the configuration will prevent further spread ...
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If one looks into a mirror, he can see a certain field of view. If he places a convex lens that magnifies (or a concave lens that does the opposite) in front of the mirror, but so that he can still ...
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### Reflection in Convex mirrors [closed]
A monkey starts chucking polished stainless bocce balls at you. The bocce balls are 6cm in radius. Where does your image form as a function of bocce balls distance and what is the size of your image ...
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### Rays in Symmetric Resonator
I'm having some trouble figuring out how to get started on this question: If I have a symmetric resonator with two concave mirrors of radii $R$ separated by a certain distance, after how many round ...
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I'm having an extremely difficult time finding an optics program that is easy to use and offers accurate physics simulations. I'm not asking for much, I just want to be able to simulate a laser going ...
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### Why does your reflection stay the same size when you move further away from the mirror?
This was an experiment I saw in my son's workbook. It said to mark out the top of your forehead and the bottom of your chin on a mirror using a whiteboard marker. Then slowly move backwards, and ...
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### Reflection of Electromagnetic Waves
Visible light - Being an Electromagnetic wave is reflected by glass (take mirror). Would all other waves in the electromagnetic spectrum be reflected in the same way by our simple mirror... For highly ...
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When a ray of ordinary light is passed on the surface of the water the reflected light will be completely polarized( vibrations in one plane). My question is what will be plane of vibration in the ...
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### Eliminating IR light reflection perceived by a steep viewing angle
I am having a problem with reflection on an acrylic surface, in the IR part of the spectrum. This reflection is interfering with an algorithm that looks at objects, as it makes two show up when only ...
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### size and distance of mirror used in projected “smart” whiteboard
Apologies in advance for complete ignorance of optics (beyond conic sections) How is the size, distance, and angle of the mirror used here calculated? The mirror is I believe the open flap in front ...
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### What longest time ever was achieved at holding light in a closed volume?
For what longest possible time it was possible to hold light in a closed volume with mirrored walls? I would be most interested for results with empty volume but results with solid-state volume may ...
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In day, when you look in the room through the window out, you can clearly see what happens outside. At night when it's dark outside but there's light inside you can look in the window but it becomes a ...
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### Is there a limit to the resolving power of a mirror telescope?
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### Distance of objects in car mirrors
We've all seen that label on our passenger side mirrors that says, "Objects in mirror are closer than they appear." Why is this? Further, why does it only apply to the passenger side mirror, and not ...
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### Light Energy Absorption In Mirror
Let the amount of energy in one pulse of (laser) light be $E$, and the wavelength be $\lambda$. This pulse goes straight to the mirror, and it is reflected by the mirror. Let the reflectivity of ...
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Let's say you have the ability to shine some light into a perfectly round sphere and the sphere's interior surface was perfectly smooth and reflective and there was no way for the light to escape. If ...
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### How come an anti-reflective coating makes glass *more* transparent?
The book I'm reading about optics says that an anti-reflective film applied on glass* makes the glass more transparent, because the air→film and film→glass reflected waves (originated from a paraxial ...
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### Why does light reflect more intensely when it hits a surface at a large angle?
I mean, what is happening at a microscopic level to cause this behavior? Here's what I got from Wikipedia: On Reflection (physics)#Reflection of light it says that "solving Maxwell's equations for a ...
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### How do you calculate the intensity of light around the focal point from a focused collimated beam of light?
Problem/Purpose of me asking this question to you people who know more than me: So I'm doing a science project where I'm collimating a beam of light to a focus point in a light medium (water vapor or ...
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### Why can't I see far when I look in a mirror?
I'm myopic. It's a fact. I understand exactly how it works because Internet told me light rays encounter themeselves too soon in my eyes... that is why I can't see far objects even if I see near ...
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### Virtual images in (plane) mirrors?
The following image is taken from teaching physics lecture Was man aus virtuellen Bildern lernen kann (in German): Now the cited paper claims that the left hand side is the correct picture to ...
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### Why does my watch act like a mirror under water?
I have a digital watch, rated to go underwater to $100 \rm m$. When it is underwater it can be read normally, up until you reach a certain angle, then suddenly, it becomes almost like a mirror, ...
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### How can I determine transmission/reflection coefficients for light?
When light rays reflect off a boundary between two materials with different indices of refraction, a lot of the sources I've seen (recently) don't discuss the relation between the amplitude (or ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9395608901977539, "perplexity_flag": "middle"} |
http://stats.stackexchange.com/questions/43405/item-information-in-irt | # Item information in IRT
According to item information curves, item information for a 2PL IRT model is
$I(\theta)=a^2_i p_i(\theta) q_i(\theta)$
1. To determine $p_i(\theta)$ and $q_i(\theta)$, do you just use the observed response pattern for the item, e.g., 1110 implies $p_i(\theta)=0.75$? Do you use the responses of all individuals or just one?
2. Why doesn't item difficulty affect item information?
3. (Perhaps this is the same as 2.) Using this definition of item information, how can you plot item information as a function of individual proficiency $\theta_p$?
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## 1 Answer
1. Having fitted the IRT, you use the predicted probabilities from it. The pattern for a given item is not informative unless you know the abilities of the students who responded to it. This may have been a difficult question, but the first three students were bright enough to answer it, while the fourth one who failed it may have been a median student, but this could have been above his or her head.
2. Item difficulty appears in the expressions for $p_i(\theta)$. In fact, the product $p_i(\theta)q_i(\theta)$ is maximized when $\theta=$ difficulty, as then both $p_i(\theta)=q_i(\theta)=1/2$.
3. Uhm... you just plot $I(\theta)$ as a function of $\theta$?
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I must be missing something obvious. Could you give the formula for $p_i(\theta)$? – Jack Tanner Nov 12 '12 at 18:24
Ah, figured it out. $p_i(\theta) = \frac{1}{1 + exp(a_i(b_i - \theta_p))}$. – Jack Tanner Nov 12 '12 at 20:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 13, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9480159282684326, "perplexity_flag": "middle"} |
http://physics.stackexchange.com/questions/8846/how-deep-in-the-ocean-can-a-ping-pong-ball-go-before-it-collapses-due-to-pressur/8851 | # How deep in the ocean can a ping-pong ball go before it collapses due to pressure?
Stimulated by the calculation showing that a ping-pong ball does not pop in vacuum, I'm driven to ask how deep in the ocean a ping-pong ball can be brought before it collapses due to pressure.
This has a practical application in that boats have been refloated by the expedient of pumping ping pong balls into their hull.
Youtube ping pong ball boat refloating: http://www.youtube.com/watch?v=4MOJN07XRYw
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There is a telltale, that this Donald Duck episode was read as "prior art", leading to refusal of a patent application. Which patent office was involved, I do not remember. The story was told by the patents specialist of our company. – Georg Apr 19 '11 at 9:51
1
– Georg Apr 19 '11 at 15:07
## 2 Answers
The collapse pressure of a ping-pong ball is probably limited by the eventual buckling of the wall, instead of a compressive failure of its walls. Assuming (unrealistically) that the ball is a perfect sphere, the critical external pressure will be given by
$$P_e - P_i = \frac{2E\,\left(\frac{h}{R}\right)^2}{\sqrt{3(1-\nu^2)}}$$
where $P_e$ is the external pressure, $P_i$ the internal pressure, $E$ the Young's modulus of the material, $h$ the shell thickness, $R$ the sphere radius and $\nu$ the Poisson's ratio.
A ping-pong ball has a diameter of 40 millimeters, a weight of 2.7 grams and is made of celluloid. As celluloid's density is $1.4\,\mathrm{g\ cm^{-3}}$, its wall thickness will be
$$h = \frac{2.7\,\mathrm{g}}{4\,\pi\,(2\,\mathrm{cm})^2\,1.4\,\mathrm{g\ cm^{-3}}} \approx 0.04\,cm$$
The Young's modulus for celluloid is approximately $1400\,\mathrm{MPa}$ and its Poisson's ratio can be estimated as 0.35. Replacing these values in the expression for the critical pressure:
$$P_e - P_i = \frac{2\cdot 1400\,\mathrm{MPa}\cdot 0.02^2}{\sqrt{3(1-0.35^2)}} = \frac{1.12\,\mathrm{MPa}}{1.62} \approx 0.7\,\mathrm{MPa}$$
Assuming a compressive strength of $50\,\mathrm{MPa}$), the ball will fail "compressively" at a pressure of
$$P_e - P_i = \frac{50\,\mathrm{MPa}\cdot 0.04\,\mathrm{cm}\cdot 2\,\pi\cdot2\,\mathrm{cm}}{\pi(2\,\mathrm{cm})^2} = 2\,\mathrm{MPa}$$
so the real failure mode will be buckling. If the internal pressure of the ball is not very different from the atmospheric pressure, the crush depth of a perfect ball will be
$$d_c \approx \frac{0.7\,\mathrm{MPa}}{0.01\,\mathrm{MPa}\ \mathrm{m^{-1}}} = 70\,\mathrm{m}$$
The real crush depth will be between a half and a quarter of this value, matching the experimental value of approximately 30 meters.
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Let me assume that the answer to the previous question is right that the tensile strength is 50 MPa. I think that Georg is right that it's not correct just to equate that to the pressure. (I thought it was right at first -- dimensional analysis is often your friend!)
Here's a very heuristic argument why not. Suppose that the ball is under high pressure from the outside. Then the downward force on the upper hemisphere (or the upward force on the downward hemisphere) is $\sim PA\sim P\pi r^2$, where $r$ is the ball's radius. The part of the ball around the equator has to withstand that force. If the tensile strength is $T$, then the maximum force it can withstand is something like $T$ times the area of the ball that you'd expose if you sliced the ball in half around the equator. That is, it's $T(2\pi r)t$ where $t$ is the thickness of the material.
So the maximum pressure the ball can withstand is something like $P\sim Tt/r$. If $t/r\sim 0.01$, then $P\sim 5\times 10^5$ Pa. That's 5 atmospheres. One atmosphere is 10 meters of depth in water, so I estimate that it's of order 50 m.
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http://mathoverflow.net/questions/34070/motivation-of-moment-generating-functions/34104 | ## Motivation of Moment Generating Functions
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
What is the motivation of defining the mmoment generating function of a random variable $X$ as: $E[e^{tX}]$? I know that one can obtain the mean, second moment, etc.. after computing it. But what was the intuition in using $e^{x}$? Is it because its one-to-one and always increasing?
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## 4 Answers
Take the definition of "generating function" for a sequence. Do it where the sequence is the sequence consisting of the moments of $X$. That's it.
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### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
If X and Y are independent then $E[e^{t(X+Y)}] = E[e^{tX}] E[e^{tY}]$, so convolution corresponds to multiplication of the mgf's. Another reason: the moment generating function is actually a Fourier transform.
Now suppose $X_i$ are i.i.d. with zero mean, and define $Y_n = \sum_{i=1}^n X_i/\sqrt{n}$. Define $\phi(t) = E[e^{tX_1}]$. Then $E[e^{tY_n}] = \phi(t/\sqrt{n})^n$. Under reasonable assumptions, $\phi(t) = 1 + V[X_1]t^2/2 + O(t^3)$, and so $E[e^{tY_n}] = (1 + V[X_1]t^2/2n + O(t^3)/n^{1.5})^n \longrightarrow e^{V[X_1]t^2/2}$, and we get the central limit theorem (by continuity of the Fourier transform).
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2
Actually, it's a Laplace transform. The difference is important: it's possible to find $X_1$ such that $\phi(t)$ diverges except for $t=0$. But the Fourier transform $E(e^{itX})$ is defined for every real-valued random variable and every real $t$. – Mark Meckes Aug 1 2010 at 3:30
As you suggest, the fact that $e^x$ is increasing is a useful property here. In particular, that lets you in some cases to apply Markov's/Chebychev's inequality to the random variable $e^{tX}$ in order to get exponentially decaying bounds on the tails of $X$; see e.g. Chernoff bounds. In principle any other positive increasing function could be used in the same way, but $e^x$ is a particularly useful choice because it is especially well suited to studying sums of independent random variables, as noted already in Yuval's answer.
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I think this is an answer to "What is the use of moment-generating functions", and not to "Where does the definition of moment-generating functions come from?" – shreevatsa Aug 1 2010 at 11:56
Fair enough. I'll try to turn this into an answer to the question that was actually asked. – Mark Meckes Aug 1 2010 at 13:45
The goal is to to put all the moments in one package. Since $$e^{tx} = \sum \frac{x^n}{n!} t^n$$ the coefficients of $t^n$ in $E(e^{tx})$ are (scaled) moments. In other contexts we can use $$(1-xt)^{-1} = \sum x^n t^n$$ in place of $e^{tx}$. This gives more or less what engineers call the "z-transform" and in combinatorics it is known as "ordinary generating function". Using the exponential has the happy advantage that convolution of random variables translates to product of moment generating functions.
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http://en.wikipedia.org/wiki/Scalar_theories_of_gravitation | # Scalar theories of gravitation
Scalar theories of gravitation are field theories of gravitation in which the gravitational field is described using a scalar field, which is required to satisfy some field equation.
Note: This article focuses on relativistic classical field theories of gravitation. The best known relativistic classical field theory of gravitation, general relativity, is a tensor theory, in which the gravitational interaction is described using a tensor field.
## Newtonian gravity
The prototypical scalar theory of gravitation is Newtonian gravitation. In this theory, the gravitational interaction is completely described by the potential $\Phi$, which is required to satisfy the Poisson equation (with the mass density acting as the source of the field). To wit:
$\Delta \Phi = 4 \pi G \rho$, where
• G is the gravitational constant and
• $\rho$ is the mass density.
This field theory formulation leads directly to the familiar law of universal gravitation, $F = m_1 m_2 G/r^2$.
## Nordström's theories of gravitation
The first attempts to present a relativistic (classical) field theory of gravitation were also scalar theories. Gunnar Nordström created two such theories.
Nordström's first idea (1912) was to simply replace the divergence operator in the field equation of Newtonian gravity with the d'Alembertian operator $\square = \partial_t^2 - \nabla^2$. This gives the field equation
$\square \Phi = 4 \pi G \rho$.
However, several theoretical difficulties with this theory quickly arose, and Nordström dropped it.
A year later, Nordström tried again, presenting the field equation
$\Phi \square \Phi = -4 \pi G T$,
where $T$ is the trace of the stress-energy tensor.
Solutions of Nordström's second theory are conformally flat Lorentzian spacetimes. That is, the metric tensor can be written as $g_{\mu\nu} = A \eta_{\mu\nu}$, where
• ημν is the Minkowski metric, and
• $A$ is a scalar which is a function of position.
This suggestion signifies that the inertial mass should depend on the scalar field.
Nordström's second theory satisfies the weak equivalence principle. However:
• The theory fails to predict any deflection of light passing near a massive body (contrary to observation)
• The theory predicts an anomalous perihelion precession of Mercury, but this disagrees in both sign and magnitude with the observed anomalous precession (the part which cannot be explained using Newtonian gravitation).
Despite these disappointing results, Einstein's critiques of Nordström's second theory played an important role in his development of general relativity.
## Einstein's scalar theory
In 1913, Einstein (erroneously) concluded from his hole argument that general covariance was not viable. Inspired by Nordström's work, he proposed his own scalar theory. This theory employs a massless scalar field coupled to the stress-energy tensor, which is the sum of two terms. The first,
$T^{\mu\nu}_g = \frac{1}{4 \pi G} \left [\partial^\mu \phi \, \partial^\nu \phi \, - \frac{1}{2} \eta^{\mu\nu} \partial_\lambda \phi \, \partial^\lambda \phi \right]$
represents the stress-momentum-energy of the scalar field itself. The second represents the stress-momentum-energy of any matter which may be present:
$T^{\mu\nu}_m = \rho \phi u^\mu u^\nu$
where $u^\mu$ is the velocity vector of an observer, or tangent vector to the world line of the observer. (Einstein made no attempt, in this theory, to take account of possible gravitational effects of the field energy of the electromagnetic field.)
Unfortunately, this theory is not diffeomorphism covariant. This is an important consistency condition, so Einstein dropped this theory in late 1914. Associating the scalar field with the metric leads to Einstein's later conclusions that the theory of gravitation he sought could not be a scalar theory. Indeed, the theory he finally arrived at in 1915, general relativity, is a tensor theory, not a scalar theory, with a 2-tensor, the metric, as the potential. Unlike his 1913 scalar theory, it is generally covariant, and it does take into account the field energy-momentum-stress of the electromagnetic field (or any other nongravitational field).
## Relation to the Higgs Field
While the source of tensor gravitational fields come from components of the 4-momentum of particles $k_\mu k_\nu$, the source of scalar gravitational fields come from the trace of this which is equal to the square of the rest mass of a particle just like Higgs fields. Thus the theory of scalar gravitational fields is useful in the study of Higgs fields which give mass to stationary particles.
## Additional variations
• Kaluza–Klein theory involves the use of a scalar gravitational field in addition to the electromagnetic field potential $A^\mu$ in an attempt to create a five-dimensional unification of gravity and electromagnetism. Its generalization with a 5th variable component of the metric that leads to a variable gravitational constant was first given by Pascual Jordan [1].
• Brans–Dicke theory is a scalar-tensor theory, not a scalar theory, meaning that it represents the gravitational interaction using both a scalar field and a tensor field. We mention it here because one of the field equations of this theory involves only the scalar field and the trace of the stress-energy tensor, as in Nordström's theory. Moreover, the Brans-Dicke theory is equal to the independently derived theory of Jordan (hence it is often referred to as the Jordan-Brans-Dicke or JBD theory). The Brans-Dicke theory couples a scalar field with the curvature of space-time and is self-consistent and, assuming appropriate values for a tunable constant, this theory has not been ruled out by observation. The Brans-Dicke theory is generally regarded as a leading competitor of general relativity, which is a pure tensor theory. However, the Brans-Dicke theory seems to need too high a parameter, which favours general relativity).
• Zee combined the idea of the BD theory with the Higgs-Mechanism of Symmetry Breakdown for mass generation, which led to a scalar-tensor theory with Higgs field as scalar field, in which the scalar field is massive (short-ranged). An example of this theory was proposed by H. Dehnen and H. Frommert 1991, parting from the nature of Higgs field interacting gravitational- and Yukawa (long-ranged)-like with the particles that get mass through it (Int. J. of Theor. Phys. 29(4): 361, 1990).
• The Watt-Misner theory (1999) is a recent example of a scalar theory of gravitation. It is not intended as a viable theory of gravitation (since, as Watt and Misner point out, it is not consistent with observation), but as a toy theory which can be useful in testing numerical relativity schemes. It also has pedagogical value.
## References
• Goenner, Hubert F. M., "On the History of Unified Field Theories"; Living Rev. Relativity 7(2), 2004, lrr-2004-2. Retrieved August 10, 2005.
• Ravndal, Finn (2004). "Scalar Gravitation and Extra Dimensions". arXiv:gr-qc/0405030 [gr-qc].
• Watt, Keith, and Misner, Charles W. (1999). "Relativistic Scalar Gravity: A Laboratory for Numerical Relativity". arXiv:gr-qc/9910032 [gr-qc].
• P. Jordan, Schwerkraft und Weltall, Vieweg (Braunschweig) 1955.
• H. Dehnen and H. Frommert, "Scalar Gravity and Higgs Potential"; Int. J. of Theor. Phys. 29(4): 361, 1990.
• H. Dehnen and H. Frommert, "Higgs-Field Gravity within the Standard Model"; Int. J. of Theor. Phys.30(7): 985, 1991.
• H. Dehnen et al., "Higgs-Field and a New Scalar-Tensor Theory of Gravity"; Int. J. of Theor. Phys. 31(1): 109, 1992.
• Brans (2005). "The roots of scalar-tensor theory: an approximate history". arXiv:gr-qc/0506063 [gr-qc]. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 15, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9036915898323059, "perplexity_flag": "middle"} |
http://thevirtuosi.blogspot.com/2010/06/how-long-can-you-balance-quantum-pencil.html | # The Virtuosi
Giving some account of the undertakings, studies, and labors of the ingenious in many considerable parts of the Cornell physics department.
## Wednesday, June 16, 2010
### How Long Can You Balance A (Quantum) Pencil
Sorry for the delay between posts. Here in Virtuosi-land, we've all begun our summer research projects and I think we've just become a little bogged down in the rush that is starting a summer research project. You feel as though you have no idea what the heck is going on, and just try desperately to keep your head up as you hit the ground running, but thats a topic for another post.
In this post I'd like to address a fun physics problem.
How long can you balance a pencil on its tip? I mean in a perfect world, how long?
No really. Think about it a second. Try and come up with an answer before your proceed.
What this question will become by the end of this post is something like the following:
Given that Quantum Mechanics exists, what is the longest time you could conceivably balance a pencil, even in principle?
I will walk you through my approach to answering this question. I think it is a good problem to illustrate how to solve non-trivial physics problems.
I will try and go into some detail about how I arrived at my solution. For most of you this will probably be quite boring, so feel free to skip ahead to the last section for some numbers and plots.
### Finding an Equation of Motion
The first thing we need to do is find an equation of motion to describe our system. Lets consider the angle theta that the pencil makes with respect to the vertical. Lets treat this as a torque problem. Dealing with rotating systems is almost identical to dealing with free particles in Newtonian mechanics. Instead of Netwon's first law, relating forces to acceleration
$F = m \ddot x$
we just replace it with the rotational analogue of force - torque, the rotational analogue of acceleration - rotational acceleration, and the rotational analogue of mass - the moment of inertia.
$T = I \ddot \theta$
(I've taken the usual physics notation here, dots represent time derivatives) We need to determine the torque and moment of inertia of our pencil.
At this point I need to model the system. I need to break up the real world, rather complicated idea of a pencil, and turn it into an approximation that retains all of the important bits but enables me to actually proceed.
So, I will model a pencil as a rod, a uniform rod with a constant mass density. In doing so, I can proceed. The moment of inertia of a rod about its end is rather easy to calculate. If you are not familiar with the result I recommend you try the integral yourself.
$I = \int r^2 \ dm = \frac{1}{3} m l^2$\
where m is the total mass of my pencil and l is its length. I will take a pencil's mass to be 5 g and its length to be 10 cm.
Now the torque. The only force the pencil feels is the force due to gravity, which acts from the center of mass, which for my model of a pencil occurs at half its length. I additionally wish to express the force in terms of the parameter I decided would be useful, namely theta, the angle the pencil makes with the vertical. I obtain
$T = r \times F = \frac{1}{2} m g l \sin \theta$
Great, putting the pieces together we obtain an equation of motion for our pencil
$\frac{1}{2} m g l \sin \theta = \frac{1}{3} m l^2 \ddot \theta$
rearranging I get this into a nicer form
$\ddot \theta - \frac{3}{2} \frac{g}{l} \sin \theta = 0$
in fact, I'll utilize another time honored physics trick of the trade and simplify my expression further by making up a new symbol. Since I've done these kinds of problems before I can make a rather intelligent replacement
$\omega^2 = \frac{3}{2} \frac{g}{l}$
obtaining finally
$\ddot \theta - \omega^2 \sin \theta = 0$
And we've done it.
### Looking at the equation of motion
Now that we've found the equation of motion, lets look at it a bit.
First off, what does an equation of motion tell us? Well, it tells us all of the physics of our system of interest. That little equation contains all of the information about how our little model pencil can move. (Notice that while I haven't yet been explicit about it, in my model of the pencil, I also don't allow the tip to move at all, the pencil is only able to pivot about its tip).
Great. A useful thing to do when confronting a new equation of motion is to try and find its fixed points. I.e. try and find states in which your system can be which do not evolve in time. How can I do that? Sounds complicated. In fact, I'll sort of work backwards. I want to know the values that do not evolve in time, meaning of course that if I were to find such a solution, all of the terms that depend on time would be zero. So, if such a solution exists, for that solution the derivative term will vanish. So the solutions have to be solutions to the much simpler equation
$\sin \theta = 0$
Which we know the solutions. In fact, lets be a little smart about things and only worry about theta = 0 and theta = pi. Thinking back to our model this suggest a pencil being straight up (theta = 0) and straight down (theta = pi).
These are the stable points of the equation. The second one you are familiar with. If you instead of balancing a pencil 'up', try to balance it 'down', you know that if you start with the pencil pointing straight down it stays that way and doesn't do anything interesting.
But what about that first solution theta =0? That indicates that if you could start this model pencil exactly straight up, it would stay that way forever, and also not do anything interesting.
Oh no you cry. It seems as though we've already answered the question. How long can you balance a pencil? It looks like you could do it forever if you did it perfectly.
But you and I both know that is impossible. You can't ever balance a pencil forever. I've never done it, and tonight I've spent a lot of time trying.
So what went wrong?
### When your approximations fail
So what went wrong again? It seems like I've gotten an answer, namely in my model you could, at least in principle balance your pencil forever. But you and I both know you can't. Something is amiss.
Hopefully, the first thought that occurs to you is something along the lines of the following.
Of course you dummy! You could in principle balance a pencil forever, but in the real world, you can't set the pencil up standing perfectly straight. Even if its tilted just a little bit, its going to fall. This is exactly the problem with your physicists, you don't live in the real world!
Whoa, lets not be so harsh there. I made some rather crude approximations in order to get such a simple equation. You are allowed to make approximations provided (1) they are still right to as many digits as you care about, and (2) you keep in mind the approximations you made, and think a bit about how they could go wrong.
So, before we do anything too drastic, lets do with your gut. I agree, it seems like if the pencil would be at any small angle, it ought to fall. Lets double check that our equation does that.
So for the moment imagine theta being some small number. In fact, I will use the usual notation and call it epsilon. What does our equation say then?
