text:
Improper integral

From Wikipedia, the free encyclopedia
  (Redirected from Improper Riemann integral)
Jump to navigation Jump to search
An improper integral of the first kind. The integral may need to be defined on an unbounded domain.
An improper Riemann integral of the second kind. The integral may fail to exist because of a vertical asymptote in the function.

In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, , , or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration.

Specifically, an improper integral is a limit of the form:

in which one takes a limit in one or the other (or sometimes both) endpoints (Apostol 1967, §10.23).

By abuse of notation, improper integrals are often written symbolically just like standard definite integrals, perhaps with infinity among the limits of integration. When the definite integral exists (in the sense of either the Riemann integral or the more advanced Lebesgue integral), this ambiguity is resolved as both the proper and improper integral will coincide in value.

Often one is able to compute values for improper integrals, even when the function is not integrable in the conventional sense (as a Riemann integral, for instance) because of a singularity in the function or because one of the bounds of integration is infinite.

The original definition of the Riemann integral does not apply to a function such as on the interval [1, ∞), because in this case the domain of integration is unbounded. However, the Riemann integral can often be extended by continuity, by defining the improper integral instead as a limit

The narrow definition of the Riemann integral also does not cover the function on the interval [0, 1]. The problem here is that the integrand is unbounded in the domain of integration (the definition requires that both the domain of integration and the integrand be bounded). However, the improper integral does exist if understood as the limit

The improper integral

has unbounded intervals for both domain and range.

Sometimes integrals may have two singularities where they are improper. Consider, for example, the function 1/((x + 1)x) integrated from 0 to (shown right). At the lower bound, as x goes to 0 the function goes to , and the upper bound is itself , though the function goes to 0. Thus this is a doubly improper integral. Integrated, say, from 1 to 3, an ordinary Riemann sum suffices to produce a result of π/6. To integrate from 1 to , a Riemann sum is not possible. However, any finite upper bound, say t (with t > 1), gives a well-defined result, 2 arctan(t) − π/2. This has a finite limit as t goes to infinity, namely π/2. Similarly, the integral from 1/3 to 1 allows a Riemann sum as well, coincidentally again producing π/6. Replacing 1/3 by an arbitrary positive value s (with s < 1) is equally safe, giving π/2 − 2 arctan(s). This, too, has a finite limit as s goes to zero, namely π/2. Combining the limits of the two fragments, the result of this improper integral is

This process does not guarantee success; a limit might fail to exist, or might be infinite. For example, over the bounded interval from 0 to 1 the integral of 1/x does not converge; and over the unbounded interval from 1 to the integral of 1/x does not converge.

The improper integral

converges, since both left and right limits exist, though the integrand is unbounded near an interior point.

It might also happen that an integrand is unbounded near an interior point, in which case the integral must be split at that point. For the integral as a whole to converge, the limit integrals on both sides must exist and must be bounded. For example:

But the similar integral

cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see Cauchy principal value.)

Convergence of the integral[edit]

An improper integral converges if the limit defining it exists. Thus for example one says that the improper integral

exists and is equal to L if the integrals under the limit exist for all sufficiently large t, and the value of the limit is equal to L.

It is also possible for an improper integral to diverge to infinity. In that case, one may assign the value of ∞ (or -∞) to the integral. For instance

However, other improper integrals may simply diverge in no particular direction, such as

which does not exist, even as an extended real number. This is called divergence by oscillation.

A limitation of the technique of improper integration is that the limit must be taken with respect to one endpoint at a time. Thus, for instance, an improper integral of the form

can be defined by taking two separate limits; to wit

provided the double limit is finite. It can also be defined as a pair of distinct improper integrals of the first kind:

where c is any convenient point at which to start the integration. This definition also applies when one of these integrals is infinite, or both if they have the same sign.

An example of an improper integral where both endpoints are infinite is the Gaussian integral . An example which evaluates to infinity is . But one cannot even define other integrals of this kind unambiguously, such as , since the double limit is infinite and the two-integral method

yields . In this case, one can however define an improper integral in the sense of Cauchy principal value:

The questions one must address in determining an improper integral are:

  • Does the limit exist?
  • Can the limit be computed?

