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# Combinatorial Dominance Guarantees for Heuristic Algorithms

Abstract : An $f(n)$ $\textit{dominance bound}$ on a heuristic for some problem is a guarantee that the heuristic always returns a solution not worse than at least $f(n)$ solutions. In this paper, we analyze several heuristics for $\textit{Vertex Cover}$, $\textit{Set Cover}$, and $\textit{Knapsack}$ for dominance bounds. In particular, we show that the well-known $\textit{maximal matching}$ heuristic of $\textit{Vertex Cover}$ provides an excellent dominance bound. We introduce new general analysis techniques which apply to a wide range of problems and heuristics for this measure. Certain general results relating approximation ratio and combinatorial dominance guarantees for optimization problems over subsets are established. We prove certain limitations on the combinatorial dominance guarantees of polynomial-time approximation schemes (PTAS), and give inapproximability results for the problems above.
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Type de document :
Communication dans un congrès
Jacquet, Philippe. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), pp.79-94, 2007, DMTCS Proceedings
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https://hal.inria.fr/hal-01184785
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### Citation

Daniel Berend, Steven S. Skiena, Yochai Twitto. Combinatorial Dominance Guarantees for Heuristic Algorithms. Jacquet, Philippe. 2007 Conference on Analysis of Algorithms, AofA 07, 2007, Juan les Pins, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), pp.79-94, 2007, DMTCS Proceedings. 〈hal-01184785〉

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# Why no longitudinal electromagnetic waves?

According to wikipedia and other sources, there are no longitudinal electromagnetic waves in free space. I'm wondering why not.

Consider an oscillating charged particle as a source of EM waves. Say its position is given by $x(t) = sin(t)$. It is clear that at any point on the $x$ axis, the magnetic field is zero. But there is still a time-varing electric field (more or less sinusoidal in intensity, with a "DC offset" from zero), whose variations propagate at the speed of light. This sounds pretty wave-like to me. Why isn't it? Is there perhaps a reason that it can't transmit energy?

A very similar question has already been asked, but it used a "rope" analogy, and I feel that the answers overlooked the point that I'm making.

-

I think this is partly a question of vocabulary, and partly a reflection of the fact that the longitudinal Coulomb oscillations you describe fall off so rapidly with distance. (Basically $1/r^2$ instead of $1/r$.) Therefore they are usually called "near field effects" and are totally dominated by the transverse "waves" after a distance of only a very few wavelengths. Nevertheless, they do exist, even in a vacuum, and they do extend to infinity, just very, very weakly.

-

http://en.wikipedia.org/wiki/Longitudinal_wave#Electromagnetic has a good summary of the situation. There are no longitudinal solutions of the Maxwell equations in a vacuum, but you can get such solutions in a plasma.

-
Then, can EM waves be longitudinal in plasma? –  Sachin Shekhar Apr 13 at 5:03
Yes, but they're really sound waves in a charged gas not EM waves. –  John Rennie Apr 13 at 5:51

Once you get far enough away from a radiating source, your field will look approximately like a plane wave.

If you look at a plane wave, where $\vec{E}(\vec{x},t)=\vec{E}_0(\vec{k}\cdot\vec{x}-\omega t)$ and $\vec{B}(\vec{x},t)=\vec{B}_0(\vec{k}\cdot\vec{x}-\omega t)$ (for fixed functions of a single variable $\vec{E}_0$, $\vec{B}_0$), you will find that satisfying Maxwell's equations in empty space requires that $\vec{k}\cdot\vec{E}_0=\vec{k}\cdot\vec{B}_0=0$. That is, the electric and magnetic fields must be perpendicular to the direction of propagation.

Why? Because variation along the direction of propagation would lead to a non-zero divergence in $\vec{E}$ or $\vec{B}$, which is strictly forbidden. Unless, of course, you have non-zero charge density, in which case $\vec{E}$ can have a corresponding divergence. This is why longitudinal waves are possible in plasmas.

-

I don't know if this really qualifies as an answer, but if I read your question rightly I think you might find this quote interesting:

"The original forms of quantum mechanics ... [quantized] ... the electromagnetic field ... by Fourier transformation, as a superposition of plane waves having transverse, longitudinal, and timelike polarizations ... The combination of longitudinal and timelike oscillators was shown to provide the (instantaneous) Coulomb interaction of the particles, while the transverse oscillators were equivalent to photons." [1]

[1] Laurie M. Brown, Feynman's Thesis, pp. xi-xii. World Scientific (2005), paperback edition.

-
Transverse waves are not obligatory propagating. Consider a moving uniformly charge. Its electric field has longitudinal and transverse components, but nothing is a radiation. –  Vladimir Kalitvianski Mar 11 '12 at 18:23

Is this not related to the fact that the massless photon cannot have a longitudinal mode? It would have to satisfy,

$$k_\mu \epsilon^\mu = -\vec k \cdot \vec \epsilon = 0$$ If it were longitudinal, $\vec k = \vec \epsilon\times|\vec k|$ so that $\vec k \cdot \vec \epsilon=|\vec k|\ne0$.

Notice that if the photon were massive we would be allowed its rest frame in which $\vec k =0$, but it isn't, so we're not.

-
If you look at a light wave as a rotating x and y axis which propagates

forward in the z direction, the equation which might result takes the appearance of a screw [or helix]. The equation of the wave is not only a function of time, but also in z.

y = A e^ (i( B*z + w*t )) , i is the square root of -1

Note an equation of a helix which is:

X = A sin B*z

y = A cos B*z

z = z

It seems that the helix is formed by rotating the polarization of

the light wave at an angular velocity. This seems like the description of a " longitudinal " wave. I hope this will help.

-

Longitudinal electromagnetic fields are required to satisfy Maxwells divE = 0 + rho_free. They always exist even in vacuum. Plane wave approximation does not hold very well outside of a few (very limited) conditions.

-

Because you are looking in the wrong parts of science, one long forgotten and never pursued. You might research Marconi and Tesla, both of which where using longitudinal electromagnetic waves in their transmission devices. Tesla was not concerned with wireless signal transmission, but wireless "power" transmission.

https://en.wikipedia.org/wiki/Nikola_Tesla

http://www.capturedlightning.com/frames/Tesla0.html

You won't find longitudinal electromagnetic waves outside of the Tesla and Marconi era, which modern science does not bother to investigate any longer.

-
Simply wrong. Longitudinal waves can be shown to not work in free propagation but they are used regularly in wave guides. –  dmckee Jul 5 '13 at 21:40

## protected by Qmechanic♦Apr 13 at 9:31

Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.

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# PSL(2,Z)

A particularly important function is Klein’s j-function. It is defined on the upper half-plane of the complex numbers, and is incredibly symmetrical. For example, it is periodic, and thus invariant under translations by integers:

It is also invariant under a larger group of Möbius transformations, namely those corresponding to matrices with integer entries. This group is called the modular group, or $PSL(2,\mathbb{Z})$, since it is the special linear group of invertible matrices with integer entries, quotiented out by its centre of order 2.

We can think of the upper half-plane as Klein’s model of the hyperbolic plane, in which case the j-function is a function on the hyperbolic plane and the Möbius transformations correspond to orientation-preserving isometries. Hence, an equivalent definition of the modular group is the orientation-preserving symmetry group of a regular tiling of the hyperbolic plane: the Voronoi tessellation of the zeros of the j-function.

Recall that the group of proper orthochronous Lorentz transformations is isomorphic to $PSL(2, \mathbb{R})$. Consequently, we can think of the modular group as a discrete version of this.

### Fundamental groups and the complement of the trefoil knot

A surprising isomorphism pertains to knot theory and homotopy. Let X be a path-connected topological space. Then, we consider closed paths in X, beginning and ending at some pre-specified point. Under concatenation of paths, these form a monoid. When we quotient out by the relation ‘two paths are equivalent if they can be continuously deformed into each other’, then we get the fundamental group of X.

The fundamental group of any simply-connected space is just the trivial group, as any two paths can be continuously deformed into each other. The once-punctured plane has the infinite cyclic group as its fundamental group, since a path is determined up to continuous deformation by the number of times it encircles the puncture. The fundamental group of the twice-punctured plane is the free group on two generators, best visualised by its Cayley graph:

Now embed the trefoil knot in $\mathbb{R}^3$ and consider its complement. It is a remarkable fact that the fundamental group of the resulting space, when quotiented by its centre, yields a group isomorphic to the modular group!

This entry was posted in Uncategorized. Bookmark the permalink.

### 0 Responses to PSL(2,Z)

1. Anonymous says:

that’s not the free group on two generators…

• apgoucher says:

What isn’t the free group on two generators? The fundamental group of the twice-punctured plane or [the group corresponding to] that Cayley graph? I’m pretty sure they are both the free group on two generators.

• Anonymous says:

The cayley graph you have isn’t the free group on two generators.

• apgoucher says:

There’s only one 4-regular tree, and this is it…


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# Talk:Three-valued logic

## Does attention belong on talk or article page?

RJFJR 22:26, Feb 5, 2005 (UTC)

What is "attention"? --Loqi T. 05:38, 11 October 2006 (UTC)

Attention = the act or state of attending especially through applying the mind to an object of sense or thought b : a condition of readiness for such attention involving especially a selective narrowing or focusing of consciousness and receptivity — Preceding unsigned comment added by 77.169.168.165 (talk) 03:32, 6 November 2011 (UTC)

## Is this article defining 'a' logic or a 'class' of logics?

The intro currently says that ternary logic is a multi-valued logic. But the article is kind of reluctant to say exactly how to implement ternary logic (such as saying that the assignment of ternary values to truth values is or a permutation). Does that mean this is really a description of many different logics?

Am I splitting semantic hairs again? RJFJR 22:24, Feb 5, 2005 (UTC)

c:No, you're absolutely right, ternary logic is a collective name for a wide variety of logics. Qwertyus 13:19, 16 August 2005 (UTC)

There are a number of Ternary logics. The article provides a truth table for ONE of them (Admitedly a common one). The article's underlying assumption is that the third value is "unknown". There are different ternary logics with different truth tables that coorespond to other assumed meanings. Which is the "correct" truth table is still a matter of debate. (Particularly when adding modal operators to the ternary logic). Nahaj 15:35, 17 January 2006 (UTC)

Actually, the currently described truth table is that of Kleene's logic. It is obtained by a continuity (or strictness) principle (cf. J.P.Cleave: A Study of Logics, Oxford University Press 1991). Suggestion: replace the current truth table with an overview of different ternary logics, and create subpages for these. --Tillmo 15:38, 24 March 2006 (UTC)

You are confusing the semantics of the logic (what the values mean) with the relationships of the values in the logics. These are two separate issues.

