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Talk:Construction of the real numbers

Dedekind cuts
The definition of Dedekind cuts on this page seems incorrect. According to the page, a partiton of an ordered field, $$(A,B)$$, is a Dedekind cut, where $$A$$ and $$B$$ are non-empty sets, such that $$A$$ is closed downwards and $$B$$ is closed upwards. Then, $$A$$ and $$B$$ intersect at a point - ie, they do not form a partition.

The page discusses a non-closed addition, where $$(\{a\le 0\}, \{b>0\})+(\{a<0\}, \{b\ge 0\})$$ is not a Dedekind cut. But of course, neither $$(\{a\le 0\}, \{b>0\})$$ nor $$(\{a<0\}, \{b\ge 0\})$$ is a Dedekind cut by this page's definition, which requires both sets that make the cut to be closed.
 * No, this is wrong. The definition does not require each set to be topologically closed. You're confusing topologically closed with downward- or upward-closed. (Walt, it wasn't me who wrote these first comments.) Revolver 02:03, 16 Dec 2004 (UTC)

 * Oops, sorry about that. I thought this was all one big comment by you. -- Walt Pohl 05:12, 16 Dec 2004 (UTC)

The definition I'm familiar with defines A as an open set, extended downwards. B as closed and extended upwards. The least upper bound of A is the number defined by the cut. See for a reference to one such formulation.

Other references define them as a non-empty, downwards extended set, that is bounded-above, and doesn't contain a greatest element. The the least-upper bound of this set defines the number at the cut. See for a reference to one such formulation.

Please see Talk:Dedekind cut Revolver 16:30, 15 Dec 2004 (UTC)

 * The definition on this page only requires that A be "closed downward", not closed. Likewise for B. -- Walt Pohl
 * Please read the comments at Talk:Dedekind cut. This is precisely the problem. Unless A is required to be open (equiv, B is required to be closed), addition is not a closed operation. Read the comment in the article -- it is absolutely correct. We need a definition where the operation makes sense! The best way around this for the particular case of Q seems to be to required that A be open. More general constructions in general posets may handle the situation differently. Revolver 02:01, 16 Dec 2004 (UTC)

The Dedekind cut arithmetics is erroneous
Note, that if you define $$(A_x,B_x)\,$$ as the cut corresponding to $$ 2-\sqrt2\,$$, and $$(A_y,B_y)\,$$ as corresponding to $$2+\sqrt2\,$$, and apply the rules as written, then neither $$(A_{\rm sum},B_{\rm sum})\,$$, nor $$(A_{\rm prod},B_{\rm prod})\,$$ is a Dedekind cut. The former is not even a partition, since $$4 \notin A_{\rm sum} \cup B_{\rm sum}$$; and the latter is a partition but not a correct cut, since 2 is the greatest element of $$A_{\rm prod}\,$$. The older constructions were based only on the set of positive rational numbers. The cuts may be defined for the whole set $$\mathbb{Q}\,$$; but in that case the rules for the arithmetic get immensely more complex than the thing exhibited at the moment. (The same goes for any other ordered field.) The rules for arithmetics now are not completely adapted to this, even in spite of the 'cheating' at multiplication by making a restriction to nonnegative cuts. My personal suggestion is to make another approach. I've made a longer suggestion for this at de:Diskussion:Dedekindscher_Schnitt, which I prefer not to repeat here. JoergenB 01:16, 10 October 2006 (UTC)
 * I suggest we only do it for positive rationals, as that will give an idea of what is going on, (and anyone who can understand it for positives, can work out the extension.. its just tedious..). any ideas? --TM-77 14:23, 28 October 2006 (UTC)

That is the right way to do it. I might look up Landau Foundatioons of Analysis and adapt if still needed. --Gentlemath (talk) 22:59, 28 March 2009 (UTC)

Construction by decimal expansions
''We can take the infinite decimal expansion to be the definition of a real number, considering expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally.''

I don't know whether such a definition would work; it seems reasonable. I think it should be pointed out that we could use (any base? any integer as a base?). In any case surely it doesn't need to have anything to do with the number 10. Brianjd | Why restrict HTML? | 11:46, 2005 Apr 24 (UTC)

 * Doing that is the easy part. The hard part is to prove that the newly created field is complete, that is any Cauchy sequence converges. Oleg Alexandrov 15:14, 24 Apr 2005 (UTC)

 * maybe you guys can tell me how to properly define the sum of 0.44444444 and 0.66666666 (I mean the infinite decimals, of course)....in such a way that your definition applies to any two decimal expansions...--345Kai 20:40, 3 October 2006 (UTC)

 * 0.444... + 0.666... = (0.4 + 0.6) + (0.04 + 0.06) + (0.004 + 0.006) + ... = 1 + .1 + .01 + ... = 1.111... --Ian Maxwell 04:15, 14 October 2006 (UTC)

Yet it can be done. The first to do so was Gentlemath (talk) 22:50, 28 March 2009 (UTC)

Definition of multiplication of Dedekind cuts seems wrong
We want there to be a positive real number x such that ''x. x = 2''.

In the following discussion I'm going to refer only to the left element of the Dedekind cut for conciseness.

Let's first recall that there is no rational number q such that $$q \times q = 2$$.

We would expect that the Dedekind cut

 * $$ A = \{ x\in \textbf{Q} : x < 0 \lor (x \geq 0 \land x \times x < 2) \}$$

should represent the positive square root of 2. It should certainly be clear this could be the only possible definition of positive root 2, since any other positive Dedekind cut must either contain some positive x such that $$x \times x > 2$$, or must exclude some positive x such that $$x \times x <  2$$ (recalling again that there is no rational x such that $$x \times x =  2$$).

However, given the definition in the main article, we get $$A \times A =$$

 * $$ { \{ a_\mathrm{prod} \in \textbf{Q} : a_\mathrm{prod} \, {\not \in} \, \{ b_\mathrm{prod} \in \textbf{Q} : b_\mathrm{prod} = b_x \times b_y \land b_x \in A \land b_y \in A \} \}} $$

It should be apparent that, by this definition, the value of $$A \times A $$ includes the rational number 2. Since by definition the embedding of 2 in the real numbers is $$\{x: x < 2\}$$ and thus excludes 2, we are forced to conclude that the positive square root of 2 is not a real number!

In fact, we should see that the value of $$A \times A $$ contains an upper bound, namely 2, and thus does not even fit the defintion of a real number!

I'm going to work on an alternative presentation of this section, with what I regard as a correct definition of multiplication -- I'll present it in talk before making any changes. Cheers! Grover cleveland (talk) 18:58, 29 August 2008 (UTC)

 * Wikibooks simply suggests that the left product set consisting of the products of left set elements (instead of the right product set consisting of the products of right set elements, as here). But as noted on its talk page, that doesn't really work either, because minus times minus is plus and all that. Neither the definition given here nor the one given there actually makes the cuts closed under multiplication. I'm looking forward to the refactoring. -- Jao (talk) 20:14, 29 August 2008 (UTC)

Proposed replacement for definition of Dedekind cuts
A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element. Real numbers can be constructed as Dedekind cuts of rational numbers.

For convenience we may take the lower set $$A$$ as the representative of any given Dedekind cut $$(A, B)$$, since $$A$$ completely determines $$B$$. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number $$r$$ is any subset of the set $$\textbf{Q}$$ of rational numbers that fulfils the following conditions:
 * 1) $$r$$ is not empty
 * 2) $$r \neq \textbf{Q}$$
 * 3) r is closed downwards.  In other words,  for all $$x, y \in \textbf{Q}$$ such that $$x < y$$, if $$y \in r$$ then $$x \in r$$
 * 4) r contains no greatest element.  In other words, there is no $$x \in r$$ such that for all $$y \in r$$, $$y \leq x$$

\Leftrightarrow x \subset y$$
 * We define a total ordering on the set $$ \textbf{R} $$ of real numbers as follows: $$x < y

 * We embed the rational numbers into the reals by identifying the rational number q with the set of all smaller rational numbers $$ \{ x \in \textbf{Q} : x < q \} $$. Since the rational numbers are dense, such a set can have no greatest element and thus fulfils the conditions for being a real number laid out above.

 * Addition. $$A + B := \{a + b: a \in A \land b \in B\}$$

 * Subtraction. $$A - B := \{a - b: a \in A \land b \notin B\}$$

 * Negation is a special case of subtraction: $$-B := \{a - b: a < 0 \land b \notin B\}$$

 * Defining multiplication is less straightforward.
 * if $$A, B \geq 0$$ then $$ A \times B := \{ a \times b : a \geq 0 \land a \in A \land b \geq 0 \land b \in B \} \cup \{ x : x < 0 \}$$
 * if either $$A$$ or $$B$$ is negative, we use the identities $$ A \times B = -(A \times -B) = -(-A \times B) = (-A \times -B)$$ to convert $$A$$ and/or $$B$$ to positive numbers and then apply the definition above.
 * We define division in a similar manner:
 * if $$ B > 0 $$ then $$ A / B := \{ a / b : a \in A \land b \notin B \}$$
 * If $$B$$ is negative, we use the identity $$ A / B = -(A / -B) $$ to convert $$B$$ to a positive value and then apply the definition above.

