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Topic: Peano


  
  Peano axioms - Wikipedia, the free encyclopedia
In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic).
This theory constitutes a fundamental formalism for arithmetic, and the Peano axioms form a basis for the formalisation of stronger theories, such as second-order arithmetic.
Peano arithmetic raises a number of metamathematical and philosophical issues, primarily involving questions of consistency and completeness.
en.wikipedia.org /wiki/Peano_axioms   (1864 words)

  
 Giuseppe Peano - Wikipedia, the free encyclopedia
Born on a farm near the village of Spinetta in Piedmont, Italy, Peano enrolled at the nearby University of Turin in 1876.
Peano played a key role in the axiomatization of mathematics and was a leading pioneer in the development of mathematical logic.
After his mother died in 1910, Peano divided his time between teaching, working on texts aimed for secondary schooling including a dictionary of mathematics, and developing and promoting his and other artificial languages, becoming a revered member of the international auxiliary language movement.
en.wikipedia.org /wiki/Peano   (1186 words)

  
 Peano   (Site not responding. Last check: 2007-10-21)
Peano was appointed assistant to Genocchi for 1881-82 and it was in 1882 that Peano made a discovery that would be typical of his style for many years, he discovered an error in a standard definition.
Peano was about to teach the students about the area of a curved surface when he realised that the definition in Serret's book, which was the standard text for the course, was incorrect.
Peano received his qualification to be a university professor in December 1884 and he continued to teach further courses, some for Genocchi whose health had not recovered sufficiently to allow him to return to the University.
www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Peano.html   (2345 words)

  
 Guiseppe Peano
Giuseppe Peano was an Italian mathematician and logician.
Peano is well-known for his work Formulario Mathematico in which he formulated the set of nonnegative integers on the basis of three undefined terms: 0 (zero), number and successor.
Peano devised a postulate system from which the entire arithmetic of the natural numbers can be derived.
www.engr.iupui.edu /~orr/webpages/cpt120/mathbios/gepeano.htm   (1053 words)

  
 Peano
Peano's first work on logic (1888) showed that the calculus of classes and the propositional calculus were, up to notation, the same.
Peano skipped over any attempt to define the natural numbers in logic, thus bypassing certain philosophical issues that mathematicians tend to view as being incapable of precise formulation, and concentrated on the manipulation of symbols, something mathematicians find most agreeable.
Peano's response was that the ability to give brief and precise form to mathematical theorems would make the importance of his work clear.
www.math.uwaterloo.ca /~snburris/htdocs/scav/peano/peano.html   (650 words)

  
 Peano curve   (Site not responding. Last check: 2007-10-21)
In 1880 the Italian logician Giuseppe Peano (1858-1932) constructed the Peano curve, a base motif fractal which uses a line segment as base.
The fractal dimension of the Peano curve is equal to 2
Later the name of Peano curve was given to any fractal whose fractal dimension is equal to 2.
www.2dcurves.com /fractal/fractalpe.html   (106 words)

  
 Σελίδες για τον Bertrand Russell   (Site not responding. Last check: 2007-10-21)
Peano was the founder of symbolic logic and his interests centred on the foundations of mathematics and on the development of a formal logical language.
Peano introduced the basic elements of geometric calculus and gave new definitions for the length of an arc and for the area of a curved surface.
Although Peano is a founder of mathematical logic, the German mathematical philosopher Gottlob Frege (1848-1925) is considered the father of mathematical logic.
sfr.ee.teiath.gr /htmSELIDES/Russell/Peano.htm   (275 words)

  
 Giuseppe Peano - Wikipedia, the free encyclopedia   (Site not responding. Last check: 2007-10-21)
Giuseppe Peano (August 27, 1858 – April 20, 1932) was an Italian mathematician and philosopher best known for his contributions to set theory.
Born on a farm near the village of in Piedmont, Italy, Peano enrolled at the nearby University of Turin in 1876.
In 1886, Peano starting teaching concurrently at the, and was promoted to Professor First Class in 1889.
www.eastcleveland.us /project/wikipedia/index.php/Giuseppe_Peano   (1166 words)

  
 Peano axioms -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-21)
This theory constitutes a fundamental formalism for (The branch of pure mathematics dealing with the theory of numerical calculations) arithmetic, and the Peano axioms form a basis for the formalisation of stronger theories, such as second-order arithmetic.
Using the Peano axioms, one can construct many of the most important (Any notation for the representation of numbers) number systems and structures of modern mathematics.
The Peano axioms may be interpreted in the general context of (Click link for more info and facts about category theory) category theory.
www.absoluteastronomy.com /encyclopedia/p/pe/peano_axioms.htm   (2109 words)

