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Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function $$f$$ is continuous on the closed and bounded interval $$[a,b]$$, then $$f$$ must attain a maximum and a minimum, each at least once. That is, there exist numbers $$c$$ and $$d$$ in $$[a,b]$$ such that: $$f(c) \ge f(x) \ge f(d)\quad \forall x\in [a,b]$$

The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function $$f$$ on the closed interval $$[a,b]$$ is bounded on that interval; that is, there exist real numbers $$m$$ and $$M$$ such that: $$m \le f(x) \le M\quad \forall x \in [a, b].$$ This does not say that $$M$$ and $$m$$ are necessarily the maximum and minimum values of $$f$$ on the interval $$[a,b],$$ which is what the extreme value theorem stipulates must also be the case.

The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a non-empty compact space to a subset of the real numbers attains a maximum and a minimum.

History
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value. Both proofs involved what is known today as the Bolzano–Weierstrass theorem.

Functions to which the theorem does not apply
The following examples show why the function domain must be closed and bounded in order for the theorem to apply. Each fails to attain a maximum on the given interval.

 * 1) $$f(x)=x $$ defined over $$[0, \infty)$$ is not bounded from above.
 * 2) $$f(x)= \frac{x}{1+x} $$ defined over $$[0, \infty)$$ is bounded but does not attain its least upper bound $$1$$.
 * 3) $$f(x)= \frac{1}{x}$$ defined over $$(0,1]$$ is not bounded from above.
 * 4) $$f(x) = 1-x$$ defined over $$(0,1]$$ is bounded but never attains its least upper bound $$1$$.

Defining $$f(0)=0$$ in the last two examples shows that both theorems require continuity on $$[a,b]$$.

Generalization to metric and topological spaces
When moving from the real line $$\mathbb{R}$$ to metric spaces and general topological spaces, the appropriate generalization of a closed bounded interval is a compact set. A set $$K$$ is said to be compact if it has the following property: from every collection of open sets $$U_\alpha$$ such that $\bigcup U_\alpha \supset K$, a finite subcollection $$U_{\alpha_1},\ldots,U_{\alpha_n}$$can be chosen such that $\bigcup_{i=1}^n U_{\alpha_i} \supset K$. This is usually stated in short as "every open cover of $$K$$ has a finite subcover". The Heine–Borel theorem asserts that a subset of the real line is compact if and only if it is both closed and bounded. Correspondingly, a metric space has the Heine–Borel property if every closed and bounded set is also  compact.

The concept of a continuous function can likewise be generalized. Given topological spaces $$V,\ W$$, a function $$f:V\to W$$ is said to be continuous if for every open set $$U\subset W$$, $$f^{-1}(U)\subset V$$ is also open. Given these definitions, continuous functions can be shown to preserve compactness:

Theorem. If $$V,\ W$$ are topological spaces, $$f:V\to W$$ is a continuous function, and $$K\subset V$$ is compact, then $$f(K)\subset W$$ is also compact.

In particular, if $$W = \mathbb{R}$$, then this theorem implies that $$f(K)$$ is closed and bounded for any compact set $$K$$, which in turn implies that $$f$$ attains its supremum and infimum on any (nonempty) compact set $$K$$. Thus, we have the following generalization of the extreme value theorem:

Theorem. If $$K$$ is a compact set and $$f:K\to \mathbb{R}$$ is a continuous function, then $$f$$ is bounded and there exist $$p,q\in K$$ such that $f(p)=\sup_{x\in K} f(x)$ and $f(q) = \inf_{x\in K} f(x)$ .

Slightly more generally, this is also true for an upper semicontinuous function. (see compact space).

Proving the theorems
We look at the proof for the upper bound and the maximum of $$f$$. By applying these results to the function $$-f$$, the existence of the lower bound and the result for the minimum of $$f$$ follows. Also note that everything in the proof is done within the context of the real numbers.

We first prove the boundedness theorem, which is a step in the proof of the extreme value theorem. The basic steps involved in the proof of the extreme value theorem are:

 * 1) Prove the boundedness theorem.
 * 2) Find a sequence so that its image converges to the supremum of $$f$$.
 * 3) Show that there exists a subsequence that converges to a point in the domain.
 * 4) Use continuity to show that the image of the subsequence converges to the supremum.

Proof of the boundedness theorem
Statement    If $$f(x)$$ is continuous on $$[a,b]$$ then it is bounded on $$[a,b]$$

Suppose the function $$f$$ is not bounded above on the interval $$[a,b]$$. Then, for every natural number $$n$$, there exists an $$x_n \in [a,b]$$ such that $$f(x_n)>n$$. This defines a sequence $$(x_n)_{n \in \mathbb{N}}$$. Because $$[a,b]$$ is bounded, the Bolzano–Weierstrass theorem implies that there exists a convergent subsequence $$(x_{n_k})_{k \in \mathbb{N}}$$ of $$({x_n})$$. Denote its limit by $$x$$. As $$[a,b]$$ is closed, it contains $$x$$. Because $$f$$ is continuous at $$x$$, we know that $$f(x_{{n}_{k}})$$ converges to the real number $$f(x)$$ (as $$f$$ is sequentially continuous at $$x$$). But $$f(x_{{n}_{k}}) > n_k \geq k $$ for every $$k$$, which implies that $$f(x_{{n}_{k}})$$ diverges to $+ \infty $, a contradiction. Therefore, $$f$$ is bounded above on $$[a,b]$$. $ \Box $

Alternative proof
Statement    If $$f(x)$$ is continuous on $$[a,b]$$ then it is bounded on $$[a,b]$$

Proof     Consider the set $$B$$ of points $$p$$ in $$[a,b]$$ such that $$f(x)$$ is bounded on $$[a,p]$$. We note that $$a$$ is one such point, for $$f(x)$$ is bounded on $$[a,a]$$ by the value $$f(a)$$. If $$e > a$$ is another point, then all points between $$a$$ and $$e$$ also belong to $$B$$. In other words $$B$$ is an interval closed at its left end by $$a$$.

Now $$f$$ is continuous on the right at $$a$$, hence there exists $$\delta>0$$ such that $$|f(x) - f(a)| < 1$$ for all $$x$$ in $$[a,a+\delta]$$. Thus $$f$$ is bounded by $$f(a) - 1$$ and $$f(a)+1$$ on the interval $$[a,a+\delta]$$ so that all these points belong to $$B$$.

So far, we know that $$B$$ is an interval of non-zero length, closed at its left end by $$a$$.

Next, $$B$$ is bounded above by $$b$$. Hence the set $$B$$ has a supremum in $$[a,b]$$ ; let us call it $$s$$. From the non-zero length of $$B$$ we can deduce that $$s > a$$.

Suppose $$s0$$ such that $$|f(x) - f(s)| < 1$$ for all $$x$$ in $$[s-\delta,s+\delta]$$ so that $$f$$ is bounded on this interval. But it follows from the supremacy of $$s$$ that there exists a point belonging to $$B$$, $$e$$ say, which is greater than $$s-\delta/2$$. Thus $$f$$ is bounded on $$[a,e]$$ which overlaps $$[s-\delta,s+\delta]$$ so that $$f$$ is bounded on $$[a,s+\delta]$$. This however contradicts the supremacy of $$s$$.

We must therefore have $$s=b$$. Now $$f$$ is continuous on the left at $$s$$, hence there exists $$\delta>0$$ such that $$|f(x) - f(s)| < 1$$ for all $$x$$ in $$[s-\delta,s]$$ so that $$f$$ is bounded on this interval. But it follows from the supremacy of $$s$$ that there exists a point belonging to $$B$$, $$e$$ say, which is greater than $$s-\delta/2$$. Thus $$f$$ is bounded on $$[a,e]$$ which overlaps $$[s-\delta,s]$$ so that $$f$$ is bounded on $$[a,s]$$. ∎

Proof of the extreme value theorem
By the boundedness theorem, f is bounded from above, hence, by the Dedekind-completeness of the real numbers, the least upper bound (supremum) M of f exists. It is necessary to find a point d in [a, b] such that M = f(d). Let n be a natural number. As M is the least upper bound, M – 1/n is not an upper bound for f. Therefore, there exists dn in [a, b] so that M – 1/n < f(dn). This defines a sequence {dn}. Since M is an upper bound for f, we have M – 1/n < f(dn) ≤ M for all n. Therefore, the sequence {f(dn)} converges to M.

