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Talk:Cramer's rule

When did Gabriel Cramer come up with Cramer's Rule?

ANS: Around 1750, two years before his death. He died without giving an explanation of how exactly he came up with the rule. There is some speculation that Colin Maclaurin came up with the same rule in 1729 but there must have been a paper shortage in Scotland because it was never published until after he died.

A search on google for "cramer's rule maclaurin" comes up with several pages discussing who came up with it first. This one (http://www.findarticles.com/p/articles/mi_qa3789/is_200110/ai_n8969722) says that Maclaurin published it in 1748, two years before Cramer did, but that Cramer's probably gained popularity due to his superior notation.

More precise statement
Any chance someone is willing to give a more precise statement of Cramer's rule? As in the following: Let R be a commutative ring, M a finitely generated R-module with generators $$m_i$$, and $$\phi:M\rightarrow M$$ an endomorphism of R-modules. $$\phi$$ determines a matrix

$$A=(a_{ij})$$

via $$\phi(m_i)=\sum a_{ij}m_j$$. Then Adj(A)A=det(A)I, where Adj(A) is the cofactor matrix of A.

 * That is not Cramer's rule, though it is related, but in a not so trivial way. There is no general rule for solving square linear systems over a commutative ring for the simple reason that they generally do not have unique solutions (try solving $$3x=b$$ in Z/6Z). Something can be said (if the determinant of the system happens to be an invertible element of the ring for instance) but I'm not sure this article is the right place. Marc van Leeuwen (talk) 14:55, 7 September 2009 (UTC)

Can we also get an easier to understand version of Cramer's rule? If anyone could previde that I'd be mighty grateful.Phinixx (talk) 21:46, 7 January 2014 (UTC)

 * It might help if you say what you find hard to understand. (For a WP math article, that version is pretty straight-forward!) -- Elphion (talk) 17:51, 8 January 2014 (UTC)

Does the example of comparative statics in econ make any sense?
am I right that there's nothing special about comparative statics, it's just an example of solving a system of linear equations, nothing particularly about cramer's rule. This latest edit says it's of 'further significance', but it doesn't seem to be. --Tristanreid

 * I don't find that example much useful either. Oleg Alexandrov (talk) 23:52, 17 October 2005 (UTC)

 * ok, I'm yanking it. Thanks for signing me, by the way. I forgot last time. Tristanreid 02:44, 19 October 2005 (UTC)

Where did Gaon come from?
Vilna Gaon seems to have come out of left field. I note that the main article on him includes a similar statement without attribution. Does anyone have a reference for this statement? S.N. Hillbrand 20:45, 22 December 2005 (UTC)
 * Thanks for removing that. Looked like a strange edit. Oleg Alexandrov (talk) 01:25, 23 December 2005 (UTC)

I actually remember it being printed in my algebra book as a possible etymology. I really don't know if there is a credible source for it. I think they might have cast doubt on it themselves. I have no idea how it even got started, but unless I am mistaken, it was in my HBJ algebra book issued by the Los Angeles Unified School District. PhatJew 19:18, 9 March 2006 (UTC)

The reference to the Gaon is a legitimate one that is quoted in many sources. It should be kept as an alternative entymology.

Where has the nice proof gone?
About a year ago there was a beautiful proof of the Cramer's rule on this wikipedia page. Can you, please, bring it back? Thanks Pavel.Pokorny@vscht.cz —Preceding unsigned comment added by 147.33.113.54 (talk) 17:05, 6 December 2007 (UTC)

Incorrect Example
The 4th image in the example section is incorrect. The determinant in the numerator is calculated in correctly. Sorry I cannot fix it right now,will someone else?--128.101.152.71 (talk) 21:12, 10 March 2008 (UTC)

 * Done. Michael Slone (talk) 01:32, 11 March 2008 (UTC)

Re: Relevance to eigenvalues

 * "The numerator will always be zero, since here c=0."

I can't find any previous reference to c. Is this a remnant of an earlier edit? -- 72.224.136.152 (talk) 14:41, 23 September 2009 (UTC)
 * OK, the problem came from an earlier edit. Variable c was changed to b but a later reference to it remained unchanged. -- TheMaestro (talk) 02:08, 24 September 2009 (UTC)

All right, thank you. But here is another question: what does this section have to do in this article in the first place? Cramer's rule is about the case where the coefficient determinant is non-zero, in which case it gives an explicit expression for the solution. But in the case of homogeneous linear equations this is quite unnecessary: it is obvious that the zero vector will always be a solution, so you don't need a complicated formula to tell you that in case the coefficient determinant is non-zero. The only relevant point here is that the solution will be non-unique precisely when that determinant is zero; although that could possibly be considered to be a (minor) part of the statement of the theorem called Cramer's rule, it is not even mentioned in the current article. (If it were, it should be mentioned more precisely that coefficient determinant zero implies either a non-unique solution for the whole system or no solution at all, since the article talks about non-homogeneous systems; also beware that when the determinant is zero there could still be a unique solution for some of the unknowns.) My feeling is that this subsection does not belong here, and should be removed, or possibly moved to an article is it more relevant to. Marc van Leeuwen (talk) 06:50, 24 September 2009 (UTC)
 * I agree, I think the connection to Cramer's rule is tangential at best, in more senses than one. -- TheMaestro (talk) 01:50, 25 September 2009 (UTC)

Cramer's Rule is useless
I would dispute all of the (limited) claims for useful applications for Cramer's rule. The cost of applying the rule grows exponentially with dimension and I can't imagine why anyone would want to use it in practice.

Certainly, it has no role in the solution of linear systems, however small. The claim at the start of the article that "as no pivoting is needed, it is more efficient than Gaussian elimination for small matrices" is plainly wrong...pivoting is extremely efficient and improves the quality of the solutions. Would anyone like to claim a stability result for Cramer's rule? And the best way to compute the determinants used in Cramer's rule is to use Gaussian elimination, anyway. Furthermore, why would anyone use a SIMD machine to solve small linear systems? How on earth can you do this efficiently? The phrase "This formula is, however, of limited practical value for larger matrices" would be better stated as This formula is, however, of NO practical value" wherever it appears.

"Cramer's rule is of theoretical importance in that it gives an explicit expression for the solution of the system". Why is this important? Why is this better than x = A^{-1}b? Cramer's rule can be used to give an explicit representation of an individual component independently of all other components...I have heard this used as justification for its importance but, again, this could be done much more efficiently with Gaussian elimination (exploiting the resulting LU factorization).

Can anyone substantiate the line "Cramer's rule is also extremely useful for solving problems in differential geometry"? I would expect that anything it can do can be done better another way. But I am willing to be proved wrong.

Applications to algebra/relevance to eigenvalues: is it really worth mentioning that the rule can be used to prove well known results in an ugly way?

I apologize to the Cramer family for being so negative, but I have seen no evidence that the rule is anything more than a historical curiosity.

If no-one wants to refute my claims, I will happily edit the page to conform with my world view. Not that it will stop climate change, or anything :-) —Preceding unsigned comment added by 130.159.17.136 (talk) 17:18, 10 December 2009 (UTC)

 * First, wikipedia articles should not be written to confirm with any particular world view, but should reflect general consensus. If everyone considered Cramer's rule useless, then mankind would have forgotten about it long ago and this article would not have existed.

 * Now some more specific points. Although the greatest importance of Cramer's rule is definitely not in computer programming (and Cramer was not into computer science), it seems likely that, at least when one comes across a situation where a program needs the solution of a specific 2×2 system, the easiest solution will be to code the solution by Cramer's rule directly rather than to invoke some general equation-solving procedure involving Gaussian elimination. You say writing x = A^{-1}b is a theoretical alternative to Cramer's rule, but it is of no use unless you have an expression for A^{-1}. Such an expression exists... and is a blown-up version of Cramer's rule! In fact if you imagine any system where the coefficients are not completely known numbers (say they depend on one or more unspecified parameters; the worst case is that all coefficients are independent parameters), then one cannot apply Gaussian elimination or any other method that needs to make decisions depending on the values of the coefficients. Cramer's rule on the other hand needs to make no decisions at all (all that matters is that the denominator is nonzero, which is the condition for a unique solution to exist in the first place) and can be applied to such a system. That is exactly why the rule is of theoretical importance. There are many occasions in algebra where it is important for reasoning to know that an expression of a certain kind exists, even if one has no desire to actually use such an expression in computation. Marc van Leeuwen (talk) 08:23, 11 December 2009 (UTC)

 * Thank you for your nice answer. Looking at modern texts on matrix algebra, I don't think I'm alone in branding the method useless and I'm getting impatient for consensus to move my way. I've bored my colleagues into submission, and now it's time for the rest of the world!

