- Introduction
- A Metric Based On Google Scholar
- The Eigenspectrum of an Author
- The Quadratic Eigenequation of an Author
- The Eigensignal of an Author
- Blue/Red-shifts of Authors
- Notes/References

Google Scholar can be used to construct a metric which can show the relative "merit" of scientists in their corresponding fields of research, based on the work they've done.

Assuming that an author's name is unique (which is not always the case), one can construct a *characteristic publication number* or a *publication eignevalue* for a given author, say "john doe", as follows:

- Enter "author:j-doe" in the Google scholar field and press "Search".
- a
_{0}=number of results for this author, shown on the upper right hand. - a
_{1}=number of citations under the first result. Click on these citations. A new window opens. - a
_{2}=number of citations under the first result. Click on these citations. A new window opens. - ...
- Repeat, until first result shows no citations or the sequence falls into a cycle.

The publication eigenvalue for this author then, can be the number C(john doe), which has the continued fraction expansion:

C(john doe)=[a_{0};a_{1},a_{2},...,a_{n},...].

To simplify the ordering which is present in the set {C(x):x\in author}, without loss of generality we can set a_{0}=1 and look instead at the number:

C(john doe)=[1;a_{0}',a_{1}',a_{3}',...,a_{n}',...], with a_{n-1}'=a_{n}, which maps the set {C(x):x\in author} into the interval (1,∞).

Note that in this case, sup_{x}{C(x):x\in author}=∞ and inf_{x}{C(x):x\in author}=1.

Adding a citation entry a>0 to an existent continued fraction expansion of C(x), can make C(x) either larger or smaller, depending on where a is added and the number of citations at level n^{[18]}. Specifically:

- [1;a
_{1},a_{2},a_{3},...,a_{n},a]<[1;a_{1},a_{2},a_{3},...,a_{n}], if n odd, - [1;a
_{1},a_{2},a_{3},...,a_{n},a]>[1;a_{1},a_{2},a_{3},...,a_{n}], if n even - [1;a
_{1},a_{2},a_{3},...,a_{n}+a]<[1;a_{1},a_{2},a_{3},...,a_{n}], if n odd. - [1;a
_{1},a_{2},a_{3},...,a_{n}+a]>[1;a_{1},a_{2},a_{3},...,a_{n}], if n even.

The main "weight" of the number C(x) will then be carried by the term a_{1}, which is the number of publications of author x and which provides a good approximation of C(x), as C(x)~C_{2}(x)=1+1/a_{1}, which is fairly reasonable.

The *formal* definition of C(x) is slightly more involved, mainly because one needs to define it uniquely. Here's then the formal definition:

- Let x be the name of an author in Google Scholar.
- Search on x gives rise to a
_{1}results. - Each result gives rise to a
_{2,k}citations, indexed by k. - Each of those results gives rise to a
_{3,l}citations, indexed by l, and so on. - Define C(x)=[a
_{0}=1;a_{1},...,a_{n},...], with: - a
_{1}=sup_{k}{a_{1,k}},a_{2}=sup_{l}{a_{2,l}},...,a_{n}=sup_{w}{a_{n,w}}.

It can now be seen that the definition above gives rise to a *unique* number C(x), as in the first definition for "john doe", above, because the suprema are taken over finite sets indexed by k,l,m,...,w.

__A Metric Based On Google Scholar__

The definition above gives rise to the *metric*: d(x,y)=|C(x)-C(y)|. Let's verify the metric's fundamental properties:

- d(x,y)≥0: Follows from the definition of |.|.
- d(x,y)=0 <=> x=y: Let C(x)=[a
_{0}=1;a_{1},a_{2},...,a_{n}] and C(y)=[b_{0}=1;b_{1},b_{2},...,b_{m}]. If x=y, then m=n and C(x)=C(y), so |C(x)-C(y)|=d(x,y)=0.