The Bolzano–Weierstrass theorem tells us that there exists a subsequence {$$d_{n_k}$$}, which converges to some d and, as [a, b] is closed, d is in [a, b]. Since f is continuous at d, the sequence {f($$d_{n_k}$$)} converges to f(d). But {f(dn k )} is a subsequence of {f(dn)} that converges to M, so M = f(d). Therefore, f attains its supremum M at d. ∎

Alternative proof of the extreme value theorem
The set ${y ∈ R : y = f(x) for some x ∈ [a,b]}$ is a bounded set. Hence, its least upper bound exists by least upper bound property of the real numbers. Let $M = sup(f(x))$ on $[a, b]$. If there is no point x on [a, b] so that f(x) = M, then $f(x) < M$ on [a, b]. Therefore, $1/(M − f(x))$ is continuous on [a, b].

However, to every positive number ε, there is always some x in [a, b] such that $M − f(x) < ε$ because M is the least upper bound. Hence, $1/(M − f(x)) > 1/ε$, which means that $1/(M − f(x))$ is not bounded. Since every continuous function on a [a, b] is bounded, this contradicts the conclusion that $1/(M − f(x))$ was continuous on [a, b]. Therefore, there must be a point x in [a, b] such that f(x) = M. ∎

Proof using the hyperreals
In the setting of non-standard calculus, let N  be an infinite hyperinteger. The interval [0, 1] has a natural hyperreal extension. Consider its partition into N subintervals of equal infinitesimal length 1/N, with partition points xi = i /N as i "runs" from 0 to N. The function ƒ  is also naturally extended to a function ƒ* defined on the hyperreals between 0 and 1. Note that in the standard setting (when N  is finite), a point with the maximal value of ƒ can always be chosen among the N+1 points xi, by induction. Hence, by the transfer principle, there is a hyperinteger i0 such that 0 ≤ i0 ≤ N and $$f^*(x_{i_0})\geq f^*(x_i)$$  for all i = 0, ..., N. Consider the real point $$c = \mathbf{st}(x_{i_0})$$ where st is the standard part function. An arbitrary real point x lies in a suitable sub-interval of the partition, namely $$x\in [x_i,x_{i+1}]$$, so that  st(xi) = x. Applying st to the inequality $$f^*(x_{i_0})\geq f^*(x_i)$$, we obtain $$\mathbf{st}(f^*(x_{i_0}))\geq \mathbf{st}(f^*(x_i))$$. By continuity of ƒ  we have
 * $$\mathbf{st}(f^*(x_{i_0}))= f(\mathbf{st}(x_{i_0}))=f(c)$$.

Hence ƒ(c) ≥ ƒ(x), for all real x, proving c to be a maximum of ƒ.

Proof from first principles
Statement      If $$f(x)$$ is continuous on $$[a,b]$$ then it attains its supremum on $$[a,b]$$

Proof        By the Boundedness Theorem, $$f(x)$$ is bounded above on $$[a,b]$$ and by the completeness property of the real numbers has a supremum in $$[a, b]$$. Let us call it $$M$$, or $$M[a,b]$$. It is clear that the restriction of $$f$$ to the subinterval $$[a,x]$$ where $$x\le b$$ has a supremum $$M[a, x]$$ which is less than or equal to $$M$$, and that $$M[a,x]$$ increases from $$f(a)$$ to $$M$$ as $$x$$ increases from $$a$$ to $$b$$.

If $$f(a)=M$$ then we are done. Suppose therefore that $$f(a)a$$ is another point in $$L$$ then all points between $$a$$ and $$e$$ also belong to $$L$$ because $$M[a,x]$$ is monotonic increasing. Hence $$L$$ is a non-empty interval, closed at its left end by $$a$$.

Now $$f$$ is continuous on the right at $$a$$, hence there exists $$\delta>0$$ such that $$|f(x)-f(a)| < d/2$$ for all $$x$$ in $$[a,a+\delta]$$. Thus $$f$$ is less than $$M-d/2$$ on the interval $$[a,a+\delta]$$ so that all these points belong to $$L$$.

Next, $$L$$ is bounded above by $$b$$ and has therefore a supremum in $$[a,b]$$: let us call it $$s$$. We see from the above that $$s > a$$. We will show that $$s$$ is the point we are seeking i.e. the point where $$f$$ attains its supremum, or in other words $$f(s)=M$$.

Suppose the contrary viz. $$f(s)0$$ such that $$|f(x)-f(s)| < d/2$$ for all $$x$$ in $$[s-\delta,s+\delta]$$.    This means that $$f$$ is less than $$M-d/2$$ on the interval $$[s-\delta,s+\delta]$$.   But it follows from the supremacy of $$s$$ that there exists a point, $$e$$ say, belonging to $$L$$ which is greater than $$s-\delta$$.  By the definition of $$L$$, $$M[a,e]< M$$.  Let  $$d_1=M-M[a,e]$$ then for all $$x$$ in $$[a,e]$$, $$f(x)\le M-d_1$$.  Taking $$d_2$$ to be the minimum of $$d/2$$ and $$d_1$$, we have $$f(x)\le M-d_2$$ for all $$x$$ in $$[a,s+\delta]$$.  Hence $$M[a,s+\delta]0$$ such that $$|f(x)-f(s)| < d/2$$ for all $$x$$ in $$[s-\delta,s]$$.    This means that $$f$$ is less than $$M-d/2$$ on the interval $$[s-\delta,s]$$.   But it follows from the supremacy of $$s$$ that there exists a point, $$e$$ say, belonging to $$L$$ which is greater than $$s-\delta$$.  By the definition of $$L$$, $$M[a,e]< M$$.  Let  $$d_1=M-M[a,e]$$ then for all $$x$$ in $$[a,e]$$, $$f(x)\le M-d_1$$.  Taking $$d_2$$ to be the minimum of $$d/2$$ and $$d_1$$, we have $$f(x)\le M-d_2$$ for all $$x$$ in $$[a,b]$$.  This contradicts the supremacy of $$M$$ and completes the proof.

Extension to semi-continuous functions
If the continuity of the function f is weakened to semi-continuity, then the corresponding half of the boundedness theorem and the extreme value theorem hold and the values –∞ or +∞, respectively, from the extended real number line can be allowed as possible values. More precisely:

Theorem: If a function $f : [a, b] → [–∞, ∞)$ is upper semi-continuous, meaning that $$\limsup_{y\to x} f(y) \le f(x)$$ for all x in [a,b], then f is bounded above and attains its supremum.

Proof: If f(x) = –∞ for all x in [a,b], then the supremum is also –∞ and the theorem is true. In all other cases, the proof is a slight modification of the proofs given above. In the proof of the boundedness theorem, the upper semi-continuity of f at x only implies that the limit superior of the subsequence {f(xn k )} is bounded above by f(x) < ∞, but that is enough to obtain the contradiction. In the proof of the extreme value theorem, upper semi-continuity of f at d implies that the limit superior of the subsequence {f(dn k )} is bounded above by f(d), but this suffices to conclude that f(d) = M. ∎

Applying this result to −f proves:

Theorem: If a function $f : [a, b] → (–∞, ∞]$ is lower semi-continuous, meaning that $$\liminf_{y\to x} f(y) \geq f(x)$$ for all x in [a,b], then f is bounded below and attains its infimum.

A real-valued function is upper as well as lower semi-continuous, if and only if it is continuous in the usual sense. Hence these two theorems imply the boundedness theorem and the extreme value theorem.

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

title: Extreme value theorem


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Benchrest rifle

A benchrest rifle, also colloquially called a "rail gun", is a rifle with its barrel and action mechanism built into a machine rest, used mainly for benchrest shooting. The rifle has no proper stock and its base uses adjustable feet to provide a stable position on the bench, and the rifle is finely aimed with horizontal and vertical adjustments built into the rest.

Benchrest shooting
Aside from their use for testing ammunition or components by ammunition manufacturers (almost always indoors, or in a "tunnel"), benchrest rifles are used in the Unlimited Class of benchrest shooting, almost always outdoors, where the effects of the wind can be considerable.

Benchrest rifles "return to battery". That is, the repetition of the aiming point is assured by mechanical means. Secondly, in a properly built and functioning rail gun, aside from the forces used to aim the rifle, the forces generated in firing the rifle will be consistent from shot to shot.

However, the goal of most benchrest competitions is to achieve a high precision, mechanically returning the rifle to the same aiming point only removes problems caused by mirage. It is of no aid in dealing with shot deflection caused by wind. Indeed, since the usual practice in dealing with anticipated shot displacement cause by the wind is to "hold off" (i.e., use a different aiming point), some feel that the use of a return-to-battery rifle can be a disadvantage.