## 0 → 0

0 → 0 = 0, isn't it? MvR 14:47, 19 October 2005 (UTC)

It is in areas like this that people argue what the "right" truth table is. They are usually heated arguments because the answer is clear For some *specific* "meaning" of the values. And since the specific meaning being used by a speaker is the "obvious" one (to them) they assume that anyone else that doesn't agree is being dense.Bottom line: Yes, there are ternary logics that have 0 → 0 = 0. And there are ternary logics where that is not the case. Each captures a different sense of "unknown" or a different sense of "→".
Personal unsolicited opinion: Wikipedia has a bad tendency, statistically, to take ONE example of something, and write as if it were THE example of something. Nahaj 15:45, 17 January 2006 (UTC)

## Why numbers?

I find the use of 0,1,2 rather confusing. Why not use *, T and F ? --Tillmo 15:42, 24 March 2006 (UTC)

I agree. Also, I think that the conjunctions would be easier to see if they were separated each into their own table. Smmurphy(Talk) 21:54, 2 June 2006 (UTC)
I don't think it needs to be split into separate truth tables, but the values certainly need to be clarified, since (even if we're sticking to one class of ternary logic) different sources will attribute different values to 0, 1, and 2.--Rinn 16:39, 15 July 2006 (UTC)

### Truth Valuation Functions

I believe there should be a discussion of the different truth valuation functions. Definitely Strong vs. Weak Kleene logic should be addressed here. I know this is addressed slightly in George Boole's "Logical Options" to some extent, and the discussion must be carried on further elsewhere.

Yes, I've addressed this with a discussion per the truth tables derived from Kleene 1952 (see entry below). Bill Wvbailey (talk) 23:52, 6 April 2011 (UTC)

## Article is problematic

It should be a disambiguation page to different 3-valued logical systems.

## Choice of symbols

I would think that something like -, 0 and + would psychologically be the most satisfying choice to represent the different truth values. Does anyone agree? -- Dissident (Talk) 03:13, 25 August 2006 (UTC)

I disagree on the grounds that 0 is usually associated with false, thus possibly causing confusion between your - and 0. I agree with someone above who suggested that F, *, and T are good choices. Or perhaps 0, 1/2, and 1. I do agree that the current 0, 1, 2 notation is very confusing. --216.254.66.59 23:47, 28 August 2006 (UTC)
• Using the symbols { -, 0, + } is one of the common notations used in balanced ternary. I agree that { 2, 0, 1 } is confusing. And { 0, ½, 1 } is less confusing, but still represents the unknown value as the same kind of thing as a known value—do two unknowns add up to a true? I think { F, *, T } or { F, ?, T } would be a much better choice of symbols. A question mark intuitively means "unknown" while an asterisk visually scans well for finding patterns.
__ The reason we use numerical bits in binary logic is because computer memory is customarily designed to address several bits together, for efficient integer arithmetic. It's just a convenience for human engineers to denote the concepts of false and true with the numeric symbols 0 and 1 and then use the same machine for anding and oring that we use for adding and dividing. It's less cumbersome to represent the boolean true as 1 than it is to write the integer ten as FFFFTFTF. But either way, bits are not little zeros and ones or Fs and Ts inside the machine, they're just conceptualized that way by us users.
__ With the { 2, 0, 1 } symbol scheme, we're asked to re-denote 0 from false to unknown, and to re-designate false as the numeral 2, while true stays as 1. But there doesn't appear to be any corresponding integer arithmetic to go with the numerical symbol scheme. That is, a string of ternary digits (tits?) doesn't normally represent a number, so why bother using numerals to represent the concepts?
I propose using the symbols { false, unknown, true } (spelled out and italicized) whenever horizontal space is not a consideration, and the symbols { F, ?, T } (not italicized) in truth tables and similar columnized presentation formats. --Loqi T. 06:02, 29 September 2006 (UTC)
Is there any fixed usage by major journals? in the most popular textbooks? Smmurphy(Talk) 02:10, 5 October 2006 (UTC)
• I don't think there's a standard numerical notation accepted by everyone. { -1, 0, +1 } is preferred for digital hardware. { 0, ½, 1 } is popular with some, and I don't know who came up with { 2, 0, 1 } but I've seen it in more than one place. Using spelled-out words has the additional minor controversy of what word to use for the neither-true-nor-false value. I've seen unknown, maybe, possible, indeterminate, inconclusive, uncollapsed and fictive. I'd go with unknown or possible. --Loqi T. 01:51, 11 October 2006 (UTC)
This probably wouldn't be appropriate since I haven't seen it in use elsewhere, but in my own tinkering I have found { -, *, + } to be the most intuitive. It eliminates the confusion of 0 being misinterpreted as False, and * is also commonly used as a wildcard for "anything" or "unknown". Maghnus 09:05, 19 March 2007 (UTC)

## Article reduced to stub length.

The {{references}} tag has been on the article since April 16, 2006, and it has only gotten worse since then. I have solved the problem by removing the vast majority of the article's content. See WP:RS#Requesting sources. Simões (talk/contribs) 01:48, 5 November 2006 (UTC)

I have replaced the bulk of what was removed. While it is true that the article is not of high quality, it is far more informative than while it was castrated. There is nothing here that is so controversial that it can't stay until the article evolves. It is difficult to provide sources because this is a somewhat obscure topic, and everyone seems to have their own interpretations. Rather than deleting everything, however, it should be sufficient to make note of this situation within the article. And if there is a particular statement that is questionable, then {{fact}} or deletion would apply there. Don't just delete essentially the entire article. WTF. 66.191.103.64 13:17, 25 December 2006 (UTC)
The article has been restubbified. A new version is forthcoming, and the version you restored, besides being unsourced (which warrants any editor to strip it down per WP:CITE, is also misleading in places. Simões (talk/contribs) 15:15, 25 December 2006 (UTC)
Yes, I have read those guidelines and there is nothing that would warrant removing virtually the entire article, even if a better version is forthcoming. "If it is doubtful but not harmful to the whole article, use the {{fact}} tag to ask for source verification, but remember to go back and remove the claim if no source is produced within a reasonable time."
If you have problems with some of the statements, tag those for citations. Otherwise, leave it alone.
Even with its (non-harmful) flaws, this is a fairly decent introduction to the concepts of ternary logic, and you are doing a disservice to anyone who would come across this article. No, most people don't go check the history page to see if someone overzealously wiped the entire article because a few statements didn't yet have references cited. 66.191.103.64 16:03, 25 December 2006 (UTC)
There were no sources, and, as I pointed out back in November, the {{references}} had been there since April. WP:CITE says, "Any material that is challenged and for which no source is provided may be removed by any editor." I challenge all of it. Any restoration of unsourced content will be reverted. I'm sorry if you dislike this application of WP:CITE, but I am of the view that reliability is more important than "informativeness." Simões (talk/contribs) 16:12, 25 December 2006 (UTC)

## Proposed revamp

I’d like to see a section on motivations for three-valued logic (future contingents, quantum mechanics, intuitionism in mathematics, etc.), and then a section on different systems (Strong and Weak Kleene, Sobocinski, Reichenbach, etc.) There should be a brief discussion, at the end, of truth-functional completeness, including mention of Sheffer connectives. If no one objects, I’ll pull this together, incorporating as much previously posted material as I can, and heeding the above comments where possible. I'll cite books by Susan Haack and Nicholas Rescher as primary sources. Comments? --Knorlin 10:50, 25 November 2006

Sounds like a good plan. What about Łukasiewicz, though? An article on trivalent logic seems to be sorrowfully incomplete without mention of him and his work. Simões (talk/contribs) 19:51, 25 November 2006 (UTC)

Right. Actually, I think I would say, "Here's Łukas. system, and the Strong Kleene system is the same except for the following difference with the if-then connective..." --Knorlin 14:07, 25 November 2006

That's reasonable. I look forward to seeing how it turns out! Simões (talk/contribs) 23:12, 25 November 2006 (UTC)

## Ternary_logic

IMHO, You have problem with ternary logic. Plz, change back the article. We maked Setun, we understand it. Ignat99 16:39, 28 November 2006 (UTC)

Aristotlean logic which provides for true and false and ${\displaystyle \sigma \upsilon \mu \beta \varepsilon \beta \eta \kappa o\zeta }$.

It is true! Ignat99 14:10, 30 November 2006 (UTC)

I mean the Soviet Union and my schools of thought (Moscow Institute of Electronic Technology) and my teacher ([Nikolai Brusentsov http://www.icfcst.kiev.ua/MUSEUM/Brusentsov.html]) and our (russian) philosophers.
>If you think otherwise, you'll have to cite the work and section number.
Брусенцов Н.П. "Блуждание в трех соснах (Приключения диалектики в информатике)" Москва, SvR - Аргус, 2000. – 16 с. и в сборнике: "Программные системы и инструменты",Труды ф-та ВМиК МГУ, №1, Москва: МАКС Пресс, 2000, с.13-23
Васильев Н.И. Воображаемая логика. Избранные труды. - М.: "Наука,, 1989.
Карпенко А.С. Многозначные логики // Логика и компьютер. Вып.4. - М.: "Наука", 1997.
Лукасевич Я. Аристотелевская силлогистика с точки зрения современной формальной логики. - М.: ИЛ, 1959.
Пиаже Ж. Логика и психология // Ж.Пиаже. Избранные психологические труды. - М.: "Просвещение", 1969.
Порецкий П.С. О способах решения логических равенств и об обратном способе математической логики. - Казань, 1884.
Слинин Я.А. Современная модальная логика. - Л.: Изд-во Ленингр. ун-та, 1976.
Стяжкин Н.И. Формирование математической логики. - М.: "Наука", 1967.
Советская научная школа до сих пор питает все научные идеи и разработки в мире. Ваша буржуазная философия чужда диалектики и направлена только на угнетение народа посредством выдуманной истории и философии и ваших законов. Вас обманывает Ваша наука.
Read Aristotle (Organon) and find there word ${\displaystyle \sigma \upsilon \mu \beta \varepsilon \beta \eta \kappa o\zeta }$(attendant, circumstantial) and think. :-)

Ignat99 07:22, 1 December 2006 (UTC)

Greek words do not end in zeta. Obviously sigma is meant...but if we are being asked to go to the original text, you really ought to get your Greek correct in the first place. Also, I just love all the original research going on! Yay Wikipedia! DnP, 19 Feb 2007 —The preceding unsigned comment was added by 24.208.78.163 (talk) 00:03, 20 February 2007 (UTC).

## Trying to avoid an edit war

I wish Ignat99 would give a more specific citation. The Future Sea Battle example in De Interpretatione, ch. 9, is the usual reference on future contingents (see my above note). But near as I can tell (without ready access to the Greek texts and without being a specialist in this area), the word sumbebekos appears elsewhere in the Organon and doesn't directly relate to the topic at hand. I think sumbebekos relates to essential versus nonessential predication, but statements of this type are still either true or false (if I understand correctly). My source on this is http://plato.stanford.edu/entries/aristotle-logic/.

I don't sense that Simoes has a problem with three-valued logic. Just wants to see some references--which is reasonable. I'm still waiting for the libary to tell me that the Rescher book has come in, and then I'll try to get something up.

Meanwhile, let me propose that since Ignat99's interests lie in the direction of computer science applications of ternary logic, maybe we can have two articles? One can be Three-valued logic and the other can be Ternary logic (computer science). To me, tenary and binary seem like terms more current in computer science than philosophical logic, anyway.

Knorlin talk 05:51, 11 December 2006 (UTC).