We could add an example, such as the construction of root 2. Let me know what you guys think. Cheers. Grover cleveland (talk) 20:39, 29 August 2008 (UTC)

 * I haven't analyzed it in detail, but I think it looks safe. Do you have a source? Otherwise, it might look like OR. -- Jao (talk) 11:14, 30 August 2008 (UTC)

 * I've been scouring Google Books in a very frustrating search. Most textbooks just give a basic definition of cuts and possibly addition, before going on to say something like "multiplication is more complicated, and is left as an exercise for the reader" :)  However, the best ref I've found so far has been .  Google Books link here.  It gives a definition of multiplication that agrees with the one above: however it doesn't define division and gives a somewhat different (though equivalent) def of subtraction.  However, it does have a really cool diagram of a pair of scissors cutting the number line :)  The search continues.... Grover cleveland (talk) 15:40, 30 August 2008 (UTC)

 * Alright, but don't hesitate to go ahead and update the article, and give that reference for the multiplication. At any rate, a correct definition that could be accused of being partly OR is quite a few orders of magnitude better than a clearly flawed and completely unsourced definition. -- Jao (talk) 16:59, 30 August 2008 (UTC)

 * Done. Feel free to add any more references, etc.  Cheers. Grover cleveland (talk) 19:08, 30 August 2008 (UTC)

 * Looks great! -- Jao (talk) 19:09, 30 August 2008 (UTC)

Maybe the right thing to do would be to follow Landau Foundations of Analysis. I think he defines + * / and < for the positive integers and then makes the right definitions to allow signs. No free online version, alas. --Gentlemath (talk) 22:56, 28 March 2009 (UTC)

Streamlining arithmetics
It is enough to define addition and multiplication for the Dedekind cuts. To me the simplest route seems to be the following:

 * Define $$A+B := \{a+b \ | \ a\in A, b\in B\}$$. This already determines the additive inverse $$-A = \{ a-b | a<0 , b\notin B \}$$.

 * Define $$A/B := \{a/b \ | \ a\in A, b\notin B \}$$ for $$A,B>0$$. (I think this is wrong in the article.)

 * Define $$(-A)(-B) := AB := A/(1/B)$$ and $$A(-B) := (-A)B := -(AB)$$ for $$A,B>0$$ and $$AB:=0$$ if either is zero.

If one is ready to omit the details, the following definition is even more effective. Here we denote the set of Dedekind cuts by $$\mathcal D$$ and the embedding $$\mathbb Q \hookrightarrow \mathcal D$$ by $$q\mapsto q_{\mathcal D}$$.

 * Define a topology for the set $$\mathcal D$$ by introducing the open intervals $$]a,b[\subset \mathcal D$$ in the obvious way.

 * Define the addition and multiplication as the unique continuous extensions of the corresponding operators $$*:\mathbb Q\times \mathbb Q \to \mathbb Q$$ into $$*:\mathcal D\times \mathcal D \to \mathcal D$$.

In order to justify this definition one must be able to show that the continuous extensions exist and are unique, but I think that the main idea in the definition is transparent enough to pay off this drawback.Lapasotka (talk) 23:14, 19 October 2009 (UTC)

Construction from the group of integers
A recent discussion "A new (?) construction of R" (2008-09-05) on sci.math mentions a related definition of a positive real given in 1946 by A. N. Kolmogorov. — Loadmaster (talk) 18:01, 5 September 2008 (UTC)

Proposed move
I noticed an edit in which someone changed a link to construction of the real numbers to a link to construction of real numbers, with no "the". The title with the definite article is a redirect. Omitting "the" seems weird. It is as if individual real numbers were to be constructed, rather than the system as a whole. Are there opinions on moving the page to construction of the real numbers? Michael Hardy (talk) 18:56, 18 September 2008 (UTC)

 * Sounds like a flawless argument. I guess nobody just thought of it before. -- Jao (talk) 19:09, 18 September 2008 (UTC)

 * I am copying my edit from WPM: True, but it can also be read as construction of "real numbers", it being understood that everybody has heard of the famous "real numbers". It can also be read as an editorial contraction, headline-style.  I think this should be a question of wiki style guidelines rather than a semantic issue. Katzmik (talk) 05:44, 19 September 2008 (UTC)

 * Relevant guideline policy is here: WP:NAME. I can't immediately see anything which prescribes leaving out the definite article in the middle of a noun phrase. However, this sub-page, WP:COMMONNAME, may have some bearing – are people more (or less) likely to type "of the real numbers" than "of real numbers" when referring to or looking for this material? H.G. 06:34, 19 September 2008 (UTC)

 * An answer is provided by traffic statistics. construction of real numbers is hit on the average 40 times daily, whereas construction of the real numbers is hit about once.  This does not mean the former title is preferable necessarily.  The discrepancy may be due to the fact that the pages linking to it all link to the former rather than the latter.  At any rate, if we are worried about the stray bird who types in the definite article in his query, the worries are unfounded.  Katzmik (talk) 06:40, 19 September 2008 (UTC)

 * P.S. The results of all the olympiads indicate that the russians are the ones who are interested in, and good at, math, and they all tend to leave out the definite article (Russian does not have one). :) Katzmik (talk) 06:42, 19 September 2008 (UTC)
 * Do the traffic statistics distinguish between page visits by people who've typed something into the search box and page visits made by people who've clicked on an internal link? H.G. 07:30, 19 September 2008 (UTC) Forget that, I obviously can't read this morning.  H.G. 08:30, 19 September 2008 (UTC)

For what it's worth (and it's part of what I had in mind when mentioning WP:COMMONNAME), I'd expect to see this written as "construction of the real numbers", as we tend to invoke the definite article when referring to them as a group (in the non-mathematical sense). A quick gander at Google Books and Google Scholar also suggests a preference for using "the". I'm willing to bet that the proliferation of links to construction of real numbers has more to do with some editors' preference for avoiding redirects rather than common practice in real life. H.G. 08:42, 19 September 2008 (UTC)

dedekind cut the real numbers?
ok i'm just a beginner in analysis. after finishing the first chapter in rudin's principle of mathematical analysis, my first thought is, wat if we run throught the dedekind construction on the set of real numbers? i think because rudin goes on sumthin about isomorphism between Q, the rationals, and Q*, the set of rational cuts, and since each r* is the supremum of all q* such that q* is a subset of r*, we wouldn't be doing any good if we apply this construction to the reals because they already got the least upper bound property, i.e. this new set is isomorphism to the original set of reals R. if sum1 can help a poor dumbass out pls do

 * Absolutely correct. Or in simpler terms, the construction of R filled all "gaps", so there are no more "gaps" to fill. -- Jao (talk) 12:53, 12 January 2009 (UTC)

Dedekind cuts still not ok!
Proof that the details will get you in the end! I'm going to need to look up Landau and get this right!

Note that in the definition of 1/B there would be a greatest element if B is rational. One could define A/B for A,B>0 as the set of all a/b with a in A and b in the complement of B ***but not the miminimum element of that complement if there is one***. As usual, the thorniest details are for rationals when we already know the result we want.

I think that the Landau definition for 1/B with B>0 is the cut whose lower set is essentially {all x>0 | xB<1 }--Gentlemath (talk) 23:58, 28 March 2009 (UTC)

 * Not sure what the problem is. Take A = 1, B = 2.  Then A/B = {x/y : x < 1 && y >= 2}.  This has no greatest element, since (1/2) is not in A/B but all smaller rationals are.  Can you explain the worry?  Grover cleveland (talk) 14:33, 29 March 2009 (UTC)

I agree now. I'm never sure if it is legit to erase ones own talk contributions. But at least -3/10 was a problem before.--Gentlemath (talk) 17:42, 29 March 2009 (UTC)

Cauchy sequences
The last part of the construction using Cauchy sequences seems very handwaving. It is a bit confusing that the two constructed sequences $$l_n$$ and $$u_n$$ are equivalent (i.e. in the same class representing a real number), yet one of them is said to not be an upper bound, while the other one is. —Preceding unsigned comment added by 80.216.180.6 (talk) 16:20, 16 May 2009 (UTC)

schmieden-laugwitz type construction using a Frechet filter
Does anyone know if one can construct R along the lines of the hyperreal construction, but using the Frechet filter instead of the ultrafilter? This would eliminate the need for choice. Tkuvho (talk) 11:31, 21 January 2010 (UTC)
 * Not really, as far as I can see. QN modulo the cofinite filter is a rather unpleasant object (it has uncountably many square roots of 1, for example), and you'd need some way of restricting to well-behaved points. All the ways I've thought of (restricting to finite elements that are order-comparable with every rational, or restricting to points that are limits of rational sequences) more-or-less boil down to restricting to Cauchy sequences, and if you're going to do that, you might as well forget the filter and just use the Cauchy sequence construction from the beginning. By the way, article talk pages are for discussions about what should and shouldn't be in the article. Questions such as this one about actual mathematics relating to articles are better posted to WP:RD/Math, which is watched by far more people. Algebraist 18:45, 21 January 2010 (UTC)

Did Weierstrass construct the reals by means of decimals?
one more reference if someone wants to follow them up:

http://www.ams.org/bull/1909-16-02/S0002-9904-1909-01864-6/S0002-9904-1909-01864-6.pdf this is a review of the reference I gave and describes its contents

also: —Preceding unsigned comment added by Gentlemath (talk • contribs) 23:32, 14 March 2010 (UTC)

There is also http://eom.springer.de/r/r080060.htm which may actually make too strong a case. --Gentlemath (talk) 00:52, 15 March 2010 (UTC)

 * The first reference is a review of a book from 1908 by someone other than Weierstrass. It does not claim that W. defined reals by decimal expansions.  Briefly, W. defines a number as a series of positive rationals.  The second reference is inaccurate, and does not provide any sources for the claim made (that W. defined reals by decimal expansions).  With all due respect to Springer, this is simply incorrect.  The sentence following Stevin in the current version is therefore inaccurate.  Tkuvho (talk) 10:54, 16 March 2010 (UTC)

The relation to the physical operation of measurement
Dear Tkuvho

It has been seen by me that my remarks added to the topics on the connection between construction of real number system and the physical operation of measurement have been undone and removed. I think that it is a fault in that the physical operation of measurement served as a great motivation for the construction of real number system and should never be omitted because of its shear importance. If the remarks is not satisfied with, I am inclined to think that it is better improved rather than simply removed. Sorry, I am new to wikipedia and do not know much about the policy and courtesy here. I am going to undo the deletion, no implication of insulting because I do appreciate your great contribution to wikipedia. For any suggestions, maybe we could contact via e-mail: tschijnmotschau@gmail.com. Thank youTschijnmotschau (talk) 08:55, 10 December 2010 (UTC)

 * Physical operation of measurement is an important motivation. Note that all measurement is just as well done with rational numbers.  The comments you have added:

 * "An advantage of this approach is that it does not use the linear order of the rationals, only the metric. Hence it generalizes to other metric spaces. Intuitively, the construction by Cauchy sequences can be considered to be giving a measure for lengths by using coarser units of measurement first and then measuring the remaining part by progressively finer and finer units."