On the surface, this appears a valid argument, but based on match reports [IBS, NBRSA], it is not a particularly good one. While "reading the wind"—anticipating shot deflection caused by varying wind speed and direction—is the same problem faced by competitors using conventional benchrest rifles, return-to-battery rifles do, in fact, generally place higher than the "bag guns" (rifles shot off sandbags, which have to be consciously aimed for each shot). With any benchrest rifle, there are two techniques generally used to deal with the wind. The first is for the rifleman to shoot only when he considers the wind to be the same as with previous shots. Here, the return to battery rifle can be an advantage. The second is to purposefully change the point of aim, just as one would do with a conventional rifle. Due to time limits, in any given match, both techniques will usually have to be employed.

A second use of return-to-battery rifles is for the testing of components, including barrels, bullets, powder, primers, and the interaction of these and the other components of the rifle. That such testing is both useful and necessary points up that a rifle is a system, and precision shooting perforce involves all components, as well as good shooting techniques. Benchrest remove only two factors, 1) the ability to precisely return a rifle to the same point of aim, and 2) to apply constant forces to the rifle from shot to shot. A proper design and build of the rifle is difficult to achieve.

The fundamental purpose of a bench rest is to eliminate, in the shooting, inconsistent potential human interactions such as the push against the shoulder, trigger pressure, breathing, or heartbeat. This helps to provide a more consistent testing of the system components.

title: Benchrest rifle


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User talk:Fuzzywolfenburger

Your article at Dr Crippen seems to be a duplicate of the wording of the website http://www.titanic-whitestarships.com/The%20Dr.%20Crippen%20Story.htm. Are you the author of that material? If so, I apologize for deleting your material. If not, it is a violation of Wikipedia policy to allow material from other sites to be included here. RickK 03:23 25 Jul 2003 (UTC)

DO NOT delete entries on Votes for deletion without a week's discussion. RickK 07:22 26 Jul 2003 (UTC)

Hello, Fuzzywolfenburger. Welcome to Wikipedia.

A few tips for you, although it appears that they are probably superfluous, since you appear to have quite handily got the hang of the place.

 * When you have time, check out Welcome, newcomers + the links in there.
 * If puzzled, put a question on Village pump, or feel free to ask me on my talk page if it's a very general question.
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-- Cimon Avaro on a pogo-stick 01:53 28 Jul 2003 (UTC)

Hello. When you create a new page, could you begin with a complete sentence rather than a dictionary-style definition (e.g., in gentleman thief) and highlight the title word or title phrase at its first appearance? Thanks. Michael Hardy 20:00, 2 Aug 2003 (UTC)

A similar complaint. Random article produced The Fool (Zero) which I was about to place on VfD until I realized it was (sort of) notable but lacked any sort of intro whatsoever. Also, before careful to place things in your own wording--looks like cut-and-paste. Marskell 18:31, 20 August 2005 (UTC)

--If I remember correctly, I created that page to mirror the other similar tarot pages or "The Fool" was directing to the wron place. Not sure because it was over a year ago. Fuzzywolfenburger

Out of This World (card trick)
Hi. If you believe that any content on Out of This World (card trick) is a copyright violation, please follow the procedure at WP:CP, indicating exactly what existing work the material violates. Note that copyright protects a specific expression, and not an idea or method. You may also wish to contribute to the lengthy debate on Talk:Out of This World (card trick) where the various issues are discussed. Cheers, Bovlb 05:13:36, 2005-08-27 (UTC)
 * I'd also point you at Intellectual rights to magic methods, but I see you already found it. Bovlb 05:22:28, 2005-08-27 (UTC)

David Abbott
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-- Just a note for completeness...the "violation" now list the Wikipedia as the source of their article. I was the originator. Fuzzywolfenburger 06:02, 27 December 2006 (UTC)

Talk page
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Robert Todd Lincoln Beckwith
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Proposed deletion of Wolfgang Poe

The article Wolfgang Poe has been proposed for deletion  because of the following concern:
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ArbCom elections are now open!
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Nomination of Brother Matthew Luke for deletion
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The article will be discussed at Articles for deletion/Brother Matthew Luke until a consensus is reached, and anyone, including you, is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

Users may edit the article during the discussion, including to improve the article to address concerns raised in the discussion. However, do not remove the article-for-deletion notice from the top of the article. Lennart97 (talk) 17:54, 23 April 2021 (UTC)

title: User talk:Fuzzywolfenburger


text:
User talk:Egoff

Hi Egoff, welcome to the 'pedia! I hope you like the place and decide to stay. If you need any questions answered about the project then check out Help or add a question to the Village pump.

I see you've uploaded a lot of images today, and that's very appreciated. I wonder thoughif you could put a note in the image description pages saying where they came from, that would be useful for purposes of the license we operate under. Also please check Copyrights and make sure they're within guidelines. Thanks! - Hephaestos 03:25 25 Jul 2003 (UTC)

Unverified images
Hi! Thanks for uploading the following image:

 * Image:Death_of_romanus_ii.jpg

I notice it currently doesn't have an image copyright tag. Could you add one to let us know its copyright status? (You can use if you release it under the GNU Free Documentation License,  if you claim fair use, etc.) If you don't know what any of this means, just let me know at my talk page where you got the images and I'll tag them for you. Thanks so much. [[User:Poccil|Peter O. (Talk, automation script)]] 05:36, Dec 11, 2004 (UTC)

P.S. You can help tag other images at User:Yann/Untagged_Images. Thanks again.

Image Tagging for Image:Aristophanes.jpg
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text:
Simon Cowell

Simon Phillip Cowell (born 7 October 1959) is an English television personality, entrepreneur, and record executive. He has judged on the British television talent competition shows Pop Idol (2001–2003), The X Factor UK (2004–2010, 2014–2018), and Britain's Got Talent (2007–present), as well as the American television talent competition shows American Idol (2002–2010), The X Factor US (2011–2013), and America's Got Talent (2016–present). Cowell founded the British entertainment company Syco in 2005, as well as its now-defunct predecessor, Syco Music (also known as S Records) in 2002.

After some success in the 1980s and 1990s as a record producer, talent agent and consultant in the UK music industry, Cowell came to public prominence in 2001 as a judge on Pop Idol, a talent television show which he and its creator Simon Fuller successfully pitched to ITV Controller of Entertainment Claudia Rosencrantz. He subsequently created The X Factor in 2004 and Got Talent in 2006, which have become successful television franchises sold around the world.

Cowell often makes blunt and controversial comments as a television music and talent show judge, including insults and wisecracks about contestants and their singing abilities. He combines activities in both the television and music industries. Cowell has produced and promoted successful singles and albums for various recording acts whom he has signed to his record label Syco Music, including Little Mix, James Arthur, Labrinth, Leona Lewis, Fifth Harmony, Olly Murs, Noah Cyrus, Fleur East, Ella Henderson, and Susan Boyle. He has also signed successful boy bands such as Westlife, One Direction, PrettyMuch, and CNCO.

In 2004 and 2010, Time named Cowell one of the 100 most influential people in the world. In 2008, The Daily Telegraph ranked him sixth in their list of the "100 most powerful people in British culture". The same year, he received the Special Recognition Award at the National Television Awards in London. At the 2010 British Academy Television Awards, Cowell received the BAFTA Special Award for his "outstanding contribution to the entertainment industry and for his development of new talent". In 2018, he received a star on the Hollywood Walk of Fame in the television category.

Early life
Simon Phillip Cowell was born on 7 October 1959 in Lambeth, London, and raised in Elstree, Hertfordshire. His mother, Julie Brett (née Josie Dalglish, 1925–2015), was a ballet dancer and socialite, and his father, Eric Selig Phillip Cowell (1918–1999), was an estate agent, property developer, and music industry executive. Cowell's father was from a mostly Jewish family (his own mother was born in Poland), though he did not discuss his ancestry with his children. Cowell's mother was from a Christian background. He has a younger brother, Nicholas Cowell; three half brothers, John, Tony, and Michael Cowell; and a half sister, June Cowell.

Cowell attended Radlett Preparatory School, Licensed Victuallers' School in Ascot for one year, and the private Dover College, as did his brother, but left after taking GCE O levels. He passed English Language and Literature and then attended Windsor Technical College, where he gained another GCE in Sociology. Cowell took a few menial jobs—including, according to his brother Tony, working as a runner on Stanley Kubrick's 1980 horror film The Shining—but did not get along well with colleagues and bosses, until his father, who was an executive at EMI Music Publishing, managed to get him a job in the mail room. However, after failing to get a promotion, he left to try out other jobs before returning to EMI.