 * Let me address your specific points. With reference to the explicit representation of the solution of a 2x2 system, I agree that it would seem contrary not to use a formula involving the determinant. However, this solution existed well before Cramer's rule. For bigger systems, the story changes. Even an explicit expression for 3x3 systems involving expanded determinants would be inefficient and offers no insight. I would draw an (imperfect) analogy with polynomials: compare the use of the quadratic formula, which is ubiquitous in textbooks and considered useful, with the cubic and quartic formulae. So I'll modify my claim to "Cramer's rule is useless for n > 2".

 * Let me move on to the theoretical importance. Once it is known that an inverse exists, I do not see how the fact that hypothetically one could write down an explicit representation of the inverse in terms of its entries using Cramer's rule adds anything to algebraic reasoning. As you imply, one cannot prescribe a general order of pivoting which will guarantee that Gaussian elimination does not encounter a zero denominator. But for any value the entries can take, I know that such an ordering exists so long as the matrix is nonsingular. Thus, as with Cramer's rule, I know an expression of a certain kind exists, even if I have no desire to use it explicitly. In fact, I know all sorts of expressions exist simply as a consequence of the nonsingularity of the system. This simple fact is extremely useful with or without an explicit formula. What, in particular, does Cramer's rule add? I would contend that explicit use of Cramer's rule is relegated, again, to the case n = 2. —Preceding unsigned comment added by 130.159.17.136 (talk) 17:36, 14 December 2009 (UTC)

 * It would be nice if you could cite algebra text that states Cramer's rule and then says that it is plain useless (not just "not very useful for computation"). Actually even for computation it is not that bad, given that one may evaluate the determinants in any way one likes, and there are certainly methods that do not need (super)exponential time in the size to do that, at least when coefficients are numbers. But this is not the main point, most people will agree that solving an explicit linear system is usually better done by other means.

 * However for you theoretical remarks I firmly disagree. One theoretical point of huge importance is that the inverse of a square matrix exists if and only if its determinant is invertible (I say that rather than "nonzero", because I am also referring to matrices with entries in a ring rather than a field, such as matrices with polynomial entries). This fact is very closely related to Cramer's rule (some would say it is part of Cramer's rule, and in any case it can be derived immediately from it), but there is no way one could arrive at this result by just studying what happens in Gaussian elimination. So even your starting point "once it is known that an inverse exists" is using Cramer's rule: unless your knowledge that an inverse exists is based on actually knowing the inverse (in which case there is nothing left to do), I cannot see how you could be sure without knowing the determinant sufficiently well to be sure it is invertible. When you say "for any value the entries can take, I know that such an ordering exists so long as the matrix is nonsingular" this is true only if the entries take values in a field (where every nonzero element is invertible); with entries that are polynomials for instance, it could well be that the matrix is (always) invertible, but yet one cannot perform Gaussian elimination.

 * Here is a theoretical question that Gaussian elimination cannot answer for you, but Cramer's rule can: suppose you have a system depending on (one or more) parameters that can vary continuously, and its coefficient matrix is always nonsingular, so that for all parameter values there is a unique solution. Does this unique solution vary continuously with the parameters? Using Gaussian elimination, there would be discontinuous changes in the solution procedure, as coefficients becoming zero at some parameter values force you do change pivots, and one cannot see why this could not cause discontinuous changes in the solution. From Cramer's rule however continuity is immediate: the solution is expressed as quotient of two continuously varying expressions, of which the denominator never becomes zero (because this would make the coefficient matrix singular), so the result certainly varies continuously with the parameters. This is by no means the only theoretical result where Cramer's rule (or some result equivalent to it) plays a role, there are many more in abstract algebra, but it provides a simple concrete example. Marc van Leeuwen (talk) 08:01, 15 December 2009 (UTC)

 * Marc van Leeuwen pretty much says it. There is a widespread dogma that determinants in general (and Cramer's rule in particular) are "useless", seemingly perpetuated by mathematicians whose definition of usefulness seems to be proportional to their own ability to write fast Fortran code.  I have always dismissed such claims.  Determinants are used throughout mathematics, and anyone who believes that they are "useless" needs to get out more.  Sławomir Biały  (talk) 17:04, 16 December 2009 (UTC)

 * With regard to one of the questions above, regarding applications of Cramer's rule to differential geometry, many of the properties of the Hodge star operator tacitly rely on Cramer's rule and variants thereof. Sławomir Biały  (talk) 17:05, 16 December 2009 (UTC)

 * Let me approach this differently then, as I am not an abstract algebraist. None of the applications for Cramer's rule given on this page back up the claim for it's theoretical importance. If you have convincing examples, which it sounds like you do, why not add them. As a computational tool, though, Cramer's rule should be avoided and the literature that claims it is effective as an algorithm on parallel machines seems to be based on the conception that the formula for individual components offers something that the formula e_i^T A^{-1} b does not. This formula makes it clear that each individual component of the solution is, in general, dependent on every entry of the inverse matrix. Having read through a lot of papers making claims for Cramer's rule-based parallel algorithms (including misleading or unverifiable claims about stability) I certainly feel justified in rewriting the relevant sections once I've collated the appropriate references. One point of agreement is that the computational cost of Cramer's rule is over stated in many texts but, even if the determinant is computed efficiently, it is still impractical.

 * On the more theoretical side, I think that there is an undue emphasis on determinants in linear algebra. In my opinion, this is down to the historical precedence of determinants over matrices. However, I will willingly accept that determinants are a fundamental tool. But apart from anything self-referential, the fundamental results in matrix algebra (over R or C) can be stated and proved without determinants (see "Down with Determinants" in AMM, for example) by using eigenvalues, for example, as atoms. And I don't see any need to throw personal insults at people who disagree with me. I don't think anyone has lived until they've expanded a 5x5 determinant! —Preceding unsigned comment added by 130.159.17.136 (talk) 13:30, 17 December 2009 (UTC)

 * The Bareiss algorithm is a computationally efficient multistep algorithm implementing Cramer's rule. It is supposed also to be more numerically stable than ordinary Gaussian elimination for matrices that are "nearly singular", but I do not know in what sense this is true.  Most computer algebra systems include this in the symbolic toolkit because it works over arbitrary commutative rings.  Also, while it is probably true that many of the main results of matrix algebra can be achieved without determinants, at least over the complex field where eigenvalues are available, this is not a convincing argument that determinants are "useless".  Pick a linear algebra text at random and look up "characteristic polynomial" in the index, for instance.  In fact, here is an extremely instructive exercise that should explode any doubts about the usefulness of determinants: Working exclusively over the rational field (no field extensions are allowed), devise an efficient way to compute exactly the characteristic polynomial of a matrix without the use of determinants.  The restriction on not passing to a field extension is not as artificial as it might seem at first: exact representation of roots in a manner suitable for symbolic computation is itself a nontrivial task.  If I posed this as a final project in my graduate algebra course, I wonder how many of my students would be able to do it.    Sławomir Biały  (talk) 14:29, 17 December 2009 (UTC)
 * If your students have access to lambda matrices stuff or the proof of the rational canonical form stuff they are not going to have problems computing it without determinants. O(n^3) algorithms are known.   franklin   16:21, 17 December 2009 (UTC)
 * This is the general idea, at least as a corollary of the structure theorem for modules over a PID. Still, there is a rather instructive gap between knowing the proofs of these results and being able to put together a practical method, would you not agree?  Sławomir Biały  (talk) 16:28, 17 December 2009 (UTC)
 * Also there are polynomial-time algorithms for computing determinants as well. E.g., Dodgson condensation is O(n3) too.  So arguments that determinants are more computationally intensive don't really wash.   Sławomir Biały  (talk) 16:32, 17 December 2009 (UTC)
 * Characteristic polynomial and determinants are equivalent problems. I don't think any argument to rule out determinants can come out of it. About your students, they are graduate students, they will manage. Just let them have the tools.  franklin   16:59, 17 December 2009 (UTC)
 * Interesting point about determinants and the characteristic polynomial being equivalent. From that perspective, I suppose the difference is somewhat subjective.   Sławomir Biały  (talk) 17:06, 17 December 2009 (UTC)
 * Let me reiterate: I am not calling determinants "useless". I recognize they are indispensable in many situations. I personally think that their significance is over-emphasized in many areas, though. I am not suggesting that the characteristic polynomial isn't a beautiful thing. But that is not the topic here. I come here only to damn Cramer's rule! I yield to those of you who claim that Cramer's rule is theoretically important in abstract algebra, but I contend that the examples given on the enclylopedia page do not bear this out.
 * So let me return to something I still think can be emphasized, and that is that it is computationally useless. As has been pointed out, determinants can be computed efficiently in many ways (and inefficiently, too). Using Cramer's rule, an individual element of A^(-1)b requires the computation of 2 determinants. Every subsequent component requires a further n x n determinant computation. For the cost of a single determinant computation, the whole solution can be computed stably using an elimination method or through orthogonal transformations (as far as I can tell, the most efficient determinant computations are carried out using elimination techniques).
 * The stability results I know for Cramer's rule are sketchy, but I know it's instability has been highlighted for 2x2 systems. Since it's efficient computation would be on the back of some variant of Gaussian elimination, I can't see it being more stable. I am not familiar with current implementations of the Bareiss algorithm. Can you give a reference for the stability claims? I suppose these depend on definitions of "nearness to singularity".