 * and

 * "An advantage of this construction is that each real number corresponds to a unique cut. + An advantage of this construction is that each real number corresponds to a unique cut. Intuitively, the Dedekind cut construction of real number system can be view as giving a measure for lengths by using finer and finer units to measure the entire length."

 * are not helpful to understanding either Cantor's construction or Dedekind construction, and should be deleted. Tkuvho (talk) 09:10, 10 December 2010 (UTC)
 * I second that. Also, Tschijnmotschau, mind WP:BRD: since you changes have been challenged, you are supposed to discuss your addition here, not to re-add it without consensus.—Emil J. 11:09, 10 December 2010 (UTC)

I am really sorry for undoing the change without a discussion, I am new to wikipedia and is not very proficient in the regulation here. I am sorry. But I cannot agree with the idea that the rational numbers is sufficient for physical measurement. If it is indeed the case, there might have never been the introduction of real numbers into mathematics. Real numbers can be viewed as the completion (also in the sense of the convergence of Cauchy sequence) of the set of rational numbers for it to be able to be used for measurement of any physical quantity. Physically, if the measurement of some object, say, length of a line segment, is equivalent to another set of identical objects which is chosen as the unit for measurement, then the cardinality of the set of units can be taken as a measure for the physical quantity. For example, when measuring the length, if a line segment is equivalent to another set of three identical line segments, the length of the line segment can be assigned the integer value of three units. But it is not always possible to find a set of integral number of units which gives equivalent physical measurement. So rational numbers can be introduced to solve this problem to some extent. If some set with n identical objects of something can give identical measurement as the unit of measurement, then it is assigned the measurement of 1/n units. Taking a line segment with 2/3 cm as an example, no set of integral numbers of the unit of centimeter can given identical result with the 2/3 cm segment when its length is measured. But if the segment a set of three of which can give identical measurement with the unit is found first, a set of two of such segments can give equivalent measurement with the segment to be measured, then the result of 2/3 cm can be reasonably assigned. In this way, to measure a line segment with a unit, first the unit itself might be tested to see whether or not a set of integral numbers of it can give equivalent measurement as the unknown. If it cannot be so, the 1/n of the unit can be used progressively to measure the unknown until a set of integral number x of the 1/y unit segment can give equivalent measurement as the unknown, then the unknown can be assigned the measurement of x/y units. It is pointed out above that the rational numbers would be sufficient for physical measurement, but in fact, it is sufficient ONLY WHEN THE PROCESS DESCRIBED ABOVE CAN TERMINATE AT SOME POINT OF x. But is it indeed the case? Obviously not. It can well occur that when finer and finer partition of the unit is used the unknown can still not be measured exactly. Mathematically, it can be proved that the hypotenuse of an isosceles right-angled triangle can never be equal in length to a set of any number of any 1/n partition of its sides, i.e. the irrationality of $$\sqrt{2}$$. This leads, directly and quite intuitively, the Dedekind construction of real numbers, and the Cantor's construction came in another way but the essence is exactly the same. So I am inclined to think that my remarks added on the article is pertinent and helpful, hence should not be simply deleted. Thank you! Tschijnmotschau (talk) 17:54, 12 December 2010 (UTC)

 * Thanks for your comment. What you seem to be saying is that a real number (such as square root of two) can be approximated ever closer by rational numbers.  But this is a feature already of Simon Stevin's representation of numbers in terms of decimals.  Note that one cannot approximate something that does not exist yet.  Cantor and Dedekind give one a method of creating real numbers starting with the rationals, not merely approximating them by rationals.   Tkuvho (talk) 19:15, 12 December 2010 (UTC)

Sorry for having failed to make my point clear. Rather than saying that a real number can be approximated ever closer by rational numbers, what I was going to say is that mere rational numbers are not sufficient to represent the measurement of any physical quantities, hence the need for the construction of real numbers. The rational numbers can approximate ever closer to any point on a line, but it cannot specify, i. e. overlap with any point on the line. Only after the introduction of real numbers, the ideal physical measurement with infinite accuracy can be represented mathematically. This is what I want to say. Thank you!Tschijnmotschau (talk) 02:51, 13 December 2010 (UTC)

 * Measurement of physical quantities cannot possibly involve real numbers. Quantum mechanics tells us that infinite divisibility which we take for granted in any traditional approach to mathematical modeling, breaks down at a sufficiently fine level.  I find the expression "ideal physical measurement" a bit of an oxymoron.  Tkuvho (talk) 03:13, 13 December 2010 (UTC)

Sorry, I think that maybe what quantum mechanics is telling us is that we cannot make measurement of infinite accuracy without any disturbance to the state of the system, rather than that such measurement is in principle impossible. In principle, the measurement of, say, position of a particle would cause its state to collapse into a Dirac delta, a state in no conflict with the basic concepts of quantum mechanics. But this maybe is not important here. What is important is that some mathematical object is needed by physicists to represent the result of ANY POSSIBLE MEASUREMENT, including the conceptual measurement of infinite accuracy. And real numbers is a perfect construct that can make it possible. I am sorry for my improper phrase "ideal physical measurement", what I wanted to express is that the measurement is to be applied to some solid physical entity, rather than some pure mathematical structure. hence "physical", and such measurement is going to take place in imagination rather than really to be implemented in some experiment, hence "ideal". I am sorry for my improper wording. Tschijnmotschau (talk) 12:21, 16 December 2010 (UTC)
 * I agree that your "conceptual measurement" is taking place in the imagination and therefore has nothing to do with physical measurement. Attaching more "reality" to the real numbers than they really possess is a common philosophical predisposition, but it need not dominate this page.  Tkuvho (talk) 14:25, 16 December 2010 (UTC)

group or ring?
In the construction from the integers, one uses the composition in Maps(Z,Z) to define multiplication in the reals. It seems a bit misleading to claim that one uses "only" the addition. It seems almost simpler to use the multiplication in Z to define the product, instead of composition, but perhaps there are reasons not to do this. Any comments? Tkuvho (talk) 13:07, 14 December 2010 (UTC)
 * As far as I can see, almost homomorphisms are not even closed under pointwise integer multiplication.—Emil J. 13:21, 14 December 2010 (UTC)

 * Good point. Suffices to look at the identity homomorphism.  Still, you could multiply f(n) and g(n), divide by n, and take integer part.  That presumably gives the product in R, doesn't it?  Tkuvho (talk) 14:00, 14 December 2010 (UTC)
 * Why would you do it in such a complicated way when you can just compose the two functions? And what does it have to do with the article? The construction is described using composition in the sources, therefore we should report it as such, we are not supposed to report your OR. Leaving aside the fact that the general construction can be usefully applied to other abelian groups, such as Zn, that have no multiplication (see the appendix to Arthan's paper).—Emil J. 14:24, 14 December 2010 (UTC)

 * OK, thanks. Tkuvho (talk) 15:08, 14 December 2010 (UTC)

Is the information on the ordering of Cauchy sequences correct?
The article states that for some Cauchy sequences of rational numbers A = (a1, a2, ...), B = (b1, b2, ...) then A < B iff there exists N > 0 : for all n > N, an < bn. Is this correct? For example, take A := (0.9, 0.99, 0.999, 0.9999, ...), B := (1, 1.0, 1.00, 1.000, ...). For all n, an < bn, but they both converge to the same number (the number 1). Am I missing something? I am ardently not a mathematician, so I can't just say for sure that something's up, but this seems incorrect. 174.140.112.18 (talk) 06:13, 11 November 2012 (UTC)
 * I cannot find the statement you quote in the article. What I can find in the article is the correct statement that A ≤ B iff A is equivalent to B (defined earlier to mean $$\lim_{n\to\infty} a_n-b_n=0$$) or there is N such that an ≤ bn for all n > N. This means that A < B iff A is not equivalent to B and there is an N such that an < bn for all n > N. In your example, A and B are equivalent.—Emil J. 13:51, 11 November 2012 (UTC)
 * Ah, so it does. I suppose I misread it. Thanks. 174.140.112.18 (talk) 04:01, 12 November 2012 (UTC)

#Construction from Cauchy sequences
Is L a rational number we already had before construction? Or is it a rational that belongs to R?--578985s (talk) 07:28, 14 October 2014 (UTC)

the opening paragraph
I would like to improve the opening paragraph so that it more correctly introduces the subject matter. This is my first wiki edit ever, so any comments would be greatly appreciated. Something along these lines (and later on I would like to improve the article to more systematically present the constructions and related to the three aspects presented below):

The axioms of the real number system, which were colloquially well-known well before any rigorous construction of any model of the real numbers was ever constructed, are well known to essentially uniquely characterize the reals. More precisely, it follows from the axiomatic formulation of the reals that any two models satisfying these axioms are isomorphic, and thus essentially the same. This however does not rule out the possibility that the axioms are contradictory. A construction of the real numbers is a rigorous presentation of mathematical objects which are also shown to form a complete ordered field, namely the conform to the axiomatic formulation of the reals.

Constructions of the reals, and there are plenty of such constructions, may be classified based on three aspects, namely the assumed simpler model, the explicitness of the constructed objects, and the analyticity of the definition of the algebraic operations. In more detail, construction of the reals must use some agreed upon simpler entities, and most constructions appeal to the rational or the integer number systems. Then there is the issue of how tangible the constructed objects representing the real numbers are. This is of particular relevance to related efforts of implementing any particular construction on a computer, for instance for the purposes of automated proof verifiers when arguing about the the reals. Lastly, and somewhat vaguely, the amount of analytic machinery required in order to obtain the definition of the algebraic operations may vary greatly between constructions. A purist approach would demand that no analytic properties, i.e., limit-like arguments utilizing suprema or infima should be used at all when defining addition and multiplication, and such constructions exist. Other constructions may first develop certain of analytic tools as part of the construction required for the algebraic operations. — Preceding unsigned comment added by IttayWeiss (talk • contribs) 00:28, 21 February 2015 (UTC)

Assessment comment
Substituted at 01:55, 5 May 2016 (UTC)

Removed section "Stevin's Construction"
Here it is for reference: It has been known since Simon Stevin that real numbers can be represented by decimals. We can take the infinite decimal expansion to be the definition of a real number, defining expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally. This is equivalent to the constructions by Cauchy sequences or Dedekind cuts and incorporates an explicit modulus of convergence. Similarly, another radix can be used. Weierstrass attempted to construct the reals but did not entirely succeed. He pointed out that they need only be thought of as complete aggregates (sets) of units and unit fractions.