$\ddot \theta = \omega^2 \sin \epsilon$
lets make another approximation (I know I know, we've already run into trouble, but bear with me). If epsilon is going to be a really small number, then we can simplify this equation even more. That sine being in there is really bugging me. Sines are hard. So lets fix it. Can we say something about how sine behaves when the angles are super small?? In fact we can. Such an approach is super common in physics.
### A short side comment on Taylor Series
Imagine a function. Any function. Picture the graph of the function. I.e. imagine a line on a graph. No matter what function you imagine, if you zoom in far enough, at any point that function ought to look like a line. Seriously. Zoom out a little bit and it will look like a line plus a parabola. Zoom out a little more and it will look like a cubic polynomial.
You can make these statements precise, and thats the Taylor Expansion. But the idea isn't much more complicated than what I've described. Taylor expanding the sine, we obtain
$\ddot \theta \approx \omega^2 \left( \epsilon - \frac{\epsilon^3}{3!} + \cdots \right)$
So if you are at really small angles, sin(x) looks just like x. Whats really small? Well as long as x^3 is too small for you to care about. For me, for the rest of the problem that will be for angles that are less than about 0.1 radians, for which that second term is about 0.00017 radians or 0.01 degrees, which is too small for me to care.
### Coming back to the approximation bit
Anywho, for really small angles, our equation of motion is approximately
$\ddot \theta = \omega^2 \theta$
So, notice for a second that if theta is positive, since omega^2 has to be positive, then our angular acceleration is going to be positive.
So your intuition was right. If your pencil ever gets to any positive angle, even the smallest of angles, then our angular acceleration is positive and our pencil will start to fall down.
So. The next question becomes. How can we capture this bit of reality. It looks like my model has this unphysical solution. How can I make it more real worldy?
Ah, this is the real fun of physics. You could go in any number of directions. Perhaps you could try and estimate how good you can actually prepare the angle of the pencil, perhaps you could ask whether air bouncing into the pencil would make it fall, perhaps you could wonder whether adding more realism to the moment of inertia would make the pencil fall easier, maybe you could wonder whether the thermal motion of the pencil would make it fall? Maybe you could consider the pencil as an elastic object and consider it vibrating as well as pivoting. Maybe you could model the tip as being able to move? Maybe you could introduce the gravitational pull of the sun? or the moon? or you? or the nearest mountain?
The sky's the limit.
So what am I going to do? Quantum Mechanics. Seriously, bear with me a bit.
### A little preliminaries
Before proceeding any further, lets actually solve the equation of motion we just got for the smallest angles. To remind you, the equation I got for my model of a pencil in the limit of the smallest angles was
$\ddot \theta = \omega^2 \theta$
This is an equation I can solve. Its a very common differential equation, and one that we use and abuse in physics so I know the solution by heart. So lets write it down.
First just to let know you, this sort of equation, second derivative of thing being linearly proportional to thing gives you solutions that are always pairs, usually, depending on the sign of the constant, they are written as sines and cosines, or decaying and growing exponentials.
Naturally of course in order to solve a second order differential equation, we need to specify two initial conditions. In this case I will call them theta_0 and dot theta_0, representing the initial position and initial angular velocity.
In this form the solution can actually most easily be written in terms of the exponential pair associated with sine and cosine, sinh and cosh (you can read more about them here, they are really neat functions).
The solution is
$\theta(t) = \theta_0 \cosh \omega t + \dot \theta_0 /\omega \sinh \omega t$
which as you could probably convince yourself, exponentially grows for any positive theta_0 or dot theta_0.
### Abandoning Realism
At this point of considering the question, I turned down a different route. I don't really care about balancing pencils on my desk. You see, I know a curious fact. I know that in quantum mechanics there is an uncertainty principle which says that you cannot precisely know both the position and momentum of an object.
This of course means that even in principle, since our world is dominated by quantum mechanics, I could never actually balance even my model pencil forever, because I could never prepare it with perfect initial conditions.
The uncertainly principle tells us that the best possible resolution I could have in the position and momentum of an object are set by Planck's constant:
$\Delta x \Delta p \geq \frac{\hbar}{2}$
This has to be true for our pencil as well. In fact I can translate the uncertainty principle into its angular form
$\Delta \theta \Delta J \geq \frac{\hbar }{2}$
where theta is our theta and J is the angular momentum, which for our pencil we know is
$J = I \dot \theta = \frac{1}{3} m l^2$
So the uncertainty principle for our pencil is,
$\Delta \theta \Delta \dot \theta \geq \frac{3 \hbar }{2 m l^2 } \approx 3.2 \times 10^{-30} \text{ Hz}$
So what?
So, I'm going to approximate the effects the uncertainty relation has on our pencil problem by saying that when I start off the classical mechanical pencil, I'm going to require that my initial conditions satisfy the uncertainty relation:
$\theta_0 \dot\theta_0 = \frac{3 \hbar }{2 ml^2}$
which we decided is going to mean that our pencil has to fall.
The real question is? How long will it take this pseudo-quantum mechanical pencil to fall?
In order words the question I am really trying to answer is:
Assuming a completly rigid pencil which you place in a vaccum and cool down to a few millikelvin so that it is in its ground state. Roughly how long will it take this pencil to fall?
### Do it to it
So lets do it. This is going to be a bit quick, mostly because its getting late and I want to go to bed.
But the procedure is kind of straight forward now. I need to choose initial conditions subject to the above constraint, figure out how long a pencil with those initial conditions takes to get to theta = pi/2 (i.e. fall over), and then do it over and over again for different values of the initial conditions.
So, the first thing to do is figure out how to pick initial conditions that satisfy the constraint. I'll do this systematically by parameterizing the problem in terms of the ratio of the initial conditions. I.e. lets define
$\log_{10} \frac{\theta_0}{\dot \theta_0} = R$
where I've taken the log for convenience. Now, figuring how long the pencil takes to fall in principle is just numerically integrating forward the full equation of motion
$\ddot \theta = \omega^2 \sin \theta$
where I need to do it numerically because the sine makes this equation too hard to solve analytically. In order to do the numerical integration I implemented a Runge-Kutta algorithm in python.
The only problem is that I am dealing with really small numbers and my algorithm can't play well with those in any reasonable amount of time. But, I can solve analytically for the equations of motion in the small angle limit, so I actually use the solution
$\theta(t) = \theta_0 \cosh \omega t + \dot \theta_0 / \omega \sinh \omega t$
to evolve the system up to an angle of 0.1 radians, and then let the nonlinear equation and runge kutta algorithm take over.
The full python code for my problem is available here (if you want to run it, remove the .txt extension, I did that so that it would be previewable).
And what do I obtain? First looking over 20 orders of magnitude in differential initial conditions:
And second, zooming into the interesting region.
So, what is the best time you could balance a quantum mechanical pencil, i.e. what is the absolute longest time you could hope to balance a pencil in our universe? About 3.5 seconds. Seriously.
Think about that for a second. Usually you hear about the uncertainty principle, and it seems like a neat parlor trick, but something that couldn't influence your day to day lives, and here is a remarkable problem where even in the best case, the uncertainty principle puts a hard limit on championship pencil balancing which seems tantalizingly close.
And there you have a graduate student working through a somewhat non trivial problem. I probably went into way too much details with the basics, but we are still trying to feel out who our audience is. Please leave comments and let me know if I either could have left things out, or should have went into more details at parts.
### EDIT
As per request, here is how the max fall time scales with the length of the pencil assuming a pencil with uniform density.
Plotted on a log-log plot, that is a pretty darn good line. The power law dependence is
$t \sim l^{0.514}$
Neat.
Strangely enough, if I trust my numbers here, the longest you could hope to balance a 'pencil' 1 km long would be about 6 minutes. Thats a very strange mental picture.
| Labels: equations, fun, numerics, pencil balancing, quantum | |
#### 14 comments:
1. Awesome to see the uncertainty principle at work on a more tangible scale.
As for your audience, perhaps some links to explain equations and how you manipulate them for those of us not in either math or physics fields. Nevertheless, very well written and an intriguing read!
2. Forbes
sin(pi/2) = 1
3. Thank you Forbes, I goofed. Fixed
4. I like how much detail you went into, I like to see the limits of certain models and how there are more accurate (but usually more complicated) models that can be used to sharpen the prediction.
I'd like to see how the time scales with the length of the pencil.
5. Anonymous
Interestingly, I remember this exact problem from the sophomore year quantum mechanics class required of all students at Caltech.
6. Thanks, I enjoyed that a lot! I'd love to see a followup from a statistical mechanics viewpoint showing how long it would take at room temp.
Regarding audience, if you put even one single equation, no matter how simple, you will lose nearly all laypeople, even if they could understand it in principle. They just won't stick around for the explanation. Since you really do need the equations for a post of this type, you have to assume an audience of interested freshmen as a baseline. In that case you probably don't have to explain Taylor series because if they've stuck around up to that point, they probably already know them.
There are a couple of typos. One is in the solution for theta(t) -- your cos should be a cosh. You have it right in the second appearance of the equation. The other thing is that I don't understand the theta=pi fixed point. Wouldn't that put the pencil inside the table? The horizontal pencil would be theta=pi/2.
Finally, the equation of motion is the same as for a physical pendulum, so shouldn't it have a solution in terms of elliptic functions?
7. particle_person:
fixed the typo. Thanks for the input on audience.
Now, about the other stable point at pi, you're right in that this would be inside the table, but my equation of motion has no knowledge of the table. As you pointed out, this is the equation of motion for a pendulum, which if were suspended above the table could swing back and forth about theta=pi.
Finally, saying that we can write the exact solution to the differential equation in terms of elliptic integrals (while true), is kind of cheating seeing as the elliptic integrals are special functions, the elliptic integrals of the first kind are more or less defined to be the function that solves this equation.
8. my equation of motion has no knowledge of the table
Yeah, that's why I was confused by the following when I read it the first time:
If you instead of balancing a pencil 'up', try to balance it 'down', you know that if you start with the pencil pointing straight down it stays that way and doesn't do anything interesting.
Like you added, the equation doesn't know about the table so it's just not a physical solution of the original problem. It think it's less confusing to say it's unphysical than to switch the mental picture (without telling the audience) to one of someone holding the pencil by the tip, which is the only way I can make sense of balancing it down.
Anyway, that's my suggestion for improving clarity — don't switch the mental image in mid-stream without telling the audience.
9. Habit Melon: As per request, how fall time scales with length for a constant density pencil has been appended to post.
10. Dave McCloskey
I was able to integrate the equations to find a simple analytic expression.
Time to Fall = 1/k ln(4(sqrt(2)-1) / (Theta0)min)
where
k = sqrt(3 g / 2 L )
(Theta0)min = sqrt((hbar k)/ M L g))
For realistic input, I get about 4 sec.
11. Anonymous
How would balance time be affected if you add resistance by adding weight and width to the pencil, i.e. x4 adjacent 45 degree angled triangular "wings" (like if you draw an 'x' on a square page and cut out the 'x' to produce 4 equal triangle shaped cut outs), perhaps placed at the top end of the balancing pencil, like a dart or arrow, with a width of about 3mm?
12. Nice read. But half way thru its i realized that one could
Only answer this by actually trying to do it for real. Would be nice to have the time and equipment to create a vacuum and a mechanical device to actually balance the pencil 100%. Taking in account the earth's rotation ( trying to counter it some how) and in my mind I think that one could balance the pencil on its tip as long as you can counter the earths rotation. Any thoughts any one?
1. Anonymous
Noel: Earth's rotation doesn't figure into it, except to very slightly reduce the effect of gravity by generating a centrifugal force. (Yes, centrifugal forces "exist" in rotational frames.) This tiny correction is less significant than the variation of g at different parts of the earth.
2. Anonymous
Although small, I say you probably are better off trying this somewhere on the equator at noon on the spring or fall equinox when the sun's gravity, earth's gravity, and earth's centripetal force are all colinear in your frame.
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The question is how many triangles are there in the following picture? I have thought to solve it by creating a formula based on the angles of the lines starting from the bottom of each side. I ...
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### About Euclid's Elements and modern video games
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### Volume of Region in 5D Space
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### Have I made a straight line, or a circle?
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### Picture of a 4D knot
A knot is a way to put a circle into 3-space $S^1 \to \mathbb R^3$ and these are often visualized as 2D knot diagrams. Can anyone show me a diagram of a nontrivial knotted sphere \$S^2 \to \mathbb ...
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### Guaranteed Checkmate with Rooks in High-Dimensional Chess
Given an infinite (in all directions), $n$-dimensional chess board $\mathbb{Z^n}$, and a black king. What is the minimum number of white rooks necessary that can guarantee a checkmate in a finite ...
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### On nonintersecting loxodromes
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### New twist on a Putnam problem
A recent Putnam problem: Let $f$ be a real-valued function on the plane such that for every square $\square ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f$ is identically ...
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### Rolling a Sphere on the Plane
Suppose one starts with a sphere $S$ resting on a ($2$-dimensional) plane $H$ at the origin. A "move" consists of the following: Let $P$ and $Q$ be two points in $H$. Roll the sphere $S$ along a ...
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### What is the (mathematical) point of geometric constructions?
The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school ...
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### What are the dangers of visual exposition of mathematics?
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### Trying to understand why circle area is not $2 \pi r^2$
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### About translating subsets of $\Bbb R^2.$
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### A strangely connected subset of $\Bbb R^2$
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### Proof that Pi is constant (the same for all circles), without using limits
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### Ambiguous Curve: can you follow the bicycle?
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### Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?
When I was in third grade, I was playing with rectangles and diagonal lines, and discovered something very interesting with fractions. I've shown several math teachers and professors over the years, ... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 68, "mathjax_display_tex": 3, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9363614916801453, "perplexity_flag": "middle"} |
http://math.stackexchange.com/questions/193740/if-f-and-g-are-one-to-one-then-g-circ-f-is-one-to-one-and-g-circ-f | # If $F$ and $G$ are one to one, then $G \circ F$ is one to one and $(G \circ F)^\neg = F^\neg \circ G^\neg$
THEOREM: if $F$ and $G$ are one to one then $G \circ F$ is also one to one and $(G \circ F)^\neg$ = $F^\neg \circ G^\neg$
PROOF:
if $F: A\rightarrow B$, $G: B \rightarrow C$ and
$$\forall a, a' \in A \ \ F(a)=F(a') \Rightarrow a=a'$$ then F is one to one and if
$$\forall b, b' \in B \ \ G(b)=G(b') \Rightarrow b=b'$$ then G is one to one by the definition of one to one.
If $(G \circ F)(a)=(G \circ F)(a') \Rightarrow G(F(a))= G(F(a'))$ then $F(a)= F(a')$ since $G$ is one to one.
If $F(a)=F(a')$ then $a=a'$ since $F$ is one to one
Because $G \circ F$ is one to one it is also invertible so $(G \circ F)^\neg$ exist now if we compute $$\begin{align}((G\circ F)\circ (F^\neg\circ G^\neg))(a)&=\\ ((G\circ (F\circ F^\neg)\circ G^\neg))(a)&=\\ ((G\circ G^\neg))(a)&=a\end{align}$$
So, $(F^\neg\circ G^\neg)$ is the inverse of $G \circ F$ therefore $(G \circ F)^\neg=(F^\neg\circ G^\neg)$
QED
I feel like the first part is ok but the 2nd part is all messed up ... what did I do wrong?
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In order for the inverse function to exist you need bijectivity, also called "one-to-one and onto". You must prove that every point in the target space is in the image of $G\circ F$. (The statement of the theorem suggests that that's what you mean by "one to one" here, but you seem to prove injectivity only.) – Sean Eberhard Sep 10 '12 at 19:59
Please try to use more descriptive titles. – Austin Mohr Sep 10 '12 at 20:02
1
The proof of one to one has stylistic deficiencies. For one thing, the first five lines are unnecessary, and somewhat confusing. Also, symbols from logic are not quite used correctly. For invertibility in the usual sense, you need more. Otherwise, best you can do is produce an inverse on a suitably restricted subset of $C$. – André Nicolas Sep 10 '12 at 20:02
Is the notation $f^\neg$ for inverse function common? I've always only seen $f^{-1}$. – Henning Makholm Sep 10 '12 at 20:24
i couldn't figure out how to write f^{-1} – Joshua Rocky Lizardi Sep 11 '12 at 2:07
## 1 Answer
I don't see anything wrong, though you probably ought to show that $$((F^\neg\circ G^\neg)\circ(G\circ F))(a)=a,$$ or at least argue that the demonstration is similar to going the other direction.
It's also worth noting (as others have pointed out in the comments) that we're really talking about $(G\circ F)^\neg:D\to A$, where $D:=G(F(A)).$ It seems like your source (or perhaps just you) may not be overly concerned with domains/codomains of functions, and is (are) simply leaving them as understood. One should be cautious about such things, though (as evidenced by the comments above).
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 28, "mathjax_display_tex": 4, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.966856062412262, "perplexity_flag": "head"} |
http://mathhelpforum.com/advanced-algebra/195621-finding-solution-non-linear-simultaneous-equations.html | # Thread:
1. ## Finding a solution for non-linear simultaneous equations
Hi
I am not sure if the title accurately reflects my issue, apologies if this is not the case.
I am an academic engineer, I have encountered a problem where i would like to equate a circle to a triangle to simplfy a problem..... My criteria is to transform a circle to a 120, 30, 30 triangle with equivalent area and perimeter, when i quickly derived the simultaeous equations i get the following two expressions
y1 = b+2*h
y2=(b*h)/2
where:
y1=the circles perimeter
y2=the circles area
b= base of triangle
h= height of triangle
I thought i would ask as i do not want to waste time if it is not possible to solve and satisfy the 120, 30, 30 condition......... Are there any methods that can be used to tackle this issue?
2. ## Re: Finding a solution for non-linear simultaneous equations
if b,h are given, $y_1$ and $y_2$ may simply be calculated. so i assume that $y_1, y_2$ are given, and you wish to find b and h.
write: $b = y_1 - 2h = \frac{2y_2}{h}$
from $y_1 - 2h = \frac{2y_2}{h}$ we get:
$2h^2 - y_1h - 2y_2 = 0$, which is a quadratic in h.
the discriminant is $y_1^2 - 8y_2$
however, $y_1,y_2$ aren't "independent" they both depend on the radius of the circle, r:
$y_1 = 2\pi r$
$y_2 = \pi r^2$
so $y_1^2 - 8y_2 = (2\pi r)^2 - 8\pi r^2 = -4\pi r^2 < 0$, therefore: no real solutions for h.
however, in reviewing your original equations, it appears that your formula for either the area or the perimeter of the isoceles triangle is incorrect, if h is the height, then the perimeter length should be:
$b + 2(\sqrt{h^2 + \frac{b^2}{4}})$.
you might want to review that, and confirm (the calcuations get a lot messier if this is true).
3. ## Re: Finding a solution for non-linear simultaneous equations
Thanks, I did make a couple of mistakes actually, sorry about that, the equations should be
y1 = b+2*(sqrt(b^2+h^2))
y2=(b*h)/2
where:
y1=the circle z's perimeter (GIVEN)
y2=the circle z's area (GIVEN)
b= base of triangle
h= height of triangle
I want to calculate values for b, and h for a 120, 30, 30 triangle, my main question is there a solution? i.e. can you take a circle and construct a 120, 30, 30 triangle with an identicle Area and perimeter? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9022578597068787, "perplexity_flag": "middle"} |
http://physics.stackexchange.com/questions/38065/difference-between-using-displacement-and-current-configuration-as-unknown?answertab=active | # Difference between using displacement and current configuration as unknown?
We could use either the current configuration $x$ or the displacement $u$ as unknown while solving for the deformation, for example, of a solid object. I want to know what's the difference between them? Is it that there is no difference, or one is numerically better than the other?
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## 1 Answer
When you study a deformable continuum, the current configuration $x$ means very little, unless you also know what the original configuration $X$ was. Interesting things only happen when the displacement $x-X$ is different from point to point of the continuum. More precisely, when the displacement is not the sum of a rotation and a traslation, and thus the continuum is not simply undergoing rigid body motion. So in the end, you are going to be interested not in the displacement, but in the gradient of it.
There is a more relevant question when dealing with these type of problems, and is whether you are taking a Lagrangian or Eulerian view.
In the Lagrangian description of deformation, you define the current configuration as a function of the original configuration, $x = \phi(X)$, and you track what happens to 'particles' in the original configuration as the body deforms. The displacement under such description takes the form $U(X) = x-X = \phi(X)-X$, You could compute the gradient of the displacement, which is closely related to deformation, at the point that was originally at position $X$ as $\nabla U(X) = \nabla \phi(X)-I(x)$.
In the Eulerian description, you look at a fixed point in space, and consider what is going on to the particle that now is at that position. In this description, displacement is $u(x) = x-X=x - \phi^{-1}(x)$, and the gradient of the displacement for the point currently at position $x$ is $\nabla u(x) =I - \nabla \phi^{-1}(x)$.
You may already glean that the math is harder when going Eulerian, as you are having to deal with the gradient of an inverse transformation. It is nevertheless much easier from an observational perspective to handle what is going on at a certain point in space, regardless of where that chunk of matter was in a previous time.
In most elasticity studies, only infinitesimal displacements are considered, so that material points can be considered not to move, rendering the above distinction more of a formal thing. In fluid mechanics this is of paramount importance, and generally the Eulerian view prevails, even though it adds all those 'convective' terms to the equations, to account for the fact that the point under observation is moving. Link to the relevant wikipedia page.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9365861415863037, "perplexity_flag": "head"} |
http://math.stackexchange.com/questions/94285/convergence-of-integrals-of-radon-measures?answertab=votes | # Convergence of integrals of Radon measures
Let $X$ be a locally compact Hausdorff space and let $\mu_n$ be a sequence of bounded variation Radon measures on $X$ such that $\int_X g \;d\mu_n \rightarrow \int_X g \;d\mu$ for each $g \in C_0 (X)$ (ie. $g(x) \rightarrow 0$ as $x \rightarrow \infty$ in the one-point compactification of $X$) and $|\mu_n|(X) \rightarrow |\mu|(X)$. Must it hold that $\int_X f \;d\mu_n \rightarrow \int_X f \;d\mu$ for each bounded continuous function $f$, even if $f$ does not tend to zero at infinity?
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I believe this is true, and I think I have seen it in Billingsley's Convergence of Probability Measures. – Nate Eldredge Dec 26 '11 at 21:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.925011396408081, "perplexity_flag": "head"} |
http://nrich.maths.org/7431 | ### Number Round Up
Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.
### Numbers as Shapes
Use cubes to continue making the numbers from 7 to 20. Are they sticks, rectangles or squares?
### How Odd
This problem challenges you to find out how many odd numbers there are between pairs of numbers. Can you find a pair of numbers that has four odds between them?
# Largest Even
##### Stage: 1 Challenge Level:
I have a pile of nine digit cards numbered $1$ to $9$.
I take one of the cards. It is the $3$.
Which card would you choose so you could make the largest possible two-digit even number with the two cards?
We put the cards back in the pile. This time, I choose the $6$. Which card would you choose this time to make the largest possible two-digit even number?
Have a go at this with a partner. One of you chooses the first digit from the set of cards. The second person then chooses a card to make the largest possible two-digit even number. You can then swap over.
Try it several times so you are sure you have a good method. Talk about your ideas with your partner so you agree together on a 'best' method.
How would your strategy change if you had to make the largest two-digit odd number?
If you don't have a partner to work with, you could use the interactivity below. The computer selects one digit at random. You must then choose a digit to make the largest possible two-digit even number or largest possible two-digit odd number.
Enter the biggest two-digit number you can think of that uses the digit:
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 4, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9092879891395569, "perplexity_flag": "middle"} |
http://physics.stackexchange.com/questions/37877/finding-psix-t-for-a-free-particle-starting-from-a-gaussian-wave-profile?answertab=active | # Finding $\psi(x,t)$ for a free particle starting from a Gaussian wave profile $\psi(x)$
Consider a free-particle with a Gaussian wavefunction,
$$\psi(x)~=~\left(\frac{a}{\pi}\right)^{1/4}e^{-\frac12a x^2},$$ find $\psi(x,t)$.
The wavefunction is already normalized, so the next thing to find is coefficient expansion function ($\theta(k)$), where:
$$\theta(k)=\int_{-\infty}^{\infty} \psi(x)e^{-ikx} \,dx.$$ But this equation seems to be impossible to solve without error function (as maple 16 tells me).
Is there any trick to solve this?
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I am a bit confused, why are you trying to find $\psi (k)$ ? Or as you write it, $\theta (k)$ ? – DJBunk Sep 20 '12 at 23:06
Can you put what you ran through maple 16? – Magpie Apr 8 at 1:38
## 1 Answer
• First you ask for the time evolution of the wavefunction. For this you will need to use the Schrödinger equation $i \partial \psi/\partial t= \hat H \psi$ and thus will need to know the Hamiltonian ($\hat H$).
• Second you seem to want to work out the Fourier transform of the wavefunction. This will not give you the wavefunction as a function of time but will give you the wavefunction in momentum space. The integral you want to calculate is the Fourier transform of a Gaussian which is itself a Gaussian: $$\int_{-\infty}^{\infty} e^{-ax^2/2}e^{-i k x} \, dx \\ = \int_{-\infty}^{\infty} e^{-ax^2/2}\left(\cos{kx} - i \sin{kx} \right) \, dx .$$ The second term in the above integral is odd so will give zero. The first term is a known integral and gives $$=\sqrt{\frac{2\pi}{a}} e^{-k^2/2 a} ,$$ a Gaussian as promised with width inversey proportional to the original.
I am pretty certain Maple should also be able to calculate the integral for you as it is written in my fist line (Mathematica can), so I imagine you are just not entering it correctly.