  
 Peano, Giuseppe (1858-1932)
Giuseppe Peano was one of the pioneers in mathematical logic and axiomatization of mathematics.
Giuseppe Peano was born to a poor farming family in Spinetta, Italy, on August 27, 1858.
Peano's greatest contributions, however, were in the fields of axiomatization of mathematics and mathematical logic.
www.geocities.com /Athens/Olympus/2948/pgolba.html   (881 words)

  
 Giuseppe Peano   (Site not responding. Last check: 2007-10-21)
Peano continued to study pure mathematics, despite the fact that the rest of his classmates had switched to engineering after three years.
Peano was appointed assistant to Genocchi, who quite old and in relatively poor health, and so Peano took over some of his teaching.
Peano received his qualification to be a university professor in December 1884, and he continued to teach further courses, including some for Genocchi.
www.stetson.edu /%7Eefriedma/periodictable/html/Pd.html   (643 words)

  
 PlanetMath: Peano arithmetic   (Site not responding. Last check: 2007-10-21)
Peano's axioms are a definition of the set of natural numbers, denoted
Peano arithmetic consists of statements derived via these axioms.
This is version 5 of Peano arithmetic, born on 2002-03-10, modified 2004-02-25.
planetmath.org /encyclopedia/PeanoArithmetic.html   (140 words)

  
 Encyclopedia: Giuseppe Peano
Despite all this, Peano was also famous for being an accomplished wanker.
Space-filling curves or Peano curves are curves, first described by Giuseppe Peano, whose ranges contain the entire 2-dimensional unit square (or the 3-dimensional unit cube).
An artificial or constructed language (known colloquially as a conlang among aficionados), is a language whose vocabulary and grammar were specifically devised by an individual or small group, rather than having naturally evolved as part of a culture as with natural languages.
www.nationmaster.com /encyclopedia/Giuseppe-Peano   (2621 words)

  
 Giuseppe Peano   (Site not responding. Last check: 2007-10-21)
Peano Giuseppe (1858-1932) nasceva al ferma Tetto Galant presso 5km de Cuneo, Italia, in 1858.
Peano es alsi cognoscite como le creator de Latino sine Flexione, plus tarde appellate Interlingua, un lingua artificial.
Peano compilava un Vocabulario de Interlingua]] in 1915 e ille esseva pro un tempore le presidente del Academia pro Interlingua (originalmente nominate le Kadem Volapuka'').
interlingua.encyclopedia.st /Giuseppe_Peano   (601 words)