This construction has the advantage that it is close to the way we are used to thinking of real numbers and suggests series expansions for functions. A standard approach to show that all models of a complete ordered field are isomorphic is to show that any model is isomorphic to this one because we can systematically build a decimal expansion for each element.

This whole section is kind of a mess: There should really be some context and qualifications to all this before it's put back in. --Deacon Vorbis (talk) 15:23, 1 April 2017 (UTC)
 * According to the source, these notions weren't new with Stevin, although he did seem to be an important advocate.
 * It wasn't "known" since the time of Stevin, since the concept of "real number" would continue to develop over the next few centuries.
 * The bit about Weierstrass feels tacked on and makes no sense in context.
 * The second paragraph is highly dubious and sounds like personal opinion.

Citation needed
Text says "We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks."

I could not find proof in any text book, and I doubt anyone can, other than those just saying it's fact. No text books provide proof. Prove me wrong, should be easy if it's common. — Preceding unsigned comment added by 71.17.109.113 (talk) 00:17, 29 May 2017 (UTC)

Otto Stolz?
The text attributes one of the first three constructions of the real numbers to "Karl Weierstraß/Otto Stolz". I had never heard of Otto Stolz until now and it seems he was a student of Weierstraß. Is there any source for him being a co-inventor (if that is the right word)? Similarly, I thought until now that the construction using Cauchy sequences was solely due to Cantor. KarlFrei (talk) 18:34, 28 January 2019 (UTC)
 * Looking at the McTutor biography of these two authors (accessible form their WP page), it appears that both had important contributions which were independent from those Weierstraß and Cantor. So they must be mentioned here, and I'll revert your edit. However, I agree that the respective contributions must be clarified. D.Lazard (talk) 14:38, 6 March 2019 (UTC)

 * Regarding Meray, I see that he indeed apparently had the same idea as Cantor. But for Otto Stolz, it says nothing about him constructing the real numbers. (Of course he had important contributions to mathematics in general, otherwise he wouldn't have a McTutor page.) I have searched online but cannot find any reference to Stolz working on the real numbers (he obviously worked on real analysis though). KarlFrei (talk) 08:04, 7 March 2019 (UTC)

"Separates" definition could maybe use clarification.
In the informal description of one of the axioms, it states: "If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets." While it is intuitive that "separates" could certainly refer to a number appearing between both sets, the definition also allows (e.g.) X = {x|x < 0.0} and Y = {0.0}, in which case the "separator" is the only element of Y. Suggesting that 0 somehow "separates" itself from other numbers is not intuitive, particularly if you need this sort of analogy to understand the axiom. Maybe "either A has a largest element, or B has a smallest element, or there exists a z which separates A from B"...? TricksterWolf (talk) 03:31, 17 December 2021 (UTC)

A suggestion
It would be a massive help to those who do not have an advanced degree in Mathematics -- or whose knowledge of the symbology of mathematics is rusty -- if the symbol $$\mathbb{R},$$ was defined. (A quick look at R (disambiguation) shows it stands for the set of real numbers. Which means the passage "A model for the real number system consists of a set $$\mathbb {R}$$" is a circular statement.) In other words, avoid jargon so the rest of us understand the article, unless you are willing to explain it in non-technical words. -- llywrch (talk) 03:59, 22 January 2023 (UTC)

 * I have added a definition of the notation D.Lazard (talk) 14:05, 22 January 2023 (UTC)

title: Talk:Construction of the real numbers


text:
Santa Comba

Santa Comba may refer to:

People

 * Columba of Sens
 * Columba of Spain
 * Columba of Cornwall

Portugal

 * Santa Comba Dão, a city and municipality
 * Santa Comba (Ponte de Lima), a parish in the district of Ponte de Lima
 * Santa Comba (Seia), a parish in the district of Seia
 * Santa Comba (Vila Nova de Foz Côa), a parish in the district of Vila Nova de Foz Côa
 * Santa Comba de Rossas, a parish in the district of Bragança

Spain

 * Santa Comba, Galicia, a municipality in A Coruña province, Galicia

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

title: Santa Comba


text:
Shanachie Records

Shanachie Records is an American, New Jersey–based record label, founded in 1975 by Richard Nevins and Dan Collins. The label is named for the Gaelic word seanchaí (anglicised as shanachie), an Irish storyteller.

It was previously distributed by Entertainment One Distribution.

Starting as a label that specialized in fiddle music, they began releasing work by Celtic groups such as Planxty and Clannad. Other genres on the label include Latin American, African music, soul, country, and ska. In 1989 they acquired Yazoo Records from Nick Perls. This allowed them to release vintage jazz and blues recordings. Today they have another imprint, Shanachie Jazz.

In 1992 Shanachie began releasing CDs by folk singer-songwriters including Richard Shindell, Dolores Keane, John Stewart, Rod MacDonald, Richard Meyer, Karan Casey, Sue Foley, Four Bitchin' Babes, Kevin Gordon, and others.

In 1980 Shanachie released its first reggae album, King Tubbys Meets Rockers in a Firehouse by Augustus Pablo, and soon after released albums from many reggae artists including Rita Marley, Culture, Bunny Wailer, Judy Mowatt, Alpha Blondy, Lucky Dube, The Skatalites and others. Shanachie was the U.S. liaison for the UK-based reggae label, Greensleeves Records, until about 1987. Shanachie also released material for Augustus Pablo under the Message imprint of his company, Rockers International. The Grammy Award–winning Soweto Gospel Choir releases albums on Shanachie Records, as has Grammy-nominated Liquid Soul.

The Latin acts on the label include Los Jovenes del Barrio.

title: Shanachie Records


text:
Santa Comba de Rossas

Santa Comba de Rossas is a civil parish in the municipality of Bragança, Portugal. The population in 2011 was 304, in an area of 8.75 km².

title: Santa Comba de Rossas


text:
Marek Kamiński

Marek Kamiński (born 24 March 1964 in Gdańsk) is a Polish innovator, philosopher and an explorer. He is claimed to have reached both the North and the South Pole in one year without outside assistance (the North Pole on 23 May 1995; the South Pole on 27 December 1995).

Biography
Kaminski obtained his Philosophy and Physics degrees at the University of Warsaw and completed the advanced management graduate program at the IESE Business School in Barcelona. He also studied Philosophy in Hamburg.

He led the first-ever expedition to the North Pole and the South Pole with a person with a disability (Jan Mela, who was 15 at the time). He has also crossed the Gibson Desert in Australia – a journey of 800 km in 46 days.

During his "Third Pole" expedition along St. James’ trail, he travelled 4000 km in 140 days from the tomb of Immanuel Kant in Kaliningrad, Russia, to the grave of Saint James in Santiago de Compostela, Spain.

He has travelled 30,000 km by electric car from Poland to Japan through Siberia and the Gobi Desert. He was the first person to drive on the Trans-Siberian Highway in an emission-free vehicle. Kamiński is the founder of the ‘Power 4Change’ motivational method and the founder of the Marek Kamiński Foundation. The Marek Kamiński Institute and Invena.

Lectures
Kamiński has applied his experience in motivation to achieve seemingly impossible goals, as well as work-related to robots and artificial intelligence to give lectures at prestigious universities and conferences around the world.

Publications
Kamiński has authored 13 books covering his global treks and inward journeys. Some of these books have been translated in as much as 10 languages, winning numerous awards and becoming bestsellers.

The Power4Change method
An original motivational. It consists of 10 components, such as visualization, the path being more important than the end goal and getting to know oneself.

The method was created based on Kamiński's experience in achieving difficult goals. The Power4Change method is delivered through workshops, training sessions and conferences for companies and individuals and through the "Power4Change" application, available in Poland and worldwide.

Personal life
He was brought up in Połczyn-Zdrój (Pomerania). In 1982 he graduated from Władysław Broniewski high school in Koszalin. He studied Philosophy and Physics at Warsaw University and an Advanced Management Program at IESE Business School. He is the founder and co-owner of a company called Invena S.A. and founder of the Marek Kamiński Foundation.

He lives in Gdynia with his wife, daughter, and son.

title: Marek Kamiński


text:
User talk:MeddmaWamm

Good work on fixing the markup for New Hampshire, Vermont, and whatever other states you are working on. -- Kaszeta 17:12, 29 Sep 2004 (UTC)

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title: User talk:MeddmaWamm


text: User talk:82.33.48.5

title: User talk:82.33.48.5


text:
Peter Schweizer

Peter Franz Schweizer (born November 24, 1964) is an American political consultant and writer. He is the president of the Government Accountability Institute (GAI), senior editor-at-large of far-right media organization Breitbart News, and a former fellow at the conservative Hoover Institution.

Schweizer wrote Clinton Cash, a 2015 book discussing donations made to the Clinton Foundation by foreign entities and Bill and Hillary Clinton's income after they left the White House in 2001. Journalists and fact-checking organizations have criticized the book for speculation, conclusions not supported by the evidence presented, and for factual errors which were corrected in the Kindle edition.

Early life
While in high school, Schweizer attended the National Conservative Students Conference at George Washington University and was a member of Young America's Foundation (YAF). He graduated from Kentridge High School in Kent, Washington in 1983. Schweizer then attended George Washington University and, subsequently, graduate school at St Cross College of Oxford University on YAF scholarships. He was on YAF staff starting 1993 and edited its magazine.

Career
In 2012, Schweizer and Steve Bannon co-founded the Government Accountability Institute, a conservative think-tank whose stated goal is to "investigate and expose government corruption, misuse of taxpayer money and crony capitalism." Schweizer is president of the organization. It is registered as a nonpartisan organization, but largely focuses on the Democratic Party.

