The dual of Bradford's law

The Dual of Bradford’s Law

L. Egghe LUC Universitaire Campus, B-3610 Diepenbeek, Belgium*

In this article, we examine the classical law of Bradford. This law yields groups with an equal number of articles, but where the number of journals increases geometri- cally. Within each group, and starting with the last ones (the least productive journals) we examine the maximal productivity of the journals. We describe, using only y,,,, the maximal productivity (of the journal of rank one), all the possible productivities of the journals in every Brad- ford group.

The same method shows that the most productive journal in every group p (starting with the last group) produces a number of articles mP, where:

kP mP 3 -

eE

where k is the Bradford multiplicator and E is the number of Euler. Hence, the maximal journal productivity in each group forms an approximate Bradford law with fixed universal constant e-E = 0.56. We can say that the dual law of a Bradford law is an approximate Brad. ford law.

This approach is not a pure rank method (as is Brad. ford’s law), nor a pure frequency method (as is Lotka’s law), but a frequency method within a rank method.

The formula for m,, gives a theoretical formula (and hence an explanation) for k, the Bradford multiplier, which is easily applied in practical data. It also sheds more light on the Yablonsky-Goffman-Warren formula fork, which has only been established experimentally.

Introduction

The classical law of Bradford [l] is well-known and re- peated in many articles. However, for the sake of com- pleteness and also to fix the notation, we repeat it here. When we fix a certain subject and when we arrange the journals in order of decreasing productivity, then we can

Key words and phrases: Bradford law, maximal productivity, Brad-

ford groups, dual.

*Permanent address: UIA Universiteitsplein 1, B-2610 Wibrijk, Belgium

Received February 25, 1985; accepted October 30, 1985.

0 1986 by John Wiley & Sons, Inc.

form groups, starting with the most productive journals,

of equal article production y,, but with a number of journals, respectively.

ro, r,k, r,,k2, . . ., r,ki--l, . . .

where r, and k are fixed. The Bradford multiplicator is called k. This approach is obviously a rank approach.

A typical frequency approach-but formally equivalent to the above law-is the law of Lotka, stating that:

f(n) = $

wheref(n) is the relative frequency of the journals with n articles. Since f( *) must be a distribution, one has:

i f(n) = 1. iI=1

Hence:

SO:

c= l 6 ~ = - = 0.608,

.fL I?-

12 = 1 n 2

which is easily shown using Fourier-covergence theory (but this is not important here).

In an interesting article of Yablonsky [2] it is shown that:

yo = nm logk

r, = (k - l)? Ynl

where y,,, is the number of articles in the most productive journal (i.e., the journal with rank one) and n, is the

JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE. 37(4):246-255,1986 CCC 0002-8231/86/040256-05$04.00

number of journals with one article (see also [3]). Using Lotka’s equivalent law, one sees that:

n1 =yf,

So (1) and (2) become:

yo = Yi log k (1’)

r. = (k - l)y, (2’)

For k one uses the experimentally established formula:

k = $ (log ym + E)

where E = 0.5772 . . . is Euler’s number. (I.e.: Goff-

man-Warren establish experimentally that k = A/J, the average number of articles per journal; from this (3) is easily shown: cf. [3] or further on in this article). Later in this article we shall also give some critiques on this formula).

At the end of this article, we show a theoretic formula for k which contains a parameter, giving for the first time a theoretical explanation of the possible values of k. This formula for k is derived from the results we will prove in this article. A description of the method follows. (For other aspects concerning the laws of Lotka and Bradford

and for a comparison with other laws, for instance

Leimkuhler’s law, see ref. [3].) No doubt, the Bradford formulation is a rank ap-

proach of this phenomenon (starting with the lowest

ranks r = 1, 2, 3,. . .) whilst the Lotka formulation is a frequency approach (starting with the lowest frequencies

n = 1, 2, 3, . . .). In this article we start with the Bradford formulation

(i.e., the rank approach), but build in it a frequency approach in the following way: we are interested, starting from the last Bradford groups, in the possible number of articles in each journal in every Bradford group. We want

to describe it, using only the known number y,,,. We end

up with a complete classification of the possible number of articles in each journal in every Bradford group. This is the first purpose of this article.