  
 Biogragpy of Giuseppe Peano </a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> He realized that <b>Peano</b> Giuseppe was a very talented child and he took him to <a href="/topics/Turin" title="Turin" class=fl>Turin</a> in 1870 for his secondary schooling, and to prepare him for university studies. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Of course this was so against <b>Peano's</b> rigorous approach to <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a> that he argued strongly:I believe it new in the history of <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a> that authors knowingly use in their research propositions for which exceptions are known, or for which they have no proof... </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> It was a triumph for <b>Peano</b> and Russell.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.andrews.edu /~calkins/math/biograph/biopeano.htm</font>   (1680 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>Search Results for Peano</u>   <i>(Site not responding. Last check: 2007-10-21)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> was appointed as assistant to D'Ovidio at <a href="/topics/Turin" title="Turin" class=fl>Turin</a> in <a href="/topics/1880" title="1880" class=fl>1880</a> and Vailati was among the first group of students for whom he had to care. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano's</b> 1888 book Calcolo geometrico secondo l'Ausdehnungslehre di H. It was many years before this notation was to become accepted, in fact <b>Peano's</b> book seems to have had very little influence for many years. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> goes on to state the existence of a zero object 0 and says that 0a = 0, that a - b means a + (-b) and states it is easy to show that a - a = 0 and 0 + a = a.</td></tr> <tr><td></td><td colspan=2><font color=gray>www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=Peano&CONTEXT=1</font>   (3040 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.cut-the-knot.org/Curriculum/Geometry/Peano.shtml">Plane Filling Curves</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> On another occasion I mentioned that <b>Peano's</b> definition of his plane filling curve was entirely analytic. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> 36], however, that <b>Peano's</b> original curve could be obtain geometrically as the limit of a sequence of curves as was the case with Hilbert's curve. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The original <b>Peano</b> curve maps the unit interval [0,1] onto the unit square [0,1]×[0,1].</td></tr> <tr><td></td><td colspan=2><font color=gray>www.cut-the-knot.org /Curriculum/Geometry/Peano.shtml</font>   (235 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.britannica.com/eb/article-9058868">Peano, Giuseppe --  Encyclopædia Britannica</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Italian <a href="/topics/Mathematician" title="Mathematician" class=fl>mathematician</a> and a founder of <a href="/topics/symbolic-logic" title="symbolic logic" class=fl>symbolic logic</a> whose interests centred on the foundations of <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a> and on the development of a formal logical language. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> became a lecturer of <a href="/topics/Calculus" title="Calculus" class=fl>infinitesimal calculus</a> at the University of <a href="/topics/Turin" title="Turin" class=fl>Turin</a> in 1884 and a professor in <a href="/topics/1890" title="1890" class=fl>1890</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <a href="/topics/Interlingua" title="Interlingua" class=fl>Interlingua</a> was originally developed in <a href="/topics/1903" title="1903" class=fl>1903</a> by the Italian <a href="/topics/Mathematician" title="Mathematician" class=fl>mathematician</a> Giuseppe <b>Peano</b>, but lack of clarity as to what parts of Latin were to be retained and what were to be discarded led to numerous “dialects” of <a href="/topics/Interlingua" title="Interlingua" class=fl>Interlingua</a>, confusion, and its dying out among enthusiasts.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.britannica.com /eb/article-9058868</font>   (734 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://users.skynet.be/TGMDev/curvepeano.htm">Fractals: Peano Curves</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The <b>Peano</b> curves are amongst the first known fractals curves. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> On 29 September <a href="/topics/1880" title="1880" class=fl>1880</a> <b>Peano</b> graduated as doctor of <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a> joined the staff at the University of <a href="/topics/Turin" title="Turin" class=fl>Turin</a> in <a href="/topics/1880" title="1880" class=fl>1880</a>. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The project was completed in <a href="/topics/1908" title="1908" class=fl>1908</a> and one has to admire what <b>Peano</b> achieved but although the work contained a mine of information it was little used.</td></tr> <tr><td></td><td colspan=2><font color=gray>users.skynet.be /TGMDev/curvepeano.htm</font>   (651 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>Java com.orcmid.LLC.pa Peano Arithmetic</u>   <i>(Site not responding. Last check: 2007-10-21)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> It is a peculiar characteristic of PA that it is the term use for the logical theory in which numbers and logical aspects of arithmetic are formalized. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> Numbers are in some sense what PA (the theory) speaks of as theoretical entities, and <b>Peano</b> Numerals are (usually formal) manifestations of them. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> We manifest <b>Peano</b> Numbers in a subordinate section, just in case there is need for more-abstracted and theoretical framework at an intervening level.</td></tr> <tr><td></td><td colspan=2><font color=gray>nfocentrale.net /orcmid/com.orcmid/llc/pa</font>   (222 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>Read about Giuseppe Peano at WorldVillage Encyclopedia. Research Giuseppe Peano and learn about Giuseppe Peano here!</u>   <i>(Site not responding. Last check: 2007-10-21)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Giuseppe <b>Peano</b> (<a href="/topics/August-27" title="August 27" class=fl>August 27</a>, <a href="/topics/1858" title="1858" class=fl>1858 –</a>; <a href="/topics/April-20" title="April 20" class=fl>April 20</a>, <a href="/topics/1932" title="1932" class=fl>1932</a>) was an </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> International Conference of <a href="/topics/Philosophy" title="Philosophy" class=fl>Philosophy</a> where <b>Peano</b> was a member of the patronage committee. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> After his mother died in <a href="/topics/1910" title="1910" class=fl>1910</a>, <b>Peano</b> divided his time between teaching, working on texts aimed for secondary schooling including a dictionary of <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a>, and developing and promoting his and other</td></tr> <tr><td></td><td colspan=2><font color=gray>encyclopedia.worldvillage.com /s/b/Peano</font>   (1072 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://xenia.media.mit.edu/~kelly/physPix/Peano.htm">Peano</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> is a building set comprised of fifty one-inch cubes that connect to form a modular, full-color display in three-dimensions. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> A <b>Peano</b> Curve, sometimes referred to as a Hilbert Curve, is a linear structure that turns at 90-degree angles to define a three-dimensional space with Cartesian coordinates. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Because the <b>Peano</b> topology is linear, a one, two or three-dimensional geometry may be built from a set of elements that have only two connectors each.</td></tr> <tr><td></td><td colspan=2><font color=gray>xenia.media.mit.edu /~kelly/physPix/Peano.htm</font>   (372 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.andrews.edu/~calkins/math/biograph/199899/biopeano.htm">Peano</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Giuseppe <b>Peano's</b> parents worked on a farm and Giuseppe was born in the farmhouse 'Tetto Galan' about 5 km from Cuneo. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> He realized that Giuseppe was a very talented child and he took him to <a href="/topics/Turin" title="Turin" class=fl>Turin</a> in 1870 for his secondary schooling, and to prepare him for university studies. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> took exams at Ginnasio Cavour in 1873 and then was a student at Liceo Cavour from where he graduated in <a href="/topics/1876" title="1876" class=fl>1876</a>.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.andrews.edu /~calkins/math/biograph/199899/biopeano.htm</font>   (2392 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://members.aol.com/jeff570/set.html">Earliest Uses of Symbols of Set Theory and Logic</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The main contributors were Ernst Schröder (1841-1902), Giuseppe <b>Peano</b> (1858-1932), Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970). </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> had a strong influence on Whitehead and Russell and their joint work, <a href="/topics/Principia-Mathematica" title="Principia Mathematica" class=fl>Principia Mathematica</a> (1910-1913), was itself very influential. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> introduced the backwards lower-case epsilon for "such that" in his <a href="/topics/1889" title="1889" class=fl>1889</a> "Principles of arithmetic, presented by a new method," according van Heijenoort's From Frege to Gödel: A Source Book in <a href="/topics/Mathematical-logic" title="Mathematical logic" class=fl>Mathematical Logic</a>, 1879--1931 [Judy Green].</td></tr> <tr><td></td><td colspan=2><font color=gray>members.aol.com /jeff570/set.html</font>   (1626 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><u>Peano</u>   <i>(Site not responding. Last check: 2007-10-21)</i></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> The others had continued their studies at the Engineering School which <b>Peano</b> himself had originally intended to do. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano's</b> continuous space-filling curves cannot be 1-1 of course, otherwise Netto's theorem would be contradicted. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> I believe it new in the history of <a href="/topics/Mathematics" title="Mathematics" class=fl>mathematics</a> that authors knowingly use in their research propositions for which exceptions are known, or for which they have no proof...</td></tr> <tr><td></td><td colspan=2><font color=gray>www-history.mcs.st-andrews.ac.uk /history/Mathematicians/Peano.html</font>   (2345 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://www.webfract.it/FRATTALI/vitaPeano.htm">biografia vita di Giuseppe Peano - caos e oggetti frattali - Tommaso Bientinesi</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Nel <a href="/topics/1890" title="1890" class=fl>1890</a> pubblica Sur une courbe qui remplit toute une aire plane, dove per la prima volta al mondo si parla di una curva che copre tutti i punti di un quadrato. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> Il più grande contributo di <b>Peano</b>, comunque, rimane nei campi della logica e dell'assiomatizzazione della matematica. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> non è uno scienziato che vive nella sua torre d'avorio, anzi è molto attento alle problematiche sociali del suo tempo.</td></tr> <tr><td></td><td colspan=2><font color=gray>www.webfract.it /FRATTALI/vitaPeano.htm</font>   (688 words)</td></tr> </table> </td> </tr> </table><body face="Arial"> <br> <table cellpadding=0> <tr> <td>  </td> <td> <table > <tr><td> </td><td colspan=2><a href="http://nfocentrale.net/miser/astraendo/pn">Numbering Peano</a></td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> That made me think that the Numbering <b>Peano</b> project is attempting to reveal exactly this same distinction in the realm of program specifications and use. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> <b>Peano</b> Arithmetic isn't really a computational model (though it is easy to relate a computational model to it), it is a logic-based theoretical formalism [a number theory]. </td></tr> <tr><td valign=top><img style="margin-top:4px;" src=/images/a.gif></td><td></td><td> My interest, in Numbering <b>Peano</b>, is more mundane with regard to manifestation of <b>Peano</b> numerals in a real computational system and seeing what that shows us about the manifestation of abstractions via computational means.</td></tr> <tr><td></td><td colspan=2><font color=gray>nfocentrale.net /miser/astraendo/pn</font>   (1841 words)</td></tr> </table> </td> </tr> </table><script language="JavaScript"> <!-- // This function displays the ad results. // It must be defined above the script that calls show_ads.js // to guarantee that it is defined when show_ads.js makes the call-back. function google_ad_request_done(google_ads) { // Proceed only if we have ads to display! if (google_ads.length < 1 ) return; 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