Edit: Apologies for the first comment above. I had not seen that you had written this was for a free particle, so indeed you know the Hamiltonian, the potential is $V(x,t)=0$, and so from Schrödinger's equation we know the time evolution of the energy Eigenstates is $\psi(x,t)=e^{-i \omega t}\psi(x)$. For the free particle we have $\omega=k^2/2m$ and so you know the time evolution of the Fourier transform.
So taking the Fourier transform given above, applying the time evolution, and transforming back to position space we have $$\psi(x,t)=\int_{-\infty}^{\infty} e^{-k^2/2 a}e^{-i\omega t}e^{ikx} \, dk \\ =\int_{-\infty}^{\infty} e^{-\frac{k^2}{2 a}(1+iat/m)}e^{ikx}\, dk \\ \sim e^{\frac12 \frac{x^2}{1/a+imt}}$$ as #Ron pointed out in his comment. This shows how the wavepacket spreads out with time.
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1
The fourier trasnform evolves by simple phases, and a reverse fourier transform gives the time evolution, which is a spreading Gaussian, so that the a gets replaced everywhere by ${1\over {(1/a)+it}}$ – Ron Maimon Sep 21 '12 at 6:48
Oh yeah, hadn't seen the part saying this was for a free particle (doh!). Have added an edit to the answer to complete it. Thanks for pointing that out. – Mistake Ink Sep 21 '12 at 13:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 12, "mathjax_display_tex": 5, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9419617652893066, "perplexity_flag": "head"} |
http://mathoverflow.net/questions/53471?sort=oldest | ## Are there any very hard unknots?
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
Some years ago I took a long piece of string, tied it into a loop, and tried to twist it up into a tangle that I would find hard to untangle. No matter what I did, I could never cause the later me any difficulty. Ever since, I have wondered whether there is some reasonably simple algorithm for detecting the unknot. I should be more precise about what I mean by "reasonably simple": I mean that at every stage of the untangling, it would be clear that you were making the knot simpler.
I am provoked to ask this question by reading a closely related one: http://mathoverflow.net/questions/4918/can-you-fool-snappea . That question led me to a paper by Kaufmann and Lambropoulou, which appears to address exactly my question: http://www.math.uic.edu/~kauffman/IntellUnKnot.pdf , since they define a diagram of the unknot to be hard if you cannot unknot it with Reidemeister moves without making it more complicated. For the precise definition, see page 3, Definition 1.
A good way to understand why their paper does not address my question (by the way, when I say "my" question, I am not claiming priority -- it's clear that many people have thought about this basic question, undoubtedly including Kaufmann and Lambropoulou themselves) is to look at their figure 2, an example of an unknot that is hard in their sense. But it just ain't hard if you think of it as a three-dimensional object, since the bit of string round the back can be pulled round until it no longer crosses the rest of the knot. The fact that you are looking at the knot from one particular direction, and the string as it is pulled round happens to go behind a complicated part of the tangle is completely uninteresting from a 3D perspective.
So here's a first attempt at formulating what I'm actually asking: is there a generalization of the notion of Reidemeister moves that allows you to pull a piece of string past a whole chunk of knot, provided only that that chunk is all on one side, so to speak, with the property that with these generalized Reidemeister moves there is an unknotting algorithm that reduces the complexity at every stage? I'm fully expecting the answer to be no, so what I'm really asking for is a more convincing unknot than the ones provided by Kaufmann and Lambropoulou. (There's another one on the Wikipedia knot theory page, which is also easily unknotted if you allow slightly more general moves.)
I wondered about the beautiful Figure 5 in the Kaufmann-Lambropoulou paper, but then saw that one could reduce the complexity as follows. (This will be quite hard to say in words.) In that diagram there are two roughly parallel strands in the middle going from bottom left to top right. If you move the top one of these strands over the bottom one, you can reduce the number of crossings. So if this knot were given to me as a physical object, I would have no trouble in unknotting it.
With a bit of effort, I might be able to define what I mean by a generalized Reidemeister move, but I'm worried that then my response to an example might be, "Oh, but it's clear that with that example we can reduce the number of crossings by a move of the following slightly more general type," so that the example would merely be showing that my definition was defective. So instead I prefer to keep the question a little bit vaguer: is there a known unknot diagram for which it is truly the case that to disentangle it you have to make it much more complicated? A real test of success would be if one could be presented with it as a 3D object and it would be impossible to unknot it without considerable ingenuity. (It would make a great puzzle ...)
I should stress that this question is all about combinatorial algorithms: if a knot is hard to simplify but easily recognised as the unknot by Snappea, it counts as hard in my book.
Update. Very many thanks for the extremely high-quality answers and comments below: what an advertisement for Mathoverflow. By following the link provided by Agol, I arrived at Haken's "Gordian knot," which seems to be a pretty convincing counterexample to any simple proposition to the effect that a smallish class of generalized moves can undo a knot monotonically with respect to some polynomially bounded parameter. Let me see if I can insert it:
I have stared at this unknot diagram for some time, and eventually I think I understood the technique used to produce it. It is clear that Haken started by taking a loop, pulling it until it formed something close to two parallel strands, twisting those strands several times, and then threading the ends in and out of the resulting twists. The thing that is slightly mysterious is that both ends are "locked". It is easy to see how to lock up one end, but less easy to see how to do both. In the end I think I worked out a way of doing that: basically, you lock one end first, then having done so you sort of ignore the structure of that end and do the same thing to the other end with a twisted bunch of string rather than a nice tidy end of string. I don't know how much sense that makes, but anyway I tried it. The result was disappointing at first, as the tangle I created was quite easy to simplify. But towards the end, to my pleasure, it became more difficult, and as a result I have a rather small unknot diagram that looks pretty knotted. There is a simplifying move if one looks hard enough for it, but the move is very "global" in character -- that is, it involves moving several strands at once -- which suggests that searching for it algorithmically could be quite hard. I'd love to put a picture of it up here: if anyone has any suggestions about how I could do this I would be very grateful.
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I took the liberty of adding K-L's Figs.2 & 5; hope you don't mind. – Joseph O'Rourke Jan 27 2011 at 12:04
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Far from minding, I'm absolutely delighted, and would have had no idea how to do it myself. Many thanks. – gowers Jan 27 2011 at 12:12
@Tim: given that the question in your title doesn't really match the actual question you are asking, which I think is the one in paragraph 4, can you try to visually highlight the actual question? – Willie Wong Jan 27 2011 at 12:14
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Dynnikov's algorithm (mentioned in Budney's answer) is the closest known method to what you are asking. The grid number is monotonically simplified by a version of generalized Reidemeister moves suited to grid diagrams, which allow you to do the sorts of moves that you describe (move two strands past each other, even though there are other strands in front and behind). However, this method is not strictly monotonic, since there could possibly be superexponentially many unknot diagrams of a fixed grid number. As a grad student, I used this method to simplify some unknots constructed by Haken: – Agol Jan 29 2011 at 18:44
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math.uic.edu/~agol/unknots.html – Agol Jan 29 2011 at 18:44
show 8 more comments
## 10 Answers
As you suggest, a lot of people have thought about this question. It's hard to find arrangements of an unknot that are convincingly hard to untie, but there are techniques that do pretty well.
Have you ever had to untangle a marionette, especially one that a toddler has played with? They tend to become entangled in a certain way, by a series of operations where the marionette twists so that two bundles of control strings are twisted in an opposite sense, sometimes compounded with previous entanglements. It can take considerable patience and close attention to get the mess undone. The best solution: don't give marionettes to young or inattentive children!
You can apply this to the unknot, by first winding it up in a coil, then taking opposite sides of the coil and braiding them (creating inverse braids on the two ends), then treating what you have like a marionette to be tangled. Once the arrangement has a bit of complexity, you can regroup it in another pattern (as two globs of stuff connected by $2n$ strands) and do some more marionette type entanglement. In practice, unknots can become pretty hard to undo.
As far as I can tell, the Kaufmann and Lambropoulou paper you cited deals is discussing various cases of this kind of marionette-tangling operation.
I think it's entirely possible that there's a polynomial-time combinatorial algorithm to unknot an unknottable curve, but this has been a very hard question to resolve. The minimum area of a disk that an unknot bounds grows exponentially in terms of the complexity of an unknotted curve. However, such a disk can be described with data that grows as a polynomial in terms of the number of crossings or similar measure, using normal surface theory. It's unknown (to me) but plausible (to me) that unknotting can be done by an isotopy of space that has a polynomially-bounded, perhaps linearly-bounded, "complexity", suitably defined --- that is, things like the marionette untangling moves. This would not imply you can find the isotopy easily---it just says the problem is in NP, which is already known.
One point: the Smale Conjecture, proved by Allen Hatcher, says that the group of diffeomorphisms of $S^3$ is homotopy equivalent to the subgroup $O(4)$. A corollary of this is that the space of smooth unknotted curves retracts to the space of great circles, i.e., there exists a way to isotope smooth unknotted curves to round circles that is continuous as a function of the curve.
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I have had enough children to have the marionette experience several times, but had not thought of connecting that to the unknot question. – gowers Jan 27 2011 at 12:48
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I should stress that this question is all about combinatorial algorithms: if a knot is hard to simplify but easily recognised as the unknot by Snappea, it counts as hard in my book.
With respect, this comment is confused. When SnapPea "recognizes" an unknot it only uses combinatorial algorithms.
When you give SnapPea a diagram of the unknot it first uses ad hoc, but purely combinatorial, techniques to triangulate and then simplify the knot complement. After getting down to some very small number of tetrahedra (2 to 5, say) it then tries to find a finite volume hyperbolic structure on the knot complement. SnapPea uses Newton's method as applied to the Thurston gluing equations: I would not call this step combinatorial. However, that is not relevant to your question -- Snappea always fails to find such a structure because it does not exist.
You then ask SnapPea "What is the fundamental group of the knot complement?" SnapPea then uses other ad hoc, but combinatorial, techniques to simplify the fundamental group. These succeed because the number of tetrahedra is small -- SnapPea reports that $\pi_1 \cong \mathbb{Z}$ and Bob's your uncle by the Disk Theorem.
Edit: I was going to give a brief description of what SnapPea does in the retriangulation step. But Jeff Weeks' code and comments are very clear: the file to look at is simplify_triangulation.c in the SnapPea kernel.
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OK you're right that my comment is misleading, and thank you for the explanation about what SnapPea does. I should therefore say that I am asking specifically about greedy algorithms that use mild generalizations of Reidemeister moves. What you might say is that I am interested in arguments that work directly from knot diagrams and do not construct any auxiliary objects. – gowers Jan 27 2011 at 16:23
Yes, Snappea mainly recognizes unknottability and recognition of knots, rather than recognition of the unknot. There is some analytic/combinatorial interaction in this: in particular, recognizing that there are incompressible surfaces by degeneration of simplices, and also, sometimes, retriangulating based on geometry. It would be interesting to show that recognition of unknotability is in P or even NP. – Bill Thurston Jan 27 2011 at 16:52
@gowers - This is a reasonable enough. However, are you sure that you are really interested in "diagram complexity"? Per Thurston's answer I thought you wanted to allow physical moves like "grab a sub-tangle and rotate it". Similarly, in the first paragraph of your post you explicitly say that it was the physical approach (and so not attached to any particular diagram!) that led you to ask the question. --- I'm arguing here that SnapPea's messing about with triangulations appears to have access to that three-dimensional "intuition". – Sam Nead Jan 27 2011 at 17:03
@Thurston - I am now confused. What is the difference between recognizing the unknot and recognizing "unknotability"? – Sam Nead Jan 27 2011 at 17:41
@Sam, that's a good question, and the truth is that I don't have a completely clear answer to it. I think what I'm interested in is the possibility of a 2-dimensional argument that captures some of the 3-dimensional intuition that certain moves are easy even though if you break them up into Reidemeister moves then you have to increase the number of crossings. So it's more 3D than just Reidemeister moves but more 2D than constructing triangulations and the like. – gowers Jan 27 2011 at 18:35
I believe it's not known whether or not the uniform electrostatic charge potential function (as studied by Bryson, Freedman, He and Wang: http://front.math.ucdavis.edu/9301.5212) has any critical points other than the global minimum, on the space of unknots in $S^3$.
If the above were true and there were no critical points, that should in principle give you an algorithm to simplify any trivializable knot. Perhaps it could be implemented diagramatically, via a combination of things like simplifying Reidemeister moves and inversions on circles in the diagrams corresponding to conformal transformations of $S^3$.
Ivan Dynnikov has done some work on this and has a paper claiming diagrammatic "monotonic" simplification of unknot diagrams: http://arxiv.org/abs/math.GT/0208153 I went to a couple of his talks on this topic but didn't understand the core of the argument.
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Allen Hatcher, in a preprint "Topological Moduli Spaces of Knots," subsequent to his proof of Smale's conjecture, remarks that one might expect there to be a nice energy function on the space of unknots (or, for that matter, on the space of knots of any given isotopy type) with the property that gradient flow in the direction of decreased energy always terminates at a standard form for the knot (or some very low-dimensional space of standard forms.) So you could certainly ask whether one would expect the complexity of a given unknot, in whatever sense this is meant, to be monotone decreasing as you flow it towards a nice round circle.
Greg Buck gave an interesting talk here at UW where he presented a candidate for such a function, and showed several cool movies (not available) of crimped, tangled unknots resolving to round circles. Buck said that he'd never observed the flow getting caught at a local minimum. (I didn't ask him whether the knot ever "gets more tangled on its way to being less tangled," but one could define this problem away by using his energy as the definition of tangledness....!)
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Oh, as I posted this I see that Ryan posted a similar answer with a different proposed potential function! – JSE Jan 27 2011 at 16:13
Small clarification: It's a formal consequence of Hatcher's theorem that there a function on the space of unknots that has no critical points other than the global minimum being the "round unknot" subspace. The issue is whether or not it can be chosen to be a physically meaningful function, like the uniform electric charge potential function. For every component of the space of embeddings of $S^1$ in $S^3$ I give a finite-dimensional subspace where that component deformation-retracts to in my "splicing operad" paper. So it's the analogue of "round unknots" but for actual knots. – Ryan Budney Jan 27 2011 at 16:20
I don't get it -- if the existence of such a function is a corollary of Hatcher's theorem, why does he write about it in the later linked paper in a "one might hope..." kind of way? – JSE Jan 27 2011 at 16:24
or maybe I get it -- you're saying just the existence of a retract says formally that there IS a smooth function, but there's no reason that decrease in that function should correspond to anything T-Gow thinks of as "simplification," unless the function is physically meaningful. – JSE Jan 27 2011 at 16:26
Yes, knowing such a function exists is very different from having an explicit (useful) such function that you can use. It's kind of like Smale's proof that you can turn the sphere inside out -- people wanted to see it happen, beyond an abstract proof that it can be done. – Ryan Budney Jan 27 2011 at 18:06
The unknot on the Wikipedia doesn't seem trivial:
http://en.wikipedia.org/wiki/File:Thistlethwaite_unknot.svg
Lucas
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I found a generalized move that reduces the number of crossings. Again it's not that easy to describe, but here's a try: if you look at the two bottom horizontal bits of the knot, they cross once at the bottom left, and if you follow them round clockwise they cross again fairly near the top. If you ignore the rest of the knot, you can do a Reidemeister move on just those strands and it reduces the number of crossings. So it's not minimal in the sense I tried to describe. (I see my task always as proving non-minimality rather than disentangling the entire knot.) – gowers Jan 27 2011 at 23:48
I have looked at it closer, and indeed, it is less complicated than it looks like. – Lucas K. Jan 28 2011 at 23:03
A suggestion for you actual question. What about counting layers instead of crossings? I don't know the effect. When the rope crosses, it may only when the two pieces are on different layers. Between crossings, the rope can change of layer. Count the minimal required layer changes. In your figure 2, this number might not increase when simplifying. – Lucas K. Jan 28 2011 at 23:36
Maybe the slightly off the topic of whether there are hard unknots but here are a few more comments about detecting unknots. Regarding complexity theory as Bill mentioned detecting the unknot is NP. Agol has also proved that it is co-NP (i.e. proving that a knot is not the unknot is NP). My computer science friends tell me that such problem, simultaneously coNP and NP are thought to be polynomial time.
So far this thread has mentioned Haken's algorithm and its descendants, the Snappea algorithm and Dynnikov's algorithm (Marc Culler has an implementation of this algorithm in his program gridlink) link text. There is also the Birman Hirsch algorithm based on braid simplification. All of these algorithms, I think it is fair to say, involve an exhaustive search through an exponentially large set.
The various kinds of knot homologies give certificates of unknottedness of a rather different nature. A knot $K$ is unknotted if and only if $H(K)=H(U)$. In historical order the Seiberg-Witten Floer homology of the zero surgery, Ozsvath-Szabo-Rasmussen knot homology and Khovanov homology are known to detect the unknot. The later two have combinatorially definitions which are rather straightforward to code up on a computer (making the programs run fast is a non-trivial task). Khovnanov homology having been around longer has rather good algorithms at this point. All the algorithms run however in exponential time. Since Khovanov homology determines the Jones polynomial and computing the Jones polynomial is a known computationally hard problem Khonvanov homology is likely also hard (unless by some miracle computing the group without the bigrading is computationally simpler). The OSR knot group on the other hand determines Alexander polynomial which is polynomial time computable so there is some (I think extremely slim) hope that the OSR group is polynomial time computable. In any case these algorithms have a rather different flavor, they do not involve an exhaustive search on the other hand they require dealing with exponentially large matrices. They do not tell you how to unknot the knot even if you know it to be so.
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This is somewhat tangential, but it's certainly not the majority opinion that being in coNP intersect NP is strong evidence for being in P. The most prominent example is factoring, which (when phrased as a decision problem) is in coNP intersect NP, but it is not the case that the majority of computer scientists regard this fact as strong evidence for the existence of a polynomial-time algorithm for factoring. – Timothy Chow Jan 29 2011 at 21:25
There are really two questions here: (1) Can you an untangle any unknot with relatively little work, say a polynomial number of geometric moves of some kind? (2) Given a knot, can you quickly figure out that it is an unknot, say with a polynomial amount of thought? It seems that both questions are open and that neither one implies a solution to the other. You do get an elementary relation in one direction, which however is not useful for current bounds: If you have a good bound on the number of moves, then you can do an exhaustive search to find them. Except maybe for improved constants, I do not know of rigorous bounds that are better than the Hass-Lagarias result that you need at most an exponential number of moves [arXiv:math/9807012], and the Hass-Lagarias-Pippenger result that unknottedness is in NP [arXiv:math/9807016]. In fact the two results are related in the converse direction. Their certificate of unknottedness is a disk which may have exponential area and gives you the moves; but the disk has a polynomial-length description.
As Timothy Chow says, being both in NP and coNP is certainly not strong evidence by itself that unknottedness is in P. On the contrary, people believe that NP ∩ coNP is much bigger than P. But there is no strong reason to believe that unknottedness is hard or easy as far as I know. There is a theorem of Thurston and Garside that a trivial braid can be recognized in polynomial time; maybe the same is true of the unknot. One question that interests me, maybe for no good reason, is whether unknottedness is in BQP (quantum polynomial time).
As Tom says, if people do find a polynomial-time algorithm to tell if a knot is the unknot, it almost certainly won't be by computing its Khovanov homology. It's known to be #P-hard to compute the Jones polynomial of a knot, or even to derive various sorts of partial information about the Jones polynomial. For instance, usually a single value is already #P-hard. Khovanov homology, its categorification, is even more information. It is a theorem of Kronheimer and Mrowka that Khovanov homology distinguishes the unknot; this is also a conjecture for the original Jones polynomial. In other words, if you do find a fast algorithm to distinguish the unknot, there will be many large knots for which you can say, "No, that isn't unknot!" even though you won't know its Jones polynomial or its Khovanov homology in a million years. You'll just know that the latter and probably the former isn't trivial.
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There is a new paper on this topic by Allison Henrich and Louis Kauffman, "Unknotting Unknots," arXiv:1006.4176v4 [math.GT]. In particular,
we show how to obtain a quadratic upper bound for the number of crossings that must be introduced into a sequence of unknotting moves.
A more precise version of their theorem is:
Theorem 4. Suppose $K$ is a diagram (in Morse form) of the unknot with crossing number cr$(K)$ and number of maxima b$(K)$. Let $M = 2$b$(K)$ $+$ cr$(K)$. Then the diagram can be unknotted by a sequence of Reidemeister moves so that no intermediate diagram has more than $(M − 2)^2$ crossings.
For the "Culprit" (discussed in Timothy's post), their theorem yields a bound of 324, but "in actuality we only needed a diagram with 12 crossings in our unknotting sequence":
They conclude their Introduction with this disclaimer:
we warn the reader that the difference between the lower bounds and upper bounds that are known is still vast. The quest for a satisfying answer to these questions continues.
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There is a nice example of a trivial knot in Dynnikov's paper: A New Way to Represent Links. One-Dimensional Formalism and Untangling Technology.
It was verified by computer. It is big but it fits in a page. I am not sure there is an obvious simplifying move, it would be nice to know. By the way it would be interesting to have a definition of what a 'generalized Reidemeister move' is. A 'good' definition could perhaps give interesting diagrams of the unknot that are not easy to simplify.
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Thank you for this example. It's quite interesting as it is in some sense a "product" of smaller knots. I tried replacing the bundles of strands (most of the time four strands) by a single strand and obtained a picture of a knot that I can't instantly see to be the unknot, though I did find a local way of reducing the number of crossings. If this "quotient" knot is not the unknot, then it's a very interesting example. – gowers May 9 at 14:49
Reply from Panos (under a different account): Yes, it seems that the 4 strands travel together and there is some linking on the upper part of the diagram and a 'knot' on the bottom. In the previous examples it seemed that one could simplify the diagram by moving one strand by a `generalized Reidemeister move'. If I understand you correctly you say that you can simplify Dynikov's example by moving 4 strands together ? I am curious to see this. (this is a comment to the previous comment but apparently I can not post it as a comment). – S. Carnahan♦ May 13 at 8:21
I did not see any mention of this preprint by Marc Lackenby, probably because the question is quite old :
A polynomial upper bound on Reidemeister moves http://arxiv.org/abs/1302.0180
Building on the techniques introduced by Dynnikov and adding some normal surface theory, he shows that any unknot can be unknotted using only a polynomial number of Reidemeister moves.
So if we interpret the "much more complicated" by "exponentially more complicated", it gives quite a strong "no" answer to
is there a known unknot diagram for which it is truly the case that to disentangle it you have to make it much more complicated?
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http://mathoverflow.net/questions/97828?sort=votes | ## interval exchange maps and surfaces
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I apologize if the question is a well-known theorem, but I'm just starting to learn about laminations, so I don't know much.
The question is roughly, if interval exchange maps have an underlying closed smooth surface, or if not, what is known about conditions on that.
Now I try to be more precise.
Usual interval exchange functions are bijective functions $\mathbb R/\mathbb Z=S^1\to S^1$ which are piecewise translations where "piecewise" is defined using a finite partition of $S^1$ in segments $s_i$.
Given a interval exchange function $\phi:S^1\to S^1$ one can consider its suspension $S_\phi=[0,1]\times S^1/\sim$ where the nontrivial identifications are $(0,x)\sim (1,\phi(x))$. Then one has a smooth 1-dimensional structure on $S_\phi$, given by the differentiable structure on $[0,1]$, which can be extended with continuity through the identifications. In the $S^1$-direction there is just some piecewise $C^1$-structure on the pieces $]0,1[\times s_i$, since $\phi$ is not even assumed to be continuous.
Then if I understood correctly one defines $\partial S_\phi=\cup_i [0,1]\times\partial s_i/\sim$, and this set is an union of (topological but not $C^1$, since there are cusps) copies of $S^1$. Then one obtains a surface by gluing some annuli to these circles.
The question is if there is some way of obtaining a smooth surface in this way. The case where the lenghts of $s_i$ are rational and $\phi$ is piecewise equal to translations by rational numbers is simple (one refined the segments and gets a longer, smooth, boundary), so I am interested if there is a result in the other case.
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## 1 Answer
Your method certainly works, because you are just identifying boundary edges of the annulus $[0,1] \times S^1$ in pairs to form a surface. As usual, when one glues up edge pairs of a surface-with-boundary, the endpoints of the boundary edges form "vertex cycles" whose images are points of the quotient surface, and the smooth structure then extends across each of these points. This works regardless of rationality, irrationality, etc. of the lengths of the intervals.
I would describe the method differently, though, in order to isolate the "singular" behavior in a way that is easier to analyze. It is a somewhat standard method, and is more-or-less equivalent to your method.
To describe the standard method, let me set up notation for the interval exchange map. One partitions the circle $S^1$ into $M$ intervals in two ways, $S^1 = I_1 \cup \cdots \cup I_M$ and $J_1 \cup \cdots J_M$, and so that for each $m=1,\ldots,M$ the lengths of $I_m$ and $J_m$ are equal, and then one chooses an isometry $f_m : I_m \to J_m$. The map $\phi : S^1 \to S^1$ itself is then just the union of the maps $f_1,\ldots,f_m$, perhaps restricted to the interiors of the intervals $I_m$ in order to get something well-defined and continuous (or perhaps restricted to half-open intervals in some manner to get something bijective but discontinuous, something that never seems worthwhile to me).
To get a surface, one proceeds in a few steps.
First, let $X$ be the quotient space obtained from the disjoint union of $S^1$ and the $m$ rectangles $[0,1] \times I_m$, by identifying $(0,x)$ to $x$ and identifying $(1,x)$ to $f_m(x)$ for each $m$ and each $x \in I_m$. The lower corners of these rectangles are identified 2--1 with the endpoints of the $I$'s, and the upper corners are identified 2--1 with the endpoints of $J$'s. This object $X$ might be a surface-with-boundary, if you are lucky, but it won't be if there is a common endpoint of the $I$'s and the $J$'s.