Investigative projects
Schweizer's early work at Senator Jeremiah Denton's National Forum Foundation (NFF) focused on the Cold War. He co-authored a National Review article with Denton's son, James (often cited as Jim), "Murdering SDI", about the suspicious deaths of several European officials who supported the Strategic Defense Initiative. While at the NFF, Schweizer also published a report titled "The Meaning and Destiny of the Sandinista Revolution".

In 2012, journalist Steve Kroft used Schweizer's work as the basis for a report on CBS's 60 Minutes about Congressional insider trading. Titled "Insiders: The road to the STOCK act", Kroft relied heavily on Schweizer's reporting in his 2011 book Throw Them All Out, which CBS independently verified, to demonstrate how members of Congress trade stocks unethically, and inoculate themselves from prosecution. The following year, Kroft revisited Schweizer's work to create another 60 Minutes report on how members of Congress use the funds of their political action committees for private inurement.

Books
Schweizer co-authored two novels, published in 1997 and 2005, with former Secretary of Defense Caspar Weinberger.

Friendly Spies
Schweizer's second book, published in 1992, purported to reveal espionage by the French secret service on the British government between 1984 and 1987. A follow-up by two Sunday Times reporters revealed that meetings described by Schweizer in Friendly Spies did not check out, sources were named that did not exist or could not be found, and the hotel in which the meetings allegedly had taken place did not exist at the time.

Victory
In 1994, The Atlantic Monthly Press published Schweizer's book Victory: The Reagan Administration's Secret Strategy that Hastened the Collapse of the Soviet Union. The book argued that President Ronald Reagan brought an end to the Cold War through a policy of massive defense spending which led to the dissolution of the Soviet Union. Historian Stephen E. Ambrose criticized the book, writing that "there is a great deal more to the Soviet collapse than Schweizer recognizes, and in any case, it is difficult to take seriously a book as badly written as this one... The book is consistently redundant and badly out of balance." The book later enjoyed renewed popularity with foreign policy officials during the Trump administration, with one White House official telling reporters "it was 'no secret' that the book has had some influence on policy."

Clinton Cash
In 2015, HarperCollins published Clinton Cash: The Untold Story of How and Why Foreign Governments and Businesses Helped Make Bill and Hillary Rich, a 256-page book discussing the donations made to the Clinton Foundation by foreign entities. Several media outlets received advance copies, including The New York Times, The Washington Post, and Fox News, all of whom agreed to pursue stories found in the book. While promoting the book, Schweizer falsely claimed that then-Secretary of State Hillary Clinton had veto power to stop the sale of Uranium One to a Russian state-owned company.

Time wrote that "allegations are presented as questions rather than proof" but that "the book's dark suggestions reflect the growing problem Clinton faces in her run for the White House in 2016 as more and more details of the foundation's fundraising activities present the appearance of impropriety and lack of transparency during her time as Secretary of State."

Several journalists challenged the book, writing that it contains "leaps of logic," that it "draws some conclusions that go beyond the available evidence," that "[p]arts of Schweitzer's reporting fell apart under scrutiny," and that "Schweizer is trafficking in speculation."

The book was found to contain several factual errors, including the false claim that Clinton played a central role in the sale of Uranium One to a Russian state-owned company while the Clinton Foundation was receiving large donations. Several weeks after the book's initial publication, HarperCollins and the author made several corrections to the Kindle edition of the book, amending "seven or eight" passages.

In the wake of the book's publication, the Clinton Foundation admitted that it made mistakes in disclosing some of its contributions and that it implemented new rules increasing financial reporting and limiting foreign donations.

The book was funded by the Mercer Family Foundation with a $1.7 million contribution in 2015 to the Government Accountability Institute (GAI) whose researchers Schweizer used.

Secret Empires
In 2018, HarperCollins published Secret Empires: How the American Political Class Hides Corruption and Enriches Family and Friends. Secret Empires provided details about the overseas business conducted by Hunter Biden, particularly in his employment with entities such as Ukrainian energy firm Burisma. The book also describes ties between Elaine Chao's family business Foremost Group and China, which were disputed by a spokesman for Chao's husband, Senator Mitch McConnell.

The book "was perfectly timed for the Presidential campaign" and has been cited as an initial source of the Biden–Ukraine conspiracy theory. It was given significant coverage on Fox News which gained Republican nominee Donald Trump's attention and led him to send personal lawyer Rudy Giuliani to Ukraine to pressure their new government to investigate the claims in Schweizer's book. Trump's efforts eventually resulted in a major scandal that culminated in his first impeachment.

The Daily Beast reported that the book contained 14 examples of plagiarism, including from Wikipedia. The passages in question contained similar wording to those of several Wikipedia articles, particularly the articles on Patrick R. Daley, Deepwater Horizon oil spill, Tom Steyer, and Jared Kushner. In addition to the Wikipedia articles, other passages, while citing news articles properly, contained similar wording to the cited sources.

Films
Schweizer worked with Steve Bannon on In the Face of Evil: Reagan's War in Words and Deed, a 2004 documentary about former Hollywood movie star and president Ronald Reagan, which is based on Schweizer's book Reagan's War (2003); Schweizer is credited as Executive Producer.

Schweizer is credited as a writer and producer in The Creepy Line, a film that "epitomizes a popular claim that Silicon Valley is censoring American conservatives on web platforms."

Activities
During 2008 and 2009, Schweizer served as a consultant to the Office of Presidential Speechwriting in the White House. In March 2009, Schweizer and fellow White House speechwriter Marc Thiessen opened Oval Office Writers LLC, which specializes in preparation for congressional testimony as well as pitching opinion editorials and book proposals. Schweizer's notable clients have included Sarah Palin, and he advised her on foreign policy.

Schweizer is a member of the Research Advisory Council of the James Madison Institute, a free-market think tank headquartered in Tallahassee, Florida.

Criticism
Schweizer admitted he overreached in attacking Hillary Clinton's purported role in approving a Russian uranium deal and falsely claimed that then-Secretary of State Clinton "had veto power" to stop the Russian State Atomic Nuclear Agency (Rosatom) from purchasing Uranium One. During a May 5, 2015, Politico podcast interview, Schweizer admitted that "veto is probably not the best word" and "what I meant by veto power was as we explain the process, you know, if somebody objects it kicks in the special investigation."

In a 2015 NBC interview, Schweizer said that Hillary Clinton did not support a nuclear deal with India in 2006 and that she voted for it in 2008, after donations to the Clinton Foundation. PolitiFact rated Schweizer's claims false.

Personal life
Schweizer lives in Tallahassee, Florida with his wife, Rhonda, and step-children. He and his first wife, Rochelle Schweizer, co-authored books about Disney and the Bush family. They met when she was working with the National Forum Foundation (NFF), which in 1997 merged with Freedom House.

title: Peter Schweizer


text:
NYG

NYG or nyg may refer to:

 * Marine Corps Air Facility Quantico (IATA and FAA LID: NYG), a United States Marine Corps airfield located within Marine Corps Base Quantico, Virginia
 * New York Giants, a National Football League team that uses this abbreviation for box scores and television scoring displays
 * New York Guard, the State Defense Force (SDF) of New York State and one of the four branches of the New York Military Forces (NYMF)
 * Nyindu language (ISO 639-3: nyg), a Bantu language of the Democratic Republic of the Congo

title: NYG


text:
Tomasz Stańko

Tomasz Ludwik Stańko ( Polish : ; 11 July 1942 – 29 July 2018) was a Polish trumpeter and composer associated with free jazz and the avant-garde.

In 1962, Tomasz Stańko formed his first band, the Jazz Darings, with saxophonist Janusz Muniak, pianist Adam Makowicz, bassist Jacek Ostaszewski, drummer Wiktor Perelmuter. Inspired by Ornette Coleman and the innovations of John Coltrane, Miles Davis and George Russell, the group is often cited by music historians as the first European group to play free jazz. In his later years, he collaborated with pianist Krzysztof Komeda on Komeda's album Astigmatic, recorded in late 1965. In 1968, Stańko formed a quintet whose members were Janusz Muniak (tenor and soprano saxophones, flute), Zbigniew Seifert (alto sax and violin), Bronisław Suchanek (bass), Janusz Stefański (drums, percussion). In 1975, he formed the Tomasz Stańko-Adam Makowicz Unit.

Stańko established a reputation as a leading figure not only in Polish jazz, but internationally as well, working with musicians including Jack DeJohnette, Dave Holland, Reggie Workman, Rufus Reid, Lester Bowie, David Murray, Manu Katché and Chico Freeman. From 1984, he was a member of Cecil Taylor's big band.

Early years
Tomasz Stańko was born in Rzeszów, Poland, on 11 July 1942. His first encounters with jazz were through Voice of America radio programs and tours initiated by the U.S. State Department. Coming of age in Communist Poland, Stańko was impressed by the correlation jazz had with a message of freedom. In 1958 he saw his first jazz concert given by Dave Brubeck. Along with the pianist Adam Makowicz and with the saxophonist Janusz Muniak, the group took inspiration from the music of musicians such as Ornette Coleman, George Russell and Miles Davis and was considered by many critics to be the first group in Europe to perform in the free jazz idiom.

In 1963 Stańko joined the Krzysztof Komeda quintet, where he learned much about harmony, musical structure and asymmetry. During his career with Komeda, which concluded in 1967, Stańko did five tours with the pianist and recorded eleven albums with him. In 1968 Stańko formed a quintet that met critical acclaim—one that included Zbigniew Seifert on violin and alto saxophone. In 1970, he joined the Globe Unity Orchestra, and in 1971 he collaborated with Krzysztof Penderecki and Don Cherry. Not long after he formed a quartet that included himself, saxophonist Tomasz Szukalski and Finnish drummer Edward Vesala. His performances with Vesala are often considered to be some of his most important work. In 1975, he formed the Tomasz Stańko-Adam Makowicz Unit.