I don't think Ignat99 will engage in edit warring. He's been civil and is putting talk pages to good use (despite leaving sometimes confusing messages).
As for your article-splitting proposal, it doesn't really seem necessary. I think "trivalent logic" has greater mean currency in computer science and philosophy than either "ternary logic" or "three-valued logic." So maybe the article should be renamed. But since the topics in computer science and philosophy are closely related ("trivalent logic" is not an ambiguous phrase, with one sense used in philosophy and another in CS), I think it might be best to just have sections in a single article with information on specific applications and topical emphases in the respective fields. Simões (talk/contribs) 06:54, 11 December 2006 (UTC)

Okay, one article with multiple sections. Sounds like a plan. I hope to get something up on different 3-valued systems and motivation soon. Knorlin 07:21, 11 December 2006 (UTC)

I just added two basic paragraphs, not realizing that others had major revisions in mind. Reading that the article was a stub, I just assumed it had been ignored. Now I see that this is not the case. I'll leave me edits in should they prove useful in Knorlin's rewrite. If not, trash 'em. :) 2*6 10:53, 24 December 2006 (UTC)

## Draft workspace for new version.

Knorlin has offered to do a load of work offline on a new version of the article, but it's been a while since the article has been cut down. I'd like to see some progress, and there seems to be interest by a few people in collaborating on it.

I've created a page for this collaboration: Talk:Ternary_logic/Draft It's intended as a back-end workspace for the new version of the article, so please use it in place of the live page to develop the article until this sourcing controversy is ironed out.

In the meantime, I, for one, would rather have the long, unsourced version, rather than the short, uninformative version as the live article. --Loqi T. 20:22, 25 December 2006 (UTC)

## Added slight clarification of SQL 3VL

Changed the word "triggers" to "initiates an action" to avoid possible confusion with actual SQL Triggers. Also added that while Unknown and False truth values cause DML statements to disregard rows, only a False truth value will cause a row to fail the validation of a SQL Check Constraint. Both True and Unknown indicate a valid row as far as a SQL Check Constraint is concerned. Info was added to the "Ternary logic in database applications" section. The Null article also provides a more in-depth discussion of SQL's implementation of three-valued logic, so I'm adding a link to that article as well.SqlPac 14:52, 17 April 2007 (UTC)

Oops, I was so wrapped up in "triggers" I initially put trigger in there - corrected to Check Constraints.SqlPac 15:09, 17 April 2007 (UTC)

## Reworked the Ternary logic in database applications section

Are two example of SQL 3VL really necessary here? I think two examples is redundant redundancy, especially since we have several examples of SQL expressions demonstrating how SQL 3VL is handled on the Null page.SqlPac 15:20, 20 April 2007 (UTC)

I don't think the second example was necessary here. If you boil down the second explanation, it's just re-stating the explanation for the previous example again (times two since it has to describe both expressions on either side of the OR). I don't think three copies of the same explanation in the same section of the same page are necessary. I also think it's confusing. So I changed it. SqlPac 15:29, 20 April 2007 (UTC)

Without a logical operation like OR or AND, there's not actually any ternary logic going on. I tried to make the explanation as clear as possible, but for many readers it's inherently confusing material. I want to avoid readers falling into the trap of seeing a null field as a synonym for the unknown truth state—an error I've even seen some writers make in this article. And since this is an article about ternary logic, not SQL, readers must be introduced to the concept of null fields, in order for the example to make sense. Maybe there's a better way to represent an example of ternary logic in SQL, but I'm sure there's gotta be a boolean in it somewhere, otherwise it's not really demonstrating ternary logic. --Loqi T. 07:12, 13 May 2007 (UTC)

"Without a logical operation like OR or AND, there's not actually any ternary logic going on" appears, at first glance, to be a False statement. For ternary logic to be going on, one would think you need a predicate and three possible logical results. Like this:
 The predicate (10 = 10) produces TRUE
The predicate (10 < 10) produces FALSE
The predicate (10 > NULL) produces UNKNOWN

Based on your assertion, none of these examples falls under the heading of ternary logic since none have an OR or an AND, so no ternary logic is going on. The example in the article at first appears to be complex, and 3 times redundant. Here's a short introduction to SQL NULL: "SQL Null is a marker that represents a missing value." For a thorough explanation of SQL Null and SQL ternary logic (3VL), see the Good Article, Null (SQL). This article on ternary logic is not my article, so I won't be doing any more editing on it after I add your assertion to it, per Be Afraid!: "Without a logical operation like OR or AND, there's not actually any ternary logic going on." I think it's a wonderful quote that will better inform readers why the complex example is included, and why a simple SQL predicate like (10 = 10) does not fall under the scope of ternary logic (as there's no ternary logic going on). SqlPac 16:56, 13 May 2007 (UTC)
I just helped eliminate the confusion for those who might have mistakenly believed that "ternary logic" applied to simple predicates in SQL as well as compound predicates. I added your assertion that simple predicates do not fall under the scope of ternary logic in SQL (without a logical operation like OR or AND, there's not actually any ternary logic going on), and only compound predicates do. Thanks for the heads up on that fact... I really had no idea ternary logic only applied to compound predicates in SQL, and I'm sure others will be better informed by this fact. Another article might need to be created to discuss the SQL three-valued logic system and simple predicates, since simple predicates do not fall under the scope of ternary logic (or this article). BTW, can you provide a reference for this assertion? Thanks. SqlPac 20:57, 13 May 2007 (UTC)

How utterly charming, SqlPac. Did you really intend for your principal contribution toward solving this problem to be hostile comments on the talk page and vandalism of the article? WP:POINT How could you possibly think it responsible to add this passage to the text of the article? "The simple three-valued logic system employed by SQL does not constitute "ternary logic", however. Simple predicates like the example above do not fall under the scope of "ternary logic" in SQL. In fact, for "ternary logic" to be "going on" in SQL requires a compound predicate with an AND or OR keyword joining two or more simple predicate." The worst part of that contribution is not the horrible grammar or factual inaccuracies, it's that it remained on the live article for nearly two months! --Loqi T. 22:08, 5 July 2007 (UTC)

This really is quite amusing, SqlPac. First, you apply rigorous analysis to a sentence in a casual talk-page comment, thus demonstrating the casual comment is not literally true in all cases. You then re-state that casual comment into its most extreme interpretation and put quotation marks around it, implying attribution to it's original author. You then vandalize (yes, vandalize) the live article, by stating that extreme interpretation as fact, apparently on the strength of your own original research interpretation of a casual comment you found on a Wikipedia talk page. And on top of that, you ask for the casual talk page comment to be footnoted after you used it as a source in the live article. For the record, I hereby acknowledge that ternary logic is, in fact, going on in the expressions "10 = 10", "10 < 10" and "10 > NULL". --Loqi T. (talk) 16:38, 23 November 2007 (UTC)

Your concerns about simple predicates falling under the rubric of ternary logic seem to be covered by this sentence in the introductory paragraph: "For example, the SQL expression "City = 'Paris'" resolves to FALSE for a record with "Chicago" in the City field, but it resolves to UNKNOWN for a record with a NULL City field." This sentence pretty-much sums up your table above, though it admittedly omits the P=P case, "i.e. ...resolves to TRUE for a record with "Paris" in the city field..." ). I'm guessing this sentence is what you're referring to when you talk about a first example, above. When you talk about a second example, I'm pretty sure you're referring to the paragraph that introduces the OR operator. This paragraph builds on the introductory sentence that covers the simplest of cases, giving a concrete example of how UNKNOWN truth states propagate. I'm certain this paragraph could be re-worked to provide more clarity or brevity (preferably both), but deleting it doesn't help explain anything to anyone. --Loqi T. (talk) 16:38, 23 November 2007 (UTC)

## Ternary logic as a natural phenomenon

There is a very important fact: all third values are analogous. The word "unknown" does not convey in terms of known truth, while true and false do. That is okay, but the reader can known what that means in terms of truth and falsehood. Given true and false, one has as the third value "neither true nor false as both true and false". Some examples that prove the naturalness of ternary logic are: negative, positive, neutral; heavy, light-weight, weightless; light, darkness, visual detail; transparent, opague, reflective; do, don't do, multi-task; yes, no, I don't know. A third option is present for every pair of opposites that exist. With that being so, all (dualistic and monistic) phenomenon can be digitally mapped. mcampbell422

## Trivalent logic applied to AI for skeptics

If true, then accept; If false, then reject; If unknown, be skeptical. mcampbell422

## Changed False Statement

This statement was included in the section on SQL 3VL:

Comparing anything to NULL—even another NULL—results in an UNKNOWN truth state (which happens to also be represented by a NULL value).

If "NULL=UNKNOWN", then a choice needs to be made to change the 3VL truth table (or not) to reconcile the fact that SQL NULLs propagate through expressions (a basic property of NULL):

Standard SQL 3VL truth table:

( TRUE OR UNKNOWN ) → TRUE
( FALSE AND UNKNOWN ) → FALSE

"NULL=UNKNOWN" SQL 3VL truth table:

( TRUE OR NULL ) → NULL ( = UNKNOWN )
( FALSE AND NULL ) → NULL ( = UNKNOWN )

If NULL = UNKNOWN then the result of ( TRUE OR NULL ) is not TRUE as stated in this article, but rather it is UNKNOWN. In SQL, an easier way to think of it is that NULL is a "placeholder" that represents a missing value. UNKNOWN represents the result of any comparison to a missing value.

(NULL is also not a "value", it's the absence of a value, but we'll save that discussion for another day).

Good call on removing that parenthetical. Not only was it inaccurate, but also irrelevant and confusing. ==Loqi T. (talk) 20:57, 27 February 2008 (UTC)

The article referenced provides no relevant information, other than the one sentence statement that it uses three valued logic. (I.E. going to the page doesn't gain you any information.) And that statment there has the complaint that there is no citation. And I can't find any article on the language that makes the claim. So, since seeing that also provides no information, I removed it. 155.101.224.65 (talk) 16:10, 7 April 2008 (UTC)

## The section on Electronics?

The author claims that the Electronics is four- and more- valued. Ok, he has missed the topic. I would also like to ask the author to refer us to the standard that assigns the four values for the electronics simulation. The section must be removed as a mistake. --Javalenok (talk) 19:44, 10 September 2009 (UTC)

I see that this section has been moved to four-valued logic, much to the detriment of the article, and without anything on the talk page by the person who did it. Three-valued logic using 1,0, and one of either "don't care" (X,-) or "high-Z" is basic first or second semester digital electronics. The {1,0,Z} encoding is commonly used on data busses and configurable I/O pins. It is essentially a multiplexing scheme. The {1,0,X} (aka {1,0,-} in VHDL) scheme is a metalogical extention - real devices don't have such a value value, but the representation is useful for design. (The actual device will have H,L, occasionally high-Z, or even no wire at all at places marked in the design as "don't care".) Using both {Z,-} is less usual in education, but does frequently happen when dealing with real devices which frequently have tristate outputs on their configurable I/O pins in designs where one also doesn't care about the state of some of the other pins. IEEE standard 1176 defines these states as well as some others less frequently used (9 states total- 1,0,unknown, weak-signal versions of these 3, high-Z, don't' care, uninitialized), and these are used in the VHDL standard (IEEE 1076). For instance see: http://www.ece.gatech.edu/academic/courses/spring2007/ece4170/DesignDocumentation/IEEE_1076.3.pdf pages 8-9 for an older version of the standard (the current version costs money, but the basics haven't changed).Enon (talk) 20:24, 23 February 2011 (UTC)

It also might be worth noting that Thomas Fowler built one of the first digital calculators in the 1830s using three-valued logic (balanced ternary).Enon (talk) 20:37, 23 February 2011 (UTC)

## Truth table images, Kleene and Lukasiewicz logic, SQL

The intent here is to get some tables or images that compare the three logics succinctly, because right now the article is of no help whatever. (And By the way, the Łukasiewicz article is opaque, and functionally useless. There must be a better way to present it). The following is derived from Kleene 1952:

As noted here, the tables are blurry because they're pngs that are reduced in resolution. I can do these crudely in tables (see Principle of bivalence for an example of my handiwork), but to do them nicely so that they're easy to interpret and actually look like truth tables is a trauma, been there, tried it . . .. The following is what I want the tables to look like: I could create one for the SQL logic, too. I could move a table to each logic, but what's important is to get them all in one place and into a common format. Any thoughts out there before I plunge ahead?