Next one studies the "boundary cycles" of $X$ (the quotes are there to accomodate the possibility that $X$ is not actually a surface and these boundary cycles might touch where the $I$'s and $J$'s have a common endpoint). Construct each of these cycles as follows. Start with an $I$-endpoint $x_1 \in S^1$. Of the two lower corners identified with $x_1$, pick one, $p_1 \in [0,1] \times I_{m_1}$, say. Then go up a vertical side of $[0,1] \times I_{m_1}$ to an upper corner $p_2$, identified with some $J$-endpoint $y_1 \in S^1$. Then jump to the other upper corner identified with $y_1$, say $p_3 \in [0,1] \times I_{m_2}$. Then go down a vertical side of $[0,1] \times I_{m_2}$ to a lower corner $p_4$, identified with some $I$-endpoint $x_2$. Then jump to the other lower corner identified with $x_2$, say $p_5 \in [0,1] \times I_{m_3}$. Altogether, so far, you have travelled up a vertical side of $X$ from $x_1$ to $y_1$ and down another vertical side from $y_1$ to $x_2$. Now continue the cycle, which will go up-down-up-down... across a cycle of vertical sides that closes up to form a circle in $X$ subdivided into some even number of sides.
Finally, for each of these boundary cycles subdivided into $2m$ vertical sides, one subdivides each vertical side at its midpoint into two half length subarcs, and then: identify the two subarcs incident to $x_1$ to a single arc; identify the two subarcs incident to $y_1$ to a single arc; identify the two subarcs incident to $x_2$ to a single arc; and so on around the circle. The effect is to collapse the entire circle onto a $2m$-pointed star.
The quotient is a closed surface $S$. The vertical segments $[0,1] \times (point)$ glue up to form a foliation of the surface. This foliation has a $2m$-pronged singularity'' at the valence $2m$ star point associated to each boundary cycle consisting of $2m$ vertical sides. This foliation has an invariant transverse measure, coming from the $dx$-measure on each rectangle $[0,1] \times I_m$, and the fact that each gluing map $f_m$ was an isometry.
One can proceed to a smooth structure on $S$ as follows. The smooth structures on the rectangles $[0,1] \times I_m$ glue up to form a smooth structure everywhere except at the finite set of even-pronged singularities. Moreover, the Euclidean structures on these rectangles glue up to form a Euclidean metric except at the singularities. From the Euclidean metric, you get a conformal structure everywhere except at the prong singularities. But each prong singularity is removable, so the conformal structure is extendable across the prongs, which induces an extension of the smooth structure as well.
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Thanks for the detailed description. I can follow it until "is removable" in the end. What does that precisely mean? Does the removal preserve the smoothness of the foliation? Maybe there is some well-known fact here which I don't know. – Mircea May 24 at 23:52
No there's not getting around the singularity of the foliation. But that singularity has a very specific model as the horizontal foliation of the quadratic differential $z^{2m-2} dz^2$. And that $z$ coordinate provides the smooth extension of the conformal structure across the singularity. – Lee Mosher May 25 at 2:10
Thanks again, now the answer is perfectly transparent! So not even by changing the surface construction can one avoid the singularity in the foliation? If the exchange map had rational parameters in the sense that I described in the question, one has some hope of avoiding the prongs by complicating the surface. – Mircea May 25 at 5:52
The Euler-Poincare theorem says that for a foliation with isolated singularities on a compact surface, the sum of the indices of the singularities equals the Euler characteristic of the surface. So if the surface has negative Euler characteristic, singularities are unavoidable. It's best to think of prong singularities as natural features. – Lee Mosher May 25 at 12:39
I think some confusion was in my head regarding the precise meaning of the word "foliation". Is there more hope if one just looks for laminations? – Mircea May 26 at 8:35
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http://math.stackexchange.com/questions/126015/uniform-convergence-but-no-absolute-uniform-convergence | # Uniform convergence, but no absolute uniform convergence
Can someone give an example of a series of functions $f_k(x)$ for which $\sum_{k=0}^{\infty} f_k(x)$ converges uniformly, and $\sum_{k=0}^{\infty} |f_k(x)|$ converges pointwise, but $\sum_{k=0}^{\infty} |f_k(x)|$ does not converge uniformly.
I'm having trouble finding an example (or proving this is impossible), as there is no certainty the limit of the absolute and relative series are equal.
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## 1 Answer
Take $f_n(x)=(-1)^n {x^n\over n}$, for $n\ge 1$ on $[0,1)$.
Then if $n$ and $m$ are positive integers with $m\ge n$ and $x\in[0,1)$: $$|f_n(x) +f_{n+1}(x)+\cdots +f_m(x)|\le| f_n(x)| ={x^n\over n}\le {1\over n}.$$ This implies that $\sum\limits_{n=1}^\infty f_n(x)$ converges uniformly on $[0,1)$.
The series $\sum\limits_{n=1}^\infty |f_n(x)|$ converges on $[0,1)$ as comparison with the series $\sum\limits_{n=1}^\infty x^n$ will show.
But $\sum\limits_{n=1}^\infty |f_n(x)|$ does not converge uniformly on $[0,1)$; since, for any $n$, $$|f_n(x)| +|f_{n+1}(x)|+\cdots +|f_{2n}(x)| \ge {1\over 2n} \cdot nx^{2n},$$ and $\lim\limits_{x\rightarrow1^-} x^{2n} =1$.
I think that any absolutely convergent but non-uniformly convergent series of positive, decreasing terms $f_n$ with $(f_n)$ converging uniformly to 0 would provide a counterexample by making the series "alternating".
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http://www.reference.com/browse/Dewetting | Definitions
Nearby Words
# Dewetting
In fluid mechanics, dewetting is one of the processes that can occur at a solid-liquid or liquid-liquid interface. Generally, dewetting describes the rupture of a thin liquid film on the substrate (either a liquid itself, or a solid) and the formation of droplets. The opposite process—spreading of a liquid on a substrate—is called wetting. The factor determining the wetting and dewetting for a drop of oil placed on a liquid substrate (water here) with ambient gas, is the so-called spreading coefficient $S$
$S = gamma_text\left\{gw\right\} - gamma_text\left\{go\right\} - gamma_text\left\{ow\right\}$
where $gamma_text\left\{gw\right\}$ is the gas-water surface tension, $gamma_text\left\{go\right\}$ is the gas-oil surface tension and $gamma_text\left\{ow\right\}$ is the oil water surface tension (measured on the fluids before they are brought in contact with each other)
When $S>0$, the surface is considered wettable, and if $S<0$, dewetting occurs.
Wetting and dewetting are important processes for many applications, including adhesion, lubrication, painting, printing, and protective coating. For most applications, dewetting is an unwanted process, because it destroys the applied thin film. In most dewetting studies a thin polymer film is spun onto a substrate. Even in the case of $S<0$ the film does not dewet immediately if it is in a metastable state, e.g. if the temperature is below the glass transition temperature of the polymer. Annealing such a metastable film above its glass transition temperature increases the mobility of the polymer chain molecules and dewetting takes place. When starting from a continuous film, an irregular pattern of droplets is formed. The droplet size and droplet spacing may vary over several orders of magnitude, since the dewetting starts from randomly formed holes in the film. There is no spatial correlation between the dry patches that develop. These dry patches grow and the material is accumulated in the rim surrounding the growing hole. In the case where the initially homogeneous film is thin (in the range of 100 nm), a polygon network of connected strings of material is formed, like a Voronoi pattern of polygons. These strings then can break up into droplets, a process which is known as the Rayleigh-Taylor instability. At other film thicknesses, other complicated patterns of droplets on the substrate can be observed, which stem from a fingering instability of the growing rim around the dry patch. Surfactants can have a significant effect on the spreading coefficient. When a surfactant is added, it would decrease the interfacial tension, thus increasing the spreading coefficient (i.e. making S more positive) and allowing wetting to occur. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 8, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9204655885696411, "perplexity_flag": "middle"} |
http://mathoverflow.net/questions/16994?sort=oldest | Linear Algebra Texts?
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Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to the introduction of abstract vector spaces mid-way through a course. Sometimes it feels as though I've walked into class and said "Forget math. Let's learn ancient Greek instead." Sometimes the students realize that Greek is interesting too, but it can take a lot of convincing! Hence I would really like to let students know, right from the start, what they're getting themselves into.
Does anyone know of a text that might help me do this in a not-too-advanced manner? One possibility, I guess, is Linear Algebra Done Right by Axler, but are there others? Axler's book might be too advanced.
Or would anyone caution me against trying this, based on past experience?
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I actually learnt (some) ancient greek in high school and found it interesting. In my first they also introduced abstract vector spaces from the start, and I still firmly believe it's the right way to do it. I second Lang's book on linear algebra. For a general algebra course instead I strongly vote for Aluffi's Algebra Chapter 0. – babubba Mar 3 2010 at 20:27
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Dan, it might be helpful to know what the audience for your class is. Are the students math majors or not? Have they had proof-based math already or not? In particular, some textbooks are written with the assumption that students are working with proofs for the first time and try to ease the transition; some assume students are already completely comfortable with proofs; and some don't care about proofs at all and just aim to show how to do calculations, like a typical calculus book. – Mark Meckes Mar 4 2010 at 14:47
Regarding Axler, I just reread his introduction and was reminded that his book was written for a second course in linear algebra. He doesn't say what he envisions as the content of the first course, but I'd guess it would be mainly a course on matrix computations, which his book would then complement. – Mark Meckes Mar 4 2010 at 14:49
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Hi Mark, I think there will be a range of students, mostly non-math majors, and all of them writing proofs for the first time. I feel convinced by now that Axler would not be the right choice. – Dan Ramras Mar 4 2010 at 18:43
I'm curious what book you ended up picking. – Harry Gindi Mar 10 2010 at 19:07
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15 Answers
For teaching the type of course that Dan described, I'd like to recommend David Lay's "Linear algebra". It is very thoroughly thought out and well written, with uniform difficulty level, some applications, and several possible routes/courses that he explains in the instructor's edition. Vector spaces are introduced in Chapter 4, following the chapters on linear systems, matrices, and determinants. Due to built-in redundancy, you can get there earlier, but I don't see any advantage to that. The chapter on matrices has a couple of sections that "preview" abstract linear algebra by studying the subspaces of $\mathbb{R}^n$.
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In the end, this is the book I decided to go with. – Dan Ramras May 24 2010 at 19:18
Mark it as the right answer, then:) This book isn't perfect, but I liked it a lot and I hope that so will you and your students. – Victor Protsak May 24 2010 at 22:22
Now that I'm a month into the course, I think I can heartily say that I'm happy with the book. No, it isn't perfect, but quite often the complaints I have are addressed in the author's preface for instructors (in the instructor version) and several times I've become convinced that Lay has a good point, and there's a good reason for doing things the way he does. It's very tempting to lay on tons of concepts early on in a linear algebra course. Lay's book is good about introducing concepts slowly, and then reinforcing them later with new viewpoints. – Dan Ramras Sep 19 2010 at 23:49
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Hands-down, my favorite text is Hoffman and Kunze's Linear Algebra. Chapter 1 is a review of matrices. From then on, everything is integrated. The abstract definition of a vector space is introduced in chapter 2 with a review of field theory. Chapter 3 is all about abstract linear transformations as well as the representation of such transformations as matrices. I'm not going to recount all of the chapters for you, but it seems to be exactly what you want. It's also very flexible for teaching a course. It includes sections on modules and derives the determinant both classically and using the exterior algebra. Normed spaces and inner product spaces are introduced in the second half of the book, and do not depend on some of the more "algebraic" sections (like those mentioned above on modules, tensors, and the exterior algebra).
From what I've been told, H&K has been the standard linear algebra text for the past 30 or so years, although universities have been phasing it out in recent years in favor of more "colorful" books with more emphasis on applications.
Edit: One last thing. I have not heard great things about Axler. While the book achieves its goals of avoiding bases and matrices for almost the entire book, I have heard that students who have taken a course modeled on Axler have a very hard time computing determinants and don't gain a sufficient level of competence with explicit computations using bases, which are also important. Based on your question, it seems like Axler's approach would have exactly the same problems you currently have, but going in the "opposite direciton", as it were.
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I also first learned linear algebra from H&K and share fpqc's enthusiasm for it, but I don't think it counts as "relatively gentle". I think RH is right that H&K is pretty advanced in spirit if your students don't already have experience with rigorous proof-based math. Having taught from Axler, which does have many nice points, I can also confirm the problems with it that fpqc describes. – Mark Meckes Mar 4 2010 at 14:41
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The problem with Axler is that he tries to avoid algebra at all cost - when it makes things easier and when it makes thing harder. Sometimes much harder, actually. Even if one goes into physics and never has to work over a ring different from R and C, one will realize one day that in order to compute the characteristic polynomial one doesn't have to bring the matrix in upper triangular form (no joke, this is how Axler defines the characteristic polynomial), and that often, the characteristic polynomial matters and upper diagonalization doesn't. – darij grinberg Mar 10 2010 at 18:50
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For someone who plans to work in algebra or algebraic geometry, linear algebra learnt from Axler is mostly wasted time. I don't understand what he has against the notion of determinant; this notion (with the sum-over-permutations formula that he seems to hate) comes out straightforwardly if one tries to apply Gaussian elimination to a general systems where the coefficients of the system are variables. – darij grinberg Mar 10 2010 at 18:53
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From the description given (starts with a $\textit{review}$ of matrices and fields, uses modules and exterior algebra, etc), it is patently obvious that this book (HK) is unsuitable to people without abstract algebra under their belt. Quite a bit of mathematical maturity that you cannot reasonably expect from non-math majors with or without prior proof experience is required as well. – Victor Protsak May 23 2010 at 1:31
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@Harry: Bear in mind that what works for $\textit{you}$ (and other math majors at UM), doesn't necessarily work for others. And you have just confirmed that you were comfortable with abstract algebra, at the level higher than most non-majors ever see, before starting out. Learning composition of morphisms without being able to multiply matrices is $\textit{truly}$ pointless. One imperfect analogy: it's possible to learn AG from EGA or Harstshorne ("it has been done"), but as first books they are nowhere near Shafarevich, Cox-Little-O'Shea, Reid, Mumford, and any number of other texts. – Victor Protsak May 23 2010 at 6:49
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Serge Lang's Linear Algebra does not cover much material, but is very nice for a first introduction. It does not emphasize particularly matrices and computations, so one understands immediately that matrices only come as representations of linear maps, but it's also not too abstract.
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For a second there, I thought you said "Serge Lang's _Algebra_". I'm sure you can appreciate the humor in that. – Harry Gindi Mar 3 2010 at 19:46
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Wait, so you mean there IS a readable book by Lang? – darij grinberg Mar 3 2010 at 19:56
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You're too hard on Lang, darij. Algebra is good, the differential geometry book is good, the book on cyclotomic fields is pretty good. Sure, there are unreadable sections and undefined notation, but the books are generally readable, and some of them are even pretty good! – Harry Gindi Mar 3 2010 at 20:55
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I second Lang's Linear Algebra. I also found it very accessible, and it also seems to be a good preparation for the corresponding chapters of his "Algebra" (Chs. XIII - XV of the Springer edition). – unknown (google) Mar 3 2010 at 22:30
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Most of the trouble I've had with Lang is the sheer number of mistakes in the book. There's nothing worse than a line that says "of course..." followed by a typo. It leaves you feeling like an idiot when you've not done anything wrong. – Harry Gindi Mar 10 2010 at 19:06
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I rather like Linear Algebra Done Right, and depending on the type of students you are aiming the course for, I would recommend it over Hoffman and Kunze. Since you seemed worried that Axler might be too advanced, my feeling is that Hoffman and Kunze will definitely be (especially if these are students who have never been taught proof-based mathematics).
Of course, the big caveat here being that Axler avoids determinants at all costs, and this will put more on you to introduce them comprehensively.
I've never looked at it, but another one worth considering might be Halmos's Finite Dimensional Vector Spaces.
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I'm quite fond of Halmos's FDVS. It certainly takes the abstract vector spaces approach with seriousness and gusto. I was planning on writing an answer of my own pointing towards Halmos, but endorsing this might do the trick just as well. – Mikael Vejdemo-Johansson Mar 3 2010 at 20:08
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I forgot about Halmos. You should add it as a separate answer, since I'd vote it up, but as you can see, I really think Axler's book goes about things the wrong way. – Harry Gindi Mar 3 2010 at 20:48
I have taught out of "Linear Algebra Done Right" and I like it. The main drawback I saw was that I had to introduce more computational problems (for example, so that the students could explicitly compute changes-of-basis and such). The book stays with real and complex spaces, so it's not an upper level text. But the proofs are very nice. I'm planning to use it again the next time I teach a linear algebra course at that level. – Carl Mummert Jun 19 2010 at 11:49
There's also Nicholson's Elementary Linear Algebra or the slightly more advanced Linear Algebra: With Applications. If your students react negatively to the intro of abstract vector spaces, I don't think Hoffman and Kunze's book would be good for them. While I love that book myself it might be a little too daunting for your class. Also I think that if you want to introduce abstract vector spaces from the start there's no reason you can't cover the chapter on abstract vector spaces first.
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Haven't seen Nicholson's texts. I'll take a look. Thanks! – Dan Ramras Mar 4 2010 at 3:35
There were times when I was rather fond of Strang's Linear Algebra and Its Applications. I haven't looked at it for a long time, but back then I found it very clear and appealing. Even if you don't follow the book chapter by chapter, it might still give you ideas.
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Definitely the book I would recommend for non-math majors. It has plenty of examples to motivate topics, which is what non-mathematicians need in order to be interested in linear algebra. Vector space axioms are the absolute worst way to teach linear algebra to any group of people that is not wholly composed of math majors. – Rune Jun 20 2010 at 2:13
If you are looking for a gentle introduction, that uses matrices from the beginning, I would suggest you consider "Linear Algebra" by Friedberg, Insel and Spence. I haven't used this book myself, but somebody (I trust) recommended this book to me. I now own it, and it looks very nice and gentle (but covering all the topics I would like to include), and matrices are introduced in page 8.
Alvaro
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Nooooo! I used this book when I taught a 2nd linear algebra course and that book is dry as dust. It has all kinds of neat applications, but it is really boring to read. (I really mean boring, not "too elementary"). Also, despite the abstraction over general fields in the main text, there is little compelling rationale given for needing linear algebra over something besides R or C. In particular, F_2 is used in the book only for weird counterexamples, even though linear algebra over F_2 is really useful in computer science. – KConrad Mar 10 2010 at 19:33
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I think this book is perfect. Sure it's dry, but that's fine. It has worked out examples exactly where they should be, the presentation and proofs are crystal clear, and there are tons of good exercises. I'm stuck teaching calculus from a book, which I'll not name now, that tries to "sell it" by attempting to be more readable, using poorly construed applications for motivation, and filling up empty space with colour pictures. I don't find this helps to convert anyone who isn't already interested. "Selling it" is my job as a teacher. When it comes to texts, I look for simplicity and clarity. – Brendan Cordy Apr 5 2010 at 3:02
It's dry to be sure, but it works. The only complaint I have is that there are a lot of silly computational problems. BUT there's nothing that says that you have to give those problems as an instructor, and there are some good problems in there – Michael Hoffman May 23 2010 at 0:20
@KConrad WHat the heck do you mean,dry,KC?!? It's loaded with beautiful examples and applications,some of which are rarely presented in a first course,like stochastic matrices! It's RIGOROUS without being Bourbakian,that's what I love about it. The section on the Jordan canonical form is a mess,though.Use Curtis for that and it'll be fine. – Andrew L May 23 2010 at 3:45
My old mentor Nick Metas was part of the teams of graduate students who worked over the drafts of H&K when they were writing it for the linear algebra course at MIT in the 1960's.that being said,despite its' rigor and beauty,I think a "pure" linear algebra course is just as big a mistake as a pure theoretical calculus course no matter how good the students are. It's like teaching music students all about pentamer,note grammer and acuostics and never teaching them how to play a single note.I don't go for this whole pure/applied distinction,it's an idiotic consequence of this age of specialization.I love rigor,but applications should never be denied or ignored. That's why my overall favorite LA text is Friedberg,Insel and Spence-it's the only one I've seen that aims for and hits a terrific balance between algebraic theory and applications. I also love Curtis for similar reasons,but it's coverage isn't as broad. I love books that aim for that Grand Mean Balance-sadly,in America,there aren't anywhere near enough such texts.
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I find "applications" largely misleading in Linear algebra. The fact that linear algebra has been split up from a one semester course to a two semester course (matrix algebra and linear algebra) at many schools is a real shame. If you've read Emil Artin's "Geometric Algebra" monograph, you can see that it's very easy to take the isomorphism between an abstract finitely generated vector space and k^n for granted. Conceptual understanding is much more important. – Harry Gindi Mar 10 2010 at 18:49
I've had zero experience with any of the books named, but I completely support the point that Andrew made: abstract linear algebra strips the motivation and skips perhaps the most interesting part: applications. V.I.Arnold was very caustic about this "criminal Bourbakization and algebraization of mathematics". It's a harmful fallacy that conceptual understanding and applications are mutually exclusive. – Victor Protsak May 23 2010 at 1:16
@Victor: I don't agree one bit. I realize that Arnold's diatribe is very famous and loved by Bourbaki-haters everywhere, but I counter it with a famouse paragraph from Emil Artin's monograph "Geometric Algebra": – Harry Gindi Jun 19 2010 at 10:23
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Artin's view here is very much the view of Dieudonné (as expressed in his book on linear algebra). I think that Arnold simplifies the world into black and white and attacks a straw boogeyman named Bourbaki. Hoffman and Kunze gives a very nice account of these geometric aspects as well as the algebraic ones. – Harry Gindi Jun 19 2010 at 10:43
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Harry, don't get all worked up. Let me repeat the key point: $\textit{It's a harmful fallacy that conceptual understanding and applications are mutually exclusive.}$ There isn't any application in sight in your quotes, just comparisons between different formalisms. And for what it's worth, the last one is perfectly in line with Arnold's philosophy, while it doesn't conform well to Bourbaki's way of thinking. – Victor Protsak Jun 20 2010 at 1:06
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There is no ideal text for a beginning one semester course as taught in the US to first or second year college students. Older books like H&K treat only the abstract theory, in a fairly conceptual way and (if I recall correctly) with maps written on the right contrary to what students do in calculus. A later generation of books like the original Anton are also pure math books but start by overemphasizing unrealistic manipulations with small matrices and vectors; then there is an abrupt shift to abstraction. Determinants are presented in a purely computational mode, as though they were really used for this purpose; then eigenvalues occur very late and again in oversimplified small examples. Fortunately the newer texts tend to mix pure and applied throughout, but as a result they contain far too much material for a first course. And eigenvalue theory still gets introduced very late. Strang is attractive in many ways, but too loosely written down and not suitable for an inexperienced reader without a reliable guide at hand. Aside from Strang, the emphasis in most US textbooks remains placed on unrealistic integer calculations with very small matrices rather than on the geometry of subspaces, etc. The pervasive role of geometric thinking in the subject is mostly downplayed in texts, as is the role of analysis. For self-study, something like Friedberg-Insel-Spence may be the best compromise choice.
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H&K composes in the standard way. $ST(\vec{v})=S(T(\vec{v}))$. They do not cover applications, but I think that "real-world" applications have no place in a math book. – Harry Gindi Mar 10 2010 at 19:11
@Jim I think Friedberg-Insel-Spence is the best LA book out there for a general class of math majors right now-it's the most balanced between rigor and applications and it probably covers the widest range of topics at this level.Still,I agree that I don't think The Great American LA Text has been written yet. – Andrew L Jun 19 2010 at 16:23
My personal pick is I.M.Gelfand's "Lectures on linear algebra" (link to a copy on Google Books), accompanied by two warnings: (1) the part "Introduction to tensors" is far from perfect; (2) the proof of the Jordan normal form theorem is dramatically outdated (keep in mind that the only English translation of the book is that of the 1950s edition - the latest editions contain a proof that totally makes sense). Then again, many linear algebra textbooks simply avoid Jordan normal forms completely (which I think is a mild disaster).
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Matrix Analysis and Applied Linear Algebra by Meyer is very well written with clear cut examples and exercises. I think this would make an excellent first course.
I agree also that Axler's books is a great text for the more mature.
Classics
Finite-Dimensional Vector Spaces by P. R. Halmos is an absolute essential for the budding mathematician in my opinion. This is because of the exercises (My recommendation: solve all of them).
As mentioned above Linear Algebra (2nd Edition) by Kenneth M Hoffman and Ray Kunze. This may be my favorite text because of its volume of content.
More Advanced
Advanced Linear Algebra by Steven Roman
Matrix Analysis
Matrix Analysis and Topics in Matrix Analysis by Roger A. Horn and Charles R. Johnson
Matrix Analysis by Rajendra Bhatia
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Why does no one go over applied linear algebra, or more, why is there no book that actually talks seriously about the computational end and about the theory. By computational end I mean the REAL computational end, that which is actually done on a computer or at least is the background to understand those algorithms. If there were a nice undergraduate version of Demmel then I'd defer to that book, but so far as I know such a book doesn't exist. If you're going to split linear algebra at all it would seem to be
Theoretical Linear Algebra
and
Computational Linear Algebra
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Mainly because computational linear algebra by hand is frustrating and pointless. – Harry Gindi Jun 19 2010 at 11:14
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Well, there are good books on computational linear algebra, often called "numerical analysis". One example is Golub &Van Loan. But they are far too advanced to do beginning students any good. – Felix Goldberg May 30 at 9:25
The best thing about Hoffman and Kunze's book is its beautiful exposition of Jordan Forms. If a course is planning to get to Jordan Forms as a target then I can't think of any better approach than that in Hoffman and Kunze.