1980s and 1990s
During the 1980s, he traveled to India and recorded solo work in the Taj Mahal, and also worked with Chico Freeman and Howard Johnson in Vesala's solo album Heavy Life. He was featured in a profile on composer Graham Collier in the 1985 Channel 4 documentary 'Hoarded Dreams'  In the mid-1980s, he began doing extensive work with Cecil Taylor, performing in his big bands and also led various groups of his own, including COCX (with Vitold Rek and Apostolis Anthimos). Then, before returning to ECM Records, Stańko also worked in a trio that included himself, Arild Andersen and Jon Christensen. In 1993, Stańko formed a new quartet composed of the then 16-year-old drummer Michał Miśkiewicz, along with Miśkiewicz's two friends, pianist Marcin Wasilewski and bassist Sławomir Kurkiewicz. That same year he also formed an international quartet that included Bobo Stenson, Tony Oxley and Anders Jormin. in 1994 the quartet released their first ECM recording titled Matka Joanna. In 1997, Stańko formed a group which performed the songs of pianist Krzysztof Komeda, touring London, Copenhagen, Stockholm and appearing at jazz festivals like those in Nancy and Berlin. The idea for the project came from ECM president Manfred Eicher.

Stańko lost his natural teeth in the 1990s, although over time he developed a new embouchure with the help of a skilled dentist and monotonous practice. He would spend long hours playing what he deemed to be "boring" long tones which helped to strengthen his lip, in spite of playing with the disadvantage of false teeth.

2000s
The early 2000s was a time of extensive international touring. This experience led to a second album of Stańko's Polish quartet Suspended Night. Billboard magazine noted that this album was one of the bestselling jazz albums of 2004 in the United States. Together with Suspended Night ECM released a collection of Stańko's recording in its Rarum/ Selected Recordings series.

In 2005 the quartet recorded the last joint album Lontano, again for ECM. Having an established position on the world jazz stage, each album of the quartet as celebrated across Europe and the United States. In 2005 the band also made its first tour around Asia and Australia. In 2005 Stańko also created music for the Warsaw Uprising Museum titled Freedom in August.

The second half of the first decade 2000 saw a new resurgence in Stańko's career: a chapter of experiments, creating projects, searching for new sounds. As a result of these experiences, in 2007 a new band, a Scandinavian quintet composed of Alexi Tuomarila, Jakob Bro, Anders Christensen and Olavi Louhivuori, was formed, with which Stańko recorded the 2009 album Dark Eyes for ECM.

Later years
The end of the first decade of the 21st century marked the beginning of the New York period in Stańko's life. His move to Manhattan saw him give regular concerts in New York venues and clubs, such as Birdland, Jazz Standard and Merking Hall. Stańko used the opportunity to meet with local musicians and absorbed new ideas and richness of sounds. New projects were spontaneously created, including those with Lee Konitz, Craig Taborn, Thomas Morgan, Gerald Cleaver, Chris Potter and others. The idea of the New York Quartet was forming which, after various combinations, eventually formed in 2012.

Stańko's autobiography Desperado was published in Poland in 2010, a multi-part interview conducted by Rafał Księżyk. In November 2010 he took part in the "Dzień Szakala" (The Day of the Jackal) charity concert, in Warsaw, in aid of saxophonist Tomasz Szukalski. In 2011 the Smithsonian Institution, the world's largest museum and education and research complex, published the six-disc compilation Jazz: The Smithsonian Anthology, which closes with "Suspended Night Variation VIII" by Stańko.

The second decade of the 21st century saw the emergence of Stańko's new poetry-based project. Its beginnings date back to 2009 when Jerzy Ilg, editor-in-chief of Znak Publishing House, invited the trumpeter to one of the last poetry nights of Wisława Szymborska. During the meeting at the Kraków Opera, the poet read her poems, with Stańko accompanying her on the trumpet. The CD recording from this concert was incorporated with Here, the next volume of poems published by Szymborska. Meeting the poet became the key to a new stage in Stańko's career and some of her poems provided the inspiration for new compositions and titles. He dedicated to her his album Wisława, recorded with a new New York quartet featuring Thomas Morgan, Gerald Cleaver and David Virelles, and released by ECM in 2013.

in 2014 Stańko was invited to compose a suite on the occasion of the opening of the core exhibition of the POLIN Museum of the History of Polish Jews. It was released as the album POLIN, recorded in New York with Ravi Coltrane, David Virelles, Dezron Douglas and Kush Abadey.

Stańko last studio album December Avenue was released on 31 March 2017. Reviewing the album The Guardian jazz critic John Fordham wrote: "Nobody holds a single, long-blown trumpet note like the Polish pioneer Tomasz Stańko – a wearily exhaled, soberly ironic, yet oddly awestruck sound that is unique in jazz."

On 14 January 2014, Stańko received the "Polityka Passport" for his work as a cultural leader. On the same day, in Paris, the l'Académie du jazz, under the leadership of François Lacharme, awarded him the Prix du Musicien Européen – the European Music Award of the Year – for his current achievements and his lifetime achievement.

On 15 January 2014, he received one of the three honorary Preis der deutschen Schallplattenkritik (PdSK) awards, granted by an independent association of over 140 music journalists from Germany, Austria and Switzerland.

Stańko was the organizer and director of the Jazzowa Jesień festival in Bielsko-Biała, Poland. He also became a member of the honorary committee supporting Bronisław Komorowski before the presidential elections in Poland in 2015.

In March 2018, due to suspected pneumonia, his April concerts were cancelled. Stańko died of lung cancer, in the oncological hospital in Warsaw on 29 July 2018. The following day ECM Records published a video tribute to Stańko, on its YouTube channel, entitled "Remembering Tomasz Stanko". Fellow musician Dave Holland praised him as "a unique musician with deep feelings and a gentle soul".

Critical appreciation
Derek Richardson, writing for the San Francisco Chronicle said, "Tomasz Stanko is not the first jazz musician to negotiate a rapprochement between gorgeous melodies and free improvisation. But he is one of the most eloquent proponents of extemporaneous lyricism working today." Author Brian Morton compared Stanko's lyricism to that of Miles Davis, calling it a "direct but individual offshoot."

title: Tomasz Stańko


text:
2002 CONCACAF Gold Cup

The 2002 CONCACAF Gold Cup was the sixth edition of the Gold Cup, the soccer championship of North America, Central America and the Caribbean (CONCACAF).

The tournament was once again held in the United States, in Miami and Pasadena. The format of the tournament stayed the same as in 2000: twelve teams were split into four groups of three. The top two teams in each group would advance to the quarterfinals. Ecuador and South Korea were invited from outside CONCACAF.

Canada, who rode the coin toss all the way to winning the 2000 Cup, needed luck once again, as all games in Group D ended with a 2-0 result. Lots were drawn, with Canada and Haiti moving on to the next round; Ecuador did not. But the Canadian team's luck ran dry in the semifinals, as the U.S. beat them on penalties after tying 0-0. The United States then met Costa Rica in the final and topped them 2-0 behind goals by Josh Wolff and Jeff Agoos for their first tournament win since 1991.

During the tournament, Cuban players Alberto Delgado and Rey Angel Martinez defected from Cuba to the United States.

Qualification play-off
A qualification playoff to determine the final Gold Cup entrant was held in July and August 2001.

Cuba won 1–0 on aggregate.

Squads
The 12 national teams involved in the tournament were required to register a squad of 18 players; only players in these squads were eligible to take part in the tournament.

Awards
The following awards were given at the conclusion of the tournament.
 * Most Valuable Player: 🇺🇸 Brian McBride
 * Top Goalkeeper: 🇨🇦 Lars Hirschfeld
 * Fair Play Trophy: 🇨🇷 Costa Rica

Best XI
Source:

Goalscorers
4 goals
 * 🇺🇸 Brian McBride

3 goals
 * 🇨🇦 Kevin McKenna

2 goals

 * Rolando Fonseca
 * Rónald Gómez
 * Paulo Wanchope
 * 🇪🇨 Álex Aguinaga

1 goal

 * 🇨🇦 Dwayne De Rosario
 * Walter Centeno
 * Hernán Medford
 * 🇸🇻 Santos Cabrera
 * Juan Carlos Plata
 * Charles Alerte
 * Golman Pierre
 * 🇲🇶 Patrick Percin
 * 🇲🇽 Adolfo Bautista
 * 🇲🇽 Marco Garcés
 * 🇲🇽 Jair García
 * 🇲🇽 Carlos Ochoa
 * 🇰🇷 Choi Jin-cheul
 * 🇰🇷 Kim Do-hoon
 * 🇰🇷 Song Chong-gug
 * Stern John
 * 🇺🇸 Jeff Agoos
 * 🇺🇸 DaMarcus Beasley
 * 🇺🇸 Landon Donovan
 * 🇺🇸 Ante Razov
 * 🇺🇸 Josh Wolff

Own goals

 * 🇨🇦 Mark Rogers (for Martinique)
 * 🇰🇷 Kim Do-hoon (for Canada)
 * 🇪🇨 Édison Méndez (for Haiti)

title: 2002 CONCACAF Gold Cup


text:
Sé, Santa Maria e Meixedo

Sé, Santa Maria e Meixedo is a civil parish in the municipality of Bragança, Portugal. It was formed in 2013 by the merger of the former parishes Sé, Santa Maria and Meixedo. The population in 2011 was 22,016, in an area of 35.69 km2. It forms the city centre of Bragança.

title: Sé, Santa Maria e Meixedo


text:
Steese Highway

The Steese Highway (known as the Steese Expressway within Fairbanks) is a highway in the Interior region of the U.S. state of Alaska that extends 161 mi from Fairbanks to Circle, a town on the Yukon River about 50 miles (80 km) south of the Arctic Circle. The highway was completed in 1927 and is named for U.S. Army General James G. Steese, a former president of the Alaska Road Commission. It is paved for about the first 81 mi and around the town of Central it turns to dirt and gravel. Much of it is narrow and winding.

History
The highway and surrounding region has a long association with gold mining. It was built to service the Circle Mining District, which was very productive in the 1890s, before the discovery of gold in the Klondike. Both historic sites, such as Felix Pedro's 1902 gold discovery which resulted in the founding of Fairbanks, and the preserved gold camp at Chatanika, and active dredging operations line the road.