Early career
In the early 1980s, Cowell left EMI to form E&S Music with his former boss at EMI but quit in 1983. He then formed Fanfare Records with Iain Burton, initially selling exercise videos, and music from acts such as the Italian orchestra Rondò Veneziano. He had his first hit song in 1986 with "So Macho" by Sinitta. Some of Cowell's early success came through Stock Aitken Waterman, who produced a number of hits in the 1980s. However, in 1989 the company went under, with Pete Waterman forced to put a freeze on Fanfare's assets, and taking Cowell's Porsche as compensation for outstanding production and royalties payments. Amid the company's implosion Cowell nearly became bankrupt. He then worked with BMG as an A&R consultant and set up S Records under BMG. He restarted his career in the music business by creating novelty records with acts such as the puppets Zig and Zag, Power Rangers and World Wrestling Federation. In 1995, through his persistence, he persuaded two actors, Robson Green and Jerome Flynn from the UK television drama series Soldier Soldier, to sign with him and record the song "Unchained Melody", which they had performed on the show. The recording by the duo, now named Robson & Jerome, quickly reached number 1 in the UK, staying at the top of the chart for seven weeks. It became the best-selling single of 1995 (their next release, "I Believe"/"Up on the Roof", was the third-best selling single), and their self-titled album released later in the year also became the best-selling album of 1995. They released another album and two more singles before disbanding, and sold 7 million albums and 5 million singles in total. According to Cowell, they made him his first million. Later acts he signed included Five, Westlife and Teletubbies.

Idol franchise
In 2001, Cowell was given the role of judge on the first series of Pop Idol, a show that he and the show creator Simon Fuller successfully pitched to ITV Controller of Entertainment Claudia Rosencrantz. Maggie Brown in The Guardian states, "the show became a seminal reality/entertainment format once on air that autumn". Cowell's S Records signed the top two finishers of the first season of Pop Idol, Will Young and Gareth Gates, both of whom went on to have No 1 UK hits, and they were the top 2 best-selling singles of 2002 and the decade of 2000s. He also became a judge on the first season of American Idol in 2002. With his notoriously critical reputation, Cowell has been likened to TV personalities such as Judge Judy and The Weakest Link host Anne Robinson. His closest predecessor was British TV critic Nina Myskow who, in the 1980s, became known for her harsh put-downs in New Faces, a talent show that Cowell cited as an influence. Cowell's prominence grew, fed by his signature phrase, "I don't mean to be rude, but ...", inevitably followed by an unsparingly blunt appraisal of the contestant's talents, personality, or even physical appearance. A lot of these one-liners were the product of coaching that Cowell received from publicist Max Clifford. Cowell also appeared on the one-off World Idol programme in 2003, in which it became clear that each country's version of the Idol had attempted to come up with its own "Simon Cowell" type personality.

Cowell formed a new company, Syco, in 2005. The company is divided into three units: Syco Music, Syco TV and Syco Film. Cowell returned to music with his latest brainchild signed to Syco, the internationally successful operatic pop group Il Divo, consisting of three opera singers and one pop singer of four different nationalities. Inspired by the success of Il Divo, Cowell created a child version, Angelis, made up of six 11 to 14 year-olds from choirs across the UK, with their debut album released in November 2006.

On 11 January 2010, Cowell's exit from American Idol was made official. One year after Cowell left American Idol, the show eventually completed eight consecutive seasons of leadership amongst all primetime programmes in the United States, marking the longest annual winning streak in the ratings in U.S. television history. The 2010 season was Cowell's last on the show, and he was replaced by Steven Tyler. It was also announced that FOX had acquired the rights to The X Factor USA, an American version of British show The X Factor that launched in 2011.

The X Factor
In 2004, with Sharon Osbourne and Louis Walsh, Cowell was a judge on the first series of the British television music competition The X Factor, which he created using his production company, Syco TV. The X Factor was an instant success with the viewers and enjoyed its tenth series in 2013.

Leona Lewis, the winner of the third series of The X Factor, was signed to Cowell's label Syco and has had number one singles and album sales around the world. Cowell returned for a fourth series on 18 August 2007 alongside Osbourne, Walsh and new judge, Dannii Minogue. Walsh had previously been sacked from the judging panel by Cowell for the fourth series, and he was subsequently replaced by Brian Friedman, who was a judge on Grease Is the Word. Walsh was later brought back by Cowell a week into the auditions when he and Osbourne realised that they missed Walsh and that without him, there was no chemistry between the judges. Cowell returned for the fifth series in 2008, with Walsh, Minogue, and new judge Cheryl Cole, as Osbourne had decided to quit before the show began. Cowell returned for series 6 and 7 as well, although series 7 was to be his last, as he left in 2011 in order to launch The X Factor in America. After placing third in the seventh series of The X Factor in the UK, boyband One Direction signed to Cowell's label in 2011, and the group has gone on to top singles and album charts worldwide. Cowell was replaced by Gary Barlow. In 2011, UK series eight winners Little Mix signed to Cowell's label. The girl group has sold 50 million records worldwide.

Australian The X Factor launched in 2005 on Network Ten, and it featured Mark Holden, Kate Ceberano and John Reid as the show's judges, but it was cancelled after just one season due to poor ratings. However, the show returned in 2010 and performed strongly on the Seven Network until its eighth season in 2016, when its ratings declined. Subsequently, the show was axed for a second time in January 2017.

Cowell also launched American The X Factor in September 2011 on American broadcaster Fox. It was originally announced that he would be a judge both on the UK and US editions of the show, which aired at similar times of the year, but MTV officially reported on 17 April 2011 that this was not true. Cowell was no longer to be a judge on the UK version, but he would remain a major presence backstage. He was joined by Paula Abdul, L.A. Reid, Nicole Scherzinger and formerly Cheryl Cole. Cowell's act Melanie Amaro won the season, making Cowell the winning judge. Cowell and Reid returned for season 2, while Demi Lovato and Britney Spears joined the judging panel as replacements for Abdul and Scherzinger. This season launched another of Cowell's acts into worldwide fame, the group Fifth Harmony.

Cowell returned for a third season of The X Factor in September 2013 alongside Demi Lovato, while Spears and L.A. Reid announced they would not be returning and were replaced by Paulina Rubio and Kelly Rowland. Cowell's last act Alex & Sierra won the season, marking Cowell's second season as the winning mentor after he won with Melanie Amaro in 2011.

In October 2010, Cowell signed new three-year deals with ITV for both Britain's Got Talent and The X Factor, retaining them until 2013. On 15 November 2013, the three-year deal was extended by another three years, keeping it on air until 2016.

In October 2013, it was reported that Cowell might return to the UK version of The X Factor for series 11 in place of Gary Barlow, and on 7 February 2014, his return was officially confirmed. This resulted in the cancellation of the US version after three seasons by Fox. He joined judges Louis Walsh, Cheryl Fernandez-Versini, who replaced Sharon Osbourne, and new judge, former Spice Girls member Mel B, who replaced Nicole Scherzinger. For his eighth series, he was given the Over 25s category. On 13 December, Ben Haenow and Fleur East reached the final two, which meant that Cowell was the winning mentor for the first time since series 3 2006, when he had both Leona Lewis and Ray Quinn in the final. Ben Haenow became the eleventh winner on 14 December. In 2015, Cowell returned to the X Factor for its twelfth series, along with veteran judge Cheryl Fernandez-Versini and newcomers Rita Ora and BBC Radio 1 DJ Nick Grimshaw.

In December 2020 it was announced that Cowell will be a judge on The X Factor Israel in 2021. In late May 2021, Cowell announced he will be cancelling his appearance on the show in context of the Israeli aggression against Gaza.

Got Talent

Following the success of the Idol and X Factor franchises, Cowell, his company Syco, and its business partners developed a talent show format open to performers of any kind, not only singers, but also dancers, instrumentalists, magicians, comedians, novelty acts, and so on. The origins of the Got Talent format can be traced to the British talent shows Opportunity Knocks (on screen from 1956, with the winner using the now-standard method of a telephone vote) and New Faces. Immensely popular with weekly audiences of 20 million, Opportunity Knocks showcased singers, dancers and comedians in addition to non standard performers such as acrobats, animal acts and novelty acts. Cowell states, "I was a fan of variety shows Opportunity Knocks and New Faces, and to be able to update that tradition, really was a buzz".