Sections on linear algebra in Artin and Herstein's book's are also very good but then Hoffman and Kunze win hands down if the objective is Jordan Form.
Explanation of concepts like conductors and annihilators, invariant polynomials and variations/equivalence between notions of semi-simplicity and myriad of different ways to test diagonalizability of a linear transformation are I would say the claim to fame for Hoffman and Kunze's book. And all this merges beautifully in their writing of Jordan forms, as if everything else was written just to make this concept clear.
Very importantly this books gives instructive numerical examples after every bunch of concepts.
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Look at the exposition of Jordan canonical forms in both Charles Curtis' LINEAR ALGEBRA:AN INTRODUCTORY APPROACH and in Anthony Knapp's BASIC ALGEBRA,Dan. I think you'll find both superior to H&K. H & K is just too abstract to be helpful long-term for most math majors-although,to be honest,the possibility of a course based on the union of H&K and Gilbert Strang's book has always intrigued me. – Andrew L Jun 19 2010 at 22:36
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I agree with Andrew that Charles Curtis Linear Algebra book is superior to Hoffmann and Kunze. I took my first linear algebra course from Hoffmann and Kunze, and while Curtis's book has a better exposition of the theory, I do believe that the exercises in H&K are way better. – Adrián Barquero Jun 20 2010 at 3:22
Although I have not lectured from it, I like very much Klaus Jänich's Linear Algebra book.
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A very good textbook is Shilov's. It is actually the first (or perhaps Volume 0) of his textbook in Mathematical Analysis. It covers more than the standard material, but is very clear written with many examples and exercises (many solved).
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http://requestforlogic.blogspot.co.il/2012_03_01_archive.html | # Request for Logic
## Monday, March 5, 2012
### What does focusing tell us about language design?
This post is largely based on some conversations I've had about Polarized Logic/Focusing/Call-By-Push-Value recently, especially with Neel Krishnaswami and Wes Filardo, though it was originally prompted by Manuel Simoni some time ago.
I think that one of the key observations of focusing/CBPV is that programs are dealing with two different things - data and computation - and that we tend to get the most tripped up when we confuse the two.
• Data is classified by data types (a.k.a. positive types). Data is defined by how it is constructed, and the way you use data is by pattern-matching against it.
• Computation is classified by computation types (a.k.a. negative types). Computations are defined their eliminations - that is, by how they respond to signals/messages/pokes/arguments.
There are two things I want to talk about, and they're both recursive types: call-by-push-value has positive recursive types (which have the feel of inductive types and/or algebras and/or what we're used to as datatypes in functional languages) and negative recursive types (which have the feel of recursive, lazy records and/or "codata" whatever that is and/or coalgebras and/or what William Cook calls objects). Both positive and negative recursive types are treated by Paul Blain Levy in his thesis (section 5.3.2) and in the Call-By-Push Value book (section 4.2.2).
In particular, I want to claim that Call-By-Push-Value and focusing suggest two fundamental features that should be, and generally aren't (at least simultaneously) in modern programming languages:
1. Support for structured data with rich case analysis facilities (up to and beyond what are called views)
2. Support for recursive records and negative recursive types.
### Minor addition to "core Levy"
I'll be presenting with an imaginary extension to Bauer's Levy language in this post.1 I've mucked around Levy before, of course, and it would probably help to review that previous post before reading this one. I want to make one addition to Levy before we begin making big, interesting ones. The derived form that I want to add - e1 orelse e2 - is computation code with type F bool if e1 and e1 have type F bool as well. This is definable as syntactic sugar, where x is selected to not be free in e2:
``` e1 to x in
if x then return true
else e2```
One other aside - do remember that, at the cost of potential confusion, I modified Bauer's original Levy implementation to make force a prefix operator that binds more tightly than application - force g z y can be written with parentheses as (((force g) z) y).
## Positive types, positive recursive types
The functional languages have the construction of positive types just about right: they're made up of sums and products. I used a special LLF-ish construction for datatypes in my exploration of Levy#, but the more traditional way of introducing datatypes is to say that they are recursive types μt.T(t). The recursive types we're used to thinking about are naturally interpreted as positive types, because they are data and are inspected by pattern matching.
There's a tendency in programming language design to shove positive recursive types together with labeled sum types to get a familiar datatype mechanism.2 I will go along with this tendency and merge labeled sums and recursive types, writing them as μt.[L1: T1(t), L2: T2(t), ..., Ln: Tn(t)].
Here are datatype definitions for trees of ints and lists of ints:
``` type+ tree =
μtree.
[ Leaf: int,
Node: tree * tree ]
type+ list =
μlist.
[ Nil,
Cons: int * list ]```
Note from the definition of lists that we also allow types to have no arguments: it's possible to treat the definition of Nil as syntactic sugar for Nil: unit. The associated value is Nil, which is syntactic sugar for Nil ().
There are infinitely many trees and lists as we've defined them. In fact, it's just a convention of how we have to write programs that we think of positive types as being finite sums. Even though we can't write it down as a datatype declaration, it's perfectly reasonable to think of a built-in type of infinite precision integers as being defined as follows:
``` type+ int =
μint.
[ 0,
1,
~1,
2,
~2,
3,
... ]```
The same goes for built-in strings and the like - think of built-in types being positive enumerations that were provided specially by the language since the language didn't give us the ability to write down the full declaration containing all possible constants.
### Powerful views
A point that has not, in my humble opinion, been made simply enough is that the perspective of focusing says that we should think about integrating much more powerful case analysis mechanisms into our programming languages. I learned about this point from Licata, Zeilberger, and Harper's 2008 Focusing on Binding and Computation, but their setting had enough other interesting stuff going on that it probably obscured this simple point.
Standard ML (and, to the best of my knowledge, all other functional languages circa-1997) only provides a limited form of case analysis - arbitrary finite views into the outermost structure of the datatype:
``` case x of
| Leaf 9 => ...
| Leaf y => ...
| Node (Node (Leaf 5, y), z) => ...
| Node (y, z) => ...```
This limitation comes with a nice tradeoff, in that we can pretty effectively estimate how much work compiled code needs to do to handle a pattern match. However, the structure of focusing suggests that any mathematical function from a value's structure to computation is fair game. One well-studied modernization of pattern matching is views, which allows us to group computations in other ways. One use case would be allowing us to take an integer variable x and say that it is even or odd:
``` case x of
| 0 => return true
| x is even => return false
| ~1 => return false
| x is odd => return true```
The focusing-aware view of pattern matching suggests that what a pattern match is actually doing is defining a case individually for each value structure - if we could write a countably-infinite-sized program, then we could expand the view-based program above to the no-longer-view-based, but infinitely long, program below:
``` case x of
| 0 => return true
| 1 => return true
| ~1 => return false
| 2 => return false
| ~2 => return false
| 3 => return true
| ...```
So: the focusing perspective says that any effect-free (effects include nontermination!) mathematical function we can write from value structure to computations is fair game for case analysis; views are just one known way of doing this in an interesting way. In principle, however, we can consider much richer case-analysis machinery to be fair game. For instance, there is a mathematical function $$f$$ from integer values int to computation code of type F bool with the variable coll free:
• If the integer is 1, the result is return true
• If the integer is less than 1, the result is return false
• Otherwise, if the integer is divisible by 2 and the result of dividing the integer by 2 is i, then the result is force coll i
• Otherwise, let j be 3 times the integer, plus one. Then the result is force coll j
Given this mathematical function, we have the core case analysis at the heart of the Collatz function. If we expand out this case analysis into an infinitely-long mapping as before, it would look like this:
``` rec coll: int -> F bool is
fn x: int =>
case x of
| 0 => return false
| 1 => return true
| ~1 => return false
| 2 => force coll 1
| ~2 => return false
| 3 => force coll 10
| ~3 => return false
| 4 => force coll 2
| ~4 => return false
| 5 => force coll 16
| ...```
The important question here is: what is the right way to think about how we write down total functions from value structures? Are views about right, or do we need something different, clearer/more/less powerful? I don't have a sense for the answers, but I'm pretty confident that's the right question.
### Views are only over the structure of values
One critical caveat: I used the phrase value structure in the previous discussion repeatedly on purpose. Because computations can always contain effects, we cannot poke at the computations at all when we think about views - the value structure is just the part of the value that we can inspect without traversing a thunk. Therefore, if we have this definition...
``` type+ somefn =
μsomefn.
[ BoolFn: U (bool -> F bool),
IntFn: U (int -> F int) ]```
...then a case analysis on a value of type somefn can have at most two branches - one for the BoolFn case and one for the IntFn case. We can't pattern-match into computations, so even though it would be reasonable to say that there are many, many values of type somefn, there are only two value structures that can be inspected by case analysis.
## Negative types, negative recursive types
In call-by-push-value, the computation types (a.k.a. the negative types of polarized logic/focusing) are defined by the way we use them. Computation code of computation type does stuff in response to being used - a function does stuff in response to getting an argument, and so the most well-understood negative type is implication.
Conjunction (the product or pair type) is a bit slippery, says the CBPV/focusing methodology. We've already used conjunction as a positive type (t1 * t2), but we can also think of conjunction as being a negative type. This is because it's equally valid to think of a pair being positively defined by its construction (give me two values, that's a pair) and negatively as a suspended computation that awaits you asking it for its first or second component.
CPBV/focusing makes us choose: we can have either type of conjunction, or we can have both of them, but they are different. It will be convenient if we keep the normal tuples as positive types, and associate negative types with records - that is, with named products. Records are defined by how you project from them, and just as we tie labeled sums in with positive recursive types, we'll tie labeled products in with negative recursive types.
Negative recursive types are naturally codata in the same way that positive recursive types are naturally data: the canonical example is the infinite stream.
``` type- stream =
μstream.
{ head: F int,
tail: stream }```
It's natural to define particular streams with fixedpoints:
``` val Ones = thunk rec this: stream is
{ head = return 1,
tail = force this }
val CountUp = thunk rec this: int -> stream is
fn x: int =>
{ head = return x,
tail =
force plus x 1 to y in
force this y }```
Say I wanted to get the fourth element of an infinite stream str of type U stream. The ML-ish way of projecting from records would write this as #tail (#head (#head (#head (force str)))), but I will pointedly not use that notation in favor of a different record notation: (force str).head.head.head.tail. It nests better, and works better in the case that record elements are functions. (Because functions are negative types, like records, it is very natural to have functions be the arguments to records.)
### Negative recursive types as Cook's objects
Here's a bit of a case study to conclude. The entities that William Cook names "Objects" in his paper On Understanding Data Abstraction, Revisited are recognizably negative recursive types in the sense above.3 Cook's examples can be coded in Standard ML (see here), but the language gets in the way of this encoding in a number of places.4 To see what this would look like in a language with better support, I'll encode the examples from Cook's paper in my imaginary CBPV implementation. The recursive negative type looks much like a stream, and encodes the interface for these set objects.
``` type- iset =
μiset.
{ isEmpty: F bool,
contains: int -> F bool,
insert: int -> iset,
union: U iset -> iset }```
One difficulty: the Insert and Union operations that Cook uses involve mutual recursion. I don't have mutual recursion, so rather than pretending to have it (I've abused my imaginary language extension enough already) I'll code it up using records. The record type insertunion I define isn't actually recursive, it's just a record - this is analogous to using ML's datatype mechanism (which, as we've discussed, makes positive recursive types) to define a simple enumeration.
``` type- insertunion =
μinsertunion.
{ Insert: U iset * int -> iset,
Union: U iset * U iset -> iset }
val InsertUnion = thunk
rec x: insertunion is
{ Insert = fn (s, n) =>
(force s).contains n to b in
if b then force s else
rec self: iset is
{ isEmpty = return false,
contains = fn i => (force equal i n) orelse (force s).contains i,
insert = fn i => (force x).Insert (self, i),
union = fn s' => (force x).Union (self, s') },
Union = fn (s1, s2) =>
rec self: iset is
{ isEmpty = (force s1).isEmpty orelse (force s2).isEmpty,
contains = fn i => (force s1).contains i orelse (force s2).contains i,
insert = fn i => (force x).Insert (self, i),
union = fn s' => (force x).Union (self, s') } }
val Insert: U (U iset * int -> iset) = thunk ((force InsertUnion).Insert)
val Union: U (U iset * U iset -> iset) = thunk ((force InsertUnion).Union)```
We've defined union of sets and insertion into a set, but we haven't actually defined any sets yet! Once we give the implementation of an empty set, however, we can manipulate these sets with a Java-esque method invocation style:
``` val Empty = thunk rec self: iset is
{ isEmpty = return true,
contains = fn x => return false,
insert = fn i => force Insert (self, i),
union = fn s' => force s' }
val JustOne = thunk (force Empty).insert 1
(force Empty).insert(3).union(JustOne).insert(5).contains(4)```
As Cook noted, it's also very natural to talk about infinite sets in this style, such as the set of all even numbers and the set of all integers:
``` val Evens = thunk rec self: iset is
{ isEmpty = return false,
contains = fn i =>
force mod i 2 to x in
(case x of
| 0 => return true
| _ => return false),
insert = fn i => force Insert (self, i),
union = fn s' => force Union (self, s') }
val Full = thunk rec self: iset is
{ isEmpty = return false,
contains = fn i => return true,
insert = fn i => force self,
union = fn s' => force self }```
1 If anyone is interested in helping me implement a toy language along these lines, I'm all ears. Also, the disclaimer that since I don't have an implementation there are surely bugs bugs bugs everywhere is strongly in effect.
2 As a practical matter, this makes it easier for the language to know where to put implicit roll/unroll annotations, so that programmers doesn't have to write these explicitly, which would be a pain.
3 Cook does give the disclaimer that this definition is "recognized as valid by experts in the field, although there certainly are other valid models." I carefully call these things "Cook objects" rather than "objects" - my claim is that negative recursive types correspond to what William Cook names objects, not that they correspond to what you call objects. Note, furthermore, that I could be wrong about what William Cook calls objects! I only have his paper to go on and I have been known to be a few cookies short of a haystack.
4 There are two issues with the encoding in SML. The most basic is that we have to encode the recursive types using SML's recursive type mechanism, which is biased towards the positive encoding of recursive types as finite sums. The other issue is that the way SML ties together (mutual) recursion with closures gets in the way - the standard way of writing a "thunk" as a function from unit is all over the place, and it wan't possible to define EmptySet so that it was mutually recursive with the definition of Insert and Union as a result. So, I'm certainly not arguing that SML facilities programming in (what Cook calls) an object-oriented style in a strong sense - it's unnecessarily painful to code in that style - but the minimal semantic building blocks of what Cook presented and named Object are present and accounted for. [Update:] As gasche discusses in the comments, Ocaml's records, and the way it deals with laziness, make it quite a bit better at encoding the example, but it's still not ideal.
Posted by Rob
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http://mathoverflow.net/questions/13851/the-inverse-galois-problem-and-the-monster | ## The inverse Galois problem and the Monster
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
I have a slight interest in both the inverse Galois problem and in the Monster group. I learned some time ago that all of the sporadic simple groups, with the exception of the Mathieu group $M_{23}$, have been proven to be Galois groups over $\mathbb{Q}$.
In particular, the Monster group has been proven to be a Galois group over $\mathbb{Q}$. What techniques are used to prove such an assertion?
Is proving that $M_{23}$ is also Galois over $\mathbb{Q}$ within reach? I assume that the same techniques do not apply, for it is a much more manageable group than the Monster.
-
Who has made this community wiki? I am not the one who posted this question! – Wanderer Feb 12 2010 at 10:56
1
A nice book is Völklein's "Groups as Galois Groups", where this is discussed in a fairly elementary manner. The relevant paper by Thompson: Some finite groups which appear as ${\rm Gal}\,L/K$, where $K\subseteq Q(µ_{n})$. J. Algebra 89, 437--499, dx.doi.org/10.1016/0021-8693(84)90228-X . – Steve D Feb 12 2010 at 16:46
1
Charles, I have to admit, I think many of us are curious about why you would make it community wiki now. – Ben Webster♦ Feb 12 2010 at 17:03
## 2 Answers
Hi Charles,
the monster is a nice example of how the so-called rigidity method for the inverse Galois problem works. There is a lot of beautiful mathematics behind this, I will sketch the different steps.
A general remark: it is known that every profinite group, i.e. every group which could be a Galois group of some field extension, is indeed the Galois group of some Galois extension. This is still quite elementary (a result of Leptin, also proved by Waterhouse). To make the inverse Galois question more interesting, we should therefore consider a fixed base field $K$. We can't hope that any profinite group will still be a Galois group over $K$ - every Galois group over $K$ is a quotient of the absolute Galois group of $K$, and therefore the cardinality of a Galois group over $K$ is bounded from above, whereas it is easy to see that there are profinite groups which are "strictly bigger" in cardinality. So a very reasonable question is indeed to ask whether every finite group is a Galois group over some fixed base field $K$. The most natural case is to ask the question for $K = \mathbb{Q}$, but also other base fields can be considered - for example, for $K = \mathbb{C}(t)$ the inverse Galois conjecture is true. In fact, the inverse Galois problems for different base fields $K$ are sometimes closely linked; the method which I will sketch below is a perfect illustration for this, since we will have the consider four different base fields: $\mathbb{C}(t)$, $\overline{\mathbb{Q}}(t)$, $\mathbb{Q}(t)$, and of course $\mathbb{Q}$.
(1) Start with the fact that each finite group $G$ can be realized as a Galois group over $\mathbb{C}(t)$. This follows from the theory of coverings of Riemann-surfaces; if $G$ can be generated by $n - 1$ elements, then we can realize $G$ as a quotient of the fundamental group of the punctured Riemann sphere $\pi_1^{\text{top}}(\mathbb{P}^1(\mathbb{C}) \setminus \{P_1,P_2,\,\cdots,P_n\})$, where we choose the points $P_1,P_2,\,\cdots,P_n$ to be rational.
(2) We use the theory of the étale fundamental group to get an isomorphism $\pi_1(\mathbb{P}^1(\mathbb{C}) \setminus \{P_1,P_2,\,\cdots,P_n\}) = \pi_1(\mathbb{P}^1(\overline{\mathbb{Q}}) \setminus \{P_1,P_2,\,\cdots,P_n\})$, which allows us to realize $G$ as a Galois group over $\overline{\mathbb{Q}}(t)$. [$\pi_1$ is the étale fundamental group - here the profinite completion of the topological version.]
[Of course, this is already advanced material; see the Wikipedia article for background. For a proper introduction to the theory, there is SGA 1 by Grothendieck; and the recent book "Galois groups and fundamental groups" by Tamas Szamuely is a very gentle introduction (and does all this in detail).]
(3) There exists an exact sequence
$1 \to \pi_1(\mathbb{P}^1(\overline{\mathbb{Q}}) \setminus \{P_1,P_2,\,\cdots,P_n\}) \to \pi_1(\mathbb{P}^1(\mathbb{Q}) \setminus \{P_1,P_2,\,\cdots,P_n\}) \to \text{Gal}(\overline{\mathbb{Q}}|\mathbb{Q}) \to 1$
(this is a very fundamental result; see again the books I mentioned) and basically we now want to extend a surjective homomorphism from $\pi_1(\mathbb{P}^1(\overline{\mathbb{Q}}) \setminus \{P_1,P_2,\,\cdots,P_n\})$ to $G$ to a surjective homomorphism from $\pi_1(\mathbb{P}^1(\mathbb{Q}) \setminus \{P_1,P_2,\,\cdots,P_n\})$ to $G$. Of course this depends heavily on the structure of the group $G$. This works for many finite simple groups; the construction is quite general, but for particular groups there is always some technical work to do to show that the method applies. In particular it works for finite groups with a trivial centre, and a rigid system of rational conjugacy classes - these are quite technical conditions, of course, and I will just state the definitions. An $n$-tuple of conjugacy classes $C_1,C_2,\,\cdots,C_n$ of $G$ is rigid if there exists $(g_1,g_2,\,\cdots,g_n) \in G^n$ such that the $g_i$ generate $G$, $g_1g_2\cdots g_n = 1$ and $g_i \in C_i$, and if moreover $G$ acts transitively on the set of all such $n$-tuples $(g_1,g_2,\,\cdots,g_n)$. A conjugacy class $C$ of $G$ is rational if $g \in C$ implies $g^m \in C$ for all $m$ coprime to the order of $G$. I won't explain why precisely these conditions give you what you want, since it is really technical. Szamuely explains this very clearly. The conditions can be generalized, but that doesn't make it more readable...
[References: section 4.8 in Szamuely's book I mentioned above, and also Serre's wonderful book "Topics in Galois theory", which should maybe be called "Topics in inverse Galois theory" :)]
(4) The previous step allows us to descend from $\overline{\mathbb{Q}}(t)$ to $\mathbb{Q}(t)$, i.e. to realize $G$ as a Galois group of a regular extension - another technical notion which I won't explain, but it is not unimportant - of $\mathbb{Q}(t)$. To descend from $\mathbb{Q}(t)$ to $\mathbb{Q}$, there is Hilbert's irreducibility theorem, or some slight generalization (I don't remember exactly).
According to Thompson, the Monster has a rigid system of three rational conjugacy classes of orders 2, 3 and 29. So the method will apply; of course, I guess that it will be very hard to construct these conjugacy classes, and it is clear that the classification of finite simple groups has played a very big role in these developments. (But I am not a group theorist, so anyone who knows how this works is welcome to give additional information about this construction :))
So I hope this gives you an idea; I wrote this up in a hurry, so suggestions to make this clearer or more coherent (or of course corrections of details which I got wrong) are always welcome.
-
Thanks for all of this information. Can you go into a little more detail as to the precise condition (rigid system of rational conjugacy classes) that is being used? I know a bit about the Monster and a lot about $M_{23}$, and I'd like to see how the actual conditions come into play. – aorq Feb 2 2010 at 20:32
I added the definitions; it wouldn't be wise to add more details - that wouldn't improve the readibility, it is technical (and I am not a group theorist), and it is explained clearly in the references I mentioned. – Wanderer Feb 2 2010 at 20:55
this is great introduction!! just one side question, what does "rigid" mean? I mean, this word is used in a lot of places of mathematics. – natura Feb 2 2010 at 21:15
An unrelated question: how do I get my brackets { } right in LaTeX? The commands which I usually use, \{ and \}, don't seem to work here (at least not on my computer screen). – Wanderer Feb 2 2010 at 21:16
2
Two slashes works, I think. – Qiaochu Yuan Feb 2 2010 at 22:17
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I'll add a brief commment to Arne Semeets's thorough and useful answer. If I fix three rational conjugacy classes c_0, c_1, c_infty in a finite group G, then there are finitely many isomorphism classes of unramified G-coverings
X -> P^1 - 0,1,infty /Qbar
with the property that the image of tame inertia at 0 (resp 1,infty) lies in the class c_0 (resp c_1,c_infty) of G.
Call the set of such covers H. What is |H|? One can check (by comparison with the complex case) that the number of such covers is the number of conjugacy classes of triples (g_0,g_1,g_infty) with g_i in c_i and g_0 g_1 g_infty = 1. To say that (c_0,c_1,c_infty) is rigid is just to say that there is precisely ONE such triple.
In that case, H consists of just one cover. But H is evidently preserved by Galois conjugacy. So this unique cover is defined over Q. (One has to be slightly more careful when G has nontrivial center, in which case what I've really proved is something more like "there's a cover whose isomorphism class is defined over Q," not quite the same in general as "there's a cover defined over Q."
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http://physics.stackexchange.com/questions/5470/what-does-the-fine-structure-constant-describe/5629 | # What does the fine structure constant describe?
Feynman says in his book "QED" that the square root of the fine structure constant is the probability for a charged particle to emit a photon. But for which wavelength? Or is it an average over all wavelengths?
Note: I meant virtual photon, and I meant a stable charged particle, like the electron. One way to rephrase it would be: how many virtual photons (per unit volume) are there in the Coulomb field around an electron?
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You'll have to provide more context. The full quote from the book would be helpful. – user346 Feb 19 '11 at 8:13
"how many virtual photons are there in the Coulomb field around an electron?" You know that the coulomb field extends to infinity? – Georg Feb 19 '11 at 13:30
## 3 Answers
It's neither - or everything. Feynman means that the "vertex in the Feynman diagram" which has 2 external lines corresponding to a charged particle and 1 external line corresponding to a photon (all three particles have any energy/frequency and momentum/inverse_wavelength you want) is proportional to $e\approx \sqrt{\alpha}$. So he really means the probability amplitude.
One needs lots of (or at least several) other calculations to calculate the probability. In particular, a stable charged particle can never emit a photon because it would violate the energy or momentum conservation: in its initial rest frame, the energy is minimized, so one can't afford to increase the energy by changing the state of the motion (plus emitting a photon, which would make the final energy even higher). An unstable particle can decay into another particle and a photon. The decay rate will be proportional to the fine-structure constant (without the square root) but the detailed decay rate depends on the mass of the decaying particle as well as the other final decay product.
Also, $\alpha$ slowly depends on a scale - logarithmically. It's the value at $E=0$, or - approximately - anywhere at masses $m<m_e$ where $m_e$ is the electron mass where $\alpha$ has the familiar value $1/137.03604$. Even though I didn't tell you a particular process that is linked to $\alpha$, it's actually a good idea in the process of "renormalization" to define $\alpha$ operationally exactly as a particular probability or amplitude at particular energies, just like you suggested. However, there are many choices how to do that.
