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tweak ref to us army / add nz to mil operators
 
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{| cellpadding="2" cellspacing="0" style="margin:3px; border:3px solid #87CEEB;width:30%; " align="right"
------
!bgcolor="#87CEEB" colspan="3" align="center" style="border-bottom:3px solid"|De Havilland Otter
|-
|colspan="3" align="center"|[[Image:DeHavilland_Single_Otter_Harbour_Air.jpg|200px]] Otter in [[Harbour Air]] livery
|-
!bgcolor="#87CEEB" colspan="3"|Description
|-
|Role||colspan="2"|Transport
|-
|Crew||colspan="2"|1
|-
|First Flight||colspan="2"|Dec 12 1951
|-
|Entered Service||colspan="2"|
|-
|Manufacturer||colspan="2"|de Havilland Canada
|-
!bgcolor="#87CEEB" colspan="3"|Dimensions
|-
|Length|| 41 ft in|| 12.5 m
|-
|Wingspan|| 58 ft in|| 17.7 m
|-
|Height|| 13 ft in|| 4 m
|-
|Wing area|| ft²|| m²
|-
!bgcolor="#87CEEB" colspan="3"|Weights
|-
|Empty|| 5287 lb|| 2398 kg
|-
|Loaded|| 8000 lb|| 3628 kg
|-
!bgcolor="#87CEEB" colspan="3"|Powerplant
|-
|Engines||colspan="2"| 1 [[Pratt & Whitney Wasp|Pratt & Whitney S1H1-g Wasp]] radial
|-
|Power|| 600 hp||447 kW
|-
!bgcolor="#87CEEB" colspan="3"|Performance
|-
|Maximum speed|| 160 mph|| 258 km/h
|-
|Ferry range|| 960 miles|| 1545 km
|-
|Service ceiling|| 17900 ft|| 5460 m
|-
|Rate of climb|| 1000 ft/min|| 305 m/min
|-
|Wing loading|| lb/ft²|| kg/m²
|-
!bgcolor="#87CEEB" colspan="3"|Avionics
|-
|Avionics||colspan="2"|
|-
|}


[[Image:Turbo Otter on Wheel-Skiis.JPG|thumb|left|Turbo Otter on wheel-skiis]]
Hello! I think this article is looking very good. Over the past several months many improvements have been made. I wonder if it would be better titled "Lebesgue integral", since (if I'm not mistaken) that term is rather more common in text books. Certainly "Lebesgue integral" is more common than "Lebesgue integration" on the web as shown by Google searches. Likewise "Lebesgue-Stieltjes integral" is more common on the web than "Lebesgue-Stieltjes integration". There do exist [[Lebesgue integral]] and [[Lebesgue-Stieltjes integral]] in WP but these are redirects. However, [[Riemann integral]] is an article and [[Riemann integration]] is a redirect; also [[integral]] is an article and [[integration]] is a disambiguation page (there is no [[integration (mathematics)]] article). So: in summary, I propose that we move [[Lebesgue integration]] to [[Lebesgue integral]] and likewise with Lebesgue-Stieltjes. I look forward to your comments. Regards & happy editing, [[User:Wile E. Heresiarch|Wile E. Heresiarch]] 16:57, 21 Jun 2004 (UTC)
The '''de Havilland Canada DHC-3 Otter''' is a single engine, high wing, [[propeller]] driven [[aircraft]]. It was conceived to perform the same roles as the previously successful [[De Havilland Canada DHC-2 Beaver|Beaver]] but was overall a larger plane. Initially named the ''King Beaver'' de Havilland began design work in January, 1951. Canadian certification was given in November, 1952. The [[US Army]] soon became the largest operator of the aircraft (184 delivered with the designation U-1A Otter).