The first question is an issue of mathematical analysis. The second one can be addressed by calculus techniques, but also in some cases by contour integration, Fourier transforms and other more advanced methods.

Types of integrals[edit]

There is more than one theory of integration. From the point of view of calculus, the Riemann integral theory is usually assumed as the default theory. In using improper integrals, it can matter which integration theory is in play.

  • For the Riemann integral (or the Darboux integral, which is equivalent to it), improper integration is necessary both for unbounded intervals (since one cannot divide the interval into finitely many subintervals of finite length) and for unbounded functions with finite integral (since, supposing it is unbounded above, then the upper integral will be infinite, but the lower integral will be finite).
  • The Lebesgue integral deals differently with unbounded domains and unbounded functions, so that often an integral which only exists as an improper Riemann integral will exist as a (proper) Lebesgue integral, such as . On the other hand, there are also integrals that have an improper Riemann integral but do not have a (proper) Lebesgue integral, such as . The Lebesgue theory does not see this as a deficiency: from the point of view of measure theory, and cannot be defined satisfactorily. In some situations, however, it may be convenient to employ improper Lebesgue integrals as is the case, for instance, when defining the Cauchy principal value. The Lebesgue integral is more or less essential in the theoretical treatment of the Fourier transform, with pervasive use of integrals over the whole real line.
  • For the Henstock–Kurzweil integral, improper integration is not necessary, and this is seen as a strength of the theory: it encompasses all Lebesgue integrable and improper Riemann integrable functions.

Improper Riemann integrals and Lebesgue integrals[edit]

Figure 1
Figure 2

In some cases, the integral

can be defined as an integral (a Lebesgue integral, for instance) without reference to the limit

but cannot otherwise be conveniently computed. This often happens when the function f being integrated from a to c has a vertical asymptote at c, or if c = ∞ (see Figures 1 and 2). In such cases, the improper Riemann integral allows one to calculate the Lebesgue integral of the function. Specifically, the following theorem holds (Apostol 1974, Theorem 10.33):

  • If a function f is Riemann integrable on [a,b] for every b ≥ a, and the partial integrals
are bounded as b → ∞, then the improper Riemann integrals
both exist. Furthermore, f is Lebesgue integrable on [a, ∞), and its Lebesgue integral is equal to its improper Riemann integral.

For example, the integral

can be interpreted alternatively as the improper integral

or it may be interpreted instead as a Lebesgue integral over the set (0, ∞). Since both of these kinds of integral agree, one is free to choose the first method to calculate the value of the integral, even if one ultimately wishes to regard it as a Lebesgue integral. Thus improper integrals are clearly useful tools for obtaining the actual values of integrals.

In other cases, however, a Lebesgue integral between finite endpoints may not even be defined, because the integrals of the positive and negative parts of f are both infinite, but the improper Riemann integral may still exist. Such cases are "properly improper" integrals, i.e. their values cannot be defined except as such limits. For example,

cannot be interpreted as a Lebesgue integral, since

But is nevertheless integrable between any two finite endpoints, and its integral between 0 and ∞ is usually understood as the limit of the integral:

One can speak of the singularities of an improper integral, meaning those points of the extended real number line at which limits are used.

Cauchy principal value[edit]

Consider the difference in values of two limits:

The former is the Cauchy principal value of the otherwise ill-defined expression

Similarly, we have

The former is the principal value of the otherwise ill-defined expression

All of the above limits are cases of the indeterminate form ∞ − ∞.

These pathologies do not affect "Lebesgue-integrable" functions, that is, functions the integrals of whose absolute values are finite.

An improper integral may diverge in the sense that the limit defining it may not exist. In this case, there are more sophisticated definitions of the limit which can produce a convergent value for the improper integral. These are called summability methods.

One summability method, popular in Fourier analysis, is that of Cesàro summation. The integral

is Cesàro summable (C, α) if

exists and is finite (Titchmarsh 1948, §1.15). The value of this limit, should it exist, is the (C, α) sum of the integral.