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Ok, so the root of alpha is the probability amplitude. How does this allow to answer the note I added to the question? What is the virtual photon density around an electron? – user2107 Feb 19 '11 at 17:08
The fine structure constant $\alpha$ describes the fine structure of the Hydrogen levels of energy.
"I meant a stable charged particle, like the electron."
Is the electron really "stable" if any push makes radiation (inelastic "deformation")? I am afraid it is not. Such radiation occurs always ( probability = 1). The "Coulomb center" picture is obtained by summation of all inelastic pictures. It is called an "inclusive" picture. It is not an "elastic" (non-destructive) one.
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by Hans de Vries: a numerical elegant approach can be found here:
The fine structure constant: A radiative series leading to it’s exact value
Vries use both 'pi' and 'e' in the formula and reminds me Feynman (wikipedia - Fine-structure_constant ) '...is it related to pi or perhaps to the base of natural logarithms? ...' .
I do not know the meaning of this serie.
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http://crypto.stackexchange.com/questions/6313/is-aes-reducible-to-an-np-complete-problem/6332 | # Is AES reducible to an NP-complete problem?
Is breaking AES NP-hard? Can the security of AES be reduced to a NP-complete problem?
If it is reducible, what does the reduction look like? If it is not reducible, why do we assume it is secure?
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5
We know no such reductions for any cryptosystem we use in practice. Not AES, nor RSA. Such reduction isn't too useful in the first place, since complexity classes talk about asymptotic cost, whereas we want to know if a certain concrete key-size is secure. – CodesInChaos Feb 10 at 19:34
How does it "seems clear"? It is clear that some problems relating to some attacks lie in NP (assuming some kind of generalization which allows growing key sizes, simply guessing the right key and then verifying them solves it – the same goes for secret-key crypto), I don't see how you get that they are NP-hard. – Paŭlo Ebermann♦ Feb 10 at 20:20
2
"Being broken when $P = NP$" doesn't mean "being unbroken when $P \neq NP$". RSA is broken when factoring is easy, but neither is factoring known to be NP-complete nor is it known that RSA can only be broken by factoring the modulus (this is just the best known way nowadays). It is similar for most other public-key algorithms. (And even less for symmetric "bit-shuffling" crypto.) – Paŭlo Ebermann♦ Feb 10 at 22:18
3
Actually, it's not at all true that someone showing $P=NP$ implies that all cryptography is broken. For one, an existence proof (that is, a proof that $P=NP$ that does not give an explicit reduction) doesn't affect the practical security at all. In addition, an explicit reduction of an NP-hard problem of size $n$ that takes either $2^{1000} n$ or $n^{1000}$ operations would suffice to show $P=NP$, but would be of no practical concern. To be of concern, we would have to show that large NP-hard problems can be practically solved. – poncho Feb 11 at 18:06
1
@CodesInChaos why not make that first remark an answer? – owlstead Feb 12 at 0:44
show 4 more comments
## 1 Answer
There are some complexity-theoretic reasons to believe that cryptography can't be based on NP-completeness. For one example, see this paper by Akavia et al. Basically it all boils down to the mismatch between average-case hardness required for cryptography, and worst-case hardness required for NP-hardness. Moreover, many (most? all?) hard algebraic problems which serve as the basis for cryptography are in NP $\cap$ co-NP (for one such example, consider factoring). Such problems cannot be NP-complete unless NP = co-NP. You can see more discussion in this stackoverflow thread.
Finally, AES is a finite-domain function. It is invertible / distinguishable from a random permutation in (large but) constant time. The definitions of P, NP, etc., refer to asymptotic behavior -- that is, as the input size grows to infinity. Because of the algebraic structure of AES, it is probably possible to define a "generalized AES" for infinitely many key lengths (though there are many semi-arbitrary choices to be made). We could then ask about the asymptotic behavior of such a generalized AES. I'm not sure if anyone has done so.
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http://nrich.maths.org/2859/note?nomenu=1 | ## Odds or Sixes?
You can use the computer to see what happens when Tania and Derek are playing a game with a dice. They roll the dice. If the number is odd, Tania wins that round.
If the number is a six, Derek wins.
(It doesn't matter who throws the die.)
Who is more likely to win the game? Why? How could you make the game fair?
You might like to use the interactivity below which will roll the dice many times.
Full Screen Version
This text is usually replaced by the Flash movie.
### Why do this problem?
This problem gives learners the opportunity to describe and predict outcomes, and consider the meaning of 'fair'. The interactivity simulates the die-throwing which means that data can be collected quickly and easily.
### Possible approach
You could introduce this problem either by using the interactivity or by having two children come to the front to play it. Whichever way you choose, play the game a few times and if not using the interactivity, record the outcomes on the board. Ask the class to predict what would happen if the game was played many times, for example $100$ times. Take suggestions from the children, looking out for those who justify their answer based on the few games which have already been played.
Suggest that the group tests out their theories. Again, this could be done using the 'Run x100' button on the interactivity or by pairs throwing dice and then collating class results. Bring pupils together to talk about their findings and ask them whether the game is fair or not and why. Listen out for explanations which compare the number of possible winning throws using appropriate vocabulary. Some children might quantify the probability of throwing a six, for example, as $1$ out of $6$ or $\frac{1}{6}$ whereas throwing an odd number is $3$ out of $6$, or $\frac{1}{2}$.
It would be useful to encourage children to talk in pairs about what they understand as 'fair' - there will be different, but equally as valid, ideas about how to change the game.
### Key questions
What numbers are possible to throw on the dice?
Who would win with each number?
Can you use this to decide how to make the game fair?
### Possible extension
Learners could try Odds and Evens which extends the ideas in this problem.
### Possible support
Having dice available will help those children who are not familiar with them and playing the game for themselves would also be of benefit. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9612504839897156, "perplexity_flag": "middle"} |
http://mathoverflow.net/questions/13581?sort=votes | ## Quantum channels as categories: question 1.
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
A quantum channel is a mapping between Hilbert spaces, `$\Phi : L(\mathcal{H}_{A}) \to L(\mathcal{H}_{B})$`, where `$L(\mathcal{H}_{i})$` is the family of operators on `$\mathcal{H}_{i}$`. In general, we are interested in CPTP maps. The operator spaces can be interpreted as `$C^{*}$`-algebras and thus we can also view the channel as a mapping between `$C^{*}$`-algebras, `$\Phi : \mathcal{A} \to \mathcal{B}$`. Since quantum channels can carry classical information as well, we could write such a combination as `$\Phi : L(\mathcal{H}_{A}) \otimes C(X) \to L(\mathcal{H}_{B})$` where `$C(X)$` is the space of continuous functions on some set $X$ and is also a `$C^{*}$`-algebra. In other words, whether or not classical information is processed by the channel, it (the channel) is a mapping between `$C^{*}$`-algebras. Note, however, that these are not necessarily the same `$C^{*}$`-algebras. Since the channels are represented by square matrices, the input and output `$C^{*}$`-algebras must have the same dimension, $d$. Thus we can consider them both subsets of some $d$-dimensional `$C^{*}$`-algebra, `$\mathcal{C}$`, i.e. `$\mathcal{A} \subset \mathcal{C}$` and `$\mathcal{B} \subset \mathcal{C}$`. Thus a quantum channel is a mapping from $\mathcal{C}$ to itself.
Proposition A quantum channel given by `$t: L(\mathcal{H}) \to L(\mathcal{H})$`, together with the $d$-dimensional `$C^{*}$`-algebra, $\mathcal{C}$, on which it acts, forms a category we call $\mathrm{\mathbf{Chan}}(d)$ where $\mathcal{C}$ is the sole object and $t$ is the sole arrow.
Proof: Consider the quantum channels
```$\begin{eqnarray*}
r: L(\mathcal{H}_{\rho}) \to L(\mathcal{H}_{\sigma}) &
\qquad \textrm{where} \qquad &
\sigma=\sum_{i}A_{i}\rho A_{i}^{\dagger} \\
t: L(\mathcal{H}_{\sigma}) \to L(\mathcal{H}_{\tau}) &
\qquad \textrm{where} \qquad &
\tau=\sum_{j}B_{j}\sigma B_{j}^{\dagger}
\end{eqnarray*}$```
where the usual properties of such channels are assumed (e.g. trace preserving, etc.). We form the composite `$t \circ r: L(\mathcal{H}_{\rho}) \to L(\mathcal{H}_{\tau})$` where
```$\begin{align}
\tau & = \sum_{j}B_{j}\left(\sum_{i}A_{i}\rho A_{i}^{\dagger}\right)B_{j}^{\dagger} \notag \\
& = \sum_{i,j}B_{j}A_{i}\rho A_{i}^{\dagger}B_{j}^{\dagger} \\
& = \sum_{k}C_{k}\rho C_{k}^{\dagger} \notag
\end{align}$```
and the `$A_{i}$`, `$B_{i}$`, and `$C_{i}$` are Kraus operators.
Since $A$ and $B$ are summed over separate indices the trace-preserving property is maintained, i.e. `$$\sum_{k} C_{k}^{\dagger}C_{k}=\mathbf{1}.$$` For a similar methodology see Nayak and Sen (http://arxiv.org/abs/0605041).
We take the identity arrow, `$1_{\rho}: L(\mathcal{H}_{\rho}) \to L(\mathcal{H}_{\rho})$`, to be the time evolution of the state $\rho$ in the absence of any channel. Since this definition is suitably general we have that `$t \circ 1_{A}=t=1_{B} \circ t \quad \forall \,\, t: A \to B$`.
Consider the three unital quantum channels `$r: L(\mathcal{H}_{\rho}) \to L(\mathcal{H}_{\sigma})$`, `$t: L(\mathcal{H}_{\sigma}) \to L(\mathcal{H}_{\tau})$`, and `$v: L(\mathcal{H}_{\tau}) \to L(\mathcal{H}_{\upsilon})$` where `$\sigma=\sum_{i}A_{i}\rho A_{i}^{\dagger}$`, `$\tau=\sum_{j}B_{j}\sigma B_{j}^{\dagger}$`, and `$\eta=\sum_{k}C_{k}\tau C_{k}^{\dagger}$`. We have
```$\begin{align}
v \circ (t \circ r) & = v \circ \left(\sum_{i,j}B_{j}A_{i}\rho A_{i}^{\dagger}B_{j}^{\dagger}\right) = \sum_{k}C_{k} \left(\sum_{i,j}B_{j}A_{i}\rho A_{i}^{\dagger}B_{j}^{\dagger}\right) C_{k}^{\dagger} \notag \\
& = \sum_{i,j,k}C_{k}B_{j}A_{i}\rho A_{i}^{\dagger}B_{j}^{\dagger}C_{k}^{\dagger} = \sum_{i,j,k}C_{k}B_{j}\left(A_{i}\rho A_{i}^{\dagger}\right)B_{j}^{\dagger}C_{k}^{\dagger} \notag \\
& = \left(\sum_{i,j,k}C_{k}B_{j}\tau B_{j}^{\dagger}C_{k}^{\dagger}\right) \circ r = (v \circ t) \circ r \notag
\end{align}$```
and thus we have associativity. Note that similar arguments may be made for the inverse process of the channel if it exists (it is not necessary for the channel here to be reversible). $\square$
Question 1: Am I doing the last line in the associativity argument correct and/or are there any other problems here? Is there a clearer or more concise proof? I have another question I am going to ask as a separate post about a construction I did with categories and groups that assumes the above is correct but I didn't want to post it until I made sure this is correct.
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I might be misunderstanding your terminology: according to the Wikipedia article, a quantum channel is a completely-positive trace-preserving map on an ambient finite-dimensional C*-algebra ${\mathcal C}$. (I don't think calling $d$ the dimension of the $C^*$-algebra is standard, but could be wrong here.) Now it seems that what you are asking is whether the set of all such channels forms the set of arrows for a 1-object category whose object is ${\mathcal C}$. Is that right? – Yemon Choi Jan 31 2010 at 21:21
Yes. I reread your question a couple of times and I'm pretty sure that the answer is yes. Just to clarify, $\mathcal{C}$ has a specific "dimension" (I use this since it's a relic of the idea of talking about a $d$-dimensional Hilbert space - originally I was using Hilbert spaces as the objects, but then generalized to C$^{*}$-algebra. I felt it better covered any possible classical inputs if that makes sense. – Ian Durham Jan 31 2010 at 21:45
## 4 Answers
Phrasing this in terms of categories is kind of misleading: A category with a single object is just a monoid (associative binary operation with identity). So, per Yemon Choi's correction, you are just trying to demonstrate that the set of quantum channels $L(\mathcal{H}) \to L(\mathcal{H})$ forms a monoid. [Here I'm assuming that "channel" implies CPTP.]
This requires three things:
1. Closure: The product of two channels is a channel.
2. Identity: The identity operator is a channel.
3. Associativity: Multiplication of channels is associative.
1:
There are two different ways to prove closure. You used the characterization of channels as maps given by that Kraus operator form ($\sum_{i} A_{i}\rho A_{i}^{\dagger}$). This works, although I don't feel you made the construction of the $C_k$ from the $A_i$ and $B_j$ clear enough. It also requires that you've already proven this characterization (CPTP linear map $\Leftrightarrow$ Kraus operator form).
You could instead directly use the characterization of a channel as a CPTP linear map. This way is probably easier: It is immediately clear that if r is (a linear map which takes positive matrices to positive matrices and preserves trace) and t is (a linear map which takes positive matrices to positive matrices and preserves trace) then $t\circ r$ will be (a linear map which takes positive matrices to positive matrices and preserves trace). That does it.
2:
I really wish you hadn't said "time evolution". :-) But you basically have the right idea: the identity in this situation is, well, the identity map, which is obviously linear and CPTP.
3:
Practically speaking, you almost never have to prove associativity. This is because as long as your maps are functions deep down on the inside, associativity is an immediate consequence of associativity of function composition. This is one of those cases.
So in response to your question "Is there a clearer or more concise proof?" I would say "Absolutely."
But again, I think it's unnecessary to put this in the context of categories. UNLESS you plan on generalizing to channels between distinct spaces. Then the concept of a category gains its power.
Good luck!
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The reason I'm using categories has to do with what I do with these things (which will be the subject of another question here as soon as I'm satisfied I've done this part right). Regarding "time evolution," sorry. I'm a quantum physicist. :-) I'm trying to use category theory to solve a particularly intractable problem in quantum information theory while simultaneously keeping the "physical" nature of the problem in mind. Your comment on the Kraus operators was particularly helpful and is much appreciated. – Ian Durham Feb 1 2010 at 1:45
Ian's follow-up question appeared at mathoverflow.net/questions/16077/… – Scott Morrison♦ Feb 24 2010 at 6:48
2
Dear Joshua, if you have five minutes for me, I would like to ask you for a favor: Ian has tried to start an nLab entry on quantum channels here: ncatlab.org/nlab/show/quantum+channel , but there is a general feeling that this is quite in need of some improvement. Myself, I don't know the literature on "quantum channels". Just judging from your reply above, I am getting the impression that you know what you are talking. So: could you maybe do us and the world a favor, hit the "edit" button on that nLab entry, move all the discussion you find the below some horizontal line and then... – Urs Schreiber Feb 25 2010 at 15:13
2
... give a clean definition of quantum channels? I gather there is a general abstract definition and then various concrete realizations. I also gather all in all it is not a terribly mysterious idea. If you could just write out a nice clean definition and maybe give a pointer to some good authoragtive reference. That would be much appreciated. – Urs Schreiber Feb 25 2010 at 15:14
### You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.
I'm not sure if it answers your question, but if ${\mathcal C}$ is a finite-dimensional $C^*$-algebra, and if $f$ and $g$ are completely positive, trace-preserving, linear maps from ${\mathcal C}\to {\mathcal C}$, then the composite map $g\circ f$ is going to be completely positive, trace-preserving, and linear. So one can indeed define a certain category to have ${\mathcal C}$ as its sole object, and have as its set of morphisms the collection of all CP-TP linear maps from ${\mathcal C}$ to itself. The associativity rule comes for free just because composition of functions is an associative operation.
If, on the other hand, you define quantum channels to be maps of a certain concrete form (rather than as being maps which preserve certain structure) then probably one needs to do a direct calculation similar to yours.
This might not be quite what you were asking, but I hope it helps.
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Well, that's actually an interesting question. In one sense, I am interested in having them preserve some kind of structure. But I'm also interested in having their categorical form preserve (show?) any extremal characteristics they might possess (maybe I should post this as a separate question). – Ian Durham Feb 1 2010 at 0:04
1
Since the concrete form of quantum channels used above is equivalent to the definition of quantum channels as completely positive, trace-preserving, linear maps, then all you have to do is prove the equivalence and use Yemon's argument rather than doing the direct calculation. – Peter Shor Jun 5 2010 at 20:19
As I see it, this posted question and some aspects of the answers turn an important but straightforward fact into something needlessly complicated and less general.
Let $\mathcal{A}$ (Alice) and $\mathcal{B}$ (Bob) be $C^*$-algebras of observables, or better yet, von Neumann algebras of observables. Let `$\mathcal{A}^\#$` denote the dual space of (finite but not necessarily positive) states on $\mathcal{A}$, and in the von Neumann algebra case let `${}^\#\mathcal{A}$` denote the predual space of normal states. Then a quantum channel, to model a message from Alice to Bob, is a completely positive, unital map $$E:\mathcal{B} \longrightarrow \mathcal{A}.$$ The corresponding CPTP map on states is the transpose: `$$E^\#:\mathcal{A}^\# \longrightarrow \mathcal{B}^\#$$` in the von Neumann algebra case, $E$ should be normal and have a pre-transpose: `$${}^\#E:{}^\#\mathcal{A} \longrightarrow {}^\#\mathcal{B}$$` Yes, quantum channels should form a category, and yes they do. Yes, you can restrict to the one-object subcategory where the object is $B(\mathcal{H})$. You need to check that quantum channels include the identity (they do) and you need to check that they are closed under composition. It is immediate that preserving 1 (the unital condition) is closed under composition. As for complete positivity, the condition is that $$E \otimes I_\mathcal{C}:\mathcal{B} \otimes \mathcal{C} \longrightarrow \mathcal{A} \otimes \mathcal{C}$$ preserves positive states for all $\mathcal{C}$. Closure of this condition under composition isn't quite immediate, but it's still very easy.
Associativity is immediate because quantum channels are functions.
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Yeah, the original question was watered down because I was trying to figure out how to ask an appropriate question here. That failed miserably. My original point was I was using category theory to try to solve the quantum Birkhoff theorem problem. I've got alternate approaches now. Much of the discussion of this stuff has been transported to the nLab. – Ian Durham Jun 15 2010 at 12:23
Plenty of recent work addresses encoding quantum protocols (and hence quantum channels, I guess) using category theory. Check out the work of Bob Coecke and any one related to him. A good starting point is A categorical semantics of quantum protocols by Samson Abramsky and Bob Coecke. It has many of the ingredients you are probably looking for. Or perhaps Kindergarten Quantum Mechanics also by Bob Coecke. There's also Dagger Categories by Peter Selinger, capturing the underlying categorical structures: Dagger compact closed categories and completely positive maps.
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Yep. Bob is essentially the person who first introduced me to category theory so I'm aware of all that stuff. Actually, we've basically taken care of much of this over at the nLab. – Ian Durham Jun 15 2010 at 12:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 42, "mathjax_display_tex": 2, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9496833682060242, "perplexity_flag": "head"} |
http://mathhelpforum.com/calculus/155506-squeeze-theorem.html | # Thread:
1. ## Squeeze Theorem
During class today we started covering some stuff that had me totally lost. We were covering The squeeze theorem and I was doing really good up until
lim xsin 1/x
I haven't the faintest idea what he did or how to repeat it. Can anyone give me a thorough explanation of this? The guy speaks terrible english and after his very brief run through of it simply told us to do
lim x^2 sin 1/x
lim x^2 cos 1/x
at home and know how to do them for tomorrows quiz.
So any advise on where to begin with those would be awesome. I think i'm mostly lost bc I don't remeber the trig functions i need. (like tanx=sinx/cosx). But I may be way off. Help
2. note from the graph that $-|x| \le x \sin\left(\frac{1}{x}\right) \le |x|$
so ... what can you say about the limit of the function in question as $x \to 0$ ?
3. I don't understand where you come up with the x and -x at all. I got to the graph part and don't know what i'm looking for from there. And that doesn't help me understand what to do with the other 2 problems.
x^2sin1/x
x^2cos1/x
which are my main concern.
I also understand what the answer of xsin1/x is. i just need to know how to get there.
4. Are you taking the limit as x approaches 0?
First, we know that $|\sin \frac{1}{x}| \le 1$ thus it should be clear that $0 \le |x \sin \frac{1}{x}| \le |x| \Leftrightarrow -|x| \le x \sin \frac{1}{x} \le |x|$
Now, $\lim_{x\to 0} |x| = 0$. From here apply the squeeze theorem. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9755394458770752, "perplexity_flag": "middle"} |
http://mathoverflow.net/questions/27344/what-are-your-experiences-of-handouts-in-mathematics-lectures/27439 | ## What are your experiences of handouts in mathematics lectures?
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
There are many different styles of lecturing, and many different aspects that are blended together to give a whole "lecturing style". That said, I'm particularly interested in hearing people's experiences with so-called "handouts". At one extreme lie the lecturers who "dictate" a set of notes (usually not actual dictation, but by writing on a board) whilst at the other are lecturers who distribute complete lecture notes in advance.
As this is math overflow, I realise that it is extremely unlikely that it will be possible to answer the question "which is better", and I realise that this probably depends much more on other factors than just whether or not notes where given out or not, but to help me decide what to do then I'd like to hear people's experiences - both as lecturers and students. If anyone can point me to actual research on this in the (mathematics education) literature then that would be an unlooked-for bonus.
(Minor edit: in light of the way that comments get displayed in "short form", I'd like to make it clear that the "Andrew" referred to in many of the comments is not the same "Andrew" who edited this question! Unfortunately, if I put this remark in a comment - which is where it belongs - then it wouldn't be seen by those casually stumbling across this question and so those most prone to making that assumption!)
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13
This is drifting away from mathematical research. – Robin Chapman Jun 7 2010 at 12:54
32
@Robin: Is this board becoming obsessed with research ONLY and becoming infected with the worst elitism of the top mathematics programs? I wonder because the hammer is being put to a lot of otherwise interesting educational questions there really is no other forum for at this level. Unless,of course,the people policing this board think educational issues have no place on a serious academic mathematical forum....... – Andrew L Jun 7 2010 at 17:27
20
Andrew, although I agree with your point, is there any chance you'd be willing to express yourself a bit more, um, gently? Your response seems a bit over the top. – Deane Yang Jun 7 2010 at 17:32
15
On the other hand, I certainly think that the subject matter is "on-topic" for mathoverflow. – Scott Morrison♦ Jun 7 2010 at 18:07
17
Andrew, there's nothing wrong with having and expressing strong opinions, but it seems to me that you weaken your own arguments when you overstate your perceptions or speculations about other people's motives and then attack them in an overly harsh manner. – Deane Yang Jun 7 2010 at 22:35
show 17 more comments
## 16 Answers
Without pre-typed lecture notes (or a textbook that is being followed reasonably closely), many students often feel pressured to copy down every scrap that the lecturer writes down, in case they are missing out on something that will be vitally important later. This often comes at the cost of the student being able to comprehend what is going on in real-time. A related problem is that without the backup of official notes or textbook, a single typo in lecture can lead to hours of confusion on the student's part when reviewing his or her transcribed notes afterwards. (The problem is mitigated somewhat nowadays by the plethora of online mathematics resources, combined with modern search engines, but the situation is still less than ideal.)
Note also that while the lecturer may know in advance which portions of the lecture are important enough to remember, and which ones are more trifling, many students will not be able to make the distinction in real time, and will thus have to record everything, leading to a sub-optimal allocation of the student's mental resources.
To me, the above dangers are worse than the opposite danger that the students are lulled into complacency by the existence of official lecture notes, and thus cease to pay attention to the class. The latter problem can be fixed by a variety of means (e.g. making the classes more interactive or entertaining, or making the homework challenge the student beyond what is presented in the notes), and in any case is more a matter of the responsibilities of the student than of the lecturer. The former problem is however difficult for the student to address by himself or herself (using third-party lecture notes, for instance, is usually a terrible solution).
Ideally, the existence of lecture notes should free up lecture time to focus on other aspects of the course (e.g. one could do a simple example in class, and refer to the notes for a more detailed example; or a heuristic proof with some details partially filled in, with the more technical details left to the notes; one can also present the more improvisational and free-form side of mathematics effectively in lecture, whereas the text medium is far superior for presenting the polished and structured side). Using class time to mechanically repeat what is written in the notes or textbook is a waste, and reduces the lecturer to essentially being a fancy text-to-speech synthesiser (this is the dual problem to that of the student being reduced to essentially a fancy speech-to-text synthesiser); instead, lectures should complement and support, rather than replicate, text, and vice versa.