The Otter served as the basis for the very successful, [[Twin Otter]] which featured two wing mounted [[Pratt and Whitney]] PT-6 turboprops.
:On the whole, page names migrate towards the ''more'' abstract term. I think this is more suitable for WP, really: so that e.g. [[nerve]] is thought of under [[nervous system]] first (haven't checked the actual status of those). So I'd be happy to leave it as Lebesgue integration. [[User:Charles Matthews|Charles Matthews]] 18:08, 21 Jun 2004 (UTC)


Some aircraft were converted to turbine power using a Pratt & Whitney Canada PT6A [[turboprop]]. The Walter 601 Turboprop engine, manufactured in the Czech Republic, may also be fitted to the Otter. The PZL radial engine from the [[Antonov An-2]] (a plane that fulfills a very similar role) may also be fitted to the Otter.
::Hmm, I guess I'm not convinced. The Lebesgue integral is the object of interest, and the purpose of Lebesgue integration is to construct the Lebesgue integral, is it not? What other items are considered under the general heading of Lebesgue integration? Perhaps I don't get out enough, of course; I wouldn't be at all surprised. Happy editing, [[User:Wile E. Heresiarch|Wile E. Heresiarch]] 23:31, 22 Jun 2004 (UTC)


==Military Operators==
:Maybe one way would be to have an article like [[integration theories in mathematical analysis]], as a sort of umbrella; and then call the articles on particular theories [[Riemann integral]], [[Lebesgue integral]] and so on. Well, such an article would be useful in itself, as a survey. [[User:Charles Matthews|Charles Matthews]] 17:05, 23 Jun 2004 (UTC)
* Argentina, Australia, Bangladesh, Burma, Cambodia, Canada, Chile, Colombia, Costa Rica, Ethiopia, Ghana, India, Indonesia, New Zealand, Nicaragua, Nigeria, Norway, Panama, Paraguay, Philippines, United Kingdom, United States (Army, Navy)


==See also==
This seems to me like another instantiation of the "thing", "thing theory" problem. [[User:MarSch|MarSch]] 15:20, 11 Mar 2005 (UTC)
{{commons|DHC-3 Otter}}
* [[List of civil aircraft]]
* [[De Havilland Canada DHC-2 Beaver]]
* [[PAC 750XL]]
* [[Cessna Caravan]]


[[Image:C-FUKN-Northway-Aviation-DHC-3-Otter-2.jpg|thumb|left|Piston Otter on floats]]
----
Hello. About notation, under "Equivalent formulations" there are L<sup>1</sup> and C<sub>c</sub>. Do these want to be italicized as ''L''<sup>1</sup> and ''C''<sub>''c''</sub> or maybe ''C''<sub>c</sub> ? I'm just wondering how these are conventionally presented. Happy editing, [[User:Wile E. Heresiarch|Wile E. Heresiarch]] 14:14, 13 Apr 2004 (UTC)


==Related content==
::I think Lang and Rudin both use ''L''<sup>1</sup> and ''C<sub>c</sub>''. [[User:Loisel|Loisel]] 06:21, 16 Apr 2004 (UTC)
{{aircontent|
<!-- yes or no! -->
|has sequence=yes
|has relations=yes
|links=no
|has lists=no
|see also?=no


|related=
----
*[[De Havilland Canada DHC-6 Twin Otter|DHC-6]]
Hello, I've reworked the "Introduction" section to situate the Lebesgue integral historically, and to summarize the important differences with the Riemann integral. Then there are definitions and properties, and then there are the more detailed discussion sections. I hope this organization address the concerns with the "Rudinesque" previous revision. FWIW I'm not really happy with the "Lebesgue vs Riemann" aspect of this article, which might suggest there are exactly 2 defns of the integral; it's more "Lebesgue vs every predecessor". Also, I think "failure of the Riemann integral" gives a mistaken impression -- the Riemann integral has been a big success overall. Well, hope this helps. [[User:Wile E. Heresiarch|Wile E. Heresiarch]] 20:09, 8 Feb 2004 (UTC)