Cowell is the executive producer of America's Got Talent, which debuted in June 2006, along with Fremantle producers of the Idol series. The show was a huge success for NBC, drawing around 12 million viewers a week and beating So You Think You Can Dance on Fox (produced by rival and Idol creator Simon Fuller).

Britain's Got Talent debuted on ITV in June 2007. Cowell appeared as a judge alongside Amanda Holden and Piers Morgan. The show was a ratings success, and second and third seasons followed in 2008 and 2009. The third series featured a publicity coup when Susan Boyle made a global media impact comparable to that of any previous talent show series winner with her regional audition performance. In December 2019, Cowell signed a five-year deal with ITV ensuring Britain's Got Talent will stay on the channel until 2024.

In 2014, Got Talent was named the world's most successful reality TV format by Guinness World Records, with spin-offs in over 68 countries. Cowell stated, "I am very proud that Got Talent is a home grown British show. We owe its success to a group of very talented producers all over the world who have made this happen. And of course amazing talent."

In 2015, Cowell launched La Banda, his first US show since his stint with American The X Factor. The show, designed to find male singers to form the "ultimate Latino Boy Band", launched on Univision in the US on 13 September 2015. The winner of the show, boy band CNCO, signed to Cowell's record label Syco Music.

Cowell was announced as the replacement for Howard Stern on America's Got Talent on 22 October 2015. On 15 July 2020, it was announced that Cowell had bought out Sony Music from their joint venture, Syco Entertainment.

Other talent shows
On 16 March 2006, American Inventor debuted on ABC. Cowell co-produced the show with British entrepreneur Peter Jones, who had devised the concept. Having a similar format to the BBC's Dragons' Den, a show on which Jones was one of the investors, fledgling entrepreneurs from across the United States competed to see who could come up with the best new product concept. The 2006 winner, Janusz Liberkowski, received $1 million and the opportunity to develop his idea into a business. The show returned in 2007 for a second season.

In 2006, Cowell executive-produced Celebrity Duets, which was described as "an Idol show for Hollywood superstars". The show was hosted by Wayne Brady, and its judges were Marie Osmond, Little Richard, and David Foster.

Cowell was also the executive producer of Grease Is the Word for ITV. This show set out to find performers to play Danny and Sandy in the 2007 West End revival of Grease. It was hosted by Zoë Ball and judged by Britons David Ian and Sinitta and Americans David Gest and Brian Friedman. The musical theatre casting concept had already been introduced by the BBC with the ratings hit How Do You Solve a Problem like Maria?, but Cowell's show was not the hoped-for success. He himself said, "It has been slaughtered by the critics – and rightly so. It is far too similar to our other formats."

Cowell collaborated with UK production company Shed Media to produce 2008 ITV drama series Rock Rivals, which is based on an X Factor type show. In 2011, Cowell also created his first game show, titled Red or Black? and hosted by Ant & Dec, for ITV. Series 1 was broadcast from 3 to 10 September. The show was recommissioned by ITV for a second series in 2012, which aired weekly.

In 2013, Cowell, in partnership with YouTube, launched a video-sharing competition called The You Generation. In 2013, he was executive producer for ITV cookery series Food Glorious Food, which was hosted by Carol Vorderman. He did the same for the ITV game show Keep it in the Family, presented by Bradley Walsh in 2016.

In 2018, it was announced that Cowell created his first show to air on the BBC, titled The Greatest Dancer, which debuted on 5 January 2019. In the show, dancers competed for a £50,000 and a performance on the BBC's Strictly Come Dancing. A second series is scheduled to air in 2020.

In April 2020, Cowell judged a spin-off version of the former show Canada's Got Talent called Canadian Family's Got Talent, carried out virtually by Canadian broadcaster Citytv during the COVID-19 pandemic in Canada.

In the media
Cowell has been involved in charity work for many years. He is a patron of Together for Short Lives, the leading UK charity for all children with life-threatening and life-limiting conditions. He was patron of Children's Hospices UK between 2009 and 2011 before it merged to become Together for Short Lives. He donated money to Manchester Dogs' Home in England after it was hit by a large fire. In view of his charitable works, particularly the production of the charity single "Everybody Hurts" in aid of victims of the 2010 Haiti earthquake, as well as his services to the music industry, there was considerable media speculation as to whether Cowell would receive a knighthood in the Queen's 2010 Birthday Honours, a proposal allegedly put forward by then Prime Minister Gordon Brown.

In November 2003, Cowell appeared on the BBC's charity telethon Children in Need where viewers pledged thousands to see him get sawn in half. In December 2003, he published his autobiography titled I Don't Mean to be Rude, but.... In it, he told of his childhood, his years working in music and experiences on Pop Idol, Pop Stars Rivals, and American Idol, and finally, his tips for being successful as a pop star. In 2010, he was added as a new entry to the latest edition of the UK's Who's Who.

Cowell has appeared as a guest voice in an episode of The Simpsons ("Smart and Smarter"), in which he gets beaten up by Homer Simpson (while criticising Homer's punches). In May 2010, he portrayed himself again in an episode of The Simpsons, "Judge Me Tender". His voice was heard on an episode of Family Guy ("Lois Kills Stewie"), in which he told Stewie that his singing was so awful that he should be dead. He made an MTV Movie Award-winning cameo appearance as himself in Scary Movie 3, where he sits in judgment during a battle rap (and is subsequently killed by gunfire for criticising the rappers). He appears in the DVD version of Shrek 2 as himself in Far Far Away Idol and also provided the voice.

Cowell was once the fastest "Star in a Reasonably Priced Car" on BBC's motoring show Top Gear, driving a Suzuki Liana around the show's test track in a time of 1:47.1. When Top Gear retired the Liana along with its rankings after the eighth series, Cowell was the eighth fastest overall and the third fastest non-professional driver. On 11 November 2007 Cowell yet again appeared on Top Gear, achieving a time of 1:45.9 thus putting him ahead of chef Gordon Ramsay and back at the top of the table. His time of 1:45.9 was then tied with English rock singer Brian Johnson of AC/DC and Grand Designs host Kevin McCloud for the second fastest time, just 0.1 seconds behind Jamiroquai singer Jay Kay. In March 2015, Cowell defended Top Gear presenter Jeremy Clarkson after he was suspended by the BBC for punching a colleague, with Cowell stating on Good Morning Britain: "He's apologised. But I think what is quite obvious is that the public are behind him, but you know, that's Jeremy."

He appeared on an episode of Who Wants to Be a Millionaire? (the original British version) and Saturday Night Live in 2004. Cowell has also guest-starred (filling in for Regis Philbin) in the popular talk show Live with Regis and Kelly during American Idol's finalist week in early 2006. Cowell introduced entertainer Dick Clark at the 2006 Primetime Emmy Awards. He was seen on the British charity telethon Comic Relief, appearing on Comic Relief Does The Apprentice where he donated £25,000 for a fun fair ticket. Cowell has also appeared on the MTV shows Cribs and Punk'd. On Punk'd, Ryan Seacrest and Randy Jackson set him up to believe his $400,000 Rolls-Royce was stolen and had caused an accident by using a nearly identical car.

Cowell was a partner in the Royal Ascot Racing Club, a thoroughbred horse racing syndicate which owned the 2005 Epsom Derby winner, Motivator. Cowell was chosen as the first subject of the re-launched British edition of This Is Your Life, in an episode broadcast on ITV on 2 June 2007. He was presented with the Red Book by Sir Trevor McDonald while presenting American Idol.

On 1 July 2007, Cowell appeared as a speaker at the Concert for Diana (alongside Randy Jackson and Ryan Seacrest) held at Wembley Stadium, London, to celebrate the life of Princess Diana almost 10 years after her death. Proceeds from the concert went to Diana's charities as well as to charities of which her sons Princes William and Harry are patrons.

At the 2012 Pride of Britain Awards on 30 October, Cowell, along with Michael Caine, Elton John, Richard Branson and Stephen Fry, recited Rudyard Kipling's poem "If—" in tribute to the 2012 British Olympic and Paralympics athletes. In October 2013 Cowell took part in a fundraising event in Los Angeles in support of the Israeli Defense Forces. The event raised $20 million and Cowell made a personal donation of $150,000.

To raise money for the families of the victims of the Grenfell Tower fire in London in June 2017 and for The London Community Foundation, Cowell arranged the recording and release of a charity single "Bridge over Troubled Water". Cowell collaborated with a number of vocalists for the single, including Robbie Williams, Dua Lipa, Roger Daltrey and Rita Ora, and the song reached number one in the UK Singles Chart.