I discuss these issues more in my teaching statement,
http://www.math.ucla.edu/~tao/teaching.dvi
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All very good points and I agree with most of them,Terry. What I think lectures should be for is what's NOT in the written material,what textbook authors usually call motivation and context. Personal commentary,historical digressions,anecdotal tales of the lecturer's own experiences,etc. These are the aspects of a rich educational experience that are best delivered with a personal touch-definitions and theorums are best left to prefabricated notes. – Andrew L Jun 7 2010 at 20:02
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I completely agree with these sentiments, Terry. Such a system is extremely rewarding to students who work hard. Students all have varying levels of raw ability; the only thing you can do as an instructor is make hard work as valuable to them as possible. – Peter Luthy Jun 7 2010 at 20:35
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I see one danger with the assertion that lectures should complement the lecture notes and vice versa which is that if the instructor assumes that the students read and understood the lecture notes and makes this the starting point for his lecture this can make the lecture completely hopeless. So I would support an approach that the lectures should be self contained and stand-alone, but the lecture notes reduce the need for the students to write every word and may contain some (but not too much) additional material. – Gil Kalai Jun 8 2010 at 13:29
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(Similarly I dont like when lecturers regard what they say in a lecture as an addition to what is written in the transperancies that the audience should read and understand on their own, gilkalai.wordpress.com/2008/11/23/… ) – Gil Kalai Jun 8 2010 at 13:31
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But of course, even with this little concern, following a text or lecture notes is much better for the students than not having a text. The only problem is that writing lecture notes is a lot of work for the teacher. With or without lecture notes, going too quickly and trying to squeeze too much into a lecture is probably the most common pedagogical mistake in math lectures on all levels. – Gil Kalai Jun 8 2010 at 13:48
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The idea of handing out lecture notes in advance, thus allowing class time to be used more creatively, is indeed appealing. However, I clearly remember the classes where I learned the most in college. They were not like that at all. Rather, the professor wrote almost an entire textbook on the blackboard, which I dutifully transcribed. The simultaneous engagement of eye, hand, and brain was somehow engrossing. I cherished these notes and inscribed them on my memory. Now, when I teach, I meticulously prepare notes for myself but don't hand out copies. It is up to students to write their own editions, so to speak. This seems old-school, but I've found that it works. To be sure, the pace mustn't be so fast that students become mere stenographers. They need time to think things over. And care is needed to distinguish the essential from the more trifling (often simply by repeating the former a few times). While it isn't and shouldn't be the only way to teach, I feel there is a place for a more formal lecturing style alongside the exciting, freewheeling approach advocated by Prof. Tao.
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I'm very sympathetic to the old school approach because the benefit is that students actively formulate and absorb the notes-and this is something that gets lost when discusssing whether or not to prefabricate notes.If your students are brain-dead copying the notes and not learning anything from the act of trying to keep up with you,chances are your notes just aren't very informative.But for the reasons I've discussed at my responses to other posters and my own posting,I think writing them up beforehand & using the lectures for critical clarification and supplementation is the best way to go. – Andrew L Jun 7 2010 at 21:55
My impression is that the well-prepared lecture in which students are engaged tends to be the most stable; i.e., it is not very sensitive to minor differences in teaching ability/experience by the lecturer, and not very sensitive to small differences in classes (better or worse students, time of day, etc). Other methods seem to me, based on very limited experience, to be more sensitive to the professor's ability or to the class full of students. Moore's method seemed to work fairly well for Moore and some students, not so much for a random professor and random student. – Arturo Magidin Jun 8 2010 at 2:43
My short answer is to give students the notes in advance, and if that discourages them from coming to class and making the most of it, then I am a poor lecturer.
I used to think that I was giving a good lecture if I was more useful to the students than an hour reading the book. Now I think that I am giving a good lecture if I am more useful to the students than an hour reading my own set of notes. This is to say, if I can be entirely replaced by my notes, and if giving students my notes in advance means that they will not come to class, then I am not doing things right.
Yes, I understand that I am setting a very high standard, one by which most lectures by most professors (and me included) are not that good, but I think that this is the standard to aim for nevertheless. I like many of the things that Tao says in his teaching statement, in particular about how lectures are to complement, not reproduce or replace, the book or notes. And yes, I admit that this is easier said than done.
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In real life, there is are viable alternative explanations for students missing a class: some of them are slackers, or later risers, or learn best on their own. I also think that lecture/classwork on the one hand and individual reading/homework on the other hand should complement each other, so the dichotomy between which of them is more useful is a false one. For sure, a teacher should be inspiring, but optimizing the amount of time spent studying isn't a professor's job. – Victor Protsak Jun 8 2010 at 2:19
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Victor, I agree. What I actually mean is that I want my one hour lecture plus $n-1$ hours reading the book/notes to be more useful for the student than $n$ hours reading the book/notes. – Alfonso Gracia-Saz Jun 8 2010 at 2:27
The past few years I've prepared and handed out "notes" to my first year Calculus classes. Each set of notes I leave incomplete to various degrees.
For the first part of the notes for a particular section I'll type out ~90% of the information, I leave "blanks" for the students to fill in. I try to leave out words and symbols in a purposeful manner so when the students fills in that blank they have contributed a "key idea"; for instance when defining an increasing function I might let the students fill in the inequality between two expressions. In this way they aren't wasting time copying down everything I say and do, but they are still involved in the note taking, and they are paying attention to my lecture to try and fill in those missing pieces.
In the middle section of each of my notes (here I'm usually doing basic/classic example problems, and proofs of theorems) I'll usually type the problem we're working and add in some leading sub-questions or "hints" on how to do each step, and I'll leave space on the page for them to write in the "work" of the problem. Some students choose to fill in this part as I do the problem on the board, others just watch me and then try to re-create the work later on their own time.
The last part of my typical note sheet will have some additional problems and space to work them but often little to no prompting or hints. These are often the problems that I'll do "as time permits" and are often harder/longer/more involved. My students learn that I'd like them to be able to do these problems but realize that most of the time their tests and quizzes will have relatively few problems of this type.
Handing out notes like these I get to feel like I'm not "doing everything" for them, and yet at the end of the chapter/unit/class they have a nice concise "best of" version of their text to help them know what I think is important for them to study.
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Good strategy,Mark. Would love to see your notes sometime. – Andrew L Jun 8 2010 at 1:23
I know at least one person who actively prohibits students from taking notes during lectures: Ole Hald. See for example here. Note: though the link only talks about upper division and graduate courses, I'm fairly sure I remember that he did it in at least one lower division Calculus course later on.
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That's crazy,Arturo-and I don't care how good a teacher he is. I wonder how much they actually learn in the course-and more importantly,how much stays with them.I'm highly skeptical. – Andrew L Jun 7 2010 at 20:31
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It sounds crazy, and it makes a lot of students uncomfortable. <i>And</i> it takes a substantial amount of his time to get everything just so. But as I understand it, he is an <i>extremely</i> good and <i>effective</i> teacher, as measured by how the students do in later courses. I know I would be hesitant to try it on unsuspecting students, and it may be very sensitive to the qualities of the teacher, but there you are. (Hald is mentioned in Krantz's "The Survival Guide of a Mathematician" for other teaching techniques). – Arturo Magidin Jun 7 2010 at 20:41
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I was a TA for Hald last millenium. In that class (linear algebra for everyone) he used very simple examples and spent much energy and few words on his points. E.g. he asked someone in class for a number; when someone said 1 he wrote a row of 7 one's on the board. After more prompting, he followed it with a row of 4's and then one of 7's. He then used this toward his discussion of column rank = row rank. It was pedagogically perfect: he got his point across to an audience of over 200, without note-taking. Gerhard "Ask Me About System Design" Paseman, 2010.06.07 – Gerhard Paseman Jun 7 2010 at 21:22
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If this had been anybody but Ole Hald, I would be very skeptical. – Alfonso Gracia-Saz Jun 8 2010 at 1:10
From the point of view of (mostly) a student (full disclosure: a physics student!) I vastly prefer to have handed out notes.
I have three undergrad degrees: math, physics, and astronomy (each from different departments with distinct styles), and am now a physics grad student, so I've seen a number of styles coming from each of these areas. And I have to say, one of the absolute worst styles is math without any handouts (beaten only by the style where the professor exclusively tells stories or does not show up to classes!).
As Terry Tao says below (or, more likely, above, if you have this sorted by votes!) it can be very frustrating to spend hours being confused because of a simple mistake in the lecture, or a simple transcription mistake in my notes. But more than this, one of the (IMHO) major failings in most of my undergrad math classes was that they focused largely on discussing proofs and technical details, and not at all on why anything was like it was.
So, I could, e.g., come out of an analysis lecture being able to prove the uncountability of the reals with Cantor's diagonalization argument, but, since we went through it so fast and covering every detail, not having any understanding of why it worked, what the significant parts of the proof were, or why it is important. Even though in retrospect there was nothing tricky or difficult at all about this argument, I remember having a great deal of trouble understanding this at the time.
However, in the (few) math classes I had that handed out notes with technical details carefully spelled out in them, and in class focused on discussing why things were like they were, in developing some intuition, and trying to modify assumptions and see how things changes, I had a greatly improved understanding (and ability to correctly do the homework and exams!).
It has always seemed far easier for me to go through the technical details myself, taking my time, than to go through the qualitative arguments myself in a field with which I am not familiar.
In fact, I found that these kinds of more qualitative lectures actually significantly increased my ability to carefully prove things, since I was able to better understand how the steps in a proof were linked together. As opposed to understanding the technical details in each step, which are useful, but only useful to make a step careful after you are sure the step makes sense!
In the classes I've taught, and had enough control over that I have been able to do like I wanted (which, admittedly, is only one class, a baby-quantum field theory for undergrads class!) I typed up notes for my students, and found that the students who put extra time into understanding the class did substantially better than in classes where I have not been able to do this.
Students who did not put extra time outside of class into studying did not do any better, but claimed to like the notes better.
So I've felt that it's a good idea to hand out notes, since, at the very least, it will reward the students who are working hard and putting extra time into the class. Those students are the ones we want to continue in our fields, after all!
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The kind of lecture a students of math and a student of physics appreciate can be drastically different, though. For instance I find most of physics books unreadable, but I guess it is just a different habit. – Andrea Ferretti Jun 8 2010 at 13:31
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+1 Andrea! Personally, I think the diagonal argument is the perfect "lecture proof" : it's short, has no real technicalities, is instantly memorable, and has a shocking conclusion. Of course, you made the mistake of insulting my favorite proof in all of mathematics <grin>. One point I think we might agree on is that lectures are poor vehicles for going through the details of long or technical proofs. I often find myself substituting examples for such proofs in my own lectures. However, I have also found that undergraduates sometimes feel shortchanged if you don't "prove everything"... – Andy Putman Jun 8 2010 at 14:14
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+1 because, I assure you, most physicists find most physics books unreadable, too! Most of my carrier has involved secretly hoarding the few good texts I can find so I can actually learn things ;). But this has to do with most physics books leaving out the intuitive explanations, like many math books do, but also having no rigor and care to fall back on. So many physics texts just end up with nothing! – jeremy Jun 8 2010 at 14:16
I would like to make a point that, it seems to me, has not been discussed yet. Regardless of my opinion on whether students can get more or less from a given lecture by having a set of lecture notes in advance, I think that this habit prevents them from improving their study skills.
I think that two good skills are lost in this way: taking good notes in an effective way and being able to study from books.
As for the first one, taking notes need not be a mere exercise of copying from the blackboard. When I was a student I used to take notes of what the the lecturer said AND what he wrote at the blackboard, even adding connections on my own. In a sense I rewrote the whole thing in my words in real time. This usually resulted in a far better understanding of the lecture itself. It also forced me to keep a constant attention and not lose the pace, and prompted me to guess proof in advance to save time. It also forced me to stop the lecturer and make questions when I could not follow. All in all, it just made me more active. It was a really good exercise, and student should be encouraged to doing so, within limits.
As for the second one, it is a different thing studying from a set of notes that contains exactly what you need to know, presented in the same way as the lecture, rather than having to browse through a book which possibily contains more (or different) material, with a different emphasis. Sooner or later the students will need to resort to books, and at that time having some experience will help them not getting lost.
So, while lecture notes on WHATEVER will help you learn WHATEVER, they may hinder you learning other things later. I think a reasonable compromise may be to give lecture notes at the end of each topic, even with a small delay.
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I disagree, I think students who are going to put effort into learning will tend to have good study-skills regardless (pet peeve: not irregardless; which is not a word ;)) of taking in-class notes. I would also say taking notes from a book and taking notes from classes are completely different skills. I would also also say I would have agreed with you if you'd asked me a few years ago, when I was still taking notes in class, but I've found I actually retain much more when I don't take notes. – jeremy Jun 8 2010 at 14:33
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(actually irregardless is a word, according to wikipedia and wiktionary, although they claim it is not standard. But, well, I'm not native, and I have heard it many times, so I assumed it was fine; I will now use regardless instead. Thank you for telling me) – Andrea Ferretti Jun 8 2010 at 14:49
I suppose technically it would be slang or a regional word or something. But to most people it sounds odd because irregardless would logically mean the opposite of regardless (e.g., irrespective = not respective of) which would be not only a double negative, but would mean the opposite of what the speaker usually intends! – jeremy Jun 8 2010 at 14:55
Andrea: I think "irrespective" is the word you want. – Victor Protsak Jun 8 2010 at 16:08
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Since this is community wiki, I've taken the liberty of mooting most of the above comments -- sorry! – JBL Jun 8 2010 at 20:14
I provide my typed lectures to the students via my website. Looking at a nice, standard linear algebra course, I found that although many people told me they greatly appreciated the notes and used them extensively, they did not obviously improve the students' performance on tests or apparent retention of material from one lecture to the next. Having taught this class quite a few times now, I would say that the large structure is much more influential in teaching effectiveness than the perfection of the lectures. Sometimes the structure precludes effective lectures!
(The much more nonstandard thing on tropical geometry I once did was a different story, but that's because the notes were also the only text!)
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I've been always bothered by the discrepancy that you describe (appreciation vs effect on learning). What's going on here? Could this be an artifact of high school experience? I mean, every calculus, linear algebra, ODE etc course uses a well written standard textbook that students can read, yet they want the notes. Do they think that by getting the notes, they won't have to work as much? Do they feel privileged to get custom course materials? Or is it simply the force of a habit? – Victor Protsak Jun 7 2010 at 18:27
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@Victor I think part of it is certainly the fact it's proof to the students that we give a crap if they're learning or not. I don't know about everyone else,but there's nothing more discouraging then a professor who assigns a reading and leaves a half an hour before lecture ends and makes no pretense that he doesn't feel the students are wasting his or her time. A carefully produced set of notes makes students feel like you care about them,even if it's just to be professional. Another factor is that notes are much less cold and impersonal then a textbook. – Andrew L Jun 7 2010 at 20:10
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@Victor continued: Every set of lecture notes that aren't haphazardly prepared is an academic fingerprint of the instructor. No 2 are identical. This personality makes students feel they're learning what YOU want them to learn rather then a prepackaged syllabus and they trust this more (unless of course you give them the impression you don't give a crap).This gives them the added security come test time that they covered the material you're most likely to test them on. Lastly,some will feel like it's a joint effort if you give them these notes-like studying them makes them "do thier part". – Andrew L Jun 7 2010 at 20:14
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@Victor concluded And sadly,you're dead wrong about every basic class using a well written standard text. You'd be surprised what incomprehensible drivel many calculus,linear algebra,etc. courses choose at thier text-sometimes just out of spite because they're angry they actually have to teach classes.(Disclaimer:I have NOTHING against researchers-they're gifted people and vitally important. It's researchers who care ONLY about that and consider teaching for inferior beings that tick me off.) Lecture notes are a lifebelt for the students in such courses-even if they come from other sources. – Andrew L Jun 7 2010 at 20:22
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Andrew, you sound very bitter. Even though I've been around for a while, I cannot really relate to your experiences. What I am hoping to hear is an objective analysis to an important question facing many instructors, from someone who is qualified to offer it. Please, don't hijack the thread -- I am genuinely interested in what Ryan and others have to say. – Victor Protsak Jun 8 2010 at 2:04
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There is one more advantage to having notes in class that nobody has mentioned so far, perhaps because it applies more to graduate-level classes. I often find myself confused because some fact on the blackboard uses words that were defined a while ago, and that I forgot the precise meaning of. Also, sometimes it's hard to keep track of what bigger purpose does some technical lemma the lecturer is proving supposed to serve. When notes are available, I can often just look it up right there and then.
For this reason, I often find it helpful to come to seminars with my computer, and look up the paper that the speaker is talking about online.
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Why can't you simply ask in class "what does this word mean?" Surely if you can't follow the vocabulary, you can't follow the lecture. That could also be a clue to the lecturer/seminar speaker to adjust the pace. – Victor Protsak Jun 8 2010 at 1:10
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I try to limit myself to three or four such questions per class ;). I think slowly; sometimes I spend a while thinking about one thing the speaker said and miss what he is saying in the meantime. It happens just too often to ask every time. It's much worse when I am also taking notes myself. – Ilya Grigoriev Jun 8 2010 at 2:24
When you take an literature class devoted to, say, Shakespeare, this does not mean that the instructor reads Shakespeare line-by-line during class sessions. Rather, the students read the plays on their own time, and class time is devoted to discussion, analysis, etc.
I sometimes wonder if a similar model might work in a mathematics class. One might distribute lecture notes with basic definitions, simple examples, and easy propositions to be read by the students before lecture; the lecture time itself would be devoted to the proof of difficult theorems, higher-level insights, etc.
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That is what <a href="youtube.com/watch?v=WwslBPj8GgI">Eric Mazour</a> does in physics (but it is equally applicable in mathematics). – Alfonso Gracia-Saz Jun 8 2010 at 1:22
This is exactly what I've been doing. – Bart Snapp Jun 8 2010 at 15:22
In undergraduate lectures, Prof. Gowers didn't give out notes in advance; all was written on the blackboard (from memory!) Sadly I never saw his graduate lectures, but I think they were the same.
Prof. Korner, in both undergrad. and grad. lectures, gave out notes with only the theorems and definitions, and did the proofs in the lectures.
I personally really enjoyed both approaches. I'm sure most people who've seen them will agree that they are both great lecturers (but their methods could lead to disaster if you're not a great lecturer).
As a student, I thought lectures with complete notes given out in advance were not bad, but they just didn't fire up my enthusiasm in the same way. The mystery and anticipation added excitement!
Of course there's no right answer, and different students will probably prefer different methods; you can't please all students all of the time (even pleasing some students any of the time can be difficult!)
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I'd largely concur. Also, hi Zen. What brings you here? – Yemon Choi Jun 8 2010 at 23:37
One basic observation, as a student. A big reason for providing notes is if the class works out of more than one textbook (or none at all!) and you want to keep the narrative straight. The professors that I've seen do this seem to be doing it in order to get to interesting material through a path that requires minimal prerequisites, which can require working through a different progression of topics than any given textbook has in mind.
For example, Melrose teaches an undergrad functional analysis class at MIT which uses no measure theory; the class is based on a self-contained approach due to Mikusinski for defining the Lebesgue integral without defining Lebesgue measure. The class was taught in 2009 out of several sets of online notes written by other people and in 2010 out of Melrose's own notes.
At the other extreme, Munkres distributed only one set of notes when teaching undergrad topology, and that was a writeup of an alternate proof of Tychonoff's theorem. His textbook is more than good enough to serve in lieu of notes!
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It is common in some subjects--but not (yet?) in mathematics--to have everything on Powerpoint, prepared in advance of the lecture, and available on-line. Students bring their laptops to lecture (when they come to lecture) and follow along on the web copy. This presupposes you (the lecturer) know in advance what will happen, of course. Unexpected questions are not welcome. And the OP wants to do it at the beginning of the semester!
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Are you not free to disallow the use of laptops in class? – Peter Luthy Jun 7 2010 at 21:12
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I wish this day never comes for mathematics... it is already hard to follow Beamer slides, but Powerpoint... :-/ – Andrea Ferretti Jun 7 2010 at 23:18
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I agree, Gerald and Andrea -- this practice (1) is a very close approximation of reading verbatim from a textbook as the lecture, and (2) mainly serves to allow lecturers to go fast and "cover more material," while also ensuring that almost no one can follow what is actually being said in real time. – JBL Jun 8 2010 at 12:05
Apparently, slide or computer lectures are common in chemistry. – Victor Protsak Jun 8 2010 at 16:03
This last quarter I tried passing out a set of "skeleton" notes. This is nothing more than a list of definitions, theorems, sketches of proofs, and exercises. The first definition is numbered, say by "1" and then the exercises that follow it are 1.1, 1.2 etc. Perhaps this is followed by Theorem 2, with exercises 2.1, 2.2, etc. No theorems are completely proved, only sketches are given.
If a student wants to know if they understand a particular definition or theorem, they just work the exercises that follow the theorem.
I like this because it gives a logical structure to the course. Moreover, in some sense it makes mathematics an active activity for the students. When I lectured from them, I would work relevant exercises for the students on the fly. There is no false sense of security from these notes as the students have to do work to "read" them.
Overall it seemed to work well.
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I haven't really made up my mind on this issue yet. I think writing up lecture notes beforehand has 2 big advantages over "winging it": 1) It allows you to be much more detailed and thoughtful in presenting material in your own voice then a spontaneous lecture could be. It also allows you to catch mistakes much more readily before they pass into the students' hands. 2) It creates drafts for future courses that can later be modified,expanded or rewritten however you choose with the possibility of eventual publication. Or not. Of course,the problem with this approach is the lectures become much more "rigid" i.e. it's harder to add material or change things as you go and you rethink things or your perspective changes. I think the best answer is a compromise: Write up careful drafts,but leave blank spaces on the notes for handwritten additions or revisions you come up with on the spur of the moment-especially those that result from student input. These changes can then be incorporated into the eternally evolving draft.
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Almost anything is better than "winging it". – Deane Yang Jun 7 2010 at 17:41
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Andrew, are you speaking from student's perspective or from professor's perspective? It's hard to tell. – Victor Protsak Jun 7 2010 at 18:17
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I prepare my own lecture notes before going to the lecture; as you say, they help organize your thoughts, you can be sure the examples you are working out don't have particular issues that obscure your point, etc. And I save them for future courses, making notes if something seems particularly confusing or helpful. But this kind of notes is different from something I would be comfortable handing out to the students. Much like my 'notes' before a seminar/conference talk are not something I would necessarily hand out to the listeners. – Arturo Magidin Jun 7 2010 at 18:34
@Victor BOTH,Victor. I've been a student longer then I care to admit and fortunately,most of my mentors have been very active teachers rather then researchers.I've learned an enormous amount from them and what they do differently from either awful or apathetic teachers and I can't wait to apply what I've learned to my own style of lecturing. – Andrew L Jun 7 2010 at 20:04
Once upon a time, I taught a very non-standard, absolutely student-centered multivariable calculus course just using handouts. You may find the details here: (2012), Moore and Less!, PRIMUS: Problems,Resources, and Issues in Mathematics Undergraduate Studies, 22:7, 509-524 : http://dx.doi.org/10.1080/10511970.2011.639337 Here is the first paragraph of the paper:
This is the story of a very non-standard multivariable calculus course. I think it is worth hearing about since it surprised many of my mathematician col- leagues. At first glance, it surprised them since it was a non-lecturing course in which no black (or white) board, and no computer was ever used; yet, it was a multivariable calculus course covering all the standard materials of such a course. Thus, my colleagues’ first question was “so, what was the means of communication?” Fortunately, the answer is very simple; just handouts.
My usage of handouts somehow differs from the two ways you have introduced. Thus, I put this as an indirect answer to your question to broaden the possible ways you may use handouts in the future.
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http://mathoverflow.net/revisions/70606/list | ## Return to Answer
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To determine which potential zeta functions are actual zeta functions of curves is very difficult. The zeta function of a curve is determined by the zeta function of its Jacobian so one could instead ask which potential zeta functions are zeta functions of abelian varieties. This problem is solved (by Tate though as I recall Waterhouse was also involved in working out some details) and the answer is that essentially every $P(u)$ that could (i.e., having roots with the proper absolute values) occur with some extra restriction having to do with the endomorphism ring of the abelian variety having to be an order in a semi-simple algebra with some non-split factor. The conditions are anyway very explicit and reasonably easy to check.
The next step is to pick out the zeta functions of principally polarised abelian varieties among all of them. This involves more arithmetic but is also also feasible.
The tricky part is the Schottky problem, to pick out the Jacobians among the principally polarised abelian varieties. This is in principle solved (at least in characteristic zero, I am less sure about positive characteristic) but any of the existing solutions meshes very badly with the problem of zeta functions as the solution to the problem above is very non-explicit and only tells you about the existence of a p.p.a.v. with given zeta function not a description of all of them.
Even where all of this can be done, for instance in genus $1$ where the Schottky problem is trivial, it is non-trivial to get actual equations. The reason for this is essentially that Tate's existence argument is through construction of a characteristic $0$ abelian variety with complex multiplication and then reducing it modulo $p$. Hence, the only way to be explicit would seem to be to first get the Weierstrass equation for the CM-curve. This is certainly possible (and fairly, for some definition of that term, efficiently) but it is far from easy. Of course, it still doesn't give all of the p.p.a.v.'s with given zeta function (though in some cases, for instance the ordinary case, all of them are the reduction of a CM abelian variety).
Addendum: As Noam points out there are a (small) number of general results excluding some zeta functions for curves where the p.p.a.v. exists. These generally concern improvements of the Weil bounds exploiting the fact that given the zeta function we can compute the number of points of the curve over the field and extensions of it and that these number should be non-negative and increase as the field increases (though the actual proofs are sometimes quite sophisticated). To see if the bounds obtained are sharp a lot of effort has also been expended on constructing specific curves with many points. In some cases special arguments can be used to exclude some values. See this site for tables and these slides for a description of a particularly detailed analysis of some cases.
1
To determine which potential zeta functions are actual zeta functions of curves is very difficult. The zeta function of a curve is determined by the zeta function of its Jacobian so one could instead ask which potential zeta functions are zeta functions of abelian varieties. This problem is solved (by Tate though as I recall Waterhouse was also involved in working out some details) and the answer is that essentially every $P(u)$ that could (i.e., having roots with the proper absolute values) occur with some extra restriction having to do with the endomorphism ring of the abelian variety having to be an order in a semi-simple algebra with some non-split factor. The conditions are anyway very explicit and reasonably easy to check.