An integral is (C, 0) summable precisely when it exists as an improper integral. However, there are integrals which are (C, α) summable for α > 0 which fail to converge as improper integrals (in the sense of Riemann or Lebesgue). One example is the integral

which fails to exist as an improper integral, but is (C,α) summable for every α > 0. This is an integral version of Grandi's series.

Multivariable improper integrals[edit]

The improper integral can also be defined for functions of several variables. The definition is slightly different, depending on whether one requires integrating over an unbounded domain, such as , or is integrating a function with singularities, like .

Improper integrals over arbitrary domains[edit]

If is a non-negative function that is Riemann integrable over every compact cube of the form , for , then the improper integral of f over is defined to be the limit

provided it exists.

A function on an arbitrary domain A in is extended to a function on by zero outside of A:

The Riemann integral of a function over a bounded domain A is then defined as the integral of the extended function over a cube containing A:

More generally, if A is unbounded, then the improper Riemann integral over an arbitrary domain in is defined as the limit:

Improper integrals with singularities[edit]

If f is a non-negative function which is unbounded in a domain A, then the improper integral of f is defined by truncating f at some cutoff M, integrating the resulting function, and then taking the limit as M tends to infinity. That is for , set . Then define

provided this limit exists.

Functions with both positive and negative values[edit]

These definitions apply for functions that are non-negative. A more general function f can be decomposed as a difference of its positive part and negative part , so

with and both non-negative functions. The function f has an improper Riemann integral if each of and has one, in which case the value of that improper integral is defined by

In order to exist in this sense, the improper integral necessarily converges absolutely, since

  1. ^ Cooper 2005, p. 538: "We need to make this stronger definition of convergence in terms of |f(x)| because cancellation in the integrals can occur in so many different ways in higher dimensions."
  2. ^ Ghorpade & Limaye 2010, p. 448: "The relevant notion here is that of unconditional convergence." ... "In fact, for improper integrals of such functions, unconditional convergence turns out to be equivalent to absolute convergence."

  • Apostol, T (1974), Mathematical analysis, Addison-Wesley, ISBN 978-0-201-00288-1.
  • Apostol, T (1967), Calculus, Vol. 1 (2nd ed.), Jon Wiley & Sons.
  • Autar Kaw, Egwu Kalu (2008), Numerical Methods with Applications (1st ed.),
  • Titchmarsh, E (1948), Introduction to the theory of Fourier integrals (2nd ed.), New York, N.Y.: Chelsea Pub. Co. (published 1986), ISBN 978-0-8284-0324-5.
  • Cooper, Jeffery (2005), Working analysis, Gulf Professional
  • Ghorpade, Sudhir; Limaye, Balmohan (2010), A course in multivariable calculus and analysis, Springer

External links[edit]


Content-Type: application/http; msgtype=response

WARC-Identified-Payload-Type: text/html

WARC-Type: response

text:
Stamp collecting

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Le Philatéliste by François Barraud (1929).

Stamp collecting is the collecting of postage stamps and related objects. It is related to philately, which is the study of stamps. It has been one of the world's most popular hobbies since the late nineteenth century with the rapid growth of the postal service,[1] as a never-ending stream of new stamps was produced by countries that sought to advertise their distinctiveness through their stamps.

A young collector

Stamp collecting is generally accepted as one of the areas that make up the wider subject of philately, which is the study of stamps. A philatelist may, but does not have to, collect stamps. It is not uncommon for the term philatelist to be used to mean a stamp collector. Many casual stamp collectors accumulate stamps for sheer enjoyment and relaxation without worrying about the tiny details. The creation of a large or comprehensive collection, however, generally requires some philatelic knowledge and will usually contain areas of philatelic studies.

Postage stamps are often collected for their historical value and geographical aspects and also for the many subjects depicted on them, ranging from ships, horses, and birds to kings, queens and presidents.[2]

Sales of postage stamps are an important source of income for some countries whose stamp issues may exceed their postal needs, but have designs that appeal to many stamp collectors.

Queen Victoria's profile was a staple on 19th century stamps of the British Empire, shown here on a half-penny stamp of the Falkland Islands, 1891.