(unindent) If one believes that Bareiss' algorithm is really a condensation-based approach to Cramer's rule, then one reference is: But here I don't see any precise claims of stability (though I thought is was the case). Anyway, I agree that the article here is basically crap. Sławomir Biały (talk) 17:24, 17 December 2009 (UTC)
 * More theoretical examples of usefulness Bounding solutions of systems of equations, bounding number of solutions (discrete situations). Intense use is made in proves of transcendence of exponentials. In Roth's theorem's proof. The claim against Cramer's rule in many books  is just done to discourage students to use it for large dimensions since according to the topics of the course they will be tempted to compute determinants expanding along a row or things like that instead of doing elimination which is a central goal in many courses. I believe the statement about the (computational) usefulness of Cramer's should be made (with emphasis in its methodological character) and not to be mentioned in almost every section without references (which is like it is now).   franklin   18:34, 17 December 2009 (UTC)

Here's a small comment on some of the above issues. In the Wikipedia article on Newton's identities is described a procedure to compute the characteristic polynomial of a matrix using only matrix multiplication, traces, and other algebraic operations (but no determinants); I'm not sure about complexity but it definitely stays within the rationals. That article also gives an application of Cramer's rule, though its usefulness is a matter of taste: while a recursive procedure for expressing one family of polynomials in terms of another one is easily found, the result can only be expressed in a "closed" form by a determinant obtained from Cramer's rule.

The about the pros and cons of determinants, notably relative to eigenvalues. I think one needs to learn about both, and in my opinion determinants are slightly more elementary (they require knowing about about permutations and signs, but not about factoring polynomials). To me the most essential facts about determinants are If, as in the "Down with Determinants" paper mentioned one defines the determinant as the product of the eigenvalues, in the algebraic closure of the coefficient field and taken with appropriately defined multiplicities, then all these three points are problematic. As for the final one, it would tell us that over a field a square matrix is invertible unless the product of its eigenvalues is zero; never mind looking for them in an extended field, it does not require much genius to see that this means unless there is an eigenvalue zero, that is unless the matrix has a nontrivial kernel, so a square matrix over a field is invertible unless it isn't. See also some discussion here. Marc van Leeuwen (talk) 14:50, 26 December 2009 (UTC)
 * The determinant is given by a polynomial expression in the coefficients.
 * The determinant is multiplicative.
 * A square matrix is invertible if and only if its determinant is.

cramers rule proof
I would like to add a section to this article to provide a proof of cramers rule. The proof is on my user page. Djwinters (talk) 00:56, 16 January 2010 (UTC)
 * Do it. But please, type those equations, don't include those images. Or just put it and someone else will replace the images.  franklin   01:47, 16 January 2010 (UTC)
 * Another thing is that that proof is working only with three variables and three equations.  franklin   01:48, 16 January 2010 (UTC)

 * The proof is unclear in what it sets out to prove (the whole article could be clearer about the precise statement od Cramre's rule as well), and is phrased more like a recipe than like a proof. As far as I can tell it only aims at showing that the rule provides a solution in favorable cases (not that this is the only solution), and only works well if not only the denominator determinant is nonzero (which is part of the rule's hypotheses), but also all the individual xi in the solution. In other words I think the proof could be improved. It would also be good to provide a reference for this proof, or is it your own invention? Marc van Leeuwen (talk) 21:55, 16 January 2010 (UTC)

 * I agree with the fact that other proves (maybe better ones) can be given. Now, I disagree with the fact that it doesn't work with systems with some zeroes in the solution. These are rational functions, there is a general principle (follows from analytic continuation and many other principles in complex analysis) that if these manipulations work for non-zero (as the proof shows) then it works in general. The uniqueness of the solution can be derived easily from the very statement of Cramer's formulas.  franklin   22:29, 16 January 2010 (UTC)

 * I modified the proof such that there is no apparent problem when there are zeroes in the solution. The thing is that even being the previous proof correct, if it raised this kind of concerns that can be explained in a somewhat complicated way it means that it won't be very stable. Many others can come asking the same question and that can generate instability. Also it would mean to have to be constantly sending people to read the explanation in the talk page. There is also a geometric proof of Cramer's rule. I will see if I generate the necessary images to include it. It is very simple. The only strange is the fact that the determinant is an area (or a volume), although that (depending on the context) can be assumed by definition.  franklin   00:42, 17 January 2010 (UTC)

Mangled sentence
This appears in the article:
 * Cramer's rule can be used to prove that an integer programming problem whose constraint matrix is totally unimodular and whose right-hand side has integer basic solutions.
 * Cramer's rule can be used to prove that an integer programming problem whose constraint matrix is totally unimodular and whose right-hand side has integer basic solutions.

Someone forgot to finish the sentence. Michael Hardy (talk) 03:44, 30 March 2010 (UTC)

Finding inverse matrix
This section of the article seems to be out of place. There is apparently no use made of Cramer's rule in finding the inverse matrix and no use made of inverse matrices elsewhere in the article. Should we remove it?  Sp in ni ng  Spark  17:38, 17 July 2010 (UTC)
 * It is in an odd place. But my guess is that the majority of (at least American) college studends first last and only interaction w/ Cramer's rule will be as a means to find the inverse to 2x2 matrices.  It doesn't belong in a lvl2 heading though. Protonk (talk) 19:02, 17 July 2010 (UTC)
 * I have added a few lines to cramer's rule proof section to show the connection with the adjugate matrix. Djwinters (talk) 20:56, 12 August 2010 (UTC)

Time Complexity
There are two claims of the time complexity of the rule: is it O(n*n!) or is it O(n^3)? 161.31.73.176 (talk) 16:01, 29 January 2016 (UTC)

maths
I want to learn how to calculate complementaru angles Gavazacarol (talk) 16:22, 4 May 2016 (UTC)

Cramer's rule and the determinant
I could be very wrong about this, but isn't cramer's closely related to the history of the determinant? That should probably be mentioned or at least linked here, but preferably it could have its own section. Gyinagal (talk) 01:03, 28 September 2020 (UTC)

Proof & Cofactor coëfficients
The proof states C_j is a coefficient that does not depend on column j. The laplace/cofactor expansion is stepping through the matrix row-wise, thus this description is misleading.

C_j does not depend on column j, this is correct. C_1 (or any 'C_i' for that matter) is however different for each column (eg. D_j) considered. It seems to me more proper to call this C_j "C_i,j" instead. Both indicating it's independant of row i & column j, but also indicating C's (cofactors) are different between columns, and not identical as previously implied. 145.94.255.226 (talk) 14:18, 5 October 2023 (UTC)

 * This is true for Laplace expansion along a row, but Laplace expansion can also be done along a column (this is almost evident since a matrix and its transpose have the same determinant, and transposition exchanges rows and columns). So, the proof is correct. Moreover, the proof does not use Laplace expansion However, I have edited Laplace expansion for clarify this.

However, reading the proof again, I agree that it would be clearer to rename C_i as C_ij, and I'll do that. D.Lazard (talk)

Structure of the proof
The structure of the proof, to me, seems to be the following

1) Assume that a solution x exists.

2) Show that a solution, if it exists, satisfies a certain condition.

3) Show that any value satisfying this condition does, in fact, solve the original problem.

I think part 3) of the proof is unnecessarily confusing. I therefore suggest the following structural change:

1) Conclude that since the determinant of A is nonzero, a solution must exist.

2) If a solution exists, it satisfies a certain condition.

3) Done! We've characterized all the solutions.