Conversely, if d(x,y)=|C(x)-C(y)|=0 and m=n, then a_{i}=b_{i}, for all i \in {0,1,2,...,n}, which happens if x=y. If m=/=n, then the simplest case is [1;a]=[1;b,c], which gives: a=b-1/c, which forces c=1 because a and b are naturals, therefore c=1 and b=a-1. In other words, in the simplest case y has 1 less publication than x, but also one more citation. In general this may also happen if y has one less citation at level n and one more at level n+1. These are rare cases, and if they are excluded, then x=y. - For any w, d(x,y)=|C(x)-C(w)+C(w)-C(y)|≤|C(x)-C(w)|+|C(w)-C(y)|=d(x,w)+d(w,y), by the triangle inequality for |.|.

It is clear that a person with no publications, will have a characteristic number equal to infinity and the more publications an author has, the closer C(x) is to 1. This gives rise to a tempered distribution, and then one can define the *publication percentile* P(x) of a scientist x in this distribution to be: P(x)=100/C(x).

Fixing t=now(18/11/2010) and omitting the term a_{0}=1, let's then see these numbers for some scientists:

x | a_{n}, n≥1^{[1]} |
C(x) | P(x)(%) | Class^{[2]} |
Non-repeating Block^{[3]} |
Repeating Block^{[4]} |

Albert Einstein | 4470,6443,5293,2374,2344,770, 589,332,314,332,314 |
1.000223714 | 99.97763364 | q.i. | 7 | 2 |

Paul Erdos | 3030,1310,6984,4953,1809,1619, 332,161,16,161,16 |
1.000330033 | 99.96700760 | q.i. | 7 | 2 |

Isaac Newton | 2480,1777,1331,1762,831,1762, 831 |
1.000403226 | 99.95969368 | q.i. | 3 | 3 |

Donald Knuth | 1570,3685,3950,938,430,408, 496,5809,1743,1206,485,222, 15,15,17,4 |
1.000636943 | 99.93634629 | r | 16 | 0 |

Leonhard Euler | 1390,259,2665,1820,1619,332, 161,16,161,16 |
1.000719422 | 99.92810947 | q.i. | 6 | 2 |

Henri Poincare | 1120,827,2721,4859,4396,3809, 4396,3809 |
1.000892856 | 99.91079403 | q.i. | 4 | 2 |

John von Neumann | 1020,12150,16445,6460,5214,3361, 2470,2156,4633,3275,2725,3771, 4842,9666,34313,8985,9666,34313, 8985 |
1.000980392 | 99.90205681 | q.i. | 13 | 3 |

Robert Oppenheimer^{[5]} |
989,940,704,492,605,611, 314,195,191,195,191 |
1.001011121 | 99.89899001 | q.i. | 7 | 2 |

Benoit Mandelbrot | 896,22145,14018,6668,12394,16883, 12394 |
1.001116071 | 99.88851729 | q.i. | 6 | 2 |

Carl Friedrich Gauss | 888,607,1980,2812,3493,2812, 3493 |
1.001126124 | 99.88751427 | q.i. | 3 | 2 |

Georg Cantor | 866,407,1164,1363,1319,465, 84,465,84 |
1.001154731 | 99.88466007 | q.i. | 5 | 2 |

Werner Heisenberg | 863,1718,7160,7624,3251,2696, 3750,4792,9556,33976,8861,9556, 33976,8861 |
1.001158748 | 99.88425934 | q.i. | 8 | 3 |

Richard Feynman | 847,3914,2621,1439,2313,1845, 1156,2078,1156,2078 |
1.001180637 | 99.88207551 | q.i. | 6 | 2 |

Johannes Kepler | 791,80,3371,1219,67,38, 67,38 |
1.001264203 | 99.87373937 | r | 8 | 0 |

Max Planck | 785,351,844,706,1100,3624, 3440,823,205,128,65,26, 6,1 |
1.001273881 | 99.87277400 | r | 14 | 0 |

Andrew Wiles | 562,1127,2812,3493,2812,3493, 2812 |
1.001779357 | 99.82238039 | q.i. | 2 | 2 |

Subhash Kak | 527,130,58,64,144,30, 35,15,8,4 |
1.001897506 | 99.81060882 | r | 10 | 0 |

Johann Heinrich Lambert | 477,192,1007,2611,1028,379, 123,9,1 |
1.002096413 | 99.79079726 | r | 9 | 0 |