Personal life
Cowell dated English presenter Terri Seymour from 2002 to 2008. Cowell was engaged to make-up artist Mezhgan Hussainy from 2010 to 2011. In 2013, Cowell allegedly began dating an American woman named Lauren Silverman. In July 2013, Silverman's husband and Cowell's friend, Andrew Silverman, filed for an at-fault divorce, citing adultery by his wife and naming Cowell as a co-respondent. News of the divorce filing became public two weeks later, when it was widely reported that Silverman and Cowell were expecting a baby together.

Cowell said, "There are a lot of things I will eventually clear up when the time is right, but I really have to be sensitive because there's a lot of people's feelings involved here." The Silvermans released separate statements expressing concern for the well-being of their son during the divorce process. In August 2013, the Silvermans settled their divorce out of court, enabling Cowell to avoid being called as a witness in the divorce proceedings. He subsequently confirmed that Silverman was pregnant with his child, and she gave birth to their son Eric on 14 February 2014. The couple confirmed their engagement in January 2022.

In 2010, Cowell came out in support for then Conservative Party leader David Cameron for Prime Minister of the United Kingdom, writing in The Sun: "I believe he is the Prime Minister Britain needs at this time. He has substance and the stomach to navigate us through difficult times." In 2013, Cowell contravened his previous statement about David Cameron. According to the interview, while he has supported candidates, he has never voted in an election. Cowell was in favour of the UK remaining in the EU in the 2016 EU referendum.

In 2019, Cowell became a vegan after doctors advised him to change his diet for health reasons. He was also named one of the wealthiest people in the UK by the Sunday Times Rich List, Cowell was worth £385 million in 2019.

In 2020, Cowell announced he would be writing a seven-book series titled Wishfits with his son. The series was scheduled to launch in 2023. On 8 August 2020, Cowell broke his back after he fell off his new electric motorcycle which many press sources confused with an electric bike. The incident occurred while he was testing it at his home in Malibu, California. Cowell was taken to the hospital, where he underwent back surgery overnight. After the accident, he began eating animal-based food again in order to "rebuild his strength".

Awards and recognition
In 2004 and 2010, Time magazine named Cowell one of the 100 most influential people in the world. In 2008, he received the Special Recognition Award (presented to him by Andrew Lloyd Webber) at the National Television Awards held at the Royal Albert Hall. New Statesman listed Cowell at number 41 in a list of "50 People who Matter [in] 2010". TV Guide named him at number 10 in their 2013 list of The 60 Nastiest Villains of All Time.

In 2012, Cowell was featured in the BBC Radio 4 series The New Elizabethans to mark the diamond Jubilee of Queen Elizabeth II. A panel of seven academics, journalists and historians named Cowell among the group of people in the UK "whose actions during the reign of Elizabeth II have had a significant impact on lives in these islands and given the age its character". Cowell was announced to receive a star on the Hollywood Walk of Fame on 22 August 2018, in the Television category.

title: Simon Cowell


text:
Philatelic investment

Philatelic investment is investment in collectible postage stamps for the purpose of realizing a profit. Philatelic investment was popular during the 1970s but then fell out of favour following a speculative bubble and prices of rare stamps took many years to recover.

Overview
Investing in rare stamps requires a high degree of expertise and can be very risky for the novice. Rare stamps are among the most portable of tangible investments, take up little space but require careful storage as condition is one of the most important factors in determining the value of a stamp. Other tangible investments include art, antiques, precious metals, rare coins and many others that are all termed alternative investments. Interest in stamps as an investment tends to increase when traditional investments are not doing well, causing investors to seek alternatives. The increasing age of the population in western countries has also been credited with a resurgence in interest in stamps.

A British auctioneer warned: "Unless you know what you are doing – and this only comes with many years of experience – it is very difficult to buy sensibly and at the right prices. When you buy on a retail basis, you have to overcome the profit margins the dealer has built into his selling price, plus VAT. When you sell, you also have to overcome dealing charges, whether you sell to an auction house or a dealer. That would consume an awful lot of value. To overcome that mark-up and sell at a profit would require quite a few years of appreciation."

Before investing
A stamp investor should have a good knowledge of: The prospective investor will also benefit from attending stamp clubs, auctions, philatelic shows and having a relationship with a knowledgeable dealer.
 * Classification,
 * Condition grading,
 * Authentication,
 * Handling and storage,
 * The stamp market,
 * Philatelic literature.

Very subtle differences in colour, perforation, overprinting, and the like may be what differentiates a valuable stamp from a common one. The investor not only has to be very knowledgeable of these issues but also know (or at least have a reputable expert available to check) whether alterations have been made to make an inexpensive stamp look like a valuable one, or indeed, whether the stamp itself is simply a counterfeit.

The value of a stamp
The value of a stamp is determined by a number of factors including:
 * The number available on the philatelic market,
 * Demand from collectors inside and outside the country of origin,
 * Condition, a damaged stamp is worth only a fraction of one in fine condition,
 * Thematic appeal,
 * Perceptions as to current or future value,
 * Current events, a news event may temporarily increase values, for instance after the death of Diana, Princess of Wales,
 * The place of purchase or sale. Values vary according to where the transaction takes place. Prices vary from country to country for the same stamp and prices realised at an auction may be different from those charged by a dealer or in a private sale between collectors.
 * Historical significance and/or the rarity of the postmark on a used stamp.

Rare or unusual stamps can be submitted for Philatelic expertisation. The opinions of experts differ and have evolved over time. Conflicting expert opinions, such as on colour shade or whether a stamp has been reperforated, can have a huge effect on a stamp's value.

Buying stamps
There are a number of places where a prospective investor can buy stamps:
 * The internet,
 * Auctions,
 * Stamp dealers,
 * A few specialised stamp investment firms,
 * From a collector in a private sale.

Stamps purchased for investment are usually old classic stamps in fine condition, such as British Victorian stamps or American stamps from before 1900. These may be thought of as the equivalent of buying a blue chip share. Although future prices may vary, as long as there is a hobby of stamp collecting there is likely to be good demand for these stamps.

Stamps typically included in an investment portfolio will be rare and priced in the thousands of dollars or pounds but they will probably not be the greatest rarities as those unique items are typically sold at public auctions and may reach prices approaching or exceeding $1 million US.

Some collectors and investors also try to anticipate future trends and buy low now, this however, is difficult to get right and may take a long time to pay off. Investors may try to identify a developing country with an expanding middle class who may have the time and money to pursue a hobby like stamp collecting as the developing domestic demand may help to force prices up. Recent examples have been India and China.

Some firms are developing collective or mutual funds where money from many investors is pooled and each investor owns shares or units in the fund. The fund then invests the money in stamps.

Size of the market
Unlike stocks and shares, the majority of transactions in the philatelic or stamp market take place informally, by mail order, or in retail environments, and therefore the size of the market is hard to determine. The market is certainly much smaller than the financial markets but it is not trivial. It has been estimated at £5 Billion. The majority of these transactions, however, are likely to be low value items rather than investments. In a 2007 interview, Mike Hall of Stanley Gibbons estimated that "About $1 billion of rare stamps trade annually in the $10 billion-a-year stamp market." The number of collectors worldwide was estimated at 30 million in 2004. In 2009, Adrian Roose of Stanley Gibbons estimated the figure at 48 million including 18 million in China. It is not known how many of these are serious collectors.

Historical data
While there are long term records of retail stamp prices, the first catalogue being prepared in 1862, there is little objective historical data about the past performance of stamps as investments. No long term indices like the Dow Jones or FTSE Index exist, although some figures have been compiled by Stanley Gibbons and Stamp Magazine in the UK.

From 2002, Stanley Gibbons compiled a SG100 Stamp Index based on retail and auction prices for the "top 100 most frequently traded stamps" in the world. The index appeared on the list of the Bloomberg service but was discontinued and given heritage status. In 2004 they also launched an index of 30 rare British stamps. In 2012 the company launched the GB250 Rare Stamp Index 'to provide a broad view of the investment market for Great Britain stamps', listing the type of stamps they would recommend for investment purposes. The GB30 and GB250 indices are listed on both the Bloomberg Professional service and Thomson Reuters. According to Stanley Gibbons, rare stamps have averaged an annual compound return of 10 per cent over the past 50 years, however, it is important to remember that this figure has been calculated using backtesting as stamp price indexes are a recent innovation. In addition, the prices in the indexes are based in part on Stanley Gibbon's own retail price lists and frequently traded but low value stamps have been excluded from the index.