Additionally, I think the proof is a little confusing to read. I think it could be made simpler if you just simply replace one of the columns of A with the components of b, then write the components as linear combinations of the components of x, and then consider the determinant and simply apply properties of determinants in a straightforward manner. SchwaWolf (talk) 09:41, 9 April 2024 (UTC)

 * I agree that the proof could be improved. Your approach for proving that, if a solution exists, then it must have the form specified, is also what I had in mind. What was your idea for proving that a solution must exist? I think it's enough to say that any matrix with nonzero determinant has full column rank, so b can be represented as a linear combination of the columns of A, whose coefficients are given by x. Angegane (talk) 06:24, 22 April 2024 (UTC)
 * I agree also that the proof could be improved. But here your proof scheme is far too complicated. The proof consists into only two steps, and only the first one belongs to this article:
 * Cramer's rule provides a solution if the determinant is nonzero.
 * If the determinant is nonzero, then there is exactly one solution.
 * The second part is totally independent from Cramer's rule and belongs to the article Determinant. So the second part must be recalled, linked toDeterminant, but not proved here. The only proof that is useful here is simply that all equations become satisfied, if one replaces the variables with the values given by Cramer's rule. D.Lazard (talk) 09:01, 22 April 2024 (UTC)
 * I disagree with your approach because I think it is pedagogically unwise to come up with an x "out of thin air" and show that it satisfies the system of equations, especially when the "honest" proof (the one I suggested) is fairly easy and intuitive. SchwaWolf (talk) 14:38, 22 April 2024 (UTC)

title: Talk:Cramer's rule


text:
Mr. Moonlight (album)

Mr. Moonlight is the eighth studio album by British-American rock band Foreigner, released by Arista Records in Europe on 24 October and by BMG Entertainment in Japan on 23 November 1994. In the United States and Canada, it appeared in early 1995 on the Rhythm Safari label. Recorded at seven different studios across the States, the album was produced by Mick Jones, Lou Gramm, and Mike Stone, with an additional production by Phil and Joe Nicolo. It was Foreigner's last studio release until Can't Slow Down (2009).

The album was the first since Inside Information (1987) and the last to feature original lead singer Lou Gramm. Prior to starting work on Mr. Moonlight, he was the lead vocalist on three newly recorded tracks from Foreigner's compilation The Very Best ... and Beyond (1992). Mr. Moonlight was also the first album in fifteen years without bass guitarist Rick Wills, who joined the band in 1979, and drummer Dennis Elliott, who was a founding member.

Though it was intended to be a comeback release, Mr. Moonlight was a commercial disappointment, only peaking at number 136 in the Billboard 200 chart, and ranked as Foreigner's worst-selling studio album.

Personnel
Foreigner
 * Mick Jones – guitars, acoustic piano, backing vocals
 * Lou Gramm – lead vocals, backing vocals, percussion
 * Jeff Jacobs – acoustic piano, organ, keyboards, backing vocals
 * Bruce Turgon – bass, backing vocals
 * Mark Schulman – drums, backing vocals

Additional musicians
 * Ian Lloyd – backing vocals
 * Tawatha Agee – backing vocals (on "I Keep Hoping")
 * Robin Clark – backing vocals (on "I Keep Hoping")
 * Paulette McWilliams – backing vocals (on "I Keep Hoping")
 * Duane Eddy – lead guitar (on "Until the End of Time")
 * Billy Bremner – additional guitars (on "All I Need to Know")
 * Randy Cantor – keyboards and additional guitars (on "White Lie" and "Rain")
 * Luis Enriques – percussion (on "Real World" and "Running the Risk")
 * Scott Gilman – saxophone, recorder, backing vocals

Production
 * Mick Jones – producer
 * Lou Gramm – producer
 * Mike Stone – producer, engineer, mixing (at Bearsville Studios in Bearsville, New York, and Soundtrack Studios in New York, New York)
 * Phil and Joe Nicolo – additional production, mixing ("White Lie" and "Rain", at Studio 4 in Philadelphia, Pennsylvania)
 * David Bianco – mixing ("Real World")
 * Mark Stebbeds – mixing ("Big Dog", at Track Record Studios in North Hollywood, California)
 * Ted Jensen – mastering (at Sterling Studios in New York, New York)
 * Danny Clinch – photography
 * John Pitter – illustrations
 * Red Herring Design – design

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Mr. Moonlight (album)


text: Talk:Mr. Moonlight (album)

title: Talk:Mr. Moonlight (album)


text:
Frank Mazzilli

Frank Mazzilli (born December 7, 1962) is a former politician in Ontario, Canada. He was a Progressive Conservative member of the Legislative Assembly of Ontario from 1999 to 2003.

Background
Mazzilli has a diploma from the Ontario Police College, and worked as a member of the police force in London, Ontario from 1982 to 1999.

Politics
In the federal election of 1997, he ran as a Progressive Conservative candidate in the riding of London West, but lost to Liberal Sue Barnes by almost 14,000 votes.

He was elected to the Ontario legislature for the riding of London—Fanshawe in the provincial election of 1999, defeating Liberal Peter Mancini and New Democrat Irene Mathyssen in a close three-way race. In 2002, he served as a co-chair of the Ontario Crime Control Commission.

The Progressive Conservatives had lost much of their urban support by the time of the 2003 provincial election, and Mazzilli dropped to a third-place finish in London—Fanshawe against Mathyssen and the winner, Liberal Khalil Ramal.

His younger brother John Mazzilli ran unsuccessfully for the Conservative Party of Canada in both the 2004 and 2006 federal elections. Frank worked both times as his campaign co-chairman.

title: Frank Mazzilli


text:
User talk:Spellcheck

February 2014
Hello, I'm Arxiloxos. Your recent edit to the page Ivanpah Solar Power Facility appears to have added incorrect information, so I have removed it for now. If you believe the information was correct, please cite a reliable source or discuss your change on the article's talk page. If you think I made a mistake, or if you have any questions, you can leave me a message on my talk page. Thank you. Arxiloxos (talk) 20:15, 13 February 2014 (UTC)
 * Thank you for the note on my talk page. I also wasn't sure about some of the rewording and the apparent deletion of a source from the footnotes, but I didn't focus on those. In any case, if you haven't seen it already, you might want to look at the article in today's (March 1, 2014) Los Angeles Times: "Massive desert solar plant opens, already a threatened species". Best,--Arxiloxos (talk) 00:43, 2 March 2014 (UTC)

Speedy deletion nomination of Yes and No (game)

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MfD nomination of User:Spellcheck/Nicole Santos
User:Spellcheck/Nicole Santos, a page which you created or substantially contributed to, has been nominated for deletion. Your opinions on the matter are welcome; you may participate in the discussion by adding your comments at Wikipedia:Miscellany for deletion/User:Spellcheck/Nicole Santos and please be sure to sign your comments with four tildes ( ~ ). You are free to edit the content of User:Spellcheck/Nicole Santos during the discussion but should not remove the miscellany for deletion template from the top of the page; such a removal will not end the deletion discussion. Thank you. Adam9007 (talk) 23:26, 2 April 2019 (UTC)

April 2019

A page you created has been nominated for deletion as an attack page, according to section G10 of the criteria for speedy deletion.

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TransWomanDefinitionChart
Thank you for your Definition Chart, Spellcheck. I'd like to comment on it and I was wondering: wouldn't it be better if you place your comment at the bottom of the section Talk:Trans_woman, where it would be entirely in-topic and relevant? I don't see the point of having it in the middle of the section Talk:Trans_woman. It also interrupts the chronological and logical order of the discussion there. Thanks, Gitz (talk) (contribs) 12:00, 11 October 2022 (UTC)

 * Thanks for the suggestion! I wasn't sure what the best place to comment was, as there were a few discussions that seemed relevant - I'm happy to defer to someone who is more familiar with how the discussion flows. I've moved my comment accordingly, that does seem like a good place indeed. The Human Spellchecker (talk) 16:20, 11 October 2022 (UTC)

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AT-9

AT-9 may mean:

 * Curtiss AT-9, aeroplane.
 * AT-9 Spiral-2, missile.
 * Vienna, a state of Austria; AT-9 is its ISO 3166-2:AT country subdivision code.

title: AT-9


text:
Manual therapy

Manual therapy, or manipulative therapy, is a part of Physiotherapy, it is a physical treatment primarily used by physical therapists (a.k.a. physiotherapists), occupational therapists to treat musculoskeletal pain and disability; it mostly includes kneading and manipulation of muscles, joint mobilization and joint manipulation. It is also used by Rolfers, massage therapists, athletic trainers, osteopaths, and physicians.