Claude Shannon | 466,38281,11721,7672,3274,2724, 3768,4835,9646,34270,8975,9646, 34270,8975,9646 |
1.002145923 | 99.78586725 | q.i. | 9 | 3 |

Erwin Schroedinger | 431,1168,2238,2166,830,524, 830,524 |
1.002320181 | 99.76851898 | q.i. | 4 | 2 |

Robert Devaney^{[22]} |
382,3247,5433,6581,2552,1399, 1170,476,987,676,308,218, 581,237,382,278,382,278 |
1.002617799 | 99.73890360 | q.i. | 13 | 2 |

Srinivasa Ramanujan | 349,446,481,128,58,44, 18,44,18 |
1.002865311 | 99.71428754 | q.i. | 5 | 2 |

Kurt Goedel | 328,2024,4096,10422,3557,1496, 3557,1496 |
1.003048776 | 99.69604909 | q.i. | 4 | 2 |

Augustin-Louis Cauchy | 278,180,841,2703,1991,3083, 3459,1328,7007,10158,1879,10158, 1879 |
1.003597050 | 99.64158420 | q.i. | 9 | 2 |

Douglas R. Hofstadter | 249,821,3663,14885,1559,12637, 14885,1559,12637 |
1.004016045 | 99.60000195 | q.i. | 3 | 3 |

John Baez | 207,559,291,335,188,894, 492,808,705,159,57,5 |
1.004830876 | 99.51923490 | r | 12 | 0 |

Grigori Perelman | 172,701,522,549,191,180, 89,442,1097,463,1097,463, 1097 |
1.005813905 | 99.42197008 | q.i. | 8 | 2 |

John F. Waymouth^{[30]} |
171,348,116,162,78,43, 113,504,65,94,87,7,1 |
1.005847855 | 99.41861436 | r | 13 | 0 |

Henri Lebesgue | 148,167,788,2799,7119,8978, 7799,2557,7799,2557 |
1.006756483 | 99.32888603 | q.i. | 6 | 2 |

Gerald A. Edgar^{[10]} |
141,436,3022,6600,2558,1412, 1178,484,1001,685,309,219, 586,238,388,284,388,284 |
1.007092083 | 99.29578604 | q.i. | 14 | 2 |

Constantin Caratheodory | 131,326,7119,8978,7799,2557, 7799,2557 |
1.007633409 | 99.24244185 | q.i. | 4 | 2 |

Heinrich Begehr^{[24]} |
119,142,40,181,74,107, 23,60,22,17,10,2, 2 |
1.008402864 | 99.16671556 | q.i. | 11 | 1 |

Bernhard Riemann | 107,303,1980,2812,3493,2812, 3493 |
1.009345506 | 99.07410237 | q.i. | 3 | 2 |

A.O.L. Atkin^{[6]} |
95,359,2812,3493,2812,3493 | 1.010526007 | 98.95836356 | q.i. | 2 | 2 |

Pierre de Fermat | 85,15,177,766,160,243, 273,107,130,273,107,130,273 |
1.011755489 | 98.83810965 | q.i. | 6 | 3 |

Gottfried Leibniz | 76,6,87,3430,11657,7624, 3251,2696,3750,4792,9556,33976, 8861,9556,33976,8861 |
1.013129158 | 98.70409832 | q.i. | 10 | 3 |

Menelaos Karanikolas^{[7]} |
76,26,163,148,145,191, 84,134,114,432,92,432, 92 |
1.013151241 | 98.70194693 | q.i. | 8 | 2 |

Donald L. Shell^{[8]} |
75,145,2536,15236,10092,1870, 10092,1870 |
1.013332107 | 98.68432992 | q.i. | 4 | 2 |

Vassili Nestoridis^{[9]} |
71,291,169,56,52,119, 159,256,159,256,159 |
1.014083825 | 98.61117740 | q.i. | 6 | 2 |

Robert B. Israel^{[10]} |
70,226,1664,7903,2659,1206, 485,222,15,15,17,4 |
1.014284811 | 98.59163707 | r | 12 | 0 |

Arturo Magidin^{[28]} |
55,21,14,3 | 1.018166142 | 98.21579787 | r | 4 | 0 |

Victor D. Roberts^{[30]} |
47,41,82,99,32,149, 233,105,77,41,25,17, 11,17,11 |
1.021265563 | 97.91772442 | q.i. | 11 | 2 |

Michael S. Lambrou^{[31]} |
29,50,67,51,126,357, 693,344,79,1934,9484,14466, 10203,1882,10203,1882 |
1.034459001 | 96.66888675 | q.i. | 12 | 2 |

Donald L. Klipstein^{[23]} |
28,16,8,8,6,1, 11,20,9,4,10,4 |
1.035635350 | 96.55908328 | r | 12 | 0 |

Johan E. Mebius^{[28]} |
24,4,4,3,7,6, 3,6,4,6,4 |
1.041260390 | 96.03745703 | q.i. | 7 | 2 |

Paris Pamfilos^{[16]} |
22,3,2 | 1.044871795 | 95.70552147 | r | 3 | 0 |

Ioannis Papadoperakis^{[11]} |
22,291,169,56,52,119, 159,256,159,256 |
1.045447447 | 95.65282341 | q.i. | 6 | 2 |

This author | 13,10,4,17,3,17, 3 |
1.076349896 | 92.90659143 | q.i. | 3 | 2 |

Dave L. Renfro^{[20]} |
9,5,7,9,3,3 | 1.108760363 | 90.19081433 | q.i. | 4 | 1 |

Mikes Glinatsis^{[19]} |
8,51,289,211,384,543, 726,1900,2369,1900,2369 |
1.124694397 | 88.91304184 | q.i. | 7 | 2 |

The author's father^{[17]} |
6,9,18,29,17,13,6,2,1,1 | 1.163654584 | 85.93615441 | r | 10 | 0 |

Robert P. Munafo^{[21]} |
4,3,1 | 1.235294118 | 80.95238095 | r | 3 | 0 |

James D. Hooker^{[25]} |
3 | 1.333333333 | 75 | r | 1 | 0 |

... | ... | ... | ... | ... | ... | ... |

2 publications no citations | 2,0,0,0,0 | 3/2 | 66.66 | r | 1 | 0 |

1 publication no citations | 1,0,0,0,0 | 2 | 50 | r | 1 | 0 |

no publications | 0 | ∞ | 0 | r | 0 | 0 |

__The Eigenspectrum of an Author__

We can now define the *Google Scholar Eigenspectrum* of the author x to be the sequence of *convergents* for C(x), C_{n}(x)=[a_{0}=1;a_{1},a_{2},...,a_{n-1}].

The spectra of some authors are shown below. The dominant spectral line for each author lies approximately at C_{2}(x)=a_{0}+1/a_{1}=1+1/a_{1}. These spectra give you a rough idea of the colossal amount of work the corresponding authors have done in their fields.

x | C(x) | Eigenspectrum |

Albert Einstein | 1.000223714 | |

Paul Erdos | 1.000330033 | |

Isaac Newton | 1.000403226 | |

Donald Knuth | 1.000636943 | |

Leonhard Euler | 1.000719422 | |

Henri Poincare | 1.000892856 | |

John von Neumann | 1.000980392 | |

Robert Oppenheimer^{[5]} |
1.001011121 | |

Benoit Mandelbrot | 1.001116071 | |

Carl Friedrich Gauss | 1.001126124 | |

Georg Cantor | 1.001154731 | |

Werner Heisenberg | 1.001158748 | |

Richard Feynman | 1.001180637 | |

Johannes Kepler | 1.001264203 | |

Max Planck | 1.001273881 | |

Andrew Wiles | 1.001779357 | |

Subhash Kak | 1.001897506 | |

Johann Heinrich Lambert | 1.002096413 | |

Claude Shannon | 1.002145923 | |

Erwin Schroedinger | 1.002320181 | |

Robert Devaney | 1.002617799 | |

Srinivasa Ramanujan | 1.002865311 | |

Kurt Goedel | 1.003048776 | |

Augustin-Louis Cauchy | 1.003597050 | |

Douglas R. Hofstadter | 1.004016045 | |

John Baez | 1.004830876 | |

Grigori Perelman | 1.005813905 | |

John F. Waymouth | 1.005847855 | |

Henri Lebesgue | 1.006756483 | |

Gerald E. Edgar | 1.007092083 | |

Constantin Caratheodory | 1.007633409 | |

Heinrich Begehr | 1.008402864 | |

Bernhard Riemann | 1.009345506 | |

A.O.L. Atkin | 1.010526007 | |

Pierre de Fermat | 1.011755489 | |

Gottfried Leibniz | 1.013129158 | |

Menelaos Karanikolas | 1.013151241 | |

Donald L. Shell | 1.013332107 | |

Vassili Nestoridis | 1.014083825 | |

Robert B. Israel | 1.014284811 | |

Arturo Magidin | 1.018166142 | |

Victor D. Roberts | 1.021265563 | |

Michael S. Lambrou | 1.034459001 | |

Donand L. Klipstein | 1.035635350 | |

Johan E. Mebius^{[14]} |
1.041260390 | |

Paris Pamfilos | 1.044871795 | |

Ioannis Papadoperakis | 1.045447447 | |

This author | 1.076349896 | |

Dave L. Renfro^{[14]} |
1.108760363 | |

Mikes Glinatsis | 1.124694397 | |

The author's father^{[14]}^{[26]} |
1.163654584 | |

Robert P. Munafo^{[14]} |
1.235294118 | |

Bronze Ratio-2^{[27]} |
1.302775638 | |

James D. Hooker | 1.333333333 | |

Silver Ratio-1^{[27]} |
1.414213562 | |

[1;2,10] | 1.476190476 | |

[1;2,14,2] (Tl?) |
1.483333333 | |

[1;2] | 1.5 | |

[1;1,1] | 1.5 | |

Golden Ratio^{[27]} |
1.618033985 | |

[1;1,1,1,32,1,2] (Na?!) |
1.663333333 | |

[1;1,3,4,5,6,3,2] | 1.764064436 | |

[1;1,10] | 1.909090909 | |

[1;1] | 2 |

__The Quadratic Eigenequation of an Author__

Euler proved that whenever the sequence of the associated continued fraction is periodic, C(x) will equal a certain quadratic irrational æ of the form (P+sqrt(D))/Q. We find this æ for this author.

First we program some Maple code to calculate continued fractions.

> L2C:=proc(L)

> local l,c,n;

> l:=nops(L);c:=0;

> for n from 1 to l do

> c:=1/(c+L[l-n+1]);

> od;

> c;

> 1/c;

> end:

The above proc, takes as input a list of the form:

>L:=[a_{0}=1,a_{1},a_{2},a_{3},a_{4},...];

and calculates the corresponding continued fraction.

Consider then the simplest periodic continued fraction, with a period 2 block p:

[a_{1};a_{2},a_{1},a_{2},...,a_{1},a_{2},...].

What does it *mean* for this continued fraction to be periodic? It means *exactly*:

p=a_{1}+1/(a_{2}+1/p) (1)

Equation (1) translated into continued fraction notation, is:

p=[a_{1};a_{2},p] (2)

Equation (2) translated into Maple notation, is:

L2C([a_{1};a_{2},p])=p

The last equation can be solved quickly with Maple.

> eq:=L2C([a1,a2,p])=p;

> sol:=solve(eq,p);

The above gives two solutions. For this author the continued fraction of works and citations (including the initial 1) is:

L=[1;13,10,4,17,3,17,3,17,3,...]. Therefore, we can recover the periodic part, as:

> sol:=subs({a1=17,a2=3},sol[1]),subs({a1=17,a2=3},sol[2]);

sol:=17/2+/-2805^{(1/2)}/6

Note that this is complete quotient æ_{4}. We can then use the recursion found on that same page, to recover the full continued fraction, using Maple:

> zeta4:=sol[1];

> zeta3:=4+1/zeta4;

> zeta2:=10+1/zeta3;

> zeta1:=13+1/zeta2;

> zeta0:=1+1/zeta1;

Check:

> evalf(zeta0);

1.076349896

> L:=[1,13,10,4,17,3,17,3,17,3,17,3,17,3,17,3,17,3,17,3,17,3,17,3];

> evalf(L2C(L));

1.076349896

Check #2:

> convert(evalf(zeta0),confrac);

[1, 13, 10, 4, 17, 2, 1]

Success!

> zeta:=simplify(zeta0);

> conj:=denom(zeta)-2*op(denom(zeta))[2];#get rid of roots on denominator!

> zetan:=expand(numer(zeta)*conj);

> zetad:=expand(denom(zeta)*conj);

> zeta:=zetan/zetad;

And ζ=906371/842062-(1/2526186)*2805^{(1/2)}, so the Publication Eigenvalues for this author, as a function of t=now, are ζ and ζ*, which are Quadratic Irrationals^{[32]}.

We finally recover the quadratic equation via Viete's Expressions:

> zetac:=zeta-2*op(2,zeta);

> eq:=x^2-(zeta+zetac)*x+zeta*zetac=0;

> eq:=simplify(eq);

> eq:=denom(op(3,op(1,eq)))*eq;

1263093*x^{2}-2719113*x+1463387 = 0

Check:

> solve(eq,x);

{906371/842062+(1/2526186)*2805^{(1/2)}, 906371/842062-(1/2526186)*2805^{(1/2)}} = {ζ*,ζ}.

This equation then, is something like a *characteristic equation* or *eigenequation* for this author and the left part of the equation is something like a *publication eigenfunction* for this author as a function of this author's publications at the current time. As more works are published and more citations are shown, it is obvious that this characteristic equation changes as a function of time. It is a useful exercise for the reader to calculate the characteristic equation of other authors, higher on the table, above.

The convergents of C(x), are {C_{n}(x)}, n\in N, and they can be calculated with Maple:

> C:=proc(L,n)

> local cvgts;

> convert(L2C(L),confrac,cvgts);

> cvgts[n];

> end:

If δ is the Dirac Delta, the Google Scholar Publication Eigenspectrum of an author x shown on the above table then, is the convergent *pulse train* or Dirac Comb:

The *Google Scholar Publication Signal* or *Publication Eigensignal* of an author in the time domain then, will be the Inverse Fourier Transform of the author's Eigenspectrum:

Google publication Eigensignal

The latter evaluates to:

Author's publication Eigensignal in real time

If C(x) is rational the sum will consist of a finite number of terms and hence the Eigensignal will be periodic in the time domain^{[12]}. If C(x) is a quadratic irrational the sum will consist of infinitely many terms and the eigensignal will not be periodic in the time domain.

Let's calculate the real and imaginary components of the eigensignal for this author with Maple, by considering an approximation with the periodic part of C(x) repeating 4 times:

> L:=[1,13,10,4,17,3,17,3,17,3,17,3];

> S:=proc(L,n)

> local i;

> add(Dirac(xi-C(L,i)),i=1..nops(L));

> end:

> AS:=t->Int(S(L,xi)*exp(2*Pi*I*t*xi),xi=-infinity..infinity);

> with(plots):

> rexpr:=t->evalf(Re(AS(t)));

> imexpr:=t->evalf(Im(AS(t)));

> plot(rexp(t)r,t=0..10*T);

Real part of this author's Google Scholar Eigensignal in the time domain

> plot(imexpr(t),t=0..10*T);

Imaginary part of this author's Google Scholar Eigensignal in the time domain

The two signals can now be approximated via the Fourier Series. For the real Eigensignal the Fourier Coefficients are given as:

>a0:=2/T*Int(rexpr(t),t=0..T):

>a:=n->2/T*Int(rexpr(t)*cos(2*Pi*n*t/T),t=-T/2..T/2):

>b:=n->2/T*Int(rexpr(t)*sin(2*Pi*n*t/T),t=-T/2..T/2): #evaluates to 0

These evaluate to functions of the convergents of C(x)^{[13]}.