It has been suggested that John Bourke, Receiver General of Stamp Dues in Ireland, was the first collector. In 1774 he assembled a book of the existing embossed revenue stamps, ranging in value from 6 pounds to half a penny, as well as the hand stamped charge marks that were used with them. His collection is preserved in the Royal Irish Academy, Dublin.[3]

Postage stamp collecting began at the same time that stamps were first issued, and by 1860 thousands of collectors and stamp dealers were appearing around the world as this new study and hobby spread across Europe, European colonies, the United States and other parts of the world.

The first postage stamp, the Penny Black, was issued by Britain in May 1840 and pictured a young Queen Victoria. It was produced without perforations (imperforate) and consequently had to be cut from the sheet with scissors in order to be used. While unused examples of the Penny Black are quite scarce, used examples are quite common, and may be purchased for $20 to $200, depending upon condition.

People started to collect stamps almost immediately. One of the earliest and most notable was John Edward Gray. In 1862, Gray stated that he "began to collect postage stamps shortly after the system was established and before it had become a rage".[4]

Female stamp collectors date from the earliest days of postage stamp collecting. One of the earliest was Adelaide Lucy Fenton who wrote articles in the 1860s for the journal The Philatelist under the name Herbert Camoens.[5]

As the hobby and study of stamps began to grow, stamp albums and stamp related literature began to surface, and by the early 1880s publishers like Stanley Gibbons made a business out of this advent.

Children and teenagers were early collectors of stamps in the 1860s and 1870s. Many adults dismissed it as a childish pursuit but later many of those same collectors, as adults, began to systematically study the available postage stamps and publish books about them. Some stamps, such as the triangular issues of the Cape of Good Hope, have become legendary.

By the early 2000's stamp collecting was seen to be in decline as the digital world surpassed traditional hobbies even though by 2013, The Wall Street Journal estimated the global number of stamp collectors was around 60 million.[6]

Modern Collecting[edit]

In 2016 the American Philatelic Society released a paper on the Future of Philately. It suggested that many collectors were 'hidden.' This meant that people would collect at home but not necessarily associate themselves with a traditional stamp club. A challenge for these clubs was to connect with this hidden community. One of the major players in this role was Graham Beck of Exploring Stamps. Exploring Stamps was a three season YouTube channel exploring stamps in a modern light. It was a break-through series that connected with many hidden collectors due to the quality and freshness of the content. Beck's work spawned a number of new quality channels including Stamps with Sara, Ted Talks Stamps and Philately Dad.

In April, 2020 James Gavin of the Rhodesian Study Circle presented the Future of Philately talk with the American Philatelic Society's Stamp Chat series on YouTube. In this talk, the concept of Traditional and Modern collectors was discussed. The Traditional collector (those born prior to 1970s) was outlined as those who preferred paper publications, attending stamp meetings in person and were more focused on the technical aspects of the hobby. Modern collectors (those born after the 1970s) on the other hand, interacted through social media platforms, relied on digital publications, and were more focused on the presentation and exploration of the subjects of stamps. The purpose of the talk was to help stamp clubs modernise their websites from relics of the 1990's to 2020 standards, removing old practices such as paper application forms, outdated HTML designs and dated content.

Those clubs that have begun to modernise the hobby have seen growth in their membership. The Korea Stamp Society, on the brink of closure, went to a completely digital format and moved to free membership. They were able to see massive growth in their membership as a result. Following closely was the Spanish Philatelic Society, which also went to free membership. The Rhodesian Study Circle began updating its website in 2015 and established itself as a leader for how societies can deliver a new approach to collecting. They differ by mixing history with stamps. This approach was recommend in the American Philatelic Society's Future of Philately paper based on information from the Smithsonian National Postal Museum.

During the COVID-19 lockdown, many societies moved to Zoom to interact with their members. The American Philatelic Society was the most active during this period with their Stamp Chat series, often inviting influential philatelists to make presentations on a variety of topics. In October 2020, Stampex International will hold one of the first virtual stamp shows, with exhibitors and dealers making virtual appearances.