Route description
The Steese Highway is numbered Alaska Route 6 for most of its length, except for the first 11 miles (17 km) from Fairbanks to Fox, which are numbered Alaska Route 2. The highway has been designated as a National Scenic Byway.

There are three possible road closure barriers, so 511 Alaska should be checked before traveling its length to Circle Alaska.

title: Steese Highway


text:
Elemental (Tears for Fears album)

Elemental is the fourth studio album by English pop rock band Tears for Fears, released on 7 June 1993 by Mercury Records. It was the band's first album recorded following the departure of co-founder Curt Smith, with Roland Orzabal assuming sole leadership with the help of additional musicians.

The album peaked at No. 5 in the UK and No. 45 in the US; it also reached the top 10 in France and Italy, the top 20 in Canada and the top 30 in several European countries. It has been certified Silver in the UK, and Gold in the US, Canada and France. The single "Break It Down Again" was a hit across Europe and North America, and was the band's last single to reach the US top 40.

Elemental was the last album that Tears for Fears released under the Phonogram/Mercury label, the company to which the band had been signed since 1981.

Production
Elemental was the first new Tears for Fears album in four years, following 1989's The Seeds of Love. However, despite being released under the Tears for Fears moniker, it is essentially a solo effort by Roland Orzabal, as Curt Smith had left the band in 1991. Regarding the split with Smith, Orzabal stated in 1993,

Well, the fact is that I'd always been the songwriter, and Curt relied on me for songs, and those songs were getting less and less as the years progressed... And then during The Seeds of Love, I kind of took over as the producer as well, so it was becoming like a one-man band. And then in the end it became one in reality. We had a lot of shared assets—well, actually a lot of liabilities—after that last tour. And I took over the running of the company. This is just another phase Tears for Fears is going through."

Orzabal publicly downplayed Tears for Fears' loss of Curt Smith, stating, "The relationship I had with Curt was like that of a producer to an artist... I'd help him get his vocals right and even write his songs for him. It wasn't really a shared thing." Smith released his first solo album, Soul on Board, on 23 August 1993, over two months after the release of Elemental.

For much of the writing and recording of Elemental, Orzabal was assisted by musician Alan Griffiths (1959–2017), a longtime acquaintance who had collaborated with Tears for Fears before. In 1978, Griffiths became a founding member of the Bristol band Apartment, which became the new wave music band The Escape in 1981. In early 1983, The Escape signed with Phonogram Records, making the band label-mates with Tears for Fears. In late 1983, The Escape was Tears for Fears' support act; the band broke up in late 1984 and Griffiths and Escape bandmate Nicky Holland became session musicians for Tears for Fears' 1985 world tour. Holland played keyboards on Tears for Fears' The Seeds of Love, co-writing the majority of that album's songs, and her former bandmate Griffiths took on a similar songwriting role for the first Tears for Fears album sans Curt Smith. Orzabal later recalled, "When I split up with Curt, I wrote with Alan Griffiths and we were extremely prolific. We'd come up with a song virtually every day. Al was a bit of a genius, able to make musical sketches that were easy to jump off and finish into a song. That was an amazing period, and it's tougher to write with people you don't know yet."

Elemental was recorded at Orzabal's newly built home studio, Neptune's Kitchen, and was co-produced by Orzabal, Griffiths and Tim Palmer. Palmer's recent credits at the time included co-producing David Bowie's first Tin Machine album (1989's Tin Machine) and mixing Pearl Jam's debut album Ten (1991). Orzabal stated, "I'm kind of used to working autonomously... I would say that this album is less of a solo album than Seeds of Love, in fact. The team I worked with was amazing—Tim Palmer the co-producer, and Alan Griffiths, who co-wrote the stuff." Parts of the album have a more alternative rock sound than on previous Tears for Fears albums; this sound would be explored further in the next Tears for Fears album, Raoul and the Kings of Spain (1995), which Orzabal also recorded with Griffiths and Palmer. (Griffiths would also collaborate with Orzabal on the latter's first and only official solo album, Tomcats Screaming Outside [2001].)

Interviewed twenty years after the release of Elemental, Palmer recalled, "In the 80's, [Tears for Fears] had a reputation for spending a lot of time making records, but that's just the way they are. I tried to speed things up and got the record done in six months—not much of an achievement, but it was fast for them." Orzabal had a recording studio built at his home to record the album, stating, "It's an old photographer's studio that we converted. It was called Neptune's Kitchen, and we used a lot of copper that was doused in acid to look like it's been underwater and all the colors are sort of greens and blues like the sea. And there's a weathervane on top of the building which is a whale... It was necessary, economically, to work at home because I spend so long in the studios on albums that the bills mount up, and doing it at home was a sensible move."

Tim Palmer recalled,

It was a very important album for [Orzabal] because it was the first one he made without Curt Smith... There was a lot of pressure on him. Actually, this was an important record for me, as well, because Roland was one of the first artists who responded to my input musically. I'd come to the studio in the morning and throw a couple of ideas up, hoping he might like them, and he'd come in and say, 'That's really great. We should work on that.'... I played a lot of drums on the record, too. There was some editing involved, because I can’t claim to be the greatest drummer in the world. But I'm on there, and I played some guitars, as well. It was a great opportunity for me to add something musically to what I was producing."

Palmer enjoyed a good working relationship with Orzabal during the recording sessions for Elemental, recalling, "Roland is very into star signs. Before we started working on the record, he mapped out a composite star chart of us both, and he read it and said that he thought we would get along very well together, which we did."

Songs
Curt Smith's departure from Tears for Fears influenced Roland Orzabal's songwriting for the new album; Orzabal stated that "a lot of the songs were written while I was in a sense going through the 'divorce'... Things like "Break It Down Again" refer to that to some degree, and "Fish Out of Water", obviously. I did psychotherapy for about six years. I stopped going regularly when I'd finished [the song] "Elemental", which I think probably says something. I think I'm moving on."

The first single from Elemental, "Break It Down Again", was a relatively successful release, making both the UK and US top 25 singles charts. When Smith (who rejoined Tears for Fears in 2000) was asked in 2022 what his favorite "Curt Smith-less Tears for Fears songs" are, Smith answered, "I have come to really like 'Break It Down Again'. We play that song nowadays and I enjoy playing it."

"Cold" was the second single from Elemental to be released in the UK, but was not released in the US. According to Orzabal in the Elemental EPK, the song was inspired by an encounter with a German photographer who was trying to take pictures of him while he was on-stage. The photographer tried to get Orzabal's attention but he was not in the mood for photos and kept turning his face away. He eventually received a note from the photographer asking 'how can someone who makes such warm music to be so cold'. The first 2 verses of the song describe this event. The song's lyrics, written by Orzabal, contain also references to prior associates of Tears for Fears such as the line "Listened to my old friend Nockles, hoped that it would warm the cockles". "Nockles" is the nickname of former Tears for Fears keyboardist Nicky Holland, Orzabal's songwriting partner for much of the band's previous album, The Seeds of Love. Orzabal also takes a swipe at former Tears for Fears business manager Paul King with the lyric "King got caught with his fingers in the till. Where's your calculator, will you leave it in your will?" after alleged discrepancies were discovered in King's prior management of the band's financial affairs. King declared bankruptcy in 1990 and, after being found guilty of later fraudulent activities, was imprisoned in 2004 and disqualified from being a company director for a period of ten years. According to Orzabal, "After the breakup [with Smith], things were a complete mess on the business end as well... So I was on my own, without a manager, without anybody that was familiar. And I needed to go through that. You can hear it on a song like 'Cold', where I say 'Look, I need distance, I need to be on my own, I don't want a relationship.'"

The closing track of Elemental, "Goodnight Song", was released as the album's second single in the US (in lieu of "Cold"). It was also released as a single in some parts of Europe, although not in the UK. The "Cold" and "Goodnight Song" singles both featured the song "New Star" as a B-side; this song can be heard on Gloryland World Cup USA 94, the official album of the 1994 FIFA World Cup. It can also be heard during the opening credits of the 1994 film Threesome and appears on the film's soundtrack album, and was later included in the Tears for Fears compilation albums Saturnine Martial & Lunatic (1996) and Shout: The Very Best of Tears for Fears (2001).

The title track of Elemental was released in early 1994 as the album's third US single; the maxi CD version contains live tracks recorded from a Tears for Fears concert at Wembley Stadium on 14 December 1993.

Smith is also the subject of "Fish Out of Water", a song Orzabal referred to as his "How Do You Sleep?" (John Lennon's scathing 1971 song about his former Beatles bandmate and songwriting partner, Paul McCartney.) Of his relationship with Smith at the time, Orzabal stated, "I think it's similar to the animosity between a parent and a child; the parent wants the child to grow up and the child doesn't want to grow up. You know, for fuck’s sake, get out of the nest!" The opening lyrics of "Fish Out of Water" are a salvo against Smith ("You always said you were the compassionate one/But now you're laughing at the sun/With all your high class friends you think you've got it made/The only thing you made was that tanned look on your face/With all your cigarettes and fancy cars/You ain't a clue who or what you are/You're dreaming your life away..."). The song also makes references to "Neptune's Kitchen" (the studio in which Elemental was recorded), Arthur Janov's book The Primal Scream (a major influence on Orzabal and Smith's lives and the first Tears for Fears album The Hurting), and the 1983 Tears for Fears song "Memories Fade". Smith later responded to "Fish Out of Water" with the song "Sun King" in his album Mayfield (1998). In 2022, Smith said, "I get a real kick out of 'Fish Out of Water', which is about me in a very derogatory way. I found it highly amusing."

The song "Dog's a Best Friend's Dog" contains a reference to Samuel Beckett's play Waiting for Godot ("Tell Mr. Godot I'm walking the dog") and the album's penultimate track, "Brian Wilson Said", is in part a Beach Boys pastiche. The title is a backhanded reference to the Van Morrison song "Jackie Wilson Said (I'm in Heaven When You Smile)", which Morrison wrote about one of his idols, Jackie Wilson. Similarly, Orzabal wanted to write a song about one of his idols, Brian Wilson of the Beach Boys.