Stamp catalogue prices are not considered reliable as they are nothing more than estimates at the top end and represent a retail selling price at the bottom end of the market. Auction realisations may be more reliable but are difficult to use as the investor has to personally analyse the realisations from many auctions over a long period of time in order to come to any useful conclusions. While most trading in shares is on a recognised stock exchange and takes place transparently in public, that is not the case with stamps where only auction transactions take place in public view.

Selling stamps
There are a number of ways to sell stamps, as there are to buy, and each has its own advantages and disadvantages.
 * Auctions may achieve the highest prices but the costs are also high.
 * Dealers may be able to act quickly or pay cash but are likely to offer a price at least one third below the normal retail sale price for the stamp as the dealer needs to make a profit on the transaction. Some dealers aim to double their money on every transaction.
 * Private sales. Many sales take place between collectors, however, an investor who is not a collector is unlikely to have the personal contacts to secure such a sale.

There is no equivalent of the stock exchange for stamps.

Risks and disadvantages
Investing successfully in stamps requires a high degree of specialised knowledge. This takes time to acquire and there are many pitfalls for the inexperienced investor. Some of the risks and disadvantages are:
 * The return is not guaranteed.
 * The cost of buying is high compared to most other forms of investment.
 * The cost of selling is also relatively high.
 * Purchases may be liable to a sales tax, for instance VAT in the European Union, which the buyer may not be able to reclaim.
 * Stamps may need to be expertised, for a fee, to ensure that they are what they appear to be.
 * As tangible items, stamps may need to be insured and are at risk of physical damage or deterioration.
 * The future market for the sale of philatelic items is uncertain. The demand for philatelic items comes principally from collectors, not investors, and the majority of collectors are aged over 50 in western countries. There are relatively few younger collectors in Europe and North America that would be expected to be the buyers of the future, although anecdotal evidence suggest that may not be the case in India, China and other developing countries.
 * In the longer term, the future existence of postage stamps may be in doubt as people use electronic communications more and more and send fewer letters. If stamps are no longer sold for postage they may cease to be collected and if they are not collected, the vital collector demand that underpins the investment market may disappear.
 * Stamps have little intrinsic value, they do not have the raw material value of a gold coin, they do not represent a share in a business like equities, and they usually lack the enduring visual appeal of a great work of art.
 * Stamp investment is relatively unregulated compared with, for instance, investments in a mutual fund and investors may have little protection if things go wrong.
 * The size of the philatelic market is small compared to the value of the stock market and vulnerable to aggressive buying by speculators, which may distort prices. This happened in the 1970s when a speculative bubble was followed by a collapse in prices.
 * Stamps do not generate any interest or dividends.
 * It may be impossible to determine the current market value of the stamps without selling them.
 * Stamps packaged as "investment portfolios" may be charged at prices higher than their normal market value.
 * There is very little reliable and independent historical information about the performance of stamps as investments.
 * A long-term view is necessary. A quick purchase and sale is unlikely to be profitable.
 * When more traditional investments are doing well, interest in alternative investments like stamps may quickly wane.
 * Special instructions will need to be given to spouses or executors in the event of the owner's incapacity or death as they may be unfamiliar with philatelic items.
 * Stamps may take time to be sold unlike cash, equities or mutual funds which can usually be realised with minimal delay.

Advantages

 * Stamps are not highly correlated with other forms of investment and may therefore represent a valuable diversification within a wider portfolio.
 * Stamps are highly portable stores of wealth and are easily transported over national borders.
 * An ageing population in western countries means that investors approaching retirement may resume childhood hobbies, thus increasing demand.
 * There are millions of enthusiastic stamp collectors around the world creating a global marketplace.
 * There is a finite supply of classic stamps.
 * Stamps are not a financial asset and so may perform better than cash in times of high inflation.
 * As a tangible asset, a stamp cannot go out of business like a company quoted on the stock market.
 * Stamps are a relatively confidential investment. Unless bought at a public auction, ownership is private and there is no public register as there is for many investments in equities.
 * The investor is able to hold and admire his investment, and enjoy its aesthetic aspects.
 * Many stamps have an interesting historical background.

Regulation and investor protection
Stamps purchased for investment do not normally have any special regulatory protection for the purchaser. In the United Kingdom there is no regulation of this area at all from the Financial Conduct Authority. Rules elsewhere may vary. Where investment is collective through a mutual fund there may be some regulation of the activities of the fund depending on its location.

Stamp investment scandals
There have been a number of scandals in this area over the years.

In Ireland in the 1950s, Paul Singer, a Bratislavian Doctor of Philosophy, ran a Ponzi scheme under the name Shanahan Stamp Auctions. The scheme collapsed when a mysterious robbery took place at the company's office on 9 May 1959, the eve of a major auction, when more than £300,000 worth of stamps went missing. Singer was charged with fraud, but was acquitted and vanished.

In the 1970s, the bursting of a speculative bubble left investors unable to realise their investment at the price they had paid. Prices took decades to recover.

In 2006, two Spanish firms Afinsa and Forum Filatelico collapsed and left around 350,000 investors with investments worth as little as 10% of the price they had paid.

In November 2017 it was announced that Stanley Gibbons' stamp investment subsidiary in Guernsey had been placed in administration. The company was reported to have £12.6 million of philatelic stock but over £70m of liabilities, made up of £54m of "buy-back" guarantees, £6.5m owed to its parent company, and £11m of sundry liabilities.

title: Philatelic investment


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Uniqueness quantification

In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" or "∃=1".

For example, the formal statement

 * $$\exists! n \in \mathbb{N}\,(n - 2 = 4)$$

may be read as "there is exactly one natural number $$n$$ such that $$n - 2 =4$$".

Proving uniqueness
The most common technique to prove the unique existence of an object is to first prove the existence of the entity with the desired condition, and then to prove that any two such entities (say, $$a$$ and $$b$$) must be equal to each other (i.e. $$a = b$$).

For example, to show that the equation $$x + 2 = 5$$ has exactly one solution, one would first start by establishing that at least one solution exists, namely 3; the proof of this part is simply the verification that the equation below holds:

 * $$ 3 + 2 = 5. $$

To establish the uniqueness of the solution, one would proceed by assuming that there are two solutions, namely $$a$$ and $$b$$, satisfying $$x + 2 = 5$$. That is,

 * $$ a + 2 = 5\text{ and }b + 2 = 5. $$

Then since equality is a transitive relation,

 * $$ a + 2 = b + 2. $$

Subtracting 2 from both sides then yields
 * $$ a = b. $$

which completes the proof that 3 is the unique solution of $$x + 2 = 5$$.

In general, both existence (there exists at least one object) and uniqueness (there exists at most one object) must be proven, in order to conclude that there exists exactly one object satisfying a said condition.

An alternative way to prove uniqueness is to prove that there exists an object $$a$$ satisfying the condition, and then to prove that every object satisfying the condition must be equal to $$a$$.

Reduction to ordinary existential and universal quantification
Uniqueness quantification can be expressed in terms of the existential and universal quantifiers of predicate logic, by defining the formula $$\exists ! x P(x)$$ to mean

 * $$\exists x\,( P(x) \, \wedge \neg \exists y\,(P(y) \wedge y \ne x)),$$

which is logically equivalent to
 * $$\exists x \, ( P(x) \wedge \forall y\,(P(y) \to y = x)).$$

An equivalent definition that separates the notions of existence and uniqueness into two clauses, at the expense of brevity, is
 * $$\exists x\, P(x) \wedge \forall y\, \forall z\,[(P(y) \wedge P(z)) \to y = z].$$

Another equivalent definition, which has the advantage of brevity, is
 * $$\exists x\,\forall y\,(P(y) \leftrightarrow y = x).$$

Generalizations
The uniqueness quantification can be generalized into counting quantification (or numerical quantification ). This includes both quantification of the form "exactly k objects exist such that …" as well as "infinitely many objects exist such that …" and "only finitely many objects exist such that…". The first of these forms is expressible using ordinary quantifiers, but the latter two cannot be expressed in ordinary first-order logic.

Uniqueness depends on a notion of equality. Loosening this to a coarser equivalence relation yields quantification of uniqueness up to that equivalence (under this framework, regular uniqueness is "uniqueness up to equality"). For example, many concepts in category theory are defined to be unique up to isomorphism.