In addition, Instagram and Twitter saw a boom in collectors interacting online. Suzanne Rae of Art Stamped is one of the most prolific social media influencers in this area. Rae specialises in creating art with stamps and was noted in The Guardian's article Post modern: why millennials have fallen in love with stamp collecting. Both Instagram and Twitter have been critical in connecting with a female audience, breaking the mold of the male dominated hobby.

A few basic items of equipment are recommended for proper stamp collection. Stamp tongs help to handle stamps safely, a magnifying glass helps in viewing fine details and an album is a convenient way to store stamps. The stamps need to be attached to the pages of the album in some way, and stamp hinges are a cheap and simple way to do this. However, hinging stamps can damage them, thus reducing their value; today many collectors prefer more expensive hingeless mounts. Issued in various sizes, these are clear, chemically neutral thin plastic holders that open to receive stamps and are gummed on the back so that they stick to album pages. Another alternative is a stockbook, where the stamps drop into clear pockets without the need for a mount. Stamps should be stored away from light, heat and moisture or they will be damaged.

Stamps can be displayed according to the collector's wishes, by country, topic, or even by size, which can create a display pleasing to the eye. There are no rules and it is entirely a matter for the individual collector to decide. Albums can be commercially purchased, downloaded or created by the collector. In the latter cases, using acid free paper provides better long-term stamp protection.

Acquiring stamps[edit]

Many collectors ask their family and friends to save stamps for them from their mail. Although the stamps received by major businesses and those kept by elderly relatives may be of international and historical interest, the stamps received from family members are often of the definitive sort. Definitives seem mundane but, considering their variety of colours, watermarks, paper differences, perforations and printing errors, they can fill many pages in a collection.[7] Introducing either variety or specific focus to a collection can require the purchasing of stamps, either from a dealer or online. Online stamp collector clubs often contain a platform for buying/selling and trading.[8] Large numbers of relatively recent stamps, often still attached to fragments or envelopes, may be obtained cheaply and easily. Rare and old stamps can also be obtained, but these can be very expensive.

Duplicate stamps are those a collector already has and are not required, therefore, to fill a gap in a collection. Duplicate stamps can be sold or traded, so they are an important medium of exchange among collectors.

Many dealers sell stamps through the Internet while others have neighborhood shops which are among the best resources for beginning and intermediate collectors. Some dealers also jointly set up week-end stamp markets called "bourses" that move around a region from week to week. They also meet collectors at regional exhibitions and stamp shows.

Collecting specialties[edit]

A worldwide collection would be enormous, running to thousands of volumes, and would be incredibly expensive to acquire. Many consider that Count Philipp von Ferrary's collection at the beginning of the 20th century was the most complete ever formed. Many collectors limit their collecting to particular countries, certain time periods or particular subjects (called "topicals") like birds or aircraft.

The first UK commemorative stamps, 1924

Some of the more popular collecting areas include:

  • Postage stamps – particular countries and/or time periods
    • Airmail stamps – stamps may be required for airmail, which is typically more expensive and has special postage rates.
    • Commemorative stamps – stamps to commemorate events, anniversaries, etc., on sale for a limited time.
    • Definitive stamps – the most common type of stamps
    • Postage due stamps are special stamps applied by a post office to mail bearing insufficient postage. The stamps were issued in several denominations to make up different amounts due.
Stamp depicting birds
  • Topical stamp collecting – many collectors choose to organize their philatelic collection on the theme of the stamps, covers, or postmarks. Popular topical themes are animals, dogs, cats, butterflies, birds, flowers, art, sports, Olympics, maps, Disney, scouting, space, ships, Americana (topics relating to the US), stamps on stamps, famous people, chess, Chinese new year, and many others.
  • Postal stationery – includes government-issued postal cards, aerograms, letter card, wrappers, envelopes, etc., that have an imprinted stamp.
  • Sheets
    • Sheetlets – this is a format that is now issued regularly by postal administrations. Instead of issuing stamps in large sheets of 40, 100 or even 200 stamps, smaller sheetlets with 20 to 24 stamps are issued with a large selvedge area which may incorporate part of the stamp design or theme.
    • Souvenir sheets – many postal services sometimes release stamps in a format that look like a sheet with a big picture. Various parts of the picture can be torn out and used as postage stamps. See example with 10 stamps in one picture. (Souvenir sheets should be distinguished from souvenir cards, which are souvenirs of a philatelic meeting or exhibition but are not valid for postage.)
    • Miniature sheet – is very similar to a souvenir sheet, being in a sheetlet with a single or a number of stamps embedded in it.
    • Corner blocks or plate blocks – compose a block of stamps from one of the four corners of the stamp sheet. Collectors usually opt for a block of four stamps, complete with the selvage area which will sometimes have the printing details on it.
    • Coil strips – Pairs or more of stamps from rolls, premium ones showing the plate number or a coil line pair which shows the seam between the edges of the plate.
  • Revenue stamps – stamps issued to pay taxes.
    • Federal Duck Stamps (stamps for duck hunting licenses, mainly U.S. with some other countries such as Canada and New Zealand)
  • First day cover (FDCs) – envelopes with stamps attached and canceled on the first day that the stamp was issued. Most modern FDCs bear designs, called "cachets", related to the theme of the stamp issued.
  • Maximum cards – these are postcards where the stamp is on the same side as the picture and they have a close connection.
  • Souvenir pages – with first day canceled stamps on a page describing all design, printing and issuing details. These are similar to first day covers except that they are issued as printed sheets of paper instead of envelopes, and the specification of the stamp is printed by the official source. See picture of first souvenir page in the US.
  • Cinderella stamps – stamp-like labels that are not valid for postage.
  • Postmarks or postal markings in general.

A large stamp show of Stampex 2011 containing a bourse at which collectors and dealers meet.

There are thousands of organizations for collectors: local stamp clubs, special-interest groups, and national organizations. Most nations have a national collectors' organization, including the American Philatelic Society (APS) in the United States; the Royal Philatelic Society London and Philatelic Traders Society in United Kingdom; and the Royal Philatelic Society of Canada in Canada. The Internet has greatly expanded the availability of information and made it easier to obtain stamps and other philatelic material. The American Topical Association is now a part of the APS and promotes thematic collecting as well as encouraging sub-groups of numerous topics.

Stamp clubs and philatelic societies can add a social aspect to stamp collecting and provide a forum where novices can meet experienced collectors. Although such organizations are often advertised in stamp magazines and online, the relatively small number of collectors – especially outside urban areas – means that a club may be difficult to set up and sustain. The Internet partially solves this problem, as the association of collectors online is not limited by geographical distance. For this reason, many highly specific stamp clubs have been established on the Web, with international membership.[9]

Organizations such as the Cinderella Stamp Club (UK) retain hundreds of members interested in a specific aspect of collecting. Social organizations, such as the Lions Club and Rotary International, have also formed stamp collecting groups specific to those stamps that are issued from many countries worldwide that display the organization's logo.

Rare stamps[edit]

Rare stamps are often old and many have interesting stories attached to them. Some include:

Some of the most valuable stamps in the world[edit]

Stamp catalogues are the primary tool used by serious collectors to organize their collections, and for the identification and valuation of stamps. Most stamp shops have stamp catalogues available for purchase. A few catalogues are offered online, either free or for a fee. There are hundreds of different catalogues, most specializing in particular countries or periods. Collector clubs tend to provide free catalogues to their members.

Notable collectors[edit]

John Lennon's stamp album

The stamp collection assembled by French-Austrian aristocrat Philipp von Ferrary (1850–1917) at the beginning of the 20th century is widely considered the most complete stamp collection ever formed (or likely to be formed). It included, for example, all of the rare stamps described above that had been issued by 1917. However, as Ferrary was an Austrian citizen, the collection was broken up and sold by the French government after the First World War, as war reparations. A close rival was Thomas Tapling (1855–1891), whose Tapling Collection was donated to the British Museum.