Furthermore the same method yields an analytic expression for:

mP = the maximal number of articles in a journal in

group P,

where we start numbering the groups from the last group (with the least productive journals) onwards. We find that:

kP mp = -

eE = 0.56 kP

where E = 0.5772. . . is the number of Euler, and where

the approximation is better for larger p. We may say that this “dual” approach yields an “approximate” Bradford law with the same multiplicator k as in the original for-

mulation, but with a universal constant e-E 5: 0.56 in-

stead of r,. This result, in turn, has the already-mentioned appli-

cation-proving a theoretical formula for the Bradford multiplier k, yielding for the first time a theoretical ex-

planation of k.

II. The Mixed Approach

To simplify thgnotation, let us agree to denote with the symbol iYp the maximal number of articles in a journal in group p where group 1 is the last group in the Bradford

formulation, group 2 the second to last group, and so on. For the moment we do not determine how these Bradford groups are formed. This is discussed later in this article. Here, we just fix any Bradford division.

A. The Case That #I = 1

In this case the last group contains only journals with a productivity of one article on the fixed subject. This can only happen if:

yo 5 n1

So, using (1):

nl log k I nl

k s e = 2.718 . . .

or:

YIH = eh2/6)k s e(r2/6)e = 87.48.

So, yn, 5 87 is needed for this situation. Now #2 # 1 since this would be a contradiction of Bradford’s law: If

#z = #r = 1, the two last groups contain only journals

with one article, yielding the same number of journals (since all groups have the same number of articles), thus the contradiction of Bradford’s law. So, #* 1 2. Al- though this is always the case, we find a condition on k (hence on y,,). We must have here:

y,, = n1 - nl log k + F.2.r

where x > 0, according to Lotka’s law (now using abso- lute frequencies). Indeed, f(2) = ni/2= = ni/4 is the number of journals with 2 articles according to Lotka’s

law. From these journals, as assumed (#2 2 2), a positive fraction (x) must be in group 2. Hence, we have here (n i/4)x journals, or (n r/4) * 2 *x articles. Also in group 2

JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE-July 1986 247

are the remaining journals with one article which are not in group 1. They total a number of II, - II, log k journals (or articles). This explains the above formula. So:

2yf)=n* +$x

or:

and:

or:

x= 2(2Yzl log k - nJ

>o n1

log k > $

k > 1.65.

Hence:

Y,, > e (d/6)1.65 = e2.714... = 15.09.

We conclude that ym I 15 never yields #i = 1, and not even a Bradford law (as can be seen further on, in consid-

ering the following cases, we need higher y,, in order to satisfy Bradford’s law).

Of course, it can be that #2 2 3. For this we need, applying the same reasoning as in the case #2 1 2, that:

y,=nl-nn110gk+~*2+~~3xx

with x > 0. We find:

x= 12n, log k - 9nl > o

2n1

log k > $

k > 2.117

y, > 32.5,

so:

ym 1 33.

In the same way, we find for #2 2 4 the condition:

k > 2.501

y, > 61.2

so:

Y n, 2 62

#2 2 5 can never happen since for this to happen we find the condition (in the same way as above):

k > 2.834 > e

contradicting k I e. In conclusion, if the last group con-

tains only journals with only one article (only if ym I 87), the second to last group cannot contain only journals with one article; it contains at least journals with:

1 or 2 articles if y,,, 2 16

1, 2 or 3 articles if ym 2 33

1, 2, 3 or 4 articles if y, I 62;

it never contains a journal with 5 relevant articles.

The third to last group (and other groups) will be con-

sidered later.