|sequence=
:All I know is, after I studied the Riemann integral as an undergrad, I thought it was pretty hot stuff, so when I heard about this thing called Lebesgue integration, it took quite a bit of argument to convince me that this new-fangled thing was even necessary in the first place. The introduction is a good start, but I would take select parts of the "discussion" and merge them. As for giving the impression there are 2 defs of "integral", this is more a misunderstanding of math in general...people think there is some Platonic concept called "the integral" that mathematicians go out and "discover" its "true nature"...when there are just various definitions and theories that just are what they are. It's very difficult to explain this. Maybe it should be pointed out that the Riemann and Lebesgue integrals aren't the only theories of integration that have been developed -- just the most popular and widely used. [[User:Revolver|Revolver]] 22:56, 8 Feb 2004 (UTC)
[[de Havilland Chipmunk|DHC-1]] -
[[de Havilland Canada DHC-2 Beaver|DHC-2]] -
[[de Havilland Canada DHC-3 Otter|DHC-3]] -
[[de Havilland Canada DHC-4 Caribou|DHC-4]] -
[[de Havilland Canada DHC-5 Buffalo|DHC-5]] -
[[de Havilland Canada DHC-6 Twin Otter|DHC-6]]


|similar aircraft=
----
*[[De Havilland Canada DHC-2 Beaver|DHC-2]] -
I don't think the discussion should come after the formal definitions and results. If someone is unfamiliar with the Lebesgue integral (and most will be who read the article) then the formal definition will seem pointless without some kind of motivation or discussion. As the article is now, someone who doesn't know the Lebesgue integral is likely to just give up part-way through the formal derivations and never reach the discussion. We're not trying to imitate Rudin. [[User:Revolver|Revolver]] 02:06, 7 Feb 2004 (UTC)
*[[De Havilland Canada DHC-6 Twin Otter|DHC-6]]
*[[Murphy Moose]]


|lists=
: If the definitions are incomprehensible, the discussion is irrelevant (and likewise incomprehensible). The most you can say to someone who doesn't understand the definition is "All definitions of the integral are the same for easy cases, but the Lebesgue integral handles some more difficult cases."


|see also=
::Well, before the Lebesgue integral was developed, many mathematicians discussed why they thought a new type of integral was necessary, what problems it should attack, what specific examples motivated it. The idea that it's impossible to "discuss" a mathematical subject without going through the details of the definitions doesn't seem right to me -- again, if for no other reason than that the mathematicians did it themselves for years before arriving the modern definition (think of groups, topological spaces, non-Euclidean geometries, etc.) I think it's quite possible to explain the specific issues that eventually led to the creation of the Lebesgue integral, without defining the Lebesgue integral itself. And yes, I'll try to do this and put it on the talk page someday...after I finish my thesis, find a job, move, etc.,...;-) [[User:Revolver|Revolver]] 22:56, 8 Feb 2004 (UTC)


}}
I think the (short) section just before the definition should say that. A secondary problem is that the discussion, as it stands, is needlessly verbose; it needs a lot more focus. [[User:Wile E. Heresiarch|Wile E. Heresiarch]] 09:20, 7 Feb 2004 (UTC)


[[Category:Canadian civil utility aircraft 1960-1969]]
What should ideally be done, is that the "discussion" and the "formal definition" should become merged into a single thread of narration, not separated into two pieces. This of course, would take a good deal of thought and effort to do it right. [[User:Revolver|Revolver]] 02:08, 7 Feb 2004 (UTC)
[[Category:Canadian military utility aircraft 1960-1969]]


[[fr:DHC-3]]
: I'm not convinced that is desirable. Maybe you can put a version of this kind of presentation on your talk page and have people look at it. Just a thought. [[User:Wile E. Heresiarch|Wile E. Heresiarch]] 09:20, 7 Feb 2004 (UTC)
[[no:De Havilland Canada DHC-3 Otter]]


Wile, see below, there's a guy who claims this article is too advanced. The text now under "Discussion" was my attempt at answering that gripe.


{{aero-1950s-stub}}
[[User:Loisel|Loisel]] 21:13, 6 Feb 2004 (UTC)

: Hello Loisel, thanks for bringing up this topic. I moved the discussion sections below the definitions, etc., because the discussion is quite long winded and goes off on several tangents. That's fine, really, since the Lebesgue integral is important and there's a lot to be said about it. But the article works better as a reference if we get right to the point.