The common logical connectives as given by Kleene to handle those instances when the predicate of one or more partial recursive functions (being computed by the same computation) has yet to produce a truth value e.g. t or f, i.e. the joint-computation consists of one or more recursions, some of which may still in progress because their mu operators are unsatisfied. Kleene calls these "strong tables" to distinguish them from "weak" tables that, when u appears in a row or column, causes the entire row or column to have the value u (these instances are shown shaded). Furthermore, Kleene deems these " regular in the following sense: A given column (row) contains t in the u row (column), only if the column (row) consists entirely of t 's; and likewise for f." This "regularity" distinguishes Kleene's from Łukasiewicz's 3-valued logic. (Kleene 1952:332-335)
The logic of Łukasiewicz differs from that of Kleene in the three tables on the right. In Kleene's terminology these tables are strong, where they differ from "weak" tables is highlighted (the pink is just to highlight the instance when Q = u and R = u and Q ≅ R evaluates to t). This logic is not "regular" per Kleene's definition. Note that Kleene's logic does not offer the (logical) equivalence relation symbolized by " ≅ " (Kleene 1952:335); rather, he uses Łukasiewicz's table when he requires the relation " ≅ ". Kleene defines " ≅ " as complete equivalence to distinguish it from his weak equivalence " ≡ " .
The logic of SQL is the same as that of Kleene (Kleene 1952:335). The implication connective (i.e. IF-THEN) symbolized by " → " can be derived from the evaluation of the formula Q → R ≅ ¬Q V R (but caution is advised e.g. Q & (R V ¬R) ≅ Q fails; see Kleene 1952:336).

At this point I am a bit confused by Kleene's use of ≅ and ≡, and I need to verify the formulas where ≅ is used, so the above will be emeneded as I make progress. Note Kleene's interesting example of the failure of the LoEM: Q & (R V ¬R) ≅ Q. Bill Wvbailey (talk) 17:43, 6 April 2011 (UTC)

## 3072 such logics

Why are there exactly 3072 such logics? — Preceding unsigned comment added by 85.64.57.89 (talk) 11:11, 25 February 2013 (UTC)

## The new truth tables are needlessly opaque for neophytes

User 103.21.126.45 (talk) changed the crystal clear (although perhaps inelegant) table

A B A and B A or B
False False False False
False Unknown False Unknown
False True False True
Unknown False False Unknown
Unknown Unknown Unknown Unknown
Unknown True Unknown True
True False False True
True Unknown Unknown True
True True True True

into

AND True False Unknown
True True False Unknown
False False False False
Unknown Unknown False Unknown
OR True False Unknown
True True True True
False True False Unknown
Unknown True Unknown Unknown

but, unfortunately, provided no explanation for how to interpret the new tables, so the change is pedagogically insensitive to readers who may not know how to read them. What are the rows? The columns? What do the interior elements of the matrix represent? Those headers were useful and shouldn't be omitted for the sake of style. I propose to change it back to the prevision exposition that spelled out the truth values in a way that most everyone will understand.

Scwarebang (talk) 10:02, 20 August 2013 (UTC)

I find the two-dimensional matrices much easier to comprehend, as they allow to see patterns in the assignment of truth values for the connectives, whereas the linearized versions read like unstructured line noise. But then again I’m a logician, so I can’t tell what is better for the uninitiated.—Emil J. 11:17, 20 August 2013 (UTC)
I like the new ones better too as they separate the connectives into different tables. Both styles are used in logic textbooks, so which one is clearer for newbies will depend on which textbook you've read. QVVERTYVS (hm?) 11:34, 20 August 2013 (UTC)
I added more explicit headers to the tables to alleviate Scwarebang’s concern about missing information.—Emil J. 12:14, 20 August 2013 (UTC)

## non English ext link, which language?

Reference 9, "http://www.aymara.org/arpasi-idioma-aymara.html" isn't in English. I'm not sure what language it is in. Can someone recognize it with confidence and mark it? RJFJR (talk) 13:54, 20 August 2013 (UTC)

It’s just Spanish.—Emil J. 14:32, 20 August 2013 (UTC)
Thank you. RJFJR (talk) 17:02, 21 August 2013 (UTC)

## SQL and L3

The current text says that SQL is based on K3, and K3 and L3 differ with respect to {A AND B} and {A OR B} which are said to yield UNKNOWN in K3 and TRUE and FALSE in K3 when A and B are both UNKNOWN. However, K3 and L3 do have the very same truth tables for AND and OR (as stated previously in the article). This can't be right. --77.185.171.151 (talk) 16:06, 26 February 2014 (UTC)


text:
Citation

## Material Information

Title:
Performance of Greedy Scheduling Algorithm in Wireless Networks
Creator:
Li, Bo
Place of Publication:
[Gainesville, Fla.]
Florida
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (114 p.)

## Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering
Computer and Information Science and Engineering
Committee Chair:
Xia, Ye
Committee Co-Chair:
Thai, My Tra
Committee Members:
Chen, Shigang
Banerjee, Arunava
Dobra, Alin
Fang, Yuguang
8/11/2012

## Subjects

Subjects / Keywords:
Algorithms ( jstor )
Graph theory ( jstor )
Linear programming ( jstor )
Mathematical vectors ( jstor )
Matrices ( jstor )
Optimal solutions ( jstor )
Regional identity ( jstor )
Scheduling ( jstor )
Topology ( jstor )
Underestimates ( jstor )
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
channel-fading -- interference -- local-pooling -- longest-queue-first -- stability -- wireless-network-scheduling
Genre:
bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Computer Engineering thesis, Ph.D.

## Notes

Abstract:
One of the major challenges in wireless networking is how to optimize the link scheduling decisions under interference constraints. Recently, a few algorithms have been introduced to address the problem. However, solving the problem to optimality for general wireless interference models typically relies on a solution of an NP-hard sub-problem. To meet the challenge, one stream of research currently focuses on finding simpler sub-optimal scheduling algorithms and on characterizing the algorithm performance. In the first piece of our work, we investigate the performance of a specific scheduling policy called Longest Queue First (LQF), which has gained significant recognition lately due to its simplicity and high efficiency in empirical studies. There has been a sequence of studies characterizing the guaranteed performance of the LQF schedule, culminating at the construction of the $\sigma$-local pooling concept by Joo et al. We refine the notion of $\sigma$-local pooling and use the refinement to capture a larger region of guaranteed performance. In the second piece of our work, we deeply analyze the performance guarantee of the LQF algorithm in order to shape the stability region furthermore. The contribution of this study is to describe three new achievable rate regions, which are larger than the previously known regions. In particular, the new regions include all the extreme points of the capacity region and are not convex in general. We also discover a counter-intuitive phenomenon in which increasing the arrival rate may sometime help to stabilize the network. This phenomenon can be well explained using the theory developed in this study. In the third piece of our work, we study the performance of LQF algorithm based on a more practical wireless network model, where channel fading effect is considered. Unlike the previous discussed network model, the wireless channel state is time varying under the effect of channel fading. As a result, the corresponding interference relationship among links could change under different channel states. Moreover, a subset of links, which determined by the current channel state, may be prohibited to transmit data and hence be pre-exclude from the scheduling process. We adopt a more generic channel fading model than that studied by Reddy et al, so that the variation of underlying interference relationship is allowed. We derive a larger stability region $\Sigma^*(G) \Lambda$, compared to the existed result, where $\Sigma^*(G)$ is a diagonal matrix. We also propose an estimation algorithm of $\Sigma^*(G)$, which provides a performance lower bound of LQF under any given channel fading structure. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Local:
Statement of Responsibility:
by Bo Li.

## Record Information

Source Institution:
UFRGP
Rights Management:
Copyright Li, Bo. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
857772222 ( OCLC )
Classification:
LD1780 2012 ( lcc )


text:
1. ## Pre-calc question

This is probably something really simple and i'm just a moron but I have a problem on my homework that says "write a standerd form equation if it passes through the point (-1,24) and has roots of x=3 and x=-i(square root of)(5)". If anyone could help me out and point me in the right dierection it would be much appreciated .

2. ## Re: Pre-calc question

I am not sure what's meant by "standard form equation" but I expect it means a polynomial with real coefficients. Remember that all nonreal roots appear as complex conjugates when this is the case. So

\displaystyle \displaystyle \begin{align*} y &= a \left( x - 3 \right) \left[ x - \left( -i\sqrt{5} \right) \right] \left( x - i\sqrt{5} \right) \\ y &= a \left( x - 3 \right) \left( x^2 + 5 \right) \end{align*}

will suffice. Use the point that lies on the curve to evaluate \displaystyle \displaystyle \begin{align*} a \end{align*}.


text:
The current GATK version is 3.4-46

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# No plots generated by the BaseRecalibrator walker

Posts: 4Member

I cannot produce BQSR plots, although I can open the grp file with gsa.read.gatkreport.