Remark: After this part of the study it is already clear

that the reasoning in [4] on p. 1206 is not correct: they only consider the possibility that, supposing #i = 1, that #z = 2, which is incorrect, since in their examples, ym =

66 respectively ym = 325; the first example falls into this

category, but with # 2 = 4; the second example doesn’t fall in this category (see the next subsection): Their second example falls into the category #, = 2 and #2 = 6.

Also, it must be remarked that their examples do not re- ally follow a Bradford law, since, for instance, 12 i # yi.

Remark: The calculated values of y,,,, using the experi- mental formula ym = e(a2’6)R in Section II are only illus- trative. They are not needed to develop the theory itself. For this, only the inequalities involving k are important; which follow from reasoning. Furthermore, it will be shown in Section III that the above formula is very good.

B. The Case that #, = 2

Now the last group contains journals with one or two

relevant articles. According to the first subsection, k L e, or y,, I 88, but we must express that no journals with three articles appear in this last group. So there must be

the relation (using Lotka):

y, 5 nl + 7.2.

Hence:

3n, n1 log k 5 nl f nl = -

2 2

log k 5 ;

248 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE-July 1986

or:

or:

k I 4.48 . . .

so:

y, I e(r2’6)k = 1590.9

y, 5 1590.

What fraction CY of the journals with two articles do we

have in this last group? We have:

Hence:

nl log k = n1 + $a

CY = 2(log k - 1).

We have now #2 > 4. This can be shown as follows: The condition #2 > 4 gives the condition:

with x > 0. So, using cx = 2(log k - 1) and y,, = nl log k:

X

which is always

not always have yielding:

4 2y,-+ ( > = >o

nl

log k > $ = 0.917

the case, since k 1 e. However, we do #2 > 5. For this, we must have x = 1,

log k > ; > 1

General Reasoning, for #, = 2 We have #2 L n if:

nl y, = 4.2 - F.2.o + 7.3 + 5.4 16

+ --- + (n Tl)2(n - 1) + :*n*x

with x > 0. This follows from the same reasoning demon- strated several times above. We repeat the argument for the sake of clarity.

From above: there are CY journals with 2 articles in group 1 (#i = 2). The rest are in group 2. This stands for

n1/4 - (n1/4)a! journals (according to Lotka’s law

f(2) = n,/22), and hence for (n1/4)*2 - (n1/4)*2*cu articles. For #2 2 n(n = 3, 4, . . .) we also need all the

journals in group 2 with 3, 4, . . ., n - 1 articles, and a positive fraction (x) of the journals with n articles. Lotka’s law again states that for every i, f(i) = n,/i2, the number of journals with i articles, standing for

(n ,/i2) * i = n l/i articles. So there are:

y.3 + s.4 + . . . + (n “’ 112 (n - l)

articles in group 2, coming from the journals with 3, 4 , - f ., n - 1 articles, and:

nl

nznex

articles in a positive fraction of the journals with n arti-

cles. All these numbers add up to the above formula. Using CY = 2(log k - 1) and y, = n I log k we find the

condition:

/ n-l ,\

n 2y,- k

ni C Z- i=l i )

x= > 0. n1

so:

or:

Y, > e (~~/6)el/2,1/~.,.,l/~l~~-1)

For instance, for #2 L 6: k > 3.126 or y,,, 2 172; for #2 2 7: k > 4.325 or y m 1 1231. We can go on until 12 is so large that log k > 3/2.

The last possibility is #2 = 11 yielding log k = 1.4645

(for #2 = 12 we find the condition log k = 1.51 > 1.5, contradicting the assumption in subsection B).