Definitions
Irvin Korr, J. S. Denslow and colleagues did the original body of research on manual therapy. Korr described it as the "Application of an accurately determined and specifically directed manual force to the body, in order to improve mobility in areas that are restricted; in joints, in connective tissues or in skeletal muscles."

According to the Orthopaedic Manual Physical Therapy Description of Advanced Specialty Practice manual therapy is defined as a clinical approach utilizing specific hands-on techniques, including but not limited to manipulation/mobilization, used by the physical therapist to diagnose and treat soft tissues and joint structures for the purpose of modulating pain; increasing range of motion (ROM); reducing or eliminating soft tissue inflammation; inducing relaxation; improving contractile and non-contractile tissue repair, extensibility, and/or stability; facilitating movement; and improving function.

A consensus study of US chiropractors defined manual therapy (generally known as the "chiropractic adjustment" in the profession) as "Procedures by which the hands directly contact the body to treat the articulations and/or soft tissues."

Use and method
In Pakistan, Western Europe, North America and Australasia, manual therapy is usually practiced by members of specific health care professions (e.g. Chiropractors, Occupational Therapists, Osteopaths, Osteopathic physicians, Physiotherapists/Physical Therapists, Massage Therapists and Physiatrists). However, some lay practitioners (not members of a structured profession), such as bonesetters also provide some forms of manual therapy.

A survey released in May 2004 by the National Center for Complementary and Integrative Health focused on who used complementary and alternative medicine (CAM), what was used, and why it was used in the United States by adults during 2002. Massage was the fifth most commonly use CAM in the United States in 2007.

Techniques

 * Myofascial therapy targets the muscle and fascial systems, promotes flexibility and mobility of the body's connective tissues. It is said to mobilize adhesions and reduce severity/sensitivity of scarring. A critical analysis finds that the relevance of fascia to therapy doubtful.
 * Massage may be used as part of a treatment. Proponents claim this may reduce inflammation. Science writer Paul Ingraham notes that there is no evidence to support the claim.
 * Friction massage is said to increase mobilization of adhesions between fascial layers, muscles, compartments and other soft tissues. They are thought to create an inflammatory response and instigate focus to injured areas. A 2002 systematic review found that no additional benefit was incurred from the inclusion of deep tissue friction massage in a therapeutic regimen, although the conclusions were limited by the small sample sizes in available randomized clinical trials.
 * Soft Tissue Technique is firm, direct pressure to relax hypertonic muscles and stretch tight fascial structures. A 2015 review concluded that the technique is ineffective for lower back pain, and the quality of research testing its effectiveness is poor.
 * Trigger point techniques claim to address myofascial trigger points, though the explanation of how this works is controversial.

Stretching
From the main article's effectiveness section:
 * Apart from before running, stretching does not appear to reduce risk of injury during exercise.
 * Some evidence shows that pre-exercise stretching may increase range of movement.
 * The Mayo Clinic advises against bouncing, and to hold for thirty seconds. They suggest warming up before stretching or stretching post-exercise.

Taping
Manual therapy practitioners often use therapeutic taping to relieve pressure on injured soft tissue, alter muscle firing patterns or prevent re-injury. Some techniques are designed to enhance lymphatic fluid exchange. After a soft tissue injury to muscles or tendons from sports activities, over exertion or repetitive strain injury swelling may impede blood flow to the area and slow healing. Elastic taping methods may relieve pressure from swollen tissue and enhance circulation to the injured area.

According to the medical and skeptical community there is no known benefit from this technique and it is a pseudoscience.

Styles of manual therapy
There are many different styles of manual therapy. It is a fundamental feature of ayurvedic medicine, traditional Chinese medicine and some forms of alternative medicine as well as being used by mainstream medical practitioners. Hands-on bodywork is a feature of therapeutic interactions in traditional cultures around the world.

 * Acupressure
 * Anma
 * Bobath concept
 * Bodywork (alternative medicine)
 * Bone setting
 * Bowen Technique
 * Chiropractic
 * Cranio-sacral therapy
 * Dorn method
 * Joint manipulation
 * Joint mobilization
 * Spinal manipulation
 * Spinal mobilization
 * Massage therapy
 * Manual lymphatic drainage
 * Medical acupuncture
 * Muscle energy techniques
 * Myofascial release (MFR)
 * Myotherapy
 * Naprapathy
 * Osteopathic manipulative medicine
 * Osteopathy
 * Physiotherapy
 * Postural Integration
 * Rolfing
 * Shiatsu
 * Siddha Medicine
 * Structural Integration
 * Traction
 * Tui na
 * Zero Balancing

Efficacy
Due to the wide range of issues with various parts of the body and different techniques used, as well as a lack of modeling behavior, it can be difficult to tell just how effective manual therapy can be for a patient.

Results for migraines, headaches, and asthma are mixed due to a lack of clinical trials,   though at least one article states that manual therapy is effective for asthma.

Manual therapy was shown to be effective for treating back pain, with trigger point therapy being used for myofascial pain, and manual manipulation for lower back pain.

Journals

 * The Journal of Manual and Manipulative Therapy
 * Journal of Manipulative and Physiological Therapeutics - PubMed access found here

title: Manual therapy


text:
Mongul

Mongul is a supervillain appearing in comic books published by DC Comics. Writer Len Wein and artist Jim Starlin created the first version of the character, who debuted in DC Comics Presents #27 (November 1980). Jerry Ordway created the second version, who first appeared in The Adventures of Superman #454 as the lord of Warworld. He was later embellished by Peter Tomasi and Scot Eaton in Showcase '95, #8. He is based on the Mongol Empire's founder Genghis Khan and his successors, hence his name. The character was given an origin story in Green Lantern #23.2 by his co-creator Jim Starlin and artist Howard Porter as homage to the writers who participated in developing the character.

Debuting in the Bronze Age of Comic Books, Mongul has been featured in other DC Comics-endorsed products such as animated series, video games, a direct-to-DVD film, and merchandise such as action figures and trading cards.

Publication history
Mongul debuted in the title DC Comics Presents and was created by writer Len Wein and artist Jim Starlin. Starlin often receives credit as creator of the character, but Wein in an interview stated: "Well, [Mongul] had Starlin visuals, but he was my creation". Wein said he conceived Mongul specifically as a villain to physically challenge Superman.

Bronze Age (1980–1985)
Mongul was the ruler of his own unknown alien race until a revolution occurred by his people and a holy man named the Ancient Arkymandryte. And thus, Mongul was forced to flee the throne to save his life. In his first appearance, Mongul kidnaps Superman's friends (Lois Lane, Jimmy Olsen and Steve Lombard), threatening to kill them unless the hero brings him the key from a dead Warzoon race called Largas. There was a lone survivor from the dead race who struck a deal with Martian Manhunter to protect the key that can activate Warworld. After Superman retrieves the key, and is forced to fight Martian Manhunter who is protecting it, Mongul activates Warworld. Psychically linked with its controls, Mongul tries to destroy Superman and Supergirl. Mongul is ultimately rendered unconscious by a massive mental strain caused from using its controls, but manages to escape before the heroes destroy Warworld.

Mongul then tries to conquer Throneworld, the home planet of Prince Gavyn, one of the heroes who have used the name Starman. Mongul murders Gavyn's sister and forces Gavyn's lover into marrying him to usurp the throne of the empire for himself. He uses Throneworld's planet-destroying weapon to blackmail other planets into obedience. Superman arrives and battles Mongul, while Starman disables the weapon. Mongul retreats as soon as the weapon was successfully disabled, planning to make his next move. Now wanting revenge on Superman, Mongul kills a Controller and steals the Sun-Eater to devour the Earth's Sun. While the Justice League of America and Legion of Super-Heroes battle Mongul, Superman finally defeats him as the Legion destroys the Sun-Eater.

Mongul eventually attacks Superman on his birthday and ensnares him with a Black Mercy, an alien plant that feeds off a victim's "bio-aura" while rendering the victim incapable of fighting back, giving them a realistic vision of their own perfect "dream world" in return. Meanwhile, Mongul nearly killed Wonder Woman while mocking her views that women are equal. In the end, thanks to Batman, Robin and Wonder Woman, Mongul falls victim to the plant, receiving visions of being ruler of the universe.

Modern Age (1985–present)
After the 12-issue limited series Crisis on Infinite Earths, DC Comics retconned the histories of some major characters to update them for contemporary audiences. The original stories involving Mongul were no longer in continuity, and the character was reintroduced as the ruler of Warworld, a space empire where Mongul entertains the citizens with gladiatorial games. Mongul's ship captures a dying Superman, who was floating adrift in outer space after he exiled himself from Earth, and Mongul decides to use him in the games, but Superman joins forces with the alien warrior and Mongul's champion Draaga and makes Mongul flee. Mongul is then persuaded via torture to serve the Cyborg Superman to gain vengeance on Superman and to try to turn the Earth into another Warworld. In the process, Green Lantern Hal Jordan's home, Coast City, is destroyed, which leads to him joining Superman and his allies to defeat Mongul.