The exclamation mark $$!$$ can be also used as a separate quantification symbol, so $$(\exists ! x. P(x))\leftrightarrow ((\exists x. P(x))\land (! x. P(x)))$$, where $$(! x. P(x)) := (\forall a \forall b. P(a)\land P(b)\rightarrow a=b)$$. E.g. it can be safely used in the replacement axiom, instead of $$\exists !$$.

title: Uniqueness quantification


text:
Leeward Islands

The Leeward Islands are a group of islands situated where the northeastern Caribbean Sea meets the western Atlantic Ocean. Starting with the Virgin Islands east of Puerto Rico, they extend southeast to Guadeloupe and its dependencies. In English, the term Leeward Islands refers to the northern islands of the Lesser Antilles chain. The more southerly part of this chain, starting with Dominica, is called the Windward Islands. Dominica was originally considered a part of the Leeward Islands, but was transferred from the British Leeward Islands to the British Windward Islands in 1940.

Origin of the name

The name of this island group, Leeward Islands, dates from previous centuries, when sailing ships were the sole form of transportation across the Atlantic Ocean. In sailing terminology, "windward" means towards the source of the wind (upwind), while "leeward" is the opposite direction (downwind). In the West Indies, the prevailing winds, known as the trade winds, blow predominantly out of the northeast. Therefore, a sailing vessel departing from the British Gold Coast and the Gulf of Guinea, driven by the trade winds, would normally first encounter Dominica and Martinique, islands most to windward, in their west-northwesterly heading to the final destinations in the Caribbean, Central America, and Northern America. This location, Dominica and Martinique, becomes the rough dividing line between the Windward Islands and the Leeward Islands.

The early Spanish explorers called Puerto Rico and the islands to the west Sotavento, meaning "leeward". The islands to the south and east of Puerto Rico were then called Islas de Barlovento, meaning "windward islands". When the British gained control of many of the Lesser Antilles, they designated Antigua, Montserrat and the islands to the north as the Leeward Islands. Guadeloupe and the islands to the south were designated as the Windward Islands. Later on, all islands north of Martinique became known as the Leeward Islands. Dominica was transferred to the British Windward Islands in 1940, and is now considered part of the Windward Islands.

However, even in modern usage in languages other than English, notably, Dutch, French, and Spanish, all of the Lesser Antilles from the Virgin Islands to Trinidad and Tobago are known as the Windward Islands (Bovenwindse Eilanden in Dutch, Îles du Vent in French, and Islas de Barlovento in Spanish). The ABC islands and the other islands along the Venezuelan coast, known in English as the Leeward Antilles, are known in languages other than English as an equivalent of the Leeward Islands.

Geography

The islands were created mostly by volcanoes in the Lesser Antilles subduction zone. Some are still active. Notable eruptions occurred in Montserrat in the 1990s and in 2009 to 2010. At 1467 m, the highest point is La Grande Soufrière in Guadeloupe.

History
The Caribs, after whom the Caribbean is named, are believed to have migrated from the Orinoco River area in Venezuela in South America to settle in the Caribbean islands about 1200 AD, according to carbon dating. Over the century leading up to Columbus' arrival in the Caribbean archipelago in 1492, the Caribs mostly displaced the Maipurean-speaking Taínos, who settled the island chains earlier in history, by warfare, extermination and assimilation.

The islands were among the first parts of the Americas to fall under the control of the Spanish Empire. European contact commenced with Christopher Columbus's second voyage; many of the islands' names originate from this period: Montserrat was named in honour of Santa Maria de Montserrat (Our Lady of Montserrat), after the Blessed Virgin of the Monastery of Montserrat, which is located on the Mountain of Montserrat, the national shrine of Catalonia. Mont serrat in Catalan means "saw mountain", referring to the serrated appearance of the mountain range.

British colony of the Leeward Islands
The Leeward Islands became a British colony in 1671, with William Stapleton as its first governor.

Although comparatively much smaller than the surrounding islands in the Caribbean, the Leeward Islands posed the most significant rebellion to the British Stamp Act, though it was decidedly less severe in comparison to that of the mainland North American colonies.

In 1660, there were about 8,000 white settlers and approximately 2,000 African slaves in the Leeward Islands. However, that ratio narrowed over succeeding years. In 1678, there were 10,408 white settlers and 8,449 black slaves. By 1708, there was a huge disparity between the number of white settlers, which had declined to 7,311, while the number of black slaves was estimated at 23,500.

In 1816, the colony as a federation of islands was dissolved, and the individual islands were ruled individually. However, the colony of the Leeward Islands was re-established in 1833.

List of notable islands in the Leeward Islands
There are two countries and eleven territories in the Leeward Islands. From the northwest to the southeast, the main islands are:

 * Virgin Islands
 * Spanish Virgin Islands
 * 🇵🇷 Spanish (Puerto Rican) Islands (Puerto Rico) (United States)
 * Culebra, Puerto Rico (Municipality of Puerto Rico) (United States)
 * Vieques (Municipality of Puerto Rico) (United States)
 * U.S. Virgin Islands (United States)
 * Saint Thomas
 * Saint John
 * Saint Croix
 * Water Island
 * British Virgin Islands (United Kingdom)
 * Jost Van Dyke
 * Tortola
 * Virgin Gorda
 * Anegada
 * 🇦🇮 Anguilla (United Kingdom)
 * Saint Martin Island
 * Saint Martin (France)
 * Sint Maarten (Netherlands)
 * 🇧🇱 Saint Barthélemy (France)
 * Saba (Netherlands)
 * Sint Eustatius (Netherlands)
 * 🇰🇳 Saint Kitts and Nevis
 * Saint Kitts
 * Nevis
 * 🇦🇬 Antigua and Barbuda
 * Barbuda
 * Antigua
 * Redonda — uninhabited
 * 🇲🇸 Montserrat (United Kingdom)
 * 🇬🇵 Guadeloupe (France)
 * La Désirade (dependency of Guadeloupe) — literally 'The Desired', also called La Deseada
 * Îles des Saintes (dependency of Guadeloupe)
 * Marie-Galante (dependency of Guadeloupe)

title: Leeward Islands


text:
Windward Islands

The Windward Islands are the southern, generally larger islands of the Lesser Antilles. Part of the West Indies, they lie south of the Leeward Islands, approximately between latitudes 10° and 16° N and longitudes 60° and 62° W.

The name was also used to refer to a British colony which existed between 1833 and 1960 and originally consisted of the islands of Grenada, Saint Lucia, and Saint Vincent. Today these islands constitute three sovereign states, with the latter now known as Saint Vincent and the Grenadines.

The island of Dominica was traditionally considered a part of the Leeward Islands until 1940 when it was transferred from the British Leeward Islands colony to the British Windward Islands. It now composes the fourth sovereign state in the group.

Barbados (until 1885) and Tobago (until 1889) were also part of the British Windward Islands colony, but are not today regarded as being part of the Windward Islands grouping.

Name and geography
The prevailing trade winds in the West Indies blow east to west. The combination of trans-Atlantic currents and winds that provided the fastest route across the ocean brought sailing ships heading to the New World to the rough dividing line between two groups of islands. Those that fell to windward became the Windward Islands, to leeward the Leeward Islands.

Sailing vessels departing from Africa would first encounter the southeasternmost "Windward" islands of the Lesser Antilles in their west-northwesterly heading to the final destinations in the Caribbean, Central America, and Northern America. The chain of Windward Islands forms a part of the easternmost boundary of the Caribbean Sea.

However, even in modern usage in languages other than English, notably, Dutch, French, and Spanish, all of the Lesser Antilles from the Virgin Islands to Trinidad and Tobago are known as the 'Windward Islands' (Bovenwindse Eilanden in Dutch, Îles du Vent in French, and Islas de Barlovento in Spanish). The ABC Islands and the other islands along the Venezuelan coast, known in English as the Leeward Antilles, are known in languages other than English as the 'Leeward Islands'.

Countries and territories
There are four countries and one territory in the Windward Islands:

 * 🇩🇲 Dominica
 * 🇬🇩 Grenada
 * Carriacou and Petite Martinique
 * 🇲🇶 Martinique (France)
 * 🇱🇨 Saint Lucia
 * 🇻🇨 Saint Vincent and the Grenadines
 * Grenadines
 * Saint Vincent

title: Windward Islands


text:
Uniqueness

Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison, or is remarkable, or unusual. When used in relation to humans, it is often in relation to a person's personality, or some specific characteristics of it, signalling that it is unlike the personality traits that are prevalent in that individual's culture. When the term uniqueness is used in relation to an object, it is often within the realm of product, with the term being a factor used to publicize or market the product in order to make it stand out from other products within the same category.

The notion of American exceptionalism is premised on the uniqueness of the West, particularly its well-defined secularism.

title: Uniqueness