In conclusion, if the last group contains only journals with one or two relevant articles (this is so if ynr > 1590), we have in the second to last group #2 > 4 always. Fur- thermore, #2 2 n if:


or: with x > 0. Using (Y above, we now have:

Y.~ > e (n2/6)el/Zel/4...pl/2(n-1)

which can be satisfied up to n = 11. In this case we never have a journal with 12 articles in the second to last group; hence#2 I 11.

n - 1

n 2nllogk-nl c L i=i i >

x= > 0. ni

so: C. General Case #! = m

Now we generalize the study in supposing that the last group contains journals with 1, 2, . . ., m relevant articles. This can only be if:

and:

yO>nnl +?+2+ *a* + (m “_’ I)2 (m - 1)

(expressing that in the last group at least one journal ap-

pears with m relevant articles whilst no journal with m + 1 relevant articles appears). Using y, = n, log k we so have:

m-l

c L c log k 5 ; 4. i=i i i=l 1

This is a condition for m. We work more on this relation

further on in an even more general case. What fraction of the journals with m relevant articles

are in the last group? For this we have, with the same reasoning as explained above:

So, since y0 = n 1 log k:

(

m-l

a = m log k - iC, A- i >

Now we express the general condition that in the second last group we have journals with m, m -I- 1, . . . , and n relevant articles (studying completely the cases #, versus #2). so:

Y, = sm - sma + (m “+’ 1)2 (m + 1)

+ ni . . . + (n “’ 1)2

(n - 1) + 712.x

or:

y, > e(n*/6)e”‘e”4...e”z(fl-‘J

It is remarkable that this depends only on n and not on m.

What happens now in the third to last group, the fourth to last group, and so on? For this we work out the most general case.

D. Most General Case: #P versus #P.+$ .J

This case can be applied to every situation, (including the previous ones). The previous simpler cases contribute to the understanding of the general case. Here we con-

sider the pfh last group (p = 1, 2, . . .) and suppose that it contains journals with a number of relevant articles respectively mP-j, mpAl -I- 1, . . ., mP. We then study conditions in order to have in the (p + 1)“’ last group, journals with a number of relevant articles resp. m,,, mp + 1,. . ., mP+l. So, we have here the situation #,, = mP, p = 1, 2, 3, . . . and study mP.

We are first interested in the fraction CY~(P = 1, 2,

3 . .) of the journals with m,, relevant articles, that be-

long to the pfh last group. We have:

(*I y,=~-Ltp~,+ n1 P mp--l rnpel + 1

+...+ It’

mP

_ 1 + sap. mP

Indeed, there are (nl/mi-,)m,-, articles from journals with mp-i articles; from this, a fraction alp-i belongs to the (p - 1)“’ last group. In C, we calculated already (with m = ml in our general notation):

(**I ml-1

a = CY, = ml log k - ic, f >

We find with (*) and (**) for 02, also using y0 = n i log k


In order to have in the pih last group at least one jour-

nal with mP relevant articles (i.e. #P = m,,). Supposing #),-I = m,,-,, we have the condition:

so:

nl-1

2y,=nl C L+zcf2 i=l i m2

+...+ n’ m2 - 1

+ -L2. m2

so:

logk,~m~‘~ P i=i i

So we propose, for every p:

q--l ap = mp P log k - ?, ?.

i >

which we shall prove by induction. Suppose:

ap--1 = mp-, (p - 1) log k - “%~’ +).

We have, for the pth group:

y”=2.L- Hi ----ape] +

nl

mp-i mp-i m,-I+1

+...+ Iz’ +zLap.

mP -1 mp

so:

yu=2L- mp-i

*mu-1 m,-1-l

mp-i (p - 1) log k - i”, f

+ ni mpPl + 1

+ . . . + Iz’ mP

+Nlap -1 mP

from which it follows, using y, = n, log k, that:

(YP ZZ L . i

Since we verified the formula already for 02, we have thus shown that for every p, the fraction op of the journals with p relevant publications, appearing in the prh last group, is given by:

up = mp

So again we see that this condition is independent of m,,-1, a remarkable fact.

We now come to the most important part of this arti-

cle: finding an expression of mp. In order to express that the pth last group has at least

one journal with mP relevant articles whilst there are no journals with mp + 1 relevant articles, we have respec-

tively:

ni Yo>-------

n1 -ap-, +

ni

mp-i mp-i mpPl + 1

+...+ n’ mP -1

ni y,~----- ni

mp-i -ap-, + ni

mp-i mpwