## Google Scholar and the Spectra of the Scientists

Introduction

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:

1. Enter "author:j-doe" in the Google scholar field and press "Search".
2. a0=number of results for this author, shown on the upper right hand.
3. a1=number of citations under the first result. Click on these citations. A new window opens.
4. a2=number of citations under the first result. Click on these citations. A new window opens.
5. ...
6. 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)=[a0;a1,a2,...,an,...].

To simplify the ordering which is present in the set {C(x):x\in author}, without loss of generality we can set a0=1 and look instead at the number:

C(john doe)=[1;a0',a1',a3',...,an',...], with an-1'=an, which maps the set {C(x):x\in author} into the interval (1,∞).

Note that in this case, supx{C(x):x\in author}=∞ and infx{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. Specifically:

1. [1;a1,a2,a3,...,an,a]<[1;a1,a2,a3,...,an], if n odd,
2. [1;a1,a2,a3,...,an,a]>[1;a1,a2,a3,...,an], if n even
3. [1;a1,a2,a3,...,an+a]<[1;a1,a2,a3,...,an], if n odd.
4. [1;a1,a2,a3,...,an+a]>[1;a1,a2,a3,...,an], if n even.

The main "weight" of the number C(x) will then be carried by the term a1, which is the number of publications of author x and which provides a good approximation of C(x), as C(x)~C2(x)=1+1/a1, 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:

1. Let x be the name of an author in Google Scholar.
2. Search on x gives rise to a1 results.
3. Each result gives rise to a2,k citations, indexed by k.
4. Each of those results gives rise to a3,l citations, indexed by l, and so on.
5. Define C(x)=[a0=1;a1,...,an,...], with:
6. a1=supk{a1,k},a2=supl{a2,l},...,an=supw{an,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:

1. d(x,y)≥0: Follows from the definition of |.|.
2. d(x,y)=0 <=> x=y: Let C(x)=[a0=1;a1,a2,...,an] and C(y)=[b0=1;b1,b2,...,bm]. 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 ai=bi, 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.
3. 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 a0=1, let's then see these numbers for some scientists:

 x an, n≥1 C(x) P(x)(%) Class Non-repeating Block Repeating Block 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 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 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 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 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 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 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 76,26,163,148,145,191,84,134,114,432,92,432,92 1.013151241 98.70194693 q.i. 8 2 Donald L. Shell 75,145,2536,15236,10092,1870,10092,1870 1.013332107 98.68432992 q.i. 4 2 Vassili Nestoridis 71,291,169,56,52,119,159,256,159,256,159 1.014083825 98.61117740 q.i. 6 2 Robert B. Israel 70,226,1664,7903,2659,1206,485,222,15,15,17,4 1.014284811 98.59163707 r 12 0 Arturo Magidin 55,21,14,3 1.018166142 98.21579787 r 4 0 Victor D. Roberts 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 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 28,16,8,8,6,1,11,20,9,4,10,4 1.035635350 96.55908328 r 12 0 Johan E. Mebius 24,4,4,3,7,6,3,6,4,6,4 1.041260390 96.03745703 q.i. 7 2 Paris Pamfilos 22,3,2 1.044871795 95.70552147 r 3 0 Ioannis Papadoperakis 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 9,5,7,9,3,3 1.108760363 90.19081433 q.i. 4 1 Mikes Glinatsis 8,51,289,211,384,543,726,1900,2369,1900,2369 1.124694397 88.91304184 q.i. 7 2 The author's father 6,9,18,29,17,13,6,2,1,1 1.163654584 85.93615441 r 10 0 Robert P. Munafo 4,3,1 1.235294118 80.95238095 r 3 0 James D. Hooker 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), Cn(x)=[a0=1;a1,a2,...,an-1].

The spectra of some authors are shown below. The dominant spectral line for each author lies approximately at C2(x)=a0+1/a1=1+1/a1. 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 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 1.041260390 Paris Pamfilos 1.044871795 Ioannis Papadoperakis 1.045447447 This author 1.076349896 Dave L. Renfro 1.108760363 Mikes Glinatsis 1.124694397 The author's father 1.163654584 Robert P. Munafo 1.235294118 Bronze Ratio-2 1.302775638 James D. Hooker 1.333333333 Silver Ratio-1 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 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 ζ of the form (P+sqrt(D))/Q. We find this ζ 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:=[a0=1,a1,a2,a3,a4,...];

and calculates the corresponding continued fraction.

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

[a1;a2,a1,a2,...,a1,a2,...].

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

p=a1+1/(a2+1/p) (1)

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

p=[a1;a2,p] (2)

Equation (2) translated into Maple notation, is:

L2C([a1;a2,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),subs({a1=17,a2=3},sol);

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

Note that this is complete quotient ζ4. We can then use the recursion found on that same page, to recover the full continued fraction, using Maple:

> zeta4:=sol;
> 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));#get rid of roots on denominator!
> zetan:=expand(numer(zeta)*conj);

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.

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*x2-2719113*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 Eigensignal of an Author

The convergents of C(x), are {Cn(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: 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. 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;
> 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).

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

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 F12(x,t) (green)

The harmonics of the author's real Eigensignal will then be an and the amplitude of the harmonics will be given as cn=|an|. 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 (a1~11.95)

The 0-th harmonic a0 (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 |a1|~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 F12(x,t)

The above is an approximation of the Google Scholar Eigenspectrum for this author. The function jumps very hight exactly at the convergents Cn(x) and in particular at C2(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)

Blue/Red-shifts of Authors

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
> 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.

Notes/References

3. Length of non-periodic block in the author's continued fraction.
4. Length of periodic block in the author's continued fraction.
5. 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.
6. The author's Number Theory professor at U of I.
7. The best student in the author's high school class. Field: Medicine.
8. Designer of the shell-sort computer algorithm and Tetration researcher.
10. Moderator of newsgroup sci.math.research and participant on the newsgroup sci.math.
12. Can you think what rational or quadratic irrational imply in terms of publication events for the corresponding authors?
13. If C(x) is rational then the number of spectral lines is finite, hence the sums in a0 and an are finite. If C(x) is a quadratic irrational, because Cn(x) converges very fast, the Eigenspectrum can be approximated by a finite pulse train, consisting of the first m convergents Cm(x), in which case a0 and an can again be approximated by a finite summation. In the subsequent Maple approximation example m is the cardinality of the sequence list.
14. 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.
15. 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 T0~1/Cn(x).
16. Geometer. Creator of the program EucliDraw.
17. 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.
18. 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 an instead of the terms an themselves.
19. The author's general surgeon. Field: Medicine.
20. Has helped with the author's research on Tetration.
21. Has compiled a colossal amount of computational and mathematical data on his web site.
22. The most prolific author in the area of Complex Dynamical Systems.
23. Has compiled a colossal amount of light and engineering data on his web site.
24. Referee of the journal Complex Variables.
25. Has compiled a colossal amount of light-engineering data.
26. 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.
27. The eigenspectra of the metal ratios curiously contain resonance lines which match actual lines in their real spectra.
28. Frequent participant on the newsgroup sci.math.
29. 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.
30. Field: Light-engineering.
31. Professor at the University of Crete. Served as advisor for several of the author's papers.
32. Note that Q|(P2-D):
>Pn:=numer(zeta);
>Q:=denom(zeta);
>Dr:=op(2,Pn)^2;
>P:=op(1,Pn);
>(P^2-Dr)/Q;
292677. 