: Maybe to address the concern that the article is too abstract, we can make the "Introduction" section (just before the defn) better (not necessarily longer). Without getting technical, one can say the Lebesgue integral is more general than some other defns, and also the it's defined as the limit of a sequence of simpler integrals. Perhaps an example of a sequence of simple functions will make a good companion article. [[User:Wile E. Heresiarch|Wile E. Heresiarch]] 01:49, 7 Feb 2004 (UTC)

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

Michael, why did you remove the previous article and leave a corpse in its stead? I've now restored the article.

[[User:Loisel|Loisel]] 08:04, 3 Feb 2004 (UTC)

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


I've made a correction to the Technical Difficulties regarding improper Reimann Integrals on this page. What was defined was not the Improper Reimann Integral but the improper Cauchy Pricipal Value. The improper Reimann Integral does not exist in this example.

-joshua
----------
Look - I'm sure that this is a well-writen article, but could someone (knowledgeable) put in a 2-3 line explanation of what it actually about, and what it is used for? The opening sentence "Let m be ..." isn't really au fait in a generalist encyclopedia. I'm really glad we have this sort of advanced stuff, but in this case I don't even understand what I don't understand. - [[User:MMGB|MMGB]]

Shouldn't it be at some point made clear that ''m'' is the Lebesgue measure? Otherwise, there's no guarantee the integral will correspond to Riemann integration - if ''m'' is the counting measure, it will correspond to summation. But I'm not sure where precisely to work that in.

----

The link to [[monotone convergence theorem]] is to a different theorem.

[[User:Charles Matthews|Charles Matthews]] 14:00, 6 Sep 2003 (UTC)

Fixed. [[User:Loisel|Loisel]] 06:38, 7 Sep 2003 (UTC)

== Too advanced! ==

This encyclopedia article is too advanced; the only people who will be able to get the meaning are those who have already learned Lebesgue theory thoroughly. The modern, extremely compact, definitions of mathematics are simply a shorthand for people already familiar with the underlying concepts and some illustrative examples; without including the latter, the former is unintelligible.

In fairness, an article on the Lebesgue integral is certainly only of interest to a specialized audience. And it would perhaps be appropriate to include the existing article as an advanced appendix as a minimal, extremely formal, refresher, less formal explanation is needed for anyone who doesn't remember or hasn't learned what ideas the formal definition corresponds to.

For the main body of the article, it would be nice to follow history somewhat: (1) explanation of the problems encountered with the Riemann integral including specific functions or applications (2) elementary definition (NOT a formal argument: emphasis on underlying IDEAS, not on formal definitions) and explanation of how the problems of Riemann are resolved by Lebesgue (that is, examples) (3) perhaps a little modern history on the growth of analysis since Lebesgue. The formalities could be a conclusion of sorts. I would write such an article myself, but I don't know Lebesgue theory sufficiently well (hence my need to refer to an encyclopedia article); perhaps someone else might? (Anon.)

A page on [[improper integral]]s in the Riemann theory would be a help. But no one-page intro to the Lebesgue integral is going to be easy.

[[User:Charles Matthews|Charles Matthews]] 16:10, 23 Oct 2003 (UTC)


This is a valid point, however I'm not very good at history. I could attempt to vulgarize the concepts discussed in the "formal construction" section, but I couldn't color the text with historical anecdotes about integration theory: I don't know any.

Would that help?

[[User:Loisel|Loisel]] 16:47, 24 Oct 2003 (UTC)

Is this better? [[User:Loisel|Loisel]] 18:15, 24 Oct 2003 (UTC)

I'm sorry, this topic is advanced [[User:pdenapo|pdenapo]]
Wed Jan 14 03:09:34 UTC 2004

== Well done ==

With the addition of the new introductory material, this article takes the place of a good professor - at first giving a general indication of the questions and problems at hand, and once the ideas involved are explained a little further (with some nice examples) providing the formal framework which enables those ideas to be put into rigorous use. Great job.

--

Couple of points where I can carp (I do concur with the post above: the work on this page was very worthwhile).

*The Lebesgue approach is not the most elementary area-based integration theory; that distinction goes to the Riemann integral.

Don't think that's actually true (cf. Dieudonne's ''Treatise on Analysis'' ona sub-Riemann theory).

*Uniform convergence of Fourier series.

Rare? A couple of derivatives will do.