After his defeat, Mongul is imprisoned in a prison for intergalactic criminals, only to break out during a riot. His first target is Green Lantern; he learns that the one whom he faces, Kyle Rayner, is not the one he fought earlier. Mongul is defeated when Kyle's ring shows no weakness to yellow. Mongul is re-imprisoned.

Mongul breaks out of the lunar penal colony, killing everyone there, including prisoners who are left to die in the vacuum of space. His ship is almost wrecked and he is near death; he is teleported to a planet and saved. In return, he takes over the planet and ends up being left alone as the inhabitants prefer dying due to a virus than his tyranny, until he ends up finding two babies immune to the virus (a story started in Showcase '95 #7, with the two babies appearing in Showcase '95 #8; reprinted in DC Universe Special: Superman #1).

Mongul is later defeated on Earth by Wally West (the Flash) when Mongul tries to unearth a starship left from one of the Darkstars' enemies underneath Keystone City. The Flash easily defeats Mongul. The Flash seemingly uses Mongul to test his new upgraded powers. During the battle, Wally is only hit one time by the giant, hulking Mongul. Flash uses his super speed to quickly confuse and defeat Mongul and has him imprisoned in the Slab, a prison for supervillains (The Flash (vol. 2) #102, reprinted in DC Universe Special: Superman #1).

During the Underworld Unleashed crossover event, the demon-lord Neron offers various supervillains (including Mongul) enhanced power in exchange for their souls, all by lighting a carved black candle. Mongul's pride causes him to decline the offer and threaten Neron. In response, Neron beats Mongul to death for his defiance, taking his soul in the process.

Son of Mongul
Mongul's son, also named Mongul, appears to assist and train Superman, in preparation for the arrival of Imperiex. This Mongul seems to be more powerful than his father. He appears to have been killed later in the Our Worlds at War crossover, but returns during Infinite Crisis after learning from Despero that the Justice League has apparently been destroyed. His intention is to loot their Watchtower headquarters but he ends up fighting Batman, Superman, and Wonder Woman. He is almost killed by Wonder Woman before escaping via a working teleporter. The teleportation transports him to Earth, to menace Hal Jordan, the newly returned Green Lantern, by using the Black Mercy on him and the Green Arrow. In the meantime, he seeks his sister, Mongal, to settle family squabbles. The heroes break free and use a teleporter to transport Mongul and Mongal to their home planet. Stating family to be a weakness, Mongul spitefully kills Mongal with a single blow to her head.

Mongul's origins depicts him as a child who wanted to be like his father. He made journeys and he watched digital renderings where his father fought against Superman and his allies and the destruction of Coast City. He copies his father's actions when he encounters a group of aliens whose spaceship crashes on Arkymandryte, turning them into his slaves. Mongul's father returns, and discovering his son's slaves, he kills the aliens and tells him only one being on the planet is worthy of adoration.

Mongul receives a Yellow Power Ring after breaking a dying Sinestro Corps member's neck (a later promotional image shows Mongul with the Yellow ring as well a Green Lantern Corps ring). Mongul offers the Sinestro Corps inductees a choice: to serve him or die. He removes the ring from each one who refuses, and at one point had gained an extra five rings. He then attacks Arisia and Sodam Yat with Black Mercy plants, and takes them prisoner. He uses his ring to send thousands of Black Mercy seeds, which he had genetically engineered to bring the victims' greatest fears to life, instead of their dreams, to several unsuspecting planets. In a confrontation with several members of the Green Lantern Corps, Mongul is defeated when the fly-like Lantern Bzzd flies through his eye, and he is thrown down to the Black Mercies' planet. He is last seen buried in soil, being used as food by the Black Mercies, but soon breaks free and escapes the planet, while keeping his rings and his right arm. His left arm had been severed in the process, but, through the power of his rings, Mongul is able to control and direct it. He attacks a nearby ship to get food for himself, killing the husband of the pilot. This inadvertently causes the woman to become the first recruit of the Star Sapphires, the violet Power Ring having been drawn to her by the void in her heart created by her loss. Mongul uses his left arm to invade the planet Daxam and establish it as the new homeworld for his faction of the Sinestro Corps under his command. He is challenged for the leadership by Arkillo. Defeating him in single combat, Mongul pulls out Arkillo's tongue and wears it as a necklace. In the process, he gains the loyalty of the faction of the Sinestro Corps loyal to Arkillo and complete rule over the planet Daxam, but draws on himself the attentions of Arisia and Sodam Yat, the Daxamite host of Ion. Upon the arrival of Arisia and Yat, several members of the Sinestro Corps are swiftly defeated and killed by Yat, until his Superman-like powers fade under Daxam's red sun. Despite his power loss and Mongul's incredible strength, Yat does battle with him, using the Ion power to briefly launch Mongul into space, before entering Daxam's sun and transforming it from red to yellow, granting all Daxamites superpowers. The Daxamite's overwhelming attack forces Mongul to have the Sinestro Corps abandon Daxam, with the despot planning to make a different planet their home base.

Mongul takes the Sinestro Corps to Korugar, Sinestro's homeworld, having the inhabitants strung up along the streets. He also decides to rename the Yellow Lanterns as "the Mongul Corps", after himself. Soon after, Sinestro is brought to Korugar and confronts Mongul. First, using an override built into Mongul's rings, Sinestro defeats him, thus reclaiming the Sinestro Corps. Then, he imprisons Mongul in the Corps' Central Power Battery, intending to kill him once the Black Lantern Corps is dealt with.

The New 52
In The New 52, a 2011 reboot of the DC Comics universe, Mongul is once again reintroduced as lord and master of the planetary siege engine Warworld. In his introduction, he is laying waste to a planet he is poised to conquer as he brought a resisting general aboard his vessel to show him the devastation of his homeworld just before killing him and adding his remains to a trophy room on Warworld.

Mongul makes a brief appearance in the Superman: Doomed story arc. He manages to escape the Phantom Zone alongside similarly powerful villains such as Non, and prepares to attack Earth. Upon learning that Brainiac has arrived on Earth, Mongul, stricken with fear, immediately heads back into the Phantom Zone. Wonder Woman later enters the Phantom Zone and traps him in her Lasso of Truth, before ordering Mongul's Warworld to attack Brainiac's ship.

Sometime after his imprisonment, Mongul would escape the Phantom Zone with Warworld in tow, eventually setting his sights on revenge against Sinestro and his Fear Corps for prior injuries long past by seeking to lure him out using Black Mercies on Korugarian refugees, then draining his ring to uselessness through leftover technologies of the old universal survivor Relic, which he had integrated into Warworld's systems.

Mongul would use him as a bargaining chip to call the rest of his Corpsmen to do battle over Warworld to rescue their leader so he could cow them back into his services again while seeking to coax the Emotional Entity Parallax from within his hated rival out into the open, all to truly take command of the Sinestro Corps once again. His plan was going without a hitch so far, even with the untimely arrival of Bekka of the New Gods lending much needed assistance. But the tables were turned when the Apex League, a team of mercenaries whom he had enlisted, eventually turned on him at Sinestro's suggestion after Mongul's prisoner reversed the energy siphon. Afterwards, he would entrap his assailant in scrap metal lined with yellow ring fragments before jettisoning him into deep space, a bright and shiny lure the Pale Vicors would be drawn to.

The interstellar warlord would resurface as an enslaved thrall of the Pailing when his new masters attacked the Earth at their lord marshal's clarion call. Now in the service of anti-emotion, Mongul had returned to battle Sinestro more powerful than ever, while aiding in the subjugation of humanity.

The battle would rage and Sinestro would recruit many amongst the planet into his Corps to defend it; eventually Black Adam, yet another addition to the Yellow Lanterns, would cast him out of the fight just long enough for his brother in arms to deliver the killing blow to Mongul and the Pailing's leader, the Pale Bishop. In the aftermath of the battle, Sinestro ceded control of his Corps to his daughter Soranik due to his injuries, while the Sinestro Corpsmen helped Earth rebuild. Mongul would stir from his zombified state hellbent on getting Sinestro for feeding him to the Pail Vicors. Seeing as he was surrounded by Manhunters, however, he opted a hasty retreat to Earth to bide his time instead.