Here's the command:

java -Xmx1g -jar $shares/GenomeAnalysisTK-2.3-6-gebbba25/GenomeAnalysisTK.jar \ -T BaseRecalibrator \ -I ./0.reorder.bam \ -R$shares/ftp.broadinstitute.org/bundle/2.3/hg19/ucsc.hg19.fasta \
-knownSites $shares/ftp.broadinstitute.org/bundle/2.3/hg19/dbsnp_137.hg19.vcf \ -BQSR ./0.reorder.bam.recal.grp \ -o ./0.reorder.bam.post_recal.grp \ --plot_pdf_file ./0.reorder.bam.post_recal.grp.pdf \ -L chr1:1-1000 \ -l DEBUG \ --intermediate_csv_file ./0.reorder.bam.post_recal.grp.csv ##### ERROR stack trace java.lang.NullPointerException at org.broadinstitute.sting.utils.Utils.join(Utils.java:286) at org.broadinstitute.sting.utils.recalibration.RecalUtils.writeCSV(RecalUtils.java:450) at org.broadinstitute.sting.utils.recalibration.RecalUtils.generateRecalibrationPlot(RecalUtils.java:394) at org.broadinstitute.sting.gatk.walkers.bqsr.BaseRecalibrator.generatePlots(BaseRecalibrator.java:474) at org.broadinstitute.sting.gatk.walkers.bqsr.BaseRecalibrator.onTraversalDone(BaseRecalibrator.java:464) at org.broadinstitute.sting.gatk.walkers.bqsr.BaseRecalibrator.onTraversalDone(BaseRecalibrator.java:112) at org.broadinstitute.sting.gatk.executive.Accumulator$StandardAccumulator.finishTraversal(Accumulator.java:129)
at org.broadinstitute.sting.gatk.executive.LinearMicroScheduler.execute(LinearMicroScheduler.java:97)
at org.broadinstitute.sting.gatk.GenomeAnalysisEngine.execute(GenomeAnalysisEngine.java:281)
at org.broadinstitute.sting.gatk.CommandLineExecutable.execute(CommandLineExecutable.java:113)
at org.broadinstitute.sting.commandline.CommandLineProgram.start(CommandLineProgram.java:237)
at org.broadinstitute.sting.commandline.CommandLineProgram.start(CommandLineProgram.java:147)
at org.broadinstitute.sting.gatk.CommandLineGATK.main(CommandLineGATK.java:91)

It looks like the csv file is not being produced.

Thanks!

Tagged:

## Best Answer

• Posts: 8,434Administrator, GATK Dev admin
Answer ✓

Alright, what's failing is the method that creates the data lines to be written to the csv file. It's an extremely simple operation so there's got to be something wrong with your data. Considering the interval that you're running on is extremely short, there may not even be any valid data at all in the part of the table that fails to write. You should try running again from the first step of recalibration with a much longer interval, eg 20:10000000-20000000.

Geraldine Van der Auwera, PhD

## Answers

• Posts: 8,434Administrator, GATK Dev admin

All the data needed for the plots should be in the gatkreport; what do you see in that file? Have you tried running without specifying the intermediate csv file?

Geraldine Van der Auwera, PhD

• Posts: 4Member

The grp file looks good. I tried to use BQSR.R to make the plot, but it seems to require a csv file.
If I specify the intermediate csv file, then it contains only the header line. If not, then it is not produced.
Thanks

• Posts: 8,434Administrator, GATK Dev admin

What error do you get when you try using BQSR.R to make the plot?

Geraldine Van der Auwera, PhD

• Posts: 4Member

After making some changes for ggplot2 v0.9.3:
opts( -> theme( and theme_ -> element_

I get:

Error in distributeGraphRows(list(a, b, c), c(1, 1, 1)) :
object 'a' not found

It looks like a, b and c are created based on the lines in the csv file, and I don't have any.

Thanks
Alex

• Posts: 8,434Administrator, GATK Dev admin

OK, that makes sense. Can you please run your original command again with -l DEBUG but without specifying the intermediate csv file filename, then post the console output? I need to know if with the internal default, the csv file gets created properly for you and if not, at what point it fails exactly.

Geraldine Van der Auwera, PhD

• Posts: 4Member

Here is the output for the command above minus --intermediate_csv_file ./0.reorder.bam.post_recal.grp.csv
Thanks!

• Posts: 8,434Administrator, GATK Dev admin
Answer ✓

Alright, what's failing is the method that creates the data lines to be written to the csv file. It's an extremely simple operation so there's got to be something wrong with your data. Considering the interval that you're running on is extremely short, there may not even be any valid data at all in the part of the table that fails to write. You should try running again from the first step of recalibration with a much longer interval, eg 20:10000000-20000000.

Geraldine Van der Auwera, PhD

• ShanghaiPosts: 13Member

GATK 3.4 is used, but the option "--intermediate_csv_file" is not difined. How to generate the file *.grp.csv? Thanks a lot.

• Broad InstitutePosts: 2,192Member, GATK Dev, Broadie, Moderator, DSDE Dev admin
Sign In or Register to comment.


text:
# Math Help - Equation of line & Gradient qus.

1. ## Equation of line & Gradient qus.

Sorry for so many questions. Im just stuck on these and they are meant to be simple!

------------------------------------------------------

1) Find the equation of the line through the origin which is parallel to the line 4x - 27 - 5 = 0

------------------------------------------------------

2) The equation of a curve is y = (x-3)(x+4). Find the gradient of the curve:
a) at the point the curve crosses the y-axis
b) at each of the points where the curve crosses the x-axis

------------------------------------------------------

3) Find the coordinates of the point(s) on the curve y=3x^3 - x + 8 at which the gradient is
a) 8
b) 0

------------------------------------------------------

4) The equation of a curve s=4t^2 +5t. Find the gradient of the normal at each of the points where the curve crosses the t-axis.

------------------------------------------------------

5) If the gradient of a curve is y=2x^2 -3x -2. Find
a) the gradient at the point where x=0
b) the co-ordinates of the point where y=0
c) the gradient at each of the points found in b)

Sorry for so many questions. Im just stuck on these and they are meant to be simple!

------------------------------------------------------

1) Find the equation of the line through the origin which is parallel to the line 4x - 27 - 5 = 0

------------------------------------------------------
This is a basic line $x=k$ thus, any parallel line also has the same equation, wince it passes through the origin it must be $x=0$

2) The equation of a curve is y = (x-3)(x+4). Find the gradient of the curve:
a) at the point the curve crosses the y-axis
b) at each of the points where the curve crosses the x-axis

------------------------------------------------------
Thus, we have,
$y=(x-3)(x+4)=x^2+x-12$
If the gradient means derivative then,
$y'=2x+1$

It passes the y-axis when $x=0$ thus
$y=0^2+0-12=-12$

It passes the x-axis when $y=0$ thus,
$0=x^2+x-12=(x-3)(x+4)$
Thus,
$x=\left\{ \begin{array}{c}-4\\3\end{array}\right$

3) Find the coordinates of the point(s) on the curve y=3x^3 - x + 8 at which the gradient is
a) 8
b) 0
If,
$y=3x^3-x+8$
Then,
$y'=9x^2-1$
Yuu need it to be zero thus,
$0=9x^2-1$
Thus,
$9x^2=1$
Thus,
$x^2=1/3$
Thus,
$x=\pm \sqrt{1/3}=\pm \frac{\sqrt{3}}{3}$

1) Find the equation of the line through the origin which is parallel to the line 4x - 27 - 5 = 0
You probably have a typo here, given the relative location of the y and 7
keys I will guess this should have been:

" Find the equation of the line through the origin which is parallel to the line 4x - 2y - 5 = 0"

This may be rewritten:

$y=2x-5/2$,

and hence has slope $2$, and so the equation through the origin
parallel to this also has slope $2$ and so has equation:

$
y=2x
$
.

RonL

2) The equation of a curve is y = (x-3)(x+4). Find the gradient of the curve:
a) at the point the curve crosses the y-axis
b) at each of the points where the curve crosses the x-axis
Expanding the terms on the RHS and then differentiation gives:

$
y=x^2+x-12
$
,

so:

$
\frac{dy}{dx}=2x+1
$

a)The curve crosses the y axis when $x=0$, hence the gradient
of the curve is obtained by plugging this value of $x$ into the expression
for the derivative and so the required gradient is $1$

b)The curve crosses the x axis at $x=3$ and $x=-4$.
Plugging these into the expression for the derivative give gradients of $7$ and $-7$ respectively.

RonL

3) Find the coordinates of the point(s) on the curve y=3x^3 - x + 8 at which the gradient is
a) 8
b) 0
The derivative of $y=3x^3-x+8$ is:

$
\frac{dy}{dx}=9x^2-1
$

a) Now this is $8$ when:

$
9x^2-1=8
$
,

or:

$
x^2=1
$

or $x=\pm 1$, which may be plugged into the equation for y to
give $y=10$ when $x=1$ and $y=6$ when $x=-1$.
So the required points are $(1,10)$ and $(-1,6)$.

b)Now this is $0$ when:

$
9x^2-1=0
$
,

or:

$
x^2=1/9
$

or $x=\pm 1/3$.

Plugging these values back into the equation for $y$ gives the other
coordinate, and so we find the required points are: $(1/3,8)$ and $(-1/3,8)$.

RonL

4) The equation of a curve s=4t^2 +5t. Find the gradient of the normal at each of the points where the curve crosses the t-axis.
The points at which the curve crosses the t-axis correspond to the roots of:

$
4t^2+5t=0
$

which are $t=0$ and $t=-5/4$.

Now:

$
\frac{ds}{dt}=8t+5
$
,

so the slopes at the two points are $5$ and $-5$ respectively.

Now the gradient of the normal is -1 over the gradient of the tangent, and so
the gradients of the two normals are $-1/5$ and $1/5$ respectively.

RonL

5) If the gradient of a curve is y=2x^2 -3x -2. Find
a) the gradient at the point where x=0
b) the co-ordinates of the point where y=0
c) the gradient at each of the points found in b)
This has a confusing use of y for the gradient of the curve, and assuming
this is right there is sufficient information to find only the x-coords for the
answer to part b), and c) is redundant as it is given as 0.

RonL

10. This has a confusing use of y for the gradient of the curve, and assuming this is right there is sufficient information to find only the x-coords for the answer to part b), and c) is redundant as it is given as 0.
Sorry the question should be:

5) If the equation of a curve is y=2x^2 -3x -2. Find
a) the gradient at the point where x=0
b) the co-ordinates of the point where y=0
c) the gradient at each of the points found in b)


text:
• K R Rao

Articles written in Pramana – Journal of Physics

• Lattice dynamics of beryllium

Phonon frequencies in beryllium along the principal symmetry directions have been determined by means of the slow neutron inelastic scattering technique. The data are analysed in terms of a six-neighbour force constant model and the force constants are evaluated. It is concluded that strong tensor forces are present in beryllium and the importance of this finding to basic theories of lattice dynamics is pointed out.

• Normalisation and assessment of neutron diffraction data from liquids

A couple of normalisation criteria have been proposed to assess liquid structure data over the past decade. In this paper, a critical examination of these and other plausible criteria is made. Neutron diffraction data from liquid krypton measured by Clayton and Heaton is used to study these criteria. It is shown that if the structure factorS (Q)=1+γ (Q) satisfies the Krogh-Moe relation, namely$$\rho _{0 = \frac{1}{{2\pi ^2 }}} {\text{ }}\mathop \smallint \limits_{\text{0}}^\infty {\text{ }}Q^{\text{2}} {\text{ }}\gamma {\text{ (}}Q{\text{) }}dQ$$ whereQ is the wavevector transfer andρ0 the mean atomic number density,S (Q) is properly normalised. Subjecting such data to the other criteria would reveal, however, the quality of data in so far as systematic errors and/or termination errors are concerned. If the data is measured with reasonable accuracy to sufficiently large values of the wavevector transfer all these criteria would assess the quality of the data to almost the same extent. This is established in this paper by improving and extending the liquid krypton data by procedures similar to that of Kaplow, Strong and Averbach and subjecting this revised data to the various criteria. It is concluded, thereby, that (i) one can study the neutron diffraction data from liquids using any of the criteria mentioned in the paper to improve the quality of the data and (ii) for normalisation purposes the simple relation {ie328-2} would suffice.

• A simple white beam neutron diffraction technique

White beam neutron diffraction by time-of-flight (TOF) technique has been studied over a number of years as it is believed to be the most convenient method to investigate solids in a fixed geometry. This technique needs a pulsed neutron source and a suitable multichannel, analysing system. The entire system is in general, mechanically quite intricate and expensive.

We have investigated an alternative technique to achieve the end result of a constant geometry around the diffracting sample. This involves the use of a single crystal as an analyser to study diffraction pattern from the sample bathed in a white beam and diffracting at any fixed scattering angle. In this paper we report the results of our investigations and have compared this technique with other diffraction techniques. Taking Si, KCl and KNO3 as typical specimens we have illus. trated the results of our technique and we find that the results are comparable to those obtained by conventional neutron diffraction and TOF diffraction. The technique is simple in mechanical design and data acquisition. It can be easily adapted for high pressure diffraction which is being attempted.

• Inelastic neutron scattering from and lattice dynamics of α-KNO3

Coherent inelastic neutron scattering techniques are employed to measure several branches of the phoon dispersion relation in KNO3 in its orthorhombic (α-phase or phase II) form at room temperature. Group theoretical selection rules for external modes of the crystal have been used in the measurements along the three symmetry directions Σ(ξ00), Δ(0ξ0) and Λ(00ξ).

Theoretical investigation of the lattice dynamics of the crystal is carried out on the basis of a rigid molecular-ion model using the external mode formalism. A two-body potential consisting of the Coulombic interaction and the Born-Mayer type short range interaction is assumed. The effective charges and radii of different atoms are determined by applying the stability criterion for the crystal. Dispersion curves are calculated, representation by representation, making use of group theoretical information. Comparison of theoretical results with experimental information on elastic constants, optical data and neutron results are made. Agreement between theoretical and the various experimental results may be considered very satisfactory.

• Neutron, x-ray and lattice dynamical studies of paraelectric Sb2S3

Neutron and x-ray diffraction studies of Sb2S3 indicate extensive diffuse scattering in the plane perpendicular to the chain axis of polymer-like (Sb4S6)n molecules. The crystal structure of the paraelectric phase is said to be orthorhombic with space group D2h16 with four molecules per unit cell. The observed diffuse scattering may be due to static disorder or some dynamical effects. In this paper the authors have examined the possible dynamical origin by recourse to lattice dynamical studies. Dispersion relation of phonons along the three symmetry directionsa*,b* andc* is evaluated based on a lattice dynamical model incorporating Coulomb, covalent and a Born-Mayer-like short range interactions. Group theoretical analysis based on the group of neutral elements of crystal sites (GNES) was essential in order to examine and aid in the numerical computations. The group theoretical technique involving GNES extended to ‘pseudo-molecular’ systems is also discussed in this context.

The phonon dispersion relation shows that there are rather flat TA-TO branches of very low frequency in thea andc directions which may give rise to diffuse scattering. The branches along theb-axis are quite dissimilar to those alonga andc axes because of anisotropy. Variation of the potential parameters leads to instability of the lowest TA-TO branch. This is suggestive of a temperatures or pressure-dependent phase transition. However since these modes are optically ‘silent’ one needs to carry out either high resolution neutron scattering or ultrasonic studies to confirm various aspects of the theoretical studies.

• Lattice dynamics of MgSiO3 perovskite

A lattice dynamical study of the geophysically important mineral MgSiO3 in its orthorhombic perovskite phase, with space group Pnma (D2h16) has been carried out using a rigid ion model, with the potential consisting of Coulombic and short-ranged interactions. With the help of program DISPR, the ionic charges and radii were optimized using the equilibrium conditions. The resulting potential model is employed to predict the elastic constants and the phonon dispersion relations. The computed long wavelength optic modes are in good agreement with the corresponding experimental Raman and infrared active bands. The phonon density of states has been obtained and is used to evaluate the specific heat, the mean square displacements and thermal parameters of atoms.

• Phonon density of states of tetracyanoethylene from coherent inelastic neutron scattering at Dhruva reactor

Inelastic neutron scattering experiments to determine phonon density of states of coherent scattering samples of polycrystalline complex solids are generally intensity-limited and therefore are feasible only at high flux facilities. Phonon density of states of the monoclinic phase of tetracyanoethylene at 300 K, obtained using the medium resolution triple axis spectrometer at the new Indian medium flux reactor Dhruva are reported here. The raw data is converted to the “neutron weighted” phonon density of states by applying suitable corrections. Comparison made with results from a theoretical calculation based on a semirigid molecule model of lattice dynamics is fair. Results from Dhruva are also consistent with that obtained (to be published) at the high flux pulsed neutron source (ISIS) of the Rutherford Appleton Laboratory in United Kingdom.

• Phase transformation and electrical resistivity of tetracyanoethylene under pressure

This paper reports the phase transformation behaviour of tetracyanoethylene (TCNE) under pressure as revealed by AC electrical resistivity, its time evolution and X-ray diffraction studies. An irreversible transformation from monoclinic to cubic phase occurs at 2.1±0.1 GPa and is indicated by a sharp resistivity drop at this pressure. The time evolution of resistivity studies indicate that this transformation occurs via an intermediate phase having resistivity higher than either of the two crystalline phases. Finally, the kinetics of phase transformations obtained by time evolution of resistivity is compared with the X-ray studies on the pressure quenched TCNE.

• # Pramana – Journal of Physics

Volume 94, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019


text:
# Alpha and Beta

Number Theory Level 4

Let $$\alpha$$ and $$\beta$$ be positive integers such that $\frac{43}{197} <\frac{\alpha}{\beta}<\frac{17}{77}$ Find the minimum possible value of $$\beta$$.

×


text:
# Recent questions tagged csma-ca

0 votes
1 answer
1
Which multiple access technique is used by $\text{IEEE } 802.11$ standard for wireless $\text{LAN}$ ? $\text{CDMA}$ $\text{CSMA/CA}$ $\text{ALOHA}$ None of the options
0 votes
1 answer
2
Consider the following statements: Which of the above statement is correct? A Hidden terminals can be handled by using either of RTS or CTS control frame. B RTS/CTS exchange helps in reducing collisions and leads to efficient use of channel resources. C 802.11 frame has three address fields. D By the help of 802.11 frame, we can come to known whether encryption is used on n
0 votes
0 answers
3
Suppose that you have two nodes communicating with an AP that are hidden terminals to one another. The AP, except for acknowledgments, remains silent and there are no other transmitters in the network. The two nodes are both transmitting packets whose active ... nodes start transmitting at exactly the same time, what is the minimum number of backoffs before a packet is delivered successfully?
0 votes
0 answers
4
Suppose that you have two nodes communicating with an AP that are hidden terminals to one another. The AP, except for acknowledgments, remains silent and there are no other transmitters in the network. The two nodes are both transmitting packets whose active ... nodes start transmitting at exactly the same time, what is the minimum number of backoffs before a packet is delivered successfully?
To see more, click for the full list of questions or popular tags.


text:
# Statistics support in STACK.

The following optional packages provide statistics support in Maxima:

load("stats");

STACK provides the mode function which returns the modal value in a list.


text:
# Thirteen out of 20 emails in your inbox are junk emails. What is the experimental probability that your next email is junk?

Mar 15, 2017

$P \left(\text{junk mail}\right) = \frac{13}{20}$

#### Explanation:

Experimental probability is determined by actual results, while theoretical probability is determined by maths based on numbers.

The fraction of junk mails so far is $\frac{13}{20}$.

Therefore based on these results, there is a $13 \text{ in } 20$ chance that the next email will be junk mail. This is the same as the fraction.

$P \left(\text{junk mail}\right) = \frac{13}{20}$


text:
# Problem of first order differential equation.

Let $y$ be the solution of

$$y'+y=|x|, x \in \mathbb R , y(-1)=0$$

Then $y(1)$ is equal to

1) $\frac{2}{e}-\frac{2}{e^2}$

2) $\frac{2}{e}- {2}e^2$

3) $2-\frac{2}{e^2}$

4) $2-2e$

Now i solve it by method $\frac{dy}{dx}+Py=Q$ by finding Integrating factor

$e^x$ and then solution is given by $ye^x=\int e^x|x|dx$+c

Now further since x is in $\mathbb R$ so i make two separate cases one when $x\gt 0$ and other for $x\lt 0$, but the problem is now for $x\gt 0$ i am unable to find out the constant value. Is this correct way to deal this problem?

• You can make your life easier by using a definite integral to write down the value of the solution at 1. You can then split this definite integral as required. – Ian Mar 10 '16 at 3:31
• Which constant value? Since $y(-1)=0$, one has, for every $x$, $$e^xy(x)=\int_{-1}^x|t|\,e^t\,dt.$$ In particular, $$ey(1)=\int_{-1}^1|t|\,e^t\,dt=\int_0^1t\,(e^t+e^{-t})\,dt=\ldots$$ – Did Mar 10 '16 at 7:04
• Thanks. This thing i don't know but now i got it. – kapil Mar 10 '16 at 9:45

$$y'(x)+y(x)=|x|\Longleftrightarrow$$

Let $r(x)=\exp\left[\int1\space\text{d}x\right]=e^x$.

Multiply both sides by $r(x)$:

$$e^xy'(x)+e^xy(x)=e^x|x|\Longleftrightarrow$$

Substitute $e^x=\frac{\text{d}}{\text{d}x}\left(e^x\right)$:

$$e^xy'(x)+\frac{\text{d}}{\text{d}x}\left(e^x\right)\cdot y(x)=e^x|x|\Longleftrightarrow$$

Apply the reverse product rule to the left-hand side:

$$\frac{\text{d}}{\text{d}x}\left(e^xy(x)\right)=e^x|x|\Longleftrightarrow$$ $$\int\frac{\text{d}}{\text{d}x}\left(e^xy(x)\right)\space\text{d}x=\int e^x|x|\space\text{d}x\Longleftrightarrow$$ $$e^xy(x)=\int e^x|x|\space\text{d}x\Longleftrightarrow$$

When $x\in\mathbb{R}$:

$$e^xy(x)=\left(e^x(x-1)+1\right)\text{sgn}(x)+1+\text{C}\Longleftrightarrow$$ $$y(x)=\frac{\left(e^x(x-1)+1\right)\text{sgn}(x)+1+\text{C}}{e^x}$$

Now, when $y(-1)=0$, we can solve $\text{C}$:

$$0=\frac{\left(e^{-1}((-1)-1)+1\right)\text{sgn}(-1)+1+\text{C}}{e^{-1}}\Longleftrightarrow$$ $$0=\frac{\left(e^{-1}(-2)+1\right)\left(-1\right)+1+\text{C}}{e^{-1}}\Longleftrightarrow$$ $$0=\frac{\left(1-\frac{2}{e}\right)\left(-1\right)+1+\text{C}}{e^{-1}}\Longleftrightarrow$$ $$0=\frac{\frac{2}{e}+\text{C}}{e^{-1}}\Longleftrightarrow$$ $$0=2+e\text{C}\Longleftrightarrow$$ $$-2=e\text{C}\Longleftrightarrow$$ $$\text{C}=-\frac{2}{e}$$

So:

$$y(x)=\frac{\left(e^x(x-1)+1\right)\text{sgn}(x)+1-\frac{2}{e}}{e^x}$$

And now we can find $y(1)$:

$$y(1)=\frac{\left(e^1(1-1)+1\right)\text{sgn}(1)+1-\frac{2}{e}}{e}=\frac{1+1-\frac{2}{e}}{e}=\frac{1+1-\frac{2}{e}}{e}=\frac{2-\frac{2}{e}}{e}$$


text:
# How to generate a bounded random correlation matrix?

I'd like to know if anyone can suggest a method for generating random correlation matrix with 90% of the entries (except 1) between $[-0.3, 0.3]$. The other 10% should be larger than 0.3 or smaller than -0.3.

Thanks!

-
You have to be aware that you can't get arbitrary negative correlations between variables, for one thing. –  cardinal May 7 '11 at 1:23
Can you define what you mean by "random"? This seems related to your previous question. –  cardinal May 7 '11 at 1:29
Is the 90/10 requirement "hard"? In lower dimensions you might be able to get close by drawing from a Wishart centered at $I$, computing the correlation matrix, and rejecting samples that aren't within some tolerance. Though I suspect this won't scale well at all... –  JMS May 7 '11 at 1:52
@JMS We're starting to get some clarification in new comments to the preceding question linked to by @Cardinal: you might want to check there. –  whuber May 7 '11 at 2:19
@whuber Good, thanks. Might we close this then? It seems "duplicate in intention" as it were and doesn't contain much beyond my foolishness. –  JMS May 7 '11 at 3:25
show 1 more comment

Here's a heuristic that I coded up quickly that seems to do quite well:

1. Initialize a matrix with 1 on the diagonals.
2. Fill out the upper triangular sub-matrix according to your distribution (90% are uniform on (-.3,.3) and 10% outside that).
3. Make the matrix symmetric.
4. Now iterate between
• Project the matrix onto the PSD cone.
• Project the matrix onto the set of matrices with diagonal 1.
5. Alternating projections converges, so we just hope that the matrix we get out has values according to your distribution (see simulation for the check).
   pickone <- function(x){
if(runif(1)<.9){
return(runif(1,-.3,.3))
} else {
return(sample(c(-1,1),1)*runif(1,.3,1))
}
}

generateMat <- function(x){
X <- matrix(0,nrow=10,ncol=10)
diag(X) <- rep(1,10)
X[upper.tri(X)] <- sapply(1:45,pickone)
X <- X + t(X)-diag(rep(1,10))
Xnew <- X

for(i in 1:50){
eig <- eigen(Xnew)
##project onto the PSD cone
@cardinal Right on both counts. Thanks for the suggestions. –  ncray May 13 '11 at 0:17
(+1) For the basic idea. But, I don't think the distribution is going to be at all uniform on $[-3,3]$, especially for larger dimensions. –  cardinal May 13 '11 at 1:14

If all you care about is the proportion of entries between $\pm 0.3$ then sure - generate a random correlation matrix, compute the proportion of entries which are greater than $0.3$ in absolute value, and if there are too many pick some at random and reassign them to random values between $\pm 0.3$. Similarly if there are too few.

Edit: Never you mind, this won't work; see the comments...

-
Thanks for your answer. I have a question. First how to generate a random correlation matrix. Secondly, after reassign some numbers, the new matrix might not be a correlation matrix any more. How do I make sure it is still a correlation matrix? –  Richard May 7 '11 at 0:50
It is still positive (semi)definite if you do the reassignment in the way I describe, unless I'm terribly mistaken. There are all sorts of ways that you might generate a correlation matrix; see some of the links in the sidebar. A simple way is to draw a random matrix $A$ which is $n\times p$ with $n<p$, compute $A^TA$ and then normalize it to have ones on the diagonal. –  JMS May 7 '11 at 1:02
I am very interested in seeing this. Say I generate a random corr matrix that happens to have 20% of them between -0.3 and 0.3. Then I need to resample 70% of the entries unfiormly between -0.3 to 0.3. How do we see the new matrix is still psd? –  Richard May 7 '11 at 1:20
@JMS, Why would the matrix still be positive semidefinite necessarily? I don't believe your statement is true. –  cardinal May 7 '11 at 1:21
@cardinal, thanks for your comment. We had some discussion about the normally distributed random corr matrix. I don't think i can get the number large enough. So I want to give up normal and see if there is a general way to generate such correlation matrix . –  Richard May 7 '11 at 1:29

Here's an older answer to a similar question on SO. It has some code that you could try/modify:

Similar Question


text:
Beginner Number Theory Proofs - Common divisors and multiples

I'm taking a mathematics class where we have learned some introductory number theory - but I am having trouble with the whole 'proving this and that' component (most of it lol). Particularly with regards to exercises about common divisors and multiples. Consider the following example:

'Let a, b, d, be an element of the natural numbers. If d is a common divisor of a and b, then ab/d is a common multiple of a and b.' Prove this.

I started by supposing a, b, and d are natural numbers such that d is a common divisor of a and b. Then I get stuck - I do not know how to express d being a common divisor of a and b as a mathematical definition (what I mean by definition is that, for example, the definition of even is n = 2k for some integers n and k)

For d to be a common divisor of a and b, would it mean that:

a = dn+r for some integers n and r

b = dq+s for some integers q and s

by the quotient remainder theorem?

I'm just really lost and this is a last resort, I know you guys don't like 'homework' questions but if you can answer it generally that's fine.

• If you show some efforts, then everything is fine. – Peter Mar 31 '16 at 10:17
• Thank you very much for solutions Dhruv and Peter - I see it now! – Wharf Rat Mar 31 '16 at 10:36

Since $d$ is a common divisor of $a$ and $b$, $d|a$ and $d|b$.
$$dk_1 = a \text{ and } dk_2 = b \implies ab = d^2k_1k_2$$
$$\implies \frac{ab}{d} = \frac{d^2k_1k_2}{d} = dk_1k_2$$
The last number can be expressed as $dk_1(k_2)$ and also $dk_2(k_1)$ and thus it is a common multiple of both $a$ and $b$.
Suppose, $d|a$ and $d|b$, then there exist natural numbers $u,v$ with $a=ud$ and $b=vd$. Hence, $\frac{ab}{d}=uvd$ is a multiple of $a$ and of $b$.


text:
# where does ishmaels anger come from

where does ishmaels anger come from

##### Answers 1

This all depends on where in the book you are referring to?


text:
# Brown ring from the anion detection of NO3-

Today in our practicals I observed that when I did the wet test for nitrate (Making a 1 ml solution of salt solution+$\ce{FeSO4}$ solution+hot concentrated $\ce{H2SO4}$); The brown ring appeared for a very short time and then disappeared.

But later on I came to know that it was nitrate only. What may be the possible reason for this temporary appearance of a tinge of a brown colour and then its sudden disappearance?

• Probably transient formation of $\ce{NO2}$.
– Zhe
Oct 26 '17 at 13:21
• See this question and links therein. Nov 3 '17 at 11:19

The brown ring formed in the confirmatory test for the nitrate ion is due to the complex called nitroso ferrous sulphate, or pentaaqua nitrosyl iron (II) sulfate with the formula $\ce{[Fe(H2O)5NO]SO4}$. The concentrated sulfuric acid decomposes this complex and brown ring disappers in a few seconds when $\ce{H2SO4}$ moves to the upper layer. This is why slow addition of the acid (through the sides of the test tube) is advised.


text:
Article Contents
Article Contents

# Improvements in the computation of ideal class groups of imaginary quadratic number fields

• We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear algebra phase. We achieve a significant speed-up and are able to compute ideal class groups with discriminants of 110 decimal digits in less than a week.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:


text:
# Customized accent symbol

I would like to create a customized accent that is basically the symbol \therefore, i.e. I would like to have a letter with the "three-dot \therefore symbol" on top of it. I am new to LaTeX and I am not sure if there is an easy way to do this.

-
Please use backticks  not " to format code in questions, answers or comments. They will then be displayed in the proper font. –  Martin Scharrer Apr 27 '11 at 13:09
Sure. Thanks for editing my question. Will keep your suggestions in mind when I post my questions. Thanks. –  srikanth Apr 27 '11 at 13:19

You can use accents package.

\documentclass{article}
\usepackage{amssymb}
\usepackage{accents}

\begin{document}

$\accentset{\therefore}{A}$

\end{document}

DON'T use \stackrel, it is designed for math relations. And don't use \overset from amsmath, the postion is not correct.

-
Thank you for clearing that up. I will delete my answer with bad recommendations. –  ipavlic Apr 27 '11 at 13:14
It works. Thank you very much for your help. –  srikanth Apr 27 '11 at 13:18

I just wanted to show how trying to be really clever can get you shot in the foot:

$\dot{\ddot{a}}$
\bye

:-D`

-
Sounds like a great way to typeset the name of really, really heavy metal band. –  Emre Apr 27 '11 at 15:22


text:
# How did Pythagoras come up with a^2+b^2=c^2?

1. Feb 15, 2008

### thharrimw

I have wondered about this for a while now and can anyone tell me the proof behind A^2+B^2=C^2?

2. Feb 15, 2008

### HallsofIvy

Staff Emeritus
3. Feb 15, 2008

### LukeD

I personally like the proof that uses the simple fact that if vectors u and v are orthagonal, then the dot product of u and v is 0. Pretty much gives the result immediately without needing any pretty pictures (as nice as those pretty pictures are)

Let u and w be the vectors corresponding to the 2 legs of your triangle. The definition of a right angle, is one that makes $$u \cdot w = 0$$. Also note that for any vector v $$\lvert v \rvert^2 = v \cdot v$$ where $$\lvert v \rvert$$ means the length of the vector v. Note that the hypotenuse is the vector u+w

You also need to know that the dot product is pretty much just like multiplication, so it distributes over addition, meaning that $$u \cdot (v + w) = u \cdot v + u \cdot w$$ and $$(u + v) \cdot w = u \cdot w + v \cdot w$$

Then $$\lvert u+w \rvert^2 = (u+w) \cdot (u+w) = u \cdot (u+w) + w \cdot (u+w) = u \cdot u + 2 (u \cdot w) + w \cdot w = v \cdot v + w \cdot w = \lvert v \rvert^2 + \lvert w \rvert^2$$

If you know how to work with vectors, that will probably be the shortest, most direct proof of the pythagorean theorem you will ever see

4. Feb 15, 2008

### CRGreathouse

The Pythagoreans probably found the theorem (directly or indirectly) as a rule of thumb from the Egyptians, who were good practical geometers. Pythagoras was the first, I believe, to prove that the rule holds (exactly), unlike the various approximations of the day.

The theorem was probably first proved by dissection -- I can't remember, is this known?

5. Feb 15, 2008

### robert Ihnot

How did Pythagoras come up with a^2+b^2 = c^2?

There are the books of Euclid, often you can find something right out of the big bookstores, and so if one is really interested that is a question easily researched.

The proofs by the Greeks involved geometry rather than algebra, so they are, to us anyway, harder than proofs by algebra. However, when I took high school Geometry I can remember we covered the Pythagorean theorem.

6. Feb 15, 2008

### robert Ihnot

LukeD: I personally like the proof that uses the simple fact that if vectors u and v are orthagonal, then the dot product of u and v is 0. Pretty much gives the result immediately without needing any pretty pictures (as nice as those pretty pictures are)

I wonder how legitimate that is. Why not define a right angle such that the square of the sides equals the hypothesis?

7. Feb 15, 2008

### LukeD

Try out that definition, see what you get.
If $$\lvert u + w \rvert^2 = \lvert u \rvert^2 + \lvert w \rvert^2$$ is your definition of having a right angle between u and w, then what does that imply about $$u \cdot w$$?

8. Feb 15, 2008

### John Creighto

9. Feb 18, 2008

### dodo

Personally I like this one, but I'm not sure if its origin is Indian or Babylonian.

The triangles, while in the red position, make a^2 + b^2 (the two black squares), while moved to the pink position they make c^2.

#### Attached Files:

• ###### Pythagoras.png
File size:
912 bytes
Views:
50
10. Feb 26, 2008

### M Grandin

Simplest proof

Simplest possible proof I derived myself long ago:

According to similarity , if the big triangle area is k* c^2 then it contains two triangels of area k* a^2 resp k* b^2. Therefore k* c^2 = k* a^2 + k* b^2 and hence
c^2 = a^2 + b^2

11. Feb 26, 2008

### mathwonk

it depends what you think the theorem says. you can regard it as a statement abiout numbers, i.e. a relation between the squares of the lengths of the sides of a right triangle, but euclid did not view it this way.

he viewed it as an equivalence between the geometric figure of square erected on the hypotenuse, and the two squares erected on the sides. And he had no theory of area at that point.

what he essentially proved was that it is possible to decompose the larger square into pieces which can be reassembled to form the other two squares. actually he did not quite prove this, but his same method of proof does do this.

i.e. one can define several different equivalence relations:

1) One figure can be decomposed into pieces which can be reassembled to form another figure.

2) two figures can be decomposed into pieces, and both sets of pieces can be reassembled together with the same third set of pieces, to form the same figure.

3) one can assign an area to every figure in such A WAY that figures which can be related in the ways above have the same area, and then the two figures have the same area.

and so on. we tend to think of all these as the same, but actually defining area cREFULLY IS QUITE hard, and euclid did not do it. even showing the previous relations are transitive is some work, and eucloid did not do that either.

the proof that the numbers obtained by squaring the sides and adding are the same, does follow from the theory of similarity, but euclid did not have that theory in plACE AT THE TIME OF stating the pythagorean theorem.

so actually euclid did not prove the theorem at all, but gave only a partial proof.

i.e. a complete proof needs either a theory of similarity, or a theory of area, or the proof that the relations euclid used are transitive, none of which are in place fully in euclid, at the time he gave the theorem.

the treatment in hartshorne's geometry, euclid and beyond, is excellent.

Last edited: Feb 26, 2008


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# How do I map a 3D triangle into 2D?

The problem I'm having is mapping a 3D triangle into 2 dimensions. I have three points in x,y,z form, and want to map them onto the plane described by the normal of the triangle, such that I end up with three points in x,y form.

My guess would be it'd assigning an arbitrary up vector... and then... doing something? Finding the distance traveled along the plane from one vertex to another...? What do I do, and how do I do it?

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You have not specified the problem well enough. Do you have three points $(x_1,y_1,z_1), (x_2,y_2,z_2), (x_3,y_3,z_3)$ to map to two dimensional points? The simplest is to ignore the third coordinate. This is not as stupid as it sounds-you are projecting the triangle on the $xy$ plane. If you want to project onto another plane, how is it defined?
If your vector is normal (perpendicular) to the plane you want to project onto, the operation is well-defined. You can rotate so that the vector is along $z$, ignore the last coordinate, then rotate back. You will have three points, each with three coordinates, that lie in a plane perpendicular to the vector you have chosen. Is that what you want? –  Ross Millikan Jul 8 '11 at 3:42


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10.09 ANN Architectures¶

Apart from multiple perceptrons interconnected with each other, several neural network architectures have been developed over the years. We did only see the autoencoder but many others exist.

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Deep Architectures¶

• Deep Neural Networks are simply networks with lots of layers, but training layers deep through backpropagation turns to be hard. A vanishing gradient problem it tackled by careful selection of activation functions.

• Convolutional Neural Nets are networks with one or more convolutional layers, these layers are not fully connected providing feature selection based on parts of the input.

• Recurrent Neural Networks have connections going backwards, i.e. the output of a neuron feeds into the input of a neuron in the same or previous layer. RNNs do not need to be deep networks but excel as such.

• Long Short Term Memory (LSTM) are specifically constructed RNNs, with very specific activation functions per layer.

• Autoencoders are DNNs which can repeat the patterns they were presented with.

Other Architectures¶

• Hopfield Networks are early autoencoders, these could repeat a known pattern when presented with a similar one.

• Boltzman Machines are networks of probabilistic neurons where all neurons are connected in all directions. The input and output is done from the same neurons (visible neurons).

• Restricted Boltzman Machines are BMs in which the hidden layer neurons are not interconnected. These are much easier to train than full BMs.

• Deep Belief Networks are stacked RBMs on top of each other. Each RBM can be trained separately, and we can stack several layers or RBMs. These were the early DNNs.

• Self-Organizing Maps are unsupervised networks for data visualization and dimensionality reduction. They use the concept that connections through which data passes should be reinforced whilst all other connections should decay.

ANN Libraries¶

Today we still experience very fast evolution of NN techniques, and thanks to that also a very fast evolution of NN libraries. Some relevant names, all of which perform better than sklearn on NN building, follow.

Theano¶

Was one of the first academic libraries that allowed transparent use of GPUs as if one was working with NumPy array. Theano started the trend of using a directed acyclic graphs DAG, to quickly compute derivatives. It was developed at the university of Montreal and is still used in many places. Currently new features ceased to be added to Theano because it was considered counter productive to develop it further as many libraries with commercial backing can do the same.

Tensorflow and Keras¶

The current top library for most NN computing in industry. It really is just a DAG processing engine on top of tensors. Where tensors are pretty much NumPy arrays with a couple of tricks to allow for quick computation of derivatives. Its main selling point is tensorboard a web UI to monitor the processing of the graph, and therefore monitor the network training. Tensorflow adopted the Keras library as its front end, allowing for a user friendly API.

Issue with Tensorflow happen when one attempts to extend the library. The focus of Tensorflow is industry, and hence the quick development of known models. Research of completely new NN models can be tricky in Tensorflow.

A demo of a simple tensorflow (limited to a handful of 2-dimensional problems) interface can be found at tensorflow playground

Torch (pytorch)¶

In the Python universe known as pytorch is a library written (mostly) in Lua. The library allows for almost complete freedom of matching and mixing NN parts. It is hence often used to research new methods in NNs.

Just as its predecessors, Torch uses a DAG to compute derivatives. But it borrows from research in automatic graph building, and require less steps to preform the differentiation. Differentiation in torch is as easy as writing a function and asking for a derivative with a procedure. There is support for almost all array operations to be differentiated automatically.

JAX, FLAX and Friends¶

When one needs to go even further in the freedom of mixing and matching one goes all the way to the automatic differentiation libraries. The original library was autograd and one an still build reasonable NNs with it.

The issue with autograd use is that it does not support GPU computations. For automatic differentiation and nothing else we got JAX. And on top of JAX we have currently a rather new development of flax, which will provide a NN interface to the automatic differentiation.

Hence, for ultimate NN building freedom one would use autograd, or JAX if one has a GPU. But the API of these libraries can be quite a handful, and one need to understand NN internals very well in order to get anything out of them.


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# Complete factorisation of $x^8-2x^4\cos (4\theta)+1$ with roots of unity

Factorize completely $$x^8-2x^4\cos (4\theta)+1$$ using complex numbers and $$n$$-th root of unity.

The answer given is $$\prod_{r=0}^3 \left(x^2-2x\cos\left(\theta+\frac{r\pi}{2}\right)+1\right)\,.$$ Can anyone please help . I just want to know how to tackle this . I just want the conceptual trick to solve such sum.

• I don't know how to write product series in mathjax so i have written as statement product of.... as r runs.. – user782514 Dec 28 '20 at 13:48
• Thanks @ Jack LeGrüß for the required edit. – user782514 Dec 28 '20 at 13:56
• try this break up into $$(x⁴-(cos4θ+isin4θ))(x⁴-(cos4θ-isin4θ))$$ – A plus b Dec 28 '20 at 14:09
• @A plus b thanks to enlighten me. – user782514 Dec 28 '20 at 14:11

\begin{aligned} x^8-2x^4\cos4\theta+1 &=x^8-x^4(e^{4i\theta}+e^{-4i\theta})+1 \\ &=(x^4-e^{4i\theta})(x^4-e^{-4i\theta}) \\ &=(x^2+e^{2i\theta})(x^2-e^{2i\theta})(x^2+e^{-2i\theta})(x-e^{-2i\theta}) \\ &=(x+ie^{i\theta})(x-ie^{i\theta})(x+e^{i\theta})(x-e^{i\theta}) \\ &\cdot(x+ie^{-i\theta})(x-ie^{-i\theta})(x+e^{-i\theta})(x-e^{-i\theta}) \\ &=\prod_{r=0}^3(x-i^re^{i\theta})(x-i^{-r}e^{-i\theta})\quad{(*)} \\ &=\prod_{r=0}^3[x^2-x(i^re^{i\theta}+i^{-r}e^{-i\theta})+1] \\ \end{aligned}

in which the asterisk step follows from interchanging each linear factor.

Now, using the fact that for every integer $$i^r=e^{\pi ir/2}$$ for all integer $$r$$, we obtain

\begin{aligned} x^8-2x^4\cos4\theta+1 &=\prod_{r=0}^3\left[x^2-2x\left(e^{i\theta+i\pi r/2}+e^{-(i\theta+i\pi r/2)}\over2\right)+1\right] \\ &=\prod_{r=0}^3\left[x^2-2x\cos\left(\theta+{\pi r\over2}\right)+1\right] \end{aligned}

With the 4th root of unity $$e^{\pm i\frac{a+2\pi r}4}$$ for $$x^4 - e^{\pm ia}=0$$, factorize as follows \begin{align} x^8-2x^4\cos (4\theta)+1 =&\> (x^4-e^{i 4\theta})(x^4-e^{-i 4\theta}) \\ = &\prod_{r=0}^{3}(x-e^{i\frac{4\theta+2\pi r}4}) \prod_{r=0}^{3}(x-e^{-i\frac{4\theta+2\pi r}4}) \\ = &\prod_{r=0}^{3}(x-e^{i(\theta+ \frac{\pi r}2)}) (x-e^{-i(\theta+ \frac{i\pi r}2)}) \\ = &\prod_{r=0}^{3}(x²-2x\cos(\theta+\frac{\pi r}{2})+1) \end{align}


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A ray of light passes from air into water at an angle of incidence of $50^{\circ}$. What is the angle of refraction? (HINT: You will need to use the $\sin ^{-1}$ function on your calculator).
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