[[User:Charles Matthews|Charles Matthews]] 07:24, 18 Dec 2003 (UTC)

About the uniform convergence: it is true that twice differentiable periodic functions have uniformly convergent Fourier series. This is a very thin set in L^2. That is one of the meanings of "rare." Feel free to adjust the wording if you think you can improve it, but the goal of that passage (to show that there are many common examples where the uniform convergence theorem is insufficient) should remain, I think.

About "elementary." I meant to recognize the order in which this is usually taught. I don't know the sub-Riemann theory you mention, but I should've thought there would be a gazillion variants that claim to be more elementary than the Riemann integral. Again, feel free to adjust the wording if you think you can improve it. If you're changing it, I think it should probably say that Riemann is "more elementary" than Lebesgue, but I doubt it should refer to some obscure theory that's not generally taught.

Anyway, have a ball. [[User:Loisel|Loisel]] 02:04, 8 Jan 2004 (UTC)

:: [[User:Charles Matthews|Charles Matthews]] refers here to what Dieudonne calls regulated functions in the English translation (satisfying a condition something like: both one sided limits exist at every point = uniform norm closure of linear span of indicator functions of intervals ). It is more elementary in the sense that all its properties follow by continuity. Dieudonne disparages theRiemann integral (which in fact is the prevalent attitude among the mathematicians I know)

--------

I just removed the following text:

'''Correction: The improper Reimann integral does not exist for f or g since the improper Reimann Integral is defined as a double limit and you cannot subtract infinities, i.e. the improper Reimann integral is defined as limlim&int;<sub>a</sub><sup>b</sup>f(x) dx where the limits are taken as a goes to infinity and b goes to infinity. What is true is that the Improper Cauchy Principal value (about zero) for f does in fact exist and it is PV&int;<sub>-&infin;</sub><sup>&infin;</sup>f(x)=0.'''

I will modify the text to say something like '''(sometimes called the Improper Cauchy Principal value about zero)'''. Please note [[Wikipedia:Integrate_changes]], which is part of the Wikipedia policy, requests that you integrate your changes so that they form a seamless part of the article. [[User:Loisel|Loisel]] 02:14, 8 Jan 2004 (UTC)

:I think that maybe this whole paragraph should be junked, since with the proper definition with 2 limits, there is no preferred point and no translation noninvariance. Thus this is not a deficiency of the Riemann integral.[[User:MarSch|MarSch]] 15:30, 11 Mar 2005 (UTC)

------------
I've rewriten this article since the old version focused only on
the technical difficulties of Riemman integral, rather than
defining the concept of Lebesgue integral. I've included
the examples in the old version, though [[User:pdenapo|pdenapo]]
Wed Jan 14 03:09:34 UTC 2004

The new introduction is nice, but not consitent with the definition
that appears below.
[User:pdenapo|pdenapo]]Wed Jan 14 03:09:34 UTC 2004

I think the rewrite has destroyed a large quantity of useful information and has generally decreased the quality of the article. I'm tempted to revert.

[[User:Loisel|Loisel]] 08:05, 1 Feb 2004 (UTC)

== Equivalent Formulations ==

I think this page is very good.
Hello, Do you have references for the equivalent formulations of the integral? (unique continuous extension of a linear functional...) Thanks Sergei Vieira srgvie2000@yahoo.com.br

That's elementary functional analysis. See either Rudin's Real and Complex Analysis or Rudin's Functional Analysis, for instance. If f is a continuous functional defined on X, a dense subset of the Banach space Y, and if g and h are continuous functionals that agree with f when restricted to X, then g=h. To see this, let x be arbitrary in Y and let x_k be a Cauchy sequence in X converging to x. Then g(x_k) converges to g(x) and h(x_k) converges to h(x). But g(x_k)=h(x_k)=f(x_k), and f(x_k) is Cauchy. Hence f(x_k) has a unique limit point, and g(x)=h(x). You can get existence out of (say) the Hahn-Banach theorem (but you can also get it in an elementary way.)

The statement that L^1 is the completion of C_c in the norm given by the Riemann integral is just a rehash of the uniqueness of the completion of a metric space. We know that in the Lebesgue construction, C_c is dense in L^1 (and so L^1 is a concrete completion of C_c). If we define a norm on C_c using the Riemann integral, we know that it agrees with the norm you would get out of restricting the L^1 norm on C_c. Hence, C_c with the Riemann integral L^1 norm, is the same as C_c with the Lebesgue integral L^1 norm. C_c with the Riemann integral L^1 norm has an abstract completion, as per metric space theory, but the Lebesgue theory is just an explicit version of this completion.

[[User:Loisel|Loisel]] 19:44, 27 Jun 2004 (UTC)

== Indicators ==

Does anyone agree with me that there should be more consistency with the notation here? On the pages [[indicator function]] and [[Riemann integral]], ''I''<sub>(your set here)</sub> is used. I definitely prefer this, or a notation with &chi;s, to the current one on this page. Best, [[User:Mat cross|mat_x]] 15:59, 18 Aug 2004 (UTC)

== Mountains ==

The Riemann integral of the mountain requires lower sums and upper sums, in each vertical slice find the highest and lowest points. The integral beng described is more like the Cauchy integral. [[User:CSTAR|CSTAR]] 15:42, 19 Dec 2004 (UTC)

== Towards a better integration theory ==

Is there any objection to deleting this section? [[User:CSTAR|CSTAR]] 17:42, 22 Dec 2004 (UTC)

:I have no objections. You are doing a good job on the page, but I wonder whether you could write slightly more on the definition of measure besides refering the reader to the [[measure (mathematics)]] article. I fear that many readers may not get pass the sentence "Let &#956; be a (non-negative) measure on a sigma-algebra X of subsets of E." -- [[User:Jitse Niesen|Jitse Niesen]] 21:28, 22 Dec 2004 (UTC)

:: Yes I do plan to fix that soon.[[User:CSTAR|CSTAR]] 21:35, 22 Dec 2004 (UTC)

== Edits ==

Remarks on measurability condition.. MEasurability of a function as defined in the article is a property of the underlying sigma-algebras of source and target space. The user who made the latest edits, confused this with measurability of functions with respect to the sigma-algebra of the completion in the source space. This is not necessary and is nowehere used in the article.. Of course this means that if f = g a.e. and is measurable g may not be. There are a number of other changes which I don't see add anything to the article. For instance why add vector-valued functions here? If there is no objection I am going to revert. The only improvement was the use oof \liminf and \limsup whic I propose to keep. [[User:CSTAR|CSTAR]]

== Integral as area under curve ==

Intuitively, continuity is not necessary: Two disjoint rectangles with vertical sides and of different heights have an area and aren't representedby continuous functions. [[User:CSTAR|CSTAR]] 15:29, 11 Mar 2005 (UTC)

:The first sentence of the article talk about an area being bounded. If the fuction is discontinuous then no area is bounded. Of course I should also have made changes to say that the area is bounded from below by 0. I realize that even then you will have difficulty on the sides, so I will be bold and change it to something which mentions area below a (positive) function. [[User:MarSch|MarSch]] 15:40, 11 Mar 2005 (UTC)

::Huh? Discontinuity implies unboundedness? What about the indicator function of the interval [0,&nbsp;1]? That's a bounded function with a discontinuity. [[User:Michael Hardy|Michael Hardy]] 00:09, 14 Mar 2005 (UTC)

:::Perhaps [[User:MarSch|MarSch]] means with "unbounded" that the graph does not run along the boundary. For instance, with your indicator function, the graph looks like this
------
-------- ------
:::and the vertical sides of the square &mdash; running from (0,1) to (0,0) and from (1,1) to (1,0) &mdash; are not bounded by the graph. MarSch's change from "bounded" to "contained" solves the problem. -- [[User:Jitse Niesen|Jitse Niesen]] 14:12, 14 Mar 2005 (UTC)

:::: Referring to the region ''under the curve'' and above the x-axis is about as unambiguous as you can get in natural language. And the anti 2D bias isn't very informative in my view. [[User:CSTAR|CSTAR]] 14:26, 14 Mar 2005 (UTC)

:::I think we should try to keep the first sentence as easy as possible and I edited the article accordingly (bringing back the "2D bias"). -- [[User:Jitse Niesen|Jitse Niesen]] 15:10, 14 Mar 2005 (UTC)

::::As far as I've seen there are also articles about [[integration]] and [[Riemann integration]]. Your sentence would be perfect for integration. For those reading this article it is needlessly simplistic and ugly. -[[User:MarSch|MarSch]] 14:40, 4 Apr 2005 (UTC)

== Generalization ==

This article defines the integral for functions f: R -> R. From what I can tell from reading it, it's trivial to extend this definition to cover any function f: S -> R where S is any measurable set - should the article cover this? I'm particularly thinking that it's easy to define an integral on f: R x R -> R &mdash; [[User:ciphergoth|ciphergoth]] 22:30, 2005 Apr 28 (UTC)

: From what I see, the integral is defined exactly as you want, on a general space. The first several paragraphs deal with this. [[User:Oleg Alexandrov|Oleg Alexandrov]] 23:08, 28 Apr 2005 (UTC)

== Limitations of the Riemann integral ==

You show that the two functions f(x)=-1+2[x>0] and g(x)=-1+2[x>1] are not riemann integrable, and that the cauchy values of the integrals differ. Sadly. But it is not clear to me if the lebesgue integral improves the situation ? [[User:Bo Jacoby|Bo Jacoby]] 13:57, 22 September 2005 (UTC)

== Reversion ==
Hi CSTAR. You reverted my simplification. Your argument is incorrect. The function |f|, mapping x to |f(x)|, allows any real function to be split into a positive and negative part:
f=(f+|f|)/2+(f-|f|)/2. The ad hoc notations f+ and f- are not necessary. [[User:Bo Jacoby|Bo Jacoby]] 08:23, 27 September 2005 (UTC)

'''Reply'''.
# Your definition of the integral was expressed as a difference of two values both of which could be +&infin; You must assume that at least one of the values is finite.
# Re your comment: ''The ad hoc notations f+ and f- are not necessary and should be avoided''. On the contrary, this a very common notation and is hardly ad-hoc. See
:*P. Halmos, ''Measure Theory'', or
:*W. Rudin, ''Real and Complex Analysis''.
:Why do you believe that this notation should be avoided?
:--[[User:CSTAR|CSTAR]] 15:57, 27 September 2005 (UTC)
#Thank you for your reply.
#I agree that the finity condition should not be postponed.
#Using the better-known absolute value function |x| should make it easier to read. x+ = (x+|x|)/2 and x- = -(x-|x|)/2. You may assume that the reader knows |x|, but as you do not assume that the reader knows x+ and x-, you state that 'we need a few more definitions' and then you define x+ and x- in the text. Well, we don't need these definitions. We can use standard mathematical expressions instead.
#Please compare my version with your reverted version. You state that the integral is defined for real functions, but you actually define it for complex functions too. Why not state that it is defined for complex functions? You call the set X when you mean E. It's merely a typo. Your definition made in the 'indicator function' section was not used in the 'simple function' section , so either the former could be omitted or the latter be changed to build on the former. I made many small improvements.
#I wonder why the classical notation f(x)dx is abandoned. How do you express the integrand ax^2dx as opposed to ax^2da when you write ax^2d\mu? Only the repeated dummy variable tells what the integration is about.
[[User:Bo Jacoby|Bo Jacoby]] 11:53, 28 September 2005 (UTC)

i second CTARS's comments above. these notations are standard, not "ad hoc". [[User:Mct mht|Mct mht]] 18:01, 16 August 2006 (UTC)

== Silly quote==
I removed a silly quote about the Lebesgue integral not being useful "physically". I don't know what this means. Is there a physical usefulness to having the irrationals? The reals are to the rationals as Lebesgue is to Riemann. And, should we add a ton of pro-Lebesgue quotes? -[[User:cj67|cj67]]

: Re: ''And, should we add a ton of pro-Lebesgue quotes?'' I don't care one way or the other.
: Re ''The reals are to the rationals as Lebesgue is to Riemann.'' I don't know what this means. More specifiuically, the reals are the metric completion of the rationals; the extension of Riemann to Lebesgue is more subtle than an extension by continuity to a completion.--[[User:CSTAR|CSTAR]] 17:16, 26 May 2006 (UTC)