DC Rebirth
Mongul appears as a main villain in Trinity, where his use of The Green causes Poison Ivy to capture Superman, Batman and Wonder Woman and causes them to undergo hallucinations of their own childhoods. The three are eventually able to reduce Mongul's influence over his other world.

Mongul is briefly a member of a new version of the Superman Revenge Squad, also consisting of General Zod, Metallo, Cyborg Superman, Eradicator, and Blanque. After Superman is temporarily blinded, they are defeated when Lex Luthor, Supergirl, Superwoman, Kong Kenan, and Steel come to Superman's aid and Zod betrays the rest of the team for his own ends.

Mongul later escapes prison and somehow traps the Justice League inside of an intergalactic arena around the time they are investigating Barbatos. They construct a mecha out of Mongul's creatures and frighten him away, but upon returning to Earth discover that it has been sacked by Barbatos.

Powers and abilities
During the Bronze Age of Comic Books, Mongul was stronger than Superman and almost totally invulnerable to harm. Superman defeated him by foiling his schemes, but only once defeated Mongul in hand-to-hand combat. Even then, Superman fell unconscious immediately afterward. Mongul also demonstrated the ability to teleport; limited telepathy and telekinesis; and could project blasts of potent force via his eyes, hands, or chest. The character also used technology to shrink his enemies and place them in dimensional-inversion cubes designed to prevent escape by warping their interior reality and absorbing any power used against them from within. The Modern Age Mongul started off with less power than the Bronze Age version but was still a formidable foe and a capable match up against Superman on multiple occasions, although slightly weaker.

Mongul was able to resist half-power Kryptonian heat vision at point-blank range and, thanks to his enhanced musculature, was incredibly resilient to physical harm. Because of his impressive muscle structure, Mongul was also incredibly fast, and able to push himself forward at high speed while running or able to leap several miles into the air by his powerful legs. He also had his chest cannon with which he could fire potent energy blasts that could stagger or kill even Superman. So formidable a foe was he that Mongul could not only trade blows with Kryptonians but repeatedly battle Green Lanterns on a regular basis, especially given that his natural yellow skin gave him protection against their rings. He was still outclassed, though, by individuals of greater capability (be it physical or technological in nature), such as Hank Henshaw (before and after his acquisition of Superman's genetic template) and the demon Neron (who killed him with his bare hands).

His progeny Mongul II, however, showed himself to either be on equal footing or in greater strength than even his father. Having demonstrated enough might for defeating DC Universe heavy-hitters like Wonder Woman (Infinite Crisis #1 (December 2005)), along with killing members of both the Green Lantern and Sinestro Corps in his pursuit of power rings. He was strong enough to kill his sister Mongal in one blow. When acquiring his first Yellow Lantern ring, he decided to learn about its potential for 96 hours, hinting at a methodical mindset mostly absent from Yellow Lanterns. Mongul II also battled Superman, leading to his defeat after both delivered many blows, leading to Superman unleashing a massive combination attack required to overpower this villain (Superman (vol. 2) #152 (January 2000)). This battle displayed him having an endurance to massive blows from a being as powerful as Superman. After having obtained several Qwardian Power Rings, he showed a rather avid usage of their power for numerous effects (i.e., having used them along with his father's innate knowledge of the Black Mercy to modify the plant's parasitic behavioral patterns, inducing frightful mirages over happy fantasies; create and manipulate fear-driven constructs powerful enough to batter an Ion-empowered Sodam Yat; and reattach destroyed or removed limbs or body parts via precise fear energy material rearrangement).

In the new universe Mongul has been radically re-imagined from being the formidable adversary he once was during the Bronze-Silver Ages. Now back in control over Warworld, he again started cutting a bloody swath across the universe, conquering planet after planet with his cosmic dreadnaught. When he makes his way to Earth, he showcases extreme levels of super strength and resilience as he gets into a colossal battle between Superman and an empowered Batman who, at the time, had Kryptonian-like power sets. Kal-El once stated him to be about as strong as Darkseid; this is proven true in a number of cases where he showed his might by fighting off an entire brigade of Yellow Lanterns backed by a New God within their ranks. He still retains his chest cannon, which can still blow away multiple personages in one blast, as well as heat vision to incinerate his enemies.

This new Mongul is tougher, faster and more powerful than ever, easily able to take on individuals or groups as powerful as Lantern Corpsmen or even New Gods, as well as survive virtually unscathed. In the event that he does suffer from critical abrasion; Mongul also has a slight healing factor with which to help him recover almost from anything, the limits of which were never probed. He even showcased the natural capacity of flight with or without assisted propulsion more than once. For a time, he was also part of the Sinestro Corps, having access to all that it entailed; i.e., the ability to create fear-based energy constructs, gaining power from the fears of others and acquiring a power battery with which to recharge his ring. After he had been deposed by Sinestro, who later requisitioned Warworld for his own cause, Mongul would end up in the services of the Pale Vicors, an anti-emotional sect which razed planets of their resources and their inhabitants by nullifying their ability to feel and/or empathize before harvesting the targeted worlds resources. Having been given enormous power at the cost of his own individual freedom, the Pailing made Mongul stronger beyond imagining. With all new powers and abilities which allowed him to match and overpower the likes of Sinestro and a Yellow Ring-empowered Black Adam with ease, he could even singlehandedly stalemate the very firepower of Warworld itself, which had been commandeered at the time by the Sinestro Corpsman Ranx.

As a Pail Vicor, Mongul possessed all the natural powers which came with it; armed with his lance he could emit a form of apathetic force which negated the powers of the emotional spectrum, giving him similar construct forming abilities akin to a Lantern Ring, only angled to use other sentient beings' emotional anguish to drive the emotion out of them, rendering the afflicted unfeeling, empty shells of who and/or what they used to be. He could also use its power to fly and emit energy blasts with the same energy negating effects as the protective aura which shrouds all the Pailing.

Television

 * Mongul appears in the DC Animated Universe series Justice League and Justice League Unlimited, voiced by Eric Roberts.
 * Mongul appears in Batman: The Brave and the Bold, voiced by Gary Anthony Williams.
 * Mongul appears in Young Justice, voiced by Keith David.
 * Mongul appears in the Justice League Action episode "Galaxy Jest", voiced by John DiMaggio.

Film

 * Mongul appears in Superman/Batman: Public Enemies, voiced by Bruce Timm. This version was mind controlled by Gorilla Grodd in a plot to claim the bounty on Superman and Batman.
 * Mongul appears in the Tomorrowverse film Justice League: Warworld, voiced by Robin Atkin Downes. This version kept Warworld in line using psychic illusions created by a captured Martian Manhunter, and is served by Lobo and an army of White Martians. Batman, Superman, and Wonder Woman are trapped in Warworld and subjected to illusionary worlds, but eventually learn the truth and defeat Mongul, after which he is seemingly killed when Martian Manhunter destroys Warworld.

Video games

 * Mongul appears in Superman: The Man of Steel, voiced by J.S. Gilbert.
 * Mongul appears in Superman Returns, voiced by Todd Williams.
 * Mongul appears in Batman: The Brave and the Bold – The Videogame, voiced again by Gary Anthony Williams.
 * Mongul appears as a playable character in Lego DC Super-Villains, voiced by Fred Tatasciore.

Reception
In 2009, Mongul was ranked as IGN's 41st Greatest Comic Book Villain of All Time.

title: Mongul


text:
Hewson

Hewson is a surname. Notable people with the surname include:
 * Ali Hewson (born 1961), activist and wife of U2's Bono
 * Allan Hewson (born 1954), former New Zealand Rugby union All Black
 * Arthur Hewson (1914–1999), Australian politician
 * Ashleigh Hewson (born 1979), Australian Rugby union Wallaroo
 * Brian Hewson (1933–2022), retired British middle-distance athlete
 * David Hewson (born 1953), British novelist
 * Dave Hewson (composer) (born 1953), British television and film composer
 * David Hewson (Canadian football) (born 1982), Canadian football player
 * Dominic Hewson (born 1974), English cricketer
 * Emily Hewson (born 1982), Australian tennis player
 * Eve Hewson (born 1991), Irish actress and Bono and Ali Hewson's daughter
 * Gilbert Hewson (died 1951), Irish politician
 * Henry Hewson, rugby league footballer of the 1920s
 * John Hewson (died 1662), a soldier in the New Model Army and signatory on the death warrant of King Charles I
 * John Hewson (1744–1821), an American textile printer born in England
 * John Hewson (born 1946), Australian economist and former politician
 * Jones Hewson (1874–1902), Welsh singer and actor
 * Liv Hewson (born 1995), Australian actor
 * Marillyn Hewson (born 1954), American businesswoman, former Chairman, President, and CEO of Lockheed Martin
 * Bono (born Paul Hewson in 1960), Irish singer of U2
 * Peter Hewson, British singer, vocalist with Chicory Tip
 * Richard Anthony Hewson (born 1943), English musician
 * Sam Hewson (born 1988), English footballer
 * Sherrie Hewson (born 1950), English actress
 * Tony Hewson (1934–2020), former British racing cyclist
 * William Hewson (1806–1870), English theological writer
 * William Hewson (surgeon) (1739–1774), English surgeon

title: Hewson


text:
One Yellow Rabbit

One Yellow Rabbit Performance Theatre (OYR) is based in the Big Secret Theatre in Calgary’s Arts Commons. With its Resident Performing Ensemble, OYR creates original theatrical works each year for its home audiences and also hosts The High Performance Rodeo, Calgary’s International Festival of the Arts.