If C(x) is a quadratic irrational, the author's Google Scholar Eigensignal will be "almost" periodic, with a minimal period T=1/C_{n}(x) which is with Maple:

> T:=1/C(L,nops(L));#find minimal period! (which we used in calculations above)

Now we can construct a Fourier Series approximation for the real Eigensignal with Maple:

> F:=(m,t)->a0/2+add(a(n)*cos(2*n*Pi*t/T)+b(n)*sin(2*n*Pi*t/T),n=1..m):

The two plots:

> p1:=plot(rexpr(t),t=0..T,color=red):

> p2:=plot(F(12,t),t=0..T,color=green):

> display(p1,p2);

Real part of author's Publication Eigensignal (red) and Fourier Series approximation F

The *harmonics* of the author's real Eigensignal will then be a_{n} and the *amplitude* of the harmonics will be given as c_{n}=|a_{n}|. The author's *Harmonic Spectrum* with up to 10 harmonics is shown below.

> eps:=1e-1;

> PS:=[[[eps,0],[eps,evalf(abs(a0))]],seq([[eps+n,0],[eps+n,evalf(abs(sqrt(a(n)^2+b(n)^2)))]],n=1..10)]:

> plot(PS,n=0..10);

Harmonic Spectrum for this author's Eigensignal with up to 10 harmonics (a

The 0-th harmonic a_{0} (dc-term) is the red line (almost supressed). The *amplitude* of the *dominant harmonic* (green), which corresponds to the dominant spectral line shown on the author's Eigenspectrum on the table above, is |a_{1}|~11.95 and the author's Eigensignal *broadcasts* at a frequency f=1/T=C(x)~1.082 Hz.

We now have a Fourier Series approximation. Let's recover the Google Eigenspectrum from it.

> SSS:=xi->evalc(Re(int(F(12,t)*exp(-2*Pi*t*xi*I),t=-10..10))):

> plot(SSS(xi),xi=1..2);

Recovered Eigenspectrum for this author from F

The above is an approximation of the Google Scholar Eigenspectrum for this author. The function jumps very hight exactly at the convergents C_{n}(x) and in particular at C_{2}(x):

> evalf(SSS(C(L,2)));

119.3746313

We therefore have verified the commutativity in the following diagram:

Commutative diagram for Eigenspectrum, Eigensignal and Fourier Series

The publication Eigensignal's minimal period is roughly the time between two adjacent publications. For this author:

> evalf(T*365/30)

11.23774294 (months)^{[15]}

Accordingly, one can now define a x author's *blue-shift* relative to another author y, via the Google Scholar Metric, as d(x,y)=|C(x)-C(y)|. For example:

> LG:=[1,13,10,4,17,3,17,3,17,3,17,3,17,3,17,3,17,3,17,3,17,3,17,3,17,3,17];#author

> LP:=[1,22,291,169,56,52,119,159,256,159,256,159,256,159];#author's advisor

> BS:=evalf(abs(L2C(LG)-L2C(LP)));

BS:=0.03090244910

The author's advisor is blue-shifted 0.03090244910 more than the author. x more blue-shifted than y means x *has worked harder* than y. Notice how high are the blue-shifts of the big guns in Mathematics.

Alternatively one may define an author's *red-shift* relative to unity. For example:

> RSG:=evalf(L2C(LG)-1);

> RSP:=evalf(L2C(LP)-1);

0.7634989575e-1

0.04544744665

The author's advisor is 0.04544744665 and the author is 0.07634989575 red-shifted away from unity. Smaller red-shifts mean *more work*.