::Nope. Lebesgue integrable functions are the completion of the continuous Riemann integrable functions w.r.t. the L^1 norm (which can be defined using the Riemann integral, since you are only considering continuos functions).([[User:Cj67|Cj67]] 20:05, 30 May 2006 (UTC))

::: Nope. The elements have to be represented as functions. As abstract Banach spaces, L^1 is the completion of the continuous Riemann integrable functions w.r.t. the L^1 norm, as you point out. But representing an element of this Banach space as an equivalence class of measurable functions is another matter, because pointwise evaluation is not continuous as a functional on L^1. --[[User:CSTAR|CSTAR]] 20:24, 30 May 2006 (UTC)

:::: L^1 is a representation of the completion of Riemann integrable functions, just as the reals are a representation of the completion of the rationals. I think the analogy is clear. Of course, there is more involved, since L^1 does not equal the reals. But I think my point about the physical "usefulness" of the irrationals is clear. ([[User:Cj67|Cj67]] 22:15, 30 May 2006 (UTC))

:::: the analogy looks fine to me. completion of metric spaces are by construction equivalence classes. the Lebesgue-a.e. condition gives an explicit identification. [[User:Mct mht|Mct mht]] 18:08, 16 August 2006 (UTC)
::::: My point in saying that "''the extension of Riemann to Lebesgue is more subtle than an extension by continuity to a completion''" was that the "pointwise" properties of L^1 do not follow by continuity, since f_k converges to f in L^1 does not imply f_k converges to f almost a.e. (though it is true that some subsequence of f_k converges to f ae.) . ANyway, this is ome of those WP talk page discussions I'm sorry I ever got into.--[[User:CSTAR|CSTAR]] 18:22, 16 August 2006 (UTC)

Revision as of 18:23, 16 August 2006

De Havilland Otter
Otter in Harbour Air livery
Description
Role Transport
Crew 1
First Flight Dec 12 1951
Entered Service
Manufacturer de Havilland Canada
Dimensions
Length 41 ft in 12.5 m
Wingspan 58 ft in 17.7 m
Height 13 ft in 4 m
Wing area ft² m²
Weights
Empty 5287 lb 2398 kg
Loaded 8000 lb 3628 kg
Powerplant
Engines 1 Pratt & Whitney S1H1-g Wasp radial
Power 600 hp 447 kW
Performance
Maximum speed 160 mph 258 km/h
Ferry range 960 miles 1545 km
Service ceiling 17900 ft 5460 m
Rate of climb 1000 ft/min 305 m/min
Wing loading lb/ft² kg/m²
Avionics
Avionics
Turbo Otter on wheel-skiis

The de Havilland Canada DHC-3 Otter is a single engine, high wing, propeller driven aircraft. It was conceived to perform the same roles as the previously successful Beaver but was overall a larger plane. Initially named the King Beaver de Havilland began design work in January, 1951. Canadian certification was given in November, 1952. The US Army soon became the largest operator of the aircraft (184 delivered with the designation U-1A Otter).

The Otter served as the basis for the very successful, Twin Otter which featured two wing mounted Pratt and Whitney PT-6 turboprops.

Some aircraft were converted to turbine power using a Pratt & Whitney Canada PT6A turboprop. The Walter 601 Turboprop engine, manufactured in the Czech Republic, may also be fitted to the Otter. The PZL radial engine from the Antonov An-2 (a plane that fulfills a very similar role) may also be fitted to the Otter.

Military Operators

  • Argentina, Australia, Bangladesh, Burma, Cambodia, Canada, Chile, Colombia, Costa Rica, Ethiopia, Ghana, India, Indonesia, New Zealand, Nicaragua, Nigeria, Norway, Panama, Paraguay, Philippines, United Kingdom, United States (Army, Navy)

See also

Wikimedia Commons has media related to DHC-3 Otter.
Piston Otter on floats

Related content

Related development

Aircraft of comparable role, configuration, and era


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