OYR was founded as a collective in 1982 and coalesced into the OYR Performing Ensemble circa 1987-8. In its history, the company has created almost 80 full-length original productions. Ensemble personnel have remained relatively consistent since then, and OYR established a professional administrative staff.

The company’s philosophy and practice have been influenced by contemporary dance, the Artist-run centre movement, the Punk subculture, and the do-it-yourself creed.

In 1987, OYR produced the Secret Elevator Experimental Performance Festival, later renamed the High Performance Rodeo in 1988. The festival has grown annually in audience, venues, and impact, surpassing 10,000 attendees in 2006 and 16,000 in 2009. It now reaches over 20,000 audience members every year. The Rodeo features theatre, dance, music and multimedia presentations from across Canada and abroad. Artists presented in recent years range from hundreds of emerging artists to Philip Glass, Catalyst Theatre, Electric Company Theatre, Compagnie Marie Chouinard, Laurie Anderson, Andy Jones, members of The Kids In the Hall, Les Deux Mondes, Peggy Baker, Daniel MacIvor, and the OYR Ensemble itself.

His Excellency Jean-Daniel Lafond was the Rodeo’s Honorary Patron from 2008-2009. He and Governor General Michaëlle Jean attended performances and hosted three “Art Matters” community forums as part of the festival.

OYR has toured across North America, Australia, Asia, Mexico and Europe. The 1990s saw a large increase in the number and frequency of tours, starting with the productions The Erotic Irony of Old Glory and Ilsa, Queen of the Nazi Love Camp. Scotland's Traverse Theatre was often the launching pad for international tours.

In 1995, OYR expanded the Secret Theatre into the Big Secret Theatre, increasing its technical versatility and doubling its seating to 130. Seating was further upgraded in 2002 to a potential 249 (although most theatre shows play in the 189-seat configuration.)

In 1998, OYR's Denise Clarke founded the Summer Lab Intensive School of performance creation, which was attended by 20 students from across Canada. The 2009 Lab was the 13th annual edition. 250 “rabbits” have graduated.

In 2003, Banff Centre Press published Wild Theatre: The History of One Yellow Rabbit by Martin Morrow. The book contains prefaces by Ronnie Burkett and Factory Theatre's Ken Gass.

title: One Yellow Rabbit


text:
Hewson Consultants

Hewson Consultants were one of the smaller software companies which produced video games for home computers in the mid-1980s. They had a reputation for high-quality games which continually pushed the boundaries of what the computers were capable of and can be compared favourably with other ground-breaking software houses like Ultimate Play the Game and Beyond. Fourteen of their games were awarded "Megagame" by Your Sinclair.

Hewson was founded by Andrew Hewson in the early 1980s. He became interested in computers while working at the British Museum when their first machine arrived. After learning to program, Andrew wrote the programming guide book Hints and Tips for the ZX80. Following the publication, bedroom coders began to send Andrew the games they had programmed on cassette tape, giving Andrew the idea to publish the games. Hewson Consultants was born, and initially released games via mail order advertisements in computing magazines. Andrew was also a columnist in Sinclair User magazine throughout the 1980s.

Releases

 * Space Intruders - 1981 - Space Invaders clone for the ZX81
 * Pilot - 1982
 * Nightflite - 1982
 * Heathrow Air Traffic Control - 1983
 * Nightflite II - 1983
 * Knight Driver  - 1983
 * Quest Adventure - 1983
 * 3D Space-Wars - 1983
 * 3D Seiddab Attack - 1984
 * 3D Lunattack - 1984 - Reviewed in Crash issue 4 - 90%
 * Avalon - 1984 - Reviewed in Crash issue 10 - 91%
 * Technician Ted - 1984 - Reviewed in Crash issue 13 - 96%
 * Dragontorc - 1985
 * Astro Clone - 1985
 * Paradroid - 1985
 * Gribbly's Day Out - 1985
 * Firelord - 1986
 * Pyracurse - 1986
 * Quazatron - 1986
 * Southern Belle - 1986
 * Uridium - 1986
 * Technician Ted: The Megamix - 1986
 * Cybernoid - 1987
 * Exolon - 1987
 * Zynaps - 1987
 * Impossaball - 1987
 * Nebulus - 1987
 * Ranarama - 1987
 * Evening Star - 1987
 * Anarchy - 1987
 * Netherworld - 1988
 * Cybernoid II: The Revenge - 1988
 * Marauder - 1988
 * Zamzara - 1988
 * Eliminator - 1988
 * Battle Valley - 1988
 * Steel - 1989
 * Stormlord - 1989
 * Astaroth - 1989
 * Onslaught - 1989
 * Deliverance - 1990
 * Insects in Space - 1990
 * Zarathrusta - 1991

Legacy
Andrew and other members of the Hewson management team went on to form 21st Century Entertainment after Hewson shut down in the early 1990s, releasing several games such as Pinball Dreams, Pinball Fantasies and Pinball Illusions. Andrew was also the founder of ELSPA (European Leisure Software Publishers Association) which continues to be the European regulating body for the video games industry.

title: Hewson Consultants


text:
User talk:-=italian advisory=-

Welcome!

Hello, -=italian advisory=-, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers: I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~); this will automatically produce your name and the date. If you need help, check out Where to ask a question, ask me on my talk page, or place   on your talk page and someone will show up shortly to answer your questions. Again, welcome! --Flockmeal 04:44, Dec 8, 2004 (UTC)
 * The five pillars of Wikipedia
 * How to edit a page
 * Help pages
 * Tutorial
 * How to write a great article
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Enrico Forti
Hi -=italian advisory=-. We need you to take a look at Enrico Forti, and to try and and turn it into a neutral point of view (NPOV) article. Articles on Wikipedia are not meant to prove points, to argue a case, or to come to a conclusion. They are meant to be as encyclopedic and neutral as possible. Good places to get info on writing articles include How to write a great article and Wikipedia:NPOV. Thanks! — Asbestos | Talk 11:00, 8 Dec 2004 (UTC)

title: User talk:-=italian advisory=-


text:
User talk:Tyrell turing

Welcome!

Hello,, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are a few good links for newcomers: I hope you enjoy editing here and being a Wikipedian! Please sign your name on talk pages using four tildes (~~); this will automatically produce your name and the date. If you need help, check out Where to ask a question, ask me on my talk page, or place  on your talk page and someone will show up shortly to answer your questions. Again, welcome! --Flockmeal 04:47, Dec 8, 2004 (UTC)
 * The five pillars of Wikipedia
 * How to edit a page
 * Help pages
 * Tutorial
 * How to write a great article
 * Manual of Style

Principles and parameters
Hi, I see you started the article, I've been working on it a bit lately. It could still do with a lot of improvement, if you have time some help would be great :) - FrancisTyers 12:18, 10 December 2005 (UTC)

 * Looking good, some history might be useful, as in what came before, and why P&P was "invented". - FrancisTyers 20:06, 17 May 2006 (UTC)

Membrane potential
It's relieving that someone has some references, even if they are not exactly a book. I'm no expert in the topic either but i was sure about it not being a charge difference, thanks a lot for your support. The thing now is getting a nice book and giving the article a whole new look. Knights who say ni 20:46, 2 November 2006 (UTC)

title: User talk:Tyrell turing


text:
Cusco Declaration

The Cusco Declaration, formally titled Preamble to the Foundation Act of the South American Union, is a two-page declaration of intent signed by 12 South American countries during the Third South American Summit on 8 December 2004 in Cusco, Peru. It announces the foundation of the Union of South American Nations. It called for a regional parliament, a common market and a common currency.

title: Cusco Declaration