Tour
Tears for Fears supported Elemental with a live tour of North America that took place between August and November 1993, followed by one concert in Paris and four in the UK in December 1993. The band assembled for the tour comprised Roland Orzabal on vocals and lead guitar, Alan Griffiths on keyboards and guitar, Gail Ann Dorsey on bass, Brian MacLeod on drums, Jeffrey Trott on guitars, and Jebin Bruni on keyboards. Orzabal retained this band for the recording sessions of the next Tears for Fears album, Raoul and the Kings of Spain (1995), which was recorded between 1993 and 1994. The 1993 setlists, on occasion, featured covers (a live recording of Radiohead's "Creep" was released as a b-side in 1996, and the song remained a staple of the band's live shows until 2019) as well as new songs, some of which would appear on Raoul and the Kings of Spain and its b-sides, while "Size of Sorrow" wasn't released until 2004 with Curt Smith now singing the song (Gail Ann Dorsey had sung it in concert).

Regarding the Elemental tour, Orzabal stated at the time,

The show is a little more focused and a lot more gutsy than previously... The light show is fantastic. We mainly perform the new album, obviously, because it sounds great and it translates very well live. But we segue quite often into older songs, and that surprise element seems to be working very well, the audience really appreciates it... It's a lot easier [without Curt Smith], to be honest. I think I was very lazy in a duo. It was so much more exciting and inspiring when we had Oleta Adams with us on that last tour... I thought, at this point in my career, I had to bring everything back to its basic elements. For me to be at center stage just makes so much sense right now. And I have a lot of fun. I can take responsibility for what I'm doing."

Critical reception
The New York Times wrote that "the album finds Mr. Orzabal straining self-consciously to make grand statements that lack the concision and spontaneity of Tears for Fears' best early work. It also has an undertone of meanness." Entertainment Weekly wrote that "the sampled guitar bursts and nifty studio tricks, like Squeeze mimicking the Beatles, keep you obediently humming while Orzabal plays teen-neuroses word games." Trouser Press thought that the album "essentially upholds Orzabal’s dedication to create remarkable textures and settings for essentially ordinary pop songs sprung from his bristly, remote and self-critical personality."

Upon its release in 1993, Elemental received a positive review in Rolling Stone magazine. Rating the album four stars out of five, the review states,

Even with one Tear for Fear subtracted from the mix – Curt Smith is gone, but Roland Orzabal remains – the lush sound of Tears for Fears continues to improve... Since Smith quit after the 1991 Seeds of Love world tour, Orzabal is left to fend off existential angst on his own – which he does with amazing grace and integrity on Elemental. Orzabal, guitarist Alan Griffiths and co-producer Tim Palmer (Pearl Jam, Tin Machine) provide the production techniques, smooth segues and sweeping guitar-keyboard interplay that fans have come to expect from Tears for Fears. Tracks like "Cold," "Power" and "Break It Down Again" are replete with melodic free falls, harmonized and overlapped vocals, burbling keyboards and guitar shudders... Elemental flashes the same sort of sheen, wordplay and sound wash that flourished in the '80s and still fights for its place on modern rock charts.

Track listing
All songs written by Roland Orzabal and Alan Griffiths, except "Cold", written by Orzabal.

 * 1) "Elemental" – 5:30
 * 2) "Cold" – 5:05
 * 3) "Break It Down Again" – 4:31
 * 4) "Mr. Pessimist" – 6:16
 * 5) "Dog's a Best Friend's Dog" – 3:39
 * 6) "Fish out of Water" – 5:07
 * 7) "Gas Giants" – 2:40
 * 8) "Power" – 5:49
 * 9) "Brian Wilson Said" – 4:22
 * 10) "Goodnight Song" – 3:53

Notes
 * To date, Elemental has never been re-released with bonus tracks, although a half-dozen songs released as B-sides during the Elemental era were included in the Tears for Fears compilation album Saturnine Martial & Lunatic (1996).
 * Elemental was the last Tears for Fears studio album released by Mercury Records although it was not the last album recorded by the band for Mercury. Raoul and the Kings of Spain, recorded between 1993 and 1994, was originally scheduled to be released by Mercury, and promo editions of the album were circulated under the Mercury imprint. However, Raoul and the Kings of Spain was eventually released in 1995 by Sony's Epic Records after Tears for Fears signed with that label.

Personnel
Tears for Fears
 * Roland Orzabal – vocals, instruments

Additional personnel
 * Alan Griffiths – instruments
 * Tim Palmer – instruments
 * Mark O'Donoughue – Wurlitzer electric piano outro (track 10)
 * Guy Pratt – additional bass guitar (track 4)
 * John Baker – backing vocals (track 2), additional backing vocals (track 3)
 * Julian Orzabal – backing vocals (track 2)

Production

 * Roland Orzabal – producer
 * Alan Griffiths – producer
 * Tim Palmer – producer
 * Mark O'Donoughue – engineer
 * Julian Lavender – technician
 * Bob Ludwig – mastering at Gateway Mastering (Portland, Maine, USA)
 * Area – design
 * David Austen – collages
 * Pamela Springsteen – photography

title: Elemental (Tears for Fears album)


text:
Jacobite peerage

The Jacobite peerage includes those peerages created by James II and VII, and the subsequent Jacobite pretenders, after James's deposition from the thrones of England, Scotland and Ireland following the Glorious Revolution of 1688. These creations were not recognised in English, Scots or Irish law, but the titles were used in Jacobite circles in Continental Europe and recognised by France, Spain and the Papacy.

Jacobite peerages ceased to be created after 1760 except for a title created by the "Young Pretender", Prince Charles Edward Stuart, for his illegitimate daughter in or before 1783. The following tables list the peerages and baronetcies created by the Stuart claimants in exile.

title: Jacobite peerage


text:
Reagan's War

Reagan's War is a 2002 book by Peter Schweizer.

Synopsis
The book covers Ronald Reagan's actions during the Cold War, and credits him with the eventual end of the Cold War.

Reception
The book received mixed reviews. Publishers Weekly praised Schweizer's writing style, while criticizing the book's hagiographical approach to history. Walter Russell Mead gave the book a positive review in The Washington Post. Kirkus Reviews called the book a "tendentious and extremely partisan account" of Reagan's presidency and the Cold War. The book was later adapted into a film titled In the Face of Evil: Reagan's War in Word and Deed.

title: Reagan's War


text:
Duchess of Albany

The title Duchess of Albany may refer to:

People
The title Duchess of Albany, held by the wives of the Dukes of Albany, may refer to:
 * Isabella, Countess of Lennox, wife of Murdoch Stewart
 * Anne, Countess of Auvergne, wife of John Stewart
 * Anne Hyde, first wife of James, Duke of York
 * Mary of Modena (1658–1718), second wife of James, Duke of York, later Queen of England and Scotland
 * Charlotte Stuart, Duchess of Albany, illegitimate daughter of Bonnie Prince Charlie
 * Princess Helena of Waldeck and Pyrmont, wife of Prince Leopold, Duke of Albany, youngest son of Queen Victoria
 * Princess Victoria Adelaide of Schleswig-Holstein, wife of Charles Edward, Duke of Saxe-Coburg and Gotha, Duke of Albany

Vehicles

 * Duchess of Albany, an iron 3-masted ship named after Princess Helena of Waldeck and Pyrmont
 * PS Duchess of Albany (1889), a passenger vessel
 * Duchess of Albany, one of the GWR 3031 Class locomotives that were built for and run on the Great Western Railway between 1891 and 1915

Other

 * Clematis texensis, a plant sometimes known as 'Duchess of Albany'

title: Duchess of Albany


text:
São Julião

São Julião may refer to:

People

 * São Julião, Portuguese for Saint Julian

Brazil

 * São Julião (Piauí), a municipality in the State of Piauí

Portugal

 * São Julião (Gouveia), a civil parish in the municipality of Gouveia
 * São Julião (Portalegre), a civil parish in the municipality of Portalegre
 * São Julião (Setúbal), a civil parish in the municipality of Setúbal
 * São Julião (Valença), a civil parish in the municipality of Valença
 * São Julião da Figueira da Foz, a civil parish in the municipality of Figueira da Foz
 * São Julião de Montenegro, a civil parish in the municipality of Chaves
 * São Julião de Palácios, a civil parish in the municipality of Bragança
 * São Julião dos Passos, a civil parish in the municipality of Braga
 * São Julião do Tojal, a civil parish in the municipality of Loures

title: São Julião


text:
User:Bluecross

User Information

Avatar: Musica is taken from Groove Adventure Rave's Musica. Original image from http://www.tbs.co.jp/rave2001 and edited with Adobe Photoshop 7.0 for transparency use as an avatar.

Full Name   : Nathan Prima Suwanto Birthdate   : 23 March 1984 Residence   : Surabaya 60256, East Java Nationality : Indonesian Ethnicity   : Chinese Tall/Weight : 175 cm / 72 kgs Religion    : Christian, since August 2003 Status      : In A Relationship Education   : Bachelor of Information Science, 7th semester University  : Petra Christian University, Surabaya Interests   : Music, Poetry, Games, Cars, Soccer, Drama,  Anime, Manga, Japanese Art

Wikipedia Comments
Joined Wikipedia on 29 September 2004 and primarily used this site as an online reference center. Now interested to add contributions to Wikipedia. Had a journalist background as an Ex-contributor of an online videogames site, VideoGames Indonesia and GameFAQs. Start writing Christian articles mainly from new-born date on August 2003. Thinking about using journalist capabilities to aid Wikipedia's growth, but right now have no ideas as what to write other than editing existing pages.

Selected Links

 * http://bluecross.blogspot.com Personal Blog - The Heavenly Skies (currently on hiatus)
 * http://bluecross.multiply.com Second Personal Site on Multiply (primarily used for Photo Album)
 * http://www.the-outcasts.com The Outcasts Community, a Postmodernic Christian Community Forum
 * http://www.christianster.com Christianster, an Online Christian Networking site like Frienster

title: User:Bluecross