- The sequences were extracted on the date of publication (18/11/2010) and consequently they may be different on the date the reader reads this article. To calculate the continued fraction sequence, the author entered the author's names in Google Scholar, as "author:j-a-doe", where "j" and "a" are the initials of the author's first and middle name. When this didn't work, for example when the list of references included other people, the full name was entered as "author:john-doe". When even this didn't work, the name was entered as "author:john-a-doe". If you are listed above and disagree with your sequence, send this author a more appropriate sequence which can be verified via Google Scholar. The spectra tables are not meant to be exhaustive. Only a small selection of scientists is presented. If you are not listed above and you have a Google Scholar publication history, or you want to nominate someone else for inclusion, you can send this author your full name and a home page link or the name of the nominee and he will add a corresponding entry to both tables.
- Rational or quadratic irrational.
- Length of non-periodic block in the author's continued fraction.
- Length of periodic block in the author's continued fraction.
- This author considers Oppy's work to be of fundamental historical importance since with his work humanity graduated as a nuclear power. Accordingly, this man is a
*major critical point*in the development of human scientific intelligence. An author's x Eigenspectrum can be labelled*relativistic*when the red-shift of x is less than that of Oppy. In symbols: Rel(x) <=> C(x)≤C(Oppy). For example, the authors above Oppy all have relativistic eigenspectra. - The author's Number Theory professor at U of I.
- The best student in the author's high school class. Field: Medicine.
- Designer of the shell-sort computer algorithm and Tetration researcher.
- This author's advisor's advisor. Field: Complex Analysis.
- Moderator of newsgroup sci.math.research and participant on the newsgroup sci.math.
- This author's advisor.
- Can you think what rational or quadratic irrational imply in terms of publication
*events*for the corresponding authors? - If C(x) is rational then the number of spectral lines is finite, hence the sums in a
_{0}and a_{n}are finite. If C(x) is a quadratic irrational, because C_{n}(x) converges very fast, the Eigenspectrum can be approximated by a*finite*pulse train, consisting of the first m convergents C_{m}(x), in which case a_{0}and a_{n}can again be approximated by a finite summation. In the subsequent Maple approximation example m is the cardinality of the sequence list. - The Eigenspectrum can be characterized as
*rare*or*strange*in that it displays two spectral lines which are*widely separated*. It is instructive for the reader to try to identify the root cause of this phenomenon. - This means for example, that this author at the beginning was writing approximately 1 paper every 11 months, which checks pretty well with the publication dates of his papers on his Mathematics page. For an author x, his average publication period will be given roughly as T
_{0}~1/C_{n}(x). - Geometer. Creator of the program EucliDraw.
- The author's father is found having a lower rating than the author, which is rather silly and highly contradictory. The discrepancy can be partially explained by noticing that, first, his father's field was Applied Mathematics in Civil Engineering: Theory of Elasticity, which is a very rare field and second, that his father was a professor for only 5 years before leaving this post. The sequence was generated by his Ph.D.. Additionally, Google Scholar ignores several of his other publications, such as these which are in Greek Engineering journals. For details about these, consult the author's notes in his father's biography.
- This is fairly reasonable because citations detract/add importance to the main publications from the author and assign it to the authors of the citations depending on their nesting level. If one wants, one can arrange C(x) to have its convergents be monotone decreasing using appropriate functions of the a
_{n}instead of the terms a_{n}themselves. - The author's general surgeon. Field: Medicine.
- Has helped with the author's research on Tetration.
- Has compiled a colossal amount of computational and mathematical data on his web site.
- The most prolific author in the area of Complex Dynamical Systems.
- Has compiled a colossal amount of light and engineering data on his web site.
- Referee of the journal Complex Variables.
- Has compiled a colossal amount of light-engineering data.
- The main resonance line in the Eigenspectrum of the author's father corresponds roughly to the 436nm blue triple line in Mercury's spectrum emitted by the High Pressure Mercury Vapor Lamp, which played an important role in the scientific development of the author.
- The eigenspectra of the metal ratios curiously contain resonance lines which match actual lines in their real spectra.
- Frequent participant on the newsgroup sci.math.
- The signals are not very accurate because the duration of each frame in animated .gifs needs to be an integral multiple of 1/100 sec, while the actual durations after calculation are seen to be decimal multiples of 1/100 secs for the two authors shown.
- Field: Light-engineering.
- Professor at the University of Crete. Served as advisor for several of the author's papers.
- Note that Q|(P
^{2}-D):

>Pn:=numer(zeta);

>Q:=denom(zeta);

>Dr:=op(2,Pn)^2;

>P:=op(1,Pn);

>(P^2-Dr)/Q;

292677.