The next step is to pick out the zeta functions of principally polarised abelian varieties among all of them. This involves more arithmetic but is also also feasible.
The tricky part is the Schottky problem, to pick out the Jacobians among the principally polarised abelian varieties. This is in principle solved (at least in characteristic zero, I am less sure about positive characteristic) but any of the existing solutions meshes very badly with the problem of zeta functions as the solution to the problem above is very non-explicit and only tells you about the existence of a p.p.a.v. with given zeta function not a description of all of them.
Even where all of this can be done, for instance in genus $1$ where the Schottky problem is trivial, it is non-trivial to get actual equations. The reason for this is essentially that Tate's existence argument is through construction of a characteristic $0$ abelian variety with complex multiplication and then reducing it modulo $p$. Hence, the only way to be explicit would seem to be to first get the Weierstrass equation for the CM-curve. This is certainly possible (and fairly, for some definition of that term, efficiently) but it is far from easy. Of course, it still doesn't give all of the p.p.a.v.'s with given zeta function (though in some cases, for instance the ordinary case, all of them are the reduction of a CM abelian variety). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 8, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9507907629013062, "perplexity_flag": "head"} |
http://mathhelpforum.com/advanced-applied-math/127848-interpolation.html | # Thread:
1. ## Interpolation
suppose that $n \geq 1$. The function $f$ and its derivatives of order up to and including $2n+1$ are continuous on [a,b]. The points $x_i, i = 0,1,...n$ are distinct and lie in [a,b]. Construct polynomials $r_0 (x), h_i (x), k_i(x) , i= 1,...,n$ of degree $2n$ such that the polynomial $p_{2n} (x) = r_0 (x)f(x_0) + \sum_{i=1}^n h_i (x)f(x_i) + k_i (x)f ' (x_i)$ satisfies conditions $p_{2n} (x_i) = f(x_i), i=0,1,...n$ and $p ' _{2n} (x_i) = f ' (x_i), i=1,....n$
I can't get anywhere with this. My professor doesn't teach examples, only theorems... help is much appreciated =) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9683842062950134, "perplexity_flag": "head"} |
http://mathoverflow.net/revisions/82523/list | ## Return to Answer
2 deleted 30 characters in body; deleted 20 characters in body
This is strange. If $r$ is a $C^2$ extremal, and $F(x,r(x),r'(x))$ we'd have $F_r(x,r(x),r'(x)) - \frac{d}{dx} F_{r'}(x,r(x),r'(x)) = 0$ for $x \in (0,\pi/2)$, which then means $\int_{a}^{b} F_r(x,r(x),r'(x)) = F_{r'}(b,r(b),r'(b)) - F_{r'}(a,r(a),r'(a))$ for $0 < a < b < \pi/2$. The LHS is
$\int_a^b \left( \frac{1}{\pi } \ln \left( {\frac {\sin \left( x \right) }{1-\cos \left( x \right) }} \right) + \frac{1}{2} 1 \right) {\frac {r(x)}{\sqrt {{r(x)}^{2} + {r'(x)}^{2}}}}$
which, provided $r(0)=1$, goes to a finite value as $a \rightarrow 0^+$. However, $F_{r'}(a,r(a),r'(a)) = \left( \frac{1}{\pi } \ln \left( {\frac {\sin \left( a \right) }{1-\cos \left( a \right) }} \right) + \frac{1}{2} 1 \right) {\frac {r'(a)}{\sqrt {{r(a)}^{2} + {r'(a)}^{2}}}}$ which goes to $\pm \infty$ as $a \rightarrow 0^+$ if $r'(0) \neq 0$. Thus we are left to conclude that $r'(0) = 0$. But 'experimental' data (via discretizations of the problem), show this isn't the case! Either what I just wrote is wrong, or something is seriously wrong!
1
This is strange. If $r$ is a $C^2$ extremal, and $F(x,r(x),r'(x))$ we'd have $F_r(x,r(x),r'(x)) - \frac{d}{dx} F_{r'}(x,r(x),r'(x)) = 0$ for $x \in (0,\pi/2)$, which then means $\int_{a}^{b} F_r(x,r(x),r'(x)) = F_{r'}(b,r(b),r'(b)) - F_{r'}(a,r(a),r'(a))$ for $0 < a < b < \pi/2$. The LHS is
$\int_a^b \left( \frac{1}{\pi } \ln \left( {\frac {\sin \left( x \right) }{1-\cos \left( x \right) }} \right) + \frac{1}{2} \right) {\frac {r(x)}{\sqrt {{r(x)}^{2} + {r'(x)}^{2}}}}$
which, provided $r(0)=1$, goes to a finite value as $a \rightarrow 0^+$. However, $F_{r'}(a,r(a),r'(a)) = \left( \frac{1}{\pi } \ln \left( {\frac {\sin \left( a \right) }{1-\cos \left( a \right) }} \right) + \frac{1}{2} \right) {\frac {r'(a)}{\sqrt {{r(a)}^{2} + {r'(a)}^{2}}}}$ which goes to $\pm \infty$ as $a \rightarrow 0^+$ if $r'(0) \neq 0$. Thus we are left to conclude that $r'(0) = 0$. But 'experimental' data (via discretizations of the problem), show this isn't the case! Either what I just wrote is wrong, or something is seriously wrong! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 30, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9792595505714417, "perplexity_flag": "head"} |
http://mathoverflow.net/questions/68598/basic-questions-about-stacks-2 | ## Basic questions about stacks 2
### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)
I have again three basic questions about stacks.
1) When we consider categories fibered in groupoids, do we always mean small or essentially small groupoids? Especially I want to know if algebraic geometers always impose this condition when they talk about categories fibered in groupoids, especially stacks.
2) In the proof of Artin's criterion for algebraic spaces/stacks $X/S$ for every point $p \in X$ of finite type over $S$ a "local approximation" $X_p$ is constructed. Then $X = \coprod_p X_p$ does the job. But in order to show that this is actually a scheme in the given universe, we need that the points of finite type constitute a set. Perhaps I'm overlooking something trivial here, but I cannot see how we can use Artin's criterions to deduce this.
3) What is the current status of the book "Algebraic Stacks" by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche und Andrew Kresch? I would love to read it as soon as it is completed.
-
Regarding 1), the definition of a fibred category does not assume that the fibres are small, but in geometric examples they are (or at least are essentially small). For example, would you consider the groupoid of G-bundles on a fixed topological space to be a set? This groupoid is certainly essentially small once you have a classifying space BG. – David Roberts Jun 23 2011 at 12:42
Me again. The category GBund(X) of principal G-bundles on a fixed object X in a site (S,J) is equivalent to the hom-category Gpd_ana(X,_B_G) in the bicategory of internal groupoids, anafunctors and transformations (without loss of generality, assume S is a superextensive site - see nLab for definition - which is true in all geometric situations). Here _B_G is the groupoid with one object and morphisms G, and we consider X as a groupoid with only trivial morphisms. GBund(X) is essentially small when the axiom 'WISC' holds for J, again see nLab. But 'morally', if not actually, the 2-category.... – David Roberts Jun 23 2011 at 23:11
...of stacks in groupoids on (S,J) is equivalent to the bicategory of groupoids, anafunctors and transformations in S. Roughly speaking, all fibres of a stack are representable in this bicategory, and so given WISC, one knows that the fibres are essentially small. Oh, I should mention that really this is only expected to hold for geometric stacks (presentable by some space/scheme/what-have-you), but I guess that that is the case you are most interested in. – David Roberts Jun 23 2011 at 23:16
## 1 Answer
Regarding 3), Andrew Kresch just told me that they gave up on the project.
-
5
Oh no :-( – Martin Brandenburg Jun 23 2011 at 14:14
2
There is always the stack project which actively developed, available at math.columbia.edu/algebraic_geometry/stacks-git – Andrei Halanay Jun 23 2011 at 21:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.902167558670044, "perplexity_flag": "middle"} |
http://mathhelpforum.com/discrete-math/73337-harmonic-conjugate.html | # Thread:
1. ## Harmonic Conjugate
u = log|z|, Re(z) > 0
show that this is harmonic (i.e. satisfies laplace) and find the harmonic conjugate.
How would one do this? Thanks.
2. Originally Posted by universalsandbox
u = log|z|, Re(z) > 0
show that this is harmonic (i.e. satisfies laplace) and find the harmonic conjugate.
How would one do this? Thanks.
By, of course, using the Definitions of those things. Writing z= x+ iy, |z|= $\sqrt{x^2+ y^2}= (x^2+ y^2)^{1/2}$ so $u(x,y)= \log ((x^2+ y^2)^{1/2})= (1/2) \log(x^2+ y^2)$.
u(x,y) is harmonic if and only if it satisfies $nabla^2 u= \frac{\partial^2 u}{\partial x^2}+ \frac{\partial^2 u}{\partial y^2}= 0$. Find those partial derivatives and add!
If $f(z)= u(x,y)+ iv(x,y)$ is analytic then both u and v are harmonic functions and we say that they are "harmonic conjugates".
If f(z) is analytic it must satisfy $\frac{\partial u}{\partial x}= \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}$. Since you are given u, you can find the left sides of those and solve the resulting equations for v. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9115310311317444, "perplexity_flag": "middle"} |
http://www.physicsforums.com/showthread.php?t=199798 | Physics Forums
root question
Is there any "pencil and paper" method to find the nth root of a number?
Since multiplying a number by itself any number of times quickly yeilds extremely large numbers, trial and error might seem to pinpoint the root of a number, so long as it is a perfect square or cube or whatever.
But, is there any real way to pinpoint the root of a number without using a calculator or trial and error?
To compute the nth root of a number Y, just make some guess X and then improve that guess using the formula: $$\left(1-\frac{1}{n}\right)X + \frac{Y}{n X^{n-1}}$$ You can iterate this to make further improvements. E.g. suppose you want to estimate the cube root of 10. Then you can take X = 2. the formula gives: 4/3 + 10/(3*4) = 2 + 1/6 If you then take X = 2+1/6 and insert that in the formula to get 2.1545. Iterating again gives 2.15443469224. Now, believe it or not but: 2.15443469224^3 = 10.0000000307
Blog Entries: 2 If you want to know the general theory behind the above method, see Newton's method.
root question
So, this is still trial and error, but it converges very fast. At each step you double to correct number of digits. You go from a wild guess to a number that is correct to ten significant digits in about four iterations.
Quote by Count Iblis To compute the nth root of a number Y, just make some guess X and then improve that guess using the formula: $$\left(1-\frac{1}{n}\right)X + \frac{Y}{n X^{n-1}}$$ You can iterate this to make further improvements. E.g. suppose you want to estimate the cube root of 10. Then you can take X = 2. the formula gives: 4/3 + 10/(3*4) = 2 + 1/6 If you then take X = 2+1/6 and insert that in the formula to get 2.1545. Iterating again gives 2.15443469224. Now, believe it or not but: 2.15443469224^3 = 10.0000000307
wow, that's pretty neat. Thanks!
The case n = -1 is also very useful. In that case X = 1/Y but Newton's method gives: $$2X - X^{2} Y$$ Since there are no divisions in here, you can use it to do divisions. It's much faster than long division.
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http://mathhelpforum.com/geometry/210841-measure-ac.html | 3Thanks
• 1 Post By MarkFL
• 1 Post By skeeter
• 1 Post By MarkFL
# Thread:
1. ## Measure of AC
ABCD is rectangle. If AD = x + 1, CD= 2x -3 & AC= x+2, find AC.
MY ATTEMPT:
i used pythagorean theorems:
(x +1)^2+ (2x-3)^2 = (x+2)^2
i got the problem from our textbook ... And the answer without solution is at the back of the portion of the book. It says that the answer is 5.
how did they make it to 5, please let me understand.
thanks
2. ## Re: Measure of AC
The Pythagorean theorem is a good place to begin as it relates the 3 lengths given...now what do you get when you expand the binomials?
3. ## Re: Measure of AC
(AD)2 + ( CD)2 = (AC)2
(x+1)2 + ( 2x-3)2 = ( x + 2 )2
x^2 + 2x + 1 + 4x^2 – 12x + 9 = x^2 + 4x + 4
4x^2 -14x + 6 = 0
2x^2 -7x + 3 = 0
2x^2 - x – 6x + 3 = 0
( 2x – 1 ) ( x -3)= 0
X = ½ or x = 3. X cannot be ½ because then CD will become negative, thus discarding this value.
Now x = 3 thus AC = x + 2 = 3+2=5
4. ## Re: Measure of AC
Originally Posted by ibdutt
(AD)2 + ( CD)2 = (AC)2
(x+1)2 + ( 2x-3)2 = ( x + 2 )2
x^2 + 2x + 1 + 4x^2 – 12x + 9 = x^2 + 4x + 4
4x^2 -14x + 6 = 0
2x^2 -7x + 3 = 0
2x^2 - x – 6x + 3 = 0
( 2x – 1 ) ( x -3)= 0
X = ½ or x = 3. X cannot be ½ because then CD will become negative, thus discarding this value.
Now x = 3 thus AC = x + 2 = 3+2=5
You really should let the OP have a chance (say at least 24 hours) to respond to suggestions about how to work the problem on their own.
5. ## Re: Measure of AC
Originally Posted by MarkFL2
You really should let the OP have a chance (say at least 24 hours) to respond to suggestions about how to work the problem on their own.
why wait for the next 24 hours when it could be done today
6. ## Re: Measure of AC
Originally Posted by rcs
why wait for the next 24 hours when it could be done today
... because you are the one that needs to work it out, not someone else.
7. ## Re: Measure of AC
Originally Posted by rcs
why wait for the next 24 hours when it could be done today
Good forum etiquette dictates that one not give a solution immediately after someone else has given a suggestion on how to proceed. This devalues the preceding post.
You would have learned more had you expanded the binomials yourself, and realized only 1 of the roots made sense in the context of the problem.
8. ## Re: Measure of AC
Originally Posted by rcs
ABCD is rectangle. If AD = x + 1, CD= 2x -3 & AC= x+2, find AC.
MY ATTEMPT:
i used pythagorean theorems:
(x +1)^2+ (2x-3)^2 = (x+2)^2
i got the problem from our textbook ... And the answer without solution is at the back of the portion of the book. It says that the answer is 5.
how did they make it to 5, please let me understand.
thanks
Another way to proceed would be to let:
$\bar{AC}=u$ and so:
$\bar{AD}=u-1$
$\bar{CD}=2u-7$
We see we require $u>\frac{7}{2}$.
By Pythagoras, we have:
$(u-1)^2+(2u-7)^2=u^2$
$u^2-2u+1+4u^2-28u+49=u^2$
$4u^2-30u+50=0$
$2u^2-15u+25=0$
$(2u-5)(u-5)=0$
The only root that satisfies all of the criteria is:
$u=\bar{AC}=5$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8988066911697388, "perplexity_flag": "middle"} |
http://mathhelpforum.com/discrete-math/213078-real-analysis-question-subsequence.html | # Thread:
1. ## Real analysis question on subsequence
Hi, my question is:
Let L be a real number and let (a_n) be a sequence of real numbers that does not converge to L (that is, it is either divergent or its limit is not equal to L). Use the definition of convergence to L to show that for some e greater than 0, (a_n) has a subsequence (a_n(_k)) such that (a_n(_k)) isn't in the interval (L - e, L+e) for all natural numbers k.
Any help would be hugely appreciated
2. ## Re: Real analysis question on subsequence
Originally Posted by sakuraxkisu
Let L be a real number and let (a_n) be a sequence of real numbers that does not converge to L (that is, it is either divergent or its limit is not equal to L). Use the definition of convergence to L to show that for some e greater than 0, (a_n) has a subsequence (a_n(_k)) such that (a_n(_k)) isn't in the interval (L - e, L+e) for all natural numbers k.
To say that $\left( {a_n } \right)\not \to L$ means that
$\left( {\exists \varepsilon > 0} \right)\left( {\forall N} \right)\left( {\exists n_N > N} \right)\left[ {\left| {L - a_{N_n } } \right| \geqslant \varepsilon } \right]$.
Find the first $N_1$ then apply the above and find $N_2>N_1$ that works.
By induction, find a sequence of integers $N_1<N_2<\cdots<N_m<\cdots.$
3. ## Re: Real analysis question on subsequence
Plato, great explanation, solved my other problem in the series also. I am glad that I bumped into this thread.
4. ## Re: Real analysis question on subsequence
Originally Posted by SeirraFalcom
Plato, great explanation, solved my other problem in the series also. I am glad that I bumped into this thread.
Update: I am stuck again in the test prep of my discrete math subject. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9254666566848755, "perplexity_flag": "middle"} |
http://nrich.maths.org/1357&part= | nrich enriching mathematicsSkip over navigation
### Shades of Fermat's Last Theorem
The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?
### Upsetting Pitagoras
Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2
### BT.. Eat Your Heart Out
If the last four digits of my phone number are placed in front of the remaining three you get one more than twice my number! What is it?
# Euclid's Algorithm I
##### Stage: 5
Article by Alan and Toni Beardon
How can we solve equations like $13x+29y=42$ or $2x+4y=13$ with the solutions $x$ and $y$ being integers? Equations with integer solutions are called Diophantine equations after Diophantus who lived about 250 AD but the methods described here go back to Euclid (about 300 BC) and earlier. When people hear the name Euclid they think of geometry but the algorithm described here appeared as Proposition 2 in Euclid's Book 7 on Number Theory.
First we notice that $13x+29y=42$ has many solutions, for example $x=1$, $y=1$ and $x=30$, $y=-12$. Can you find others (it has infinitely many solutions)? We also notice that $2x+4y=13$ has no solutions because $2x+4y$ must be even and $13$ is odd. Can you find another equation that has no solutions?
If we can solve $3x+5y=1$ then we can also solve $3x+5y=456$. For example, $x=2$ and $y=-1$ is a solution of the first equation, so that $x=2\times 456$ and $y=-1\times 456$ is a solution of the second equation. The same argument works if we replace $456$ by any other number, so that we only have to consider equations with $1$ on the right hand side, for example $P x+Q y=1$. However if $P$ and $Q$ have a common factor $S$ then $P x+Q y$ must be a multiple of $S$ so we cannot have a solution of $P x+Q y=1$ unless $S=1$. This means that we should start by considering equations $P x+Q y=1$ where $P$ and $Q$ have no common factor.
Let us consider the example $83x+19y=1$. There is a standard method, called Euclid's Algorithm, for solving such equations. It involves taking the pair of numbers $P=83$ and $Q=19$ and replacing them successively by other pairs $(P_k,Q_k)$. We illustrate this by representing each pair of integers $(P_k,Q_k)$ by a rectangle with sides of length $P_k$ and $Q_k$.
Draw an $83$ by $19$ rectangle and mark off $4$ squares of side $19$, leaving a $19$ by $7$ rectangle.
This diagram represents the fact that
$83=4\times 19+7$
In a few steps we shall split this rectangle into 'compartments' to illustrate the whole procedure for solving this equation. (You may like to try the java applet Solving with Euclid's Algorithm which draws the rectangles and carries out all the steps automatically to solve equations of the form $P x+Q y=1$). We repeat this process using the $19$ by $7$ rectangle to obtain two squares of side $7$, and a $7$ by $5$ rectangle. Next, the $7$ by $5$ rectangle splits into a square of side $5$, and a $5$ by $2$ rectangle.
The $5$ by $2$ rectangle splits into two squares of side $2$, and a $2$ by $1$ rectangle. The $2$ by $1$ rectangle splits into two squares of side $1$ with nothing left over and the process finishes here as there is no residual rectangle. These diagrams illustrate the following equations:
\begin{eqnarray} 83 & = & 4\times 19 & + & 7\\ 19 & = & 2\times 7 & + & 5\\ 7 & = & 1\times 5 & + & 2\\ 5 & = & 2\times 2 & + & 1\\ 2 & = & 2\times 1 & + & 0 \end{eqnarray}
To find the solution we use the last non-zero remainder, namely $1=5-2[2]$
and successively substitute the remainders from the other equations until we get back to the first one giving a combination of the two original values $P=83$ and $Q=19$. The method in this example has the following steps with the remainders given in square brackets.
\begin{eqnarray} 1 & = & [5]-2[2]\\ & = & [5]-2([7]-[5])\\ & = & -2[7]+3[5]\\ & = & -2[7]+3(19-2[7])\\ & = & (3\times 19)-8[7]\\ & = & (3\times 19)-8(83-(4\times 19))\\ & = & (-8\times 83)+(35\times 19). \end{eqnarray}
Thus a solution of $83x+19y=1$ is $x=-8$ and $y=35$.
Can you find a solution of $83x+19y=7$?
Can you now find a solution of $827x+191y=2$? You should first solve the equation $827x+191y=1$ (using the computer if you wish).
For the next article in the series, click here .
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 73, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9141676425933838, "perplexity_flag": "head"} |
http://mathhelpforum.com/calculus/99432-how-do-you-find-zeros.html | # Thread:
1. ## How do you find the zeros?
I need to find the zeros as well as see where it is positive or negative so that I can match it to a graph for a question. My only problem (from what I can tell so far) is how to find the zeros. When I find the zeros, I think I might be able to manage the rest:
-x^5 + 5x^3 - 4x
-x(x^4 - 5x^2 + 4) <--------------What do I do next?
Any help would be greatly appreciated!
Thanks in advance!
2. Just by inspection you can see that 0 is a root. To see if there are any other roots, put $y = x^2$. Then your equation turns into $-x(x^4 - 5x^2 + 4) = -x(y^2 - 5y + 4) = 0$. Can you factor the expression that is quadratic in y?
3. Originally Posted by s3a
I need to find the zeros as well as see where it is positive or negative so that I can match it to a graph for a question. My only problem (from what I can tell so far) is how to find the zeros. When I find the zeros, I think I might be able to manage the rest:
-x^5 + 5x^3 - 4x
-x(x^4 - 5x^2 + 4) <--------------What do I do next?
Any help would be greatly appreciated!
Thanks in advance!
You can factor further
$(x^4-5x^2+4)=(x^2-4)(x^2-1)=(x-2)(x+2)(x-1)(x+1)$
So the result:
$-x(x-2)(x+2)(x-1)(x+1)=0\Rightarrow{x}=0,\pm2,\pm1$
4. Originally Posted by JG89
Just by inspection you can see that 0 is a root. To see if there are any other roots, put $y = x^2$. Then your equation turns into $-x(x^4 - 5x^2 + 4) = -x(y^2 - 5y + 4) = 0$. Can you factor the expression that is quadratic in y?
Ok so from -x(y^2 - 5y + 4) = 0 I got: x=0, y=1, y=4. Is this correct? If so, what do I do from here?
5. Originally Posted by s3a
Ok so from -x(y^2 - 5y + 4) = 0 I got: x=0, y=1, y=4. Is this correct? If so, what do I do from here?
Substitute back for x and solve. But I must say that when looking to factor polynomials like these, I find it easier to recognize the fact that it is "quadratic". Because
$(x^2)^2-5x^2+4$
Substitution is a very good way to do it too, but I tend to get confused when another variable is introduced. But, it all boils down to whichever method that you are most comfortable with.
6. Originally Posted by s3a
Ok so from -x(y^2 - 5y + 4) = 0 I got: x=0, y=1, y=4. Is this correct? If so, what do I do from here?
remember , you let x^2=y
so now $x^2=1\Rightarrow$ x=1 , -1
$x^2=4\Rightarrow$ x=2 , -2
Now you can proceed to find its zeros .
7. Originally Posted by VonNemo19
Substitute back for x and solve. But I must say that when looking to factor polynomials like these, I find it easier to recognize the fact that it is "quadratic". Because
$(x^2)^2-5x^2+4$
Substitution is a very good way to do it too, but I tend to get confused when another variable is introduced. But, it all boils down to whichever method that you are most comfortable with.
Ok now I see that
-x((sqrt(1))^4 - 5(2)^2 + 4) = 0
and
-x((2^4 - 5(2)^2 + 4) = 0
are both true.
So, what does this mean? Since y=1 and y=4, my zeros are x=1 and x=2?
Edit: I just saw the factoring further post. I think that's what I needed. I'll try to complete the problem now.
8. Originally Posted by s3a
Edit: I just saw the factoring further post. I think that's what I needed. I'll try to complete the problem now.
Great! I'm glad you see it!
Look at my first post. You should be able to decipher it now. If not, let me know.
9. Originally Posted by VonNemo19
Great! I'm glad you see it!
Look at my first post. You should be able to decipher it now. If not, let me know.
I was told the answer for #10 2) was C. (Problem I'm currently doing). I got it mostly right but I'm wondering why I have a portion wrong. I don't know what that table method is called but I tried to explain it in case it's not something universal. The portions I'm talking about are the "WTF" portions lol.
Attached Files
• mathhelp.pdf (113.4 KB, 24 views) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.95682293176651, "perplexity_flag": "head"} |
http://unapologetic.wordpress.com/2008/08/12/ | # The Unapologetic Mathematician
## Additive Functions
I’ve got internet access again, but I’m busy with a few things today, like assembling my furniture. Luckily, Tim Gowers has a post on “How to use Zorn’s lemma”. His example is the construction of additive (not linear) functions from $\mathbb{R}$ to itself.
In practice, as he points out, this is equivalent to defining linear functions (not just additive) from $\mathbb{R}$ to itself, if we’re considering $\mathbb{R}$ as a vector space over the field $\mathbb{Q}$ of rational numbers! This is a nifty little application within the sort of stuff we’ve been doing lately, and it really explains how we used Zorn’s Lemma when we showed that every vector space has a basis.
Posted by John Armstrong | Algebra, Linear Algebra | 9 Comments
## About this weblog
This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).
I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 4, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9213032722473145, "perplexity_flag": "middle"} |
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