Several European monarchs were keen stamp collectors, including King George V of the United Kingdom and King Carol II of Romania. King George V possessed one of the most valuable stamp collections in the world and became President of the Royal Philatelic Society. His collection was passed on to Queen Elizabeth II who, while not a serious philatelist, has a collection of British and Commonwealth first day covers which she started in 1952.[12]

U.S. President Franklin Delano Roosevelt was a stamp collector; he designed several American commemorative stamps during his term.[13] Late in life Ayn Rand renewed her childhood interest in stamps and became an enthusiastic collector.[14] Several entertainment and sport personalities have been known to be collectors. Freddie Mercury, lead singer of the band Queen, collected stamps as a child. His childhood stamp album is in the collection of the British Postal Museum & Archive.[15] John Lennon of The Beatles was a childhood stamp collector. His stamp album is held by the National Postal Museum.[16]

Former world chess champion Anatoly Karpov has amassed a huge stamp collection over the decades, led by stamps from Belgium and Belgian Congo, that has been estimated to be worth $15 million.[17]

See also[edit]

References and sources[edit]

  1. ^ Vickers, Marcia (15 December 1996). "Delivery Isn't Guaranteed, but Stamps Are Turning Profits". The New York Times. Retrieved 10 December 2009.
  2. ^ "What is stamp collecting?". Learn About Stamps. 2010. Retrieved 4 October 2010.
  3. ^ "The origin of stamp collecting in America, Part 1". Linn's Stamp News. 17 October 2014. Retrieved 15 July 2017.
  4. ^ Gray, John Edward, A Hand Catalogue of Postage Stamps for the use of the Collector, 1862, Robert Hardwicke, page viii
  5. ^ The American Philatelist, Volume 75, p.32
  6. ^ "Graph: The Wide World of Stamps". WSJ.MONEY. 2013. Archived from the original on 19 November 2019. Retrieved 24 September 2013.
  7. ^ Klug, Janet (2012). "A limited budget can still mean unlimited fun". Linns Stamp News. Retrieved 11 May 2012.
  8. ^ "Stamps on Colnect". Retrieved 30 March 2019.
  9. ^ "Stamp Clubs And Philatelic Societies". Retrieved 25 October 2008.
  10. ^ "Rare stamp sells for record price in Geneva auction". BBC News. 22 May 2010. Retrieved 24 May 2010.
  11. ^ a b Feldman, David (1993), Prices Realised for Public Auction November 1-6 November 1993, Prces Realised, Zurich: David Feldman, p. 1
  12. ^ Courtney, Nicholas. The Queen's Stamps – The Authorised History of the Royal Philatelic Collection. London: Methuen, 2004, p.303. ISBN 0-413-77228-4
  13. ^ "FDR–Stamp Collecting President". National Postal Museum. Retrieved 7 February 2010.
  14. ^ "Why I Like Stamp Collecting", Minkus Stamp Journal, 1971 [1], cf. "The new Ayn Rand companion", Mimi Reisel Gladstein, p. 130, Greenwood Publishing Group, 1999, ISBN 0-313-30321-5
  15. ^ "Famous Philatelists". The British Postal Museum & Archive. Retrieved 24 July 2009.
  16. ^ "John Lennon – The Lost Album". National Postal Museum. Retrieved 24 July 2009.
  17. ^ Anatoly Karpov Belgian stamp collection offered at David Feldman,, 2015.

Further reading[edit]

  • Klug, Janet. Guide to Stamp Collecting. Smithsonian/Collins. ISBN 0061341398 (Children's/Young Adult)
  • Watson, James. How to start Stamp Collecting. London; Stanley Gibbons. ISBN 0852596170 (Adult beginner)
  • Phillips, Stanley. Stamp Collecting: A guide to modern philately. Revised edition. London: Stanley Gibbons, 1983. ISBN 0-85259-047-4 (Adult beginner, more detailed but a little out of date)

External links[edit]

warcinfo:
isPartOf: CC-MAIN-2020-34
publisher: Common Crawl
description: Wide crawl of the web for August 2020
operator: Common Crawl Admin (info@commoncrawl.org)
hostname: ip-10-67-67-213.ec2.internal
software: Apache Nutch 1.17 (modified, https://github.com/commoncrawl/nutch/)
robots: checked via crawler-commons 1.2-SNAPSHOT (https://github.com/crawler-commons/crawler-commons)
format: WARC File Format 1.1
conformsTo: http://iipc.github.io/warc-specifications/specifications/warc-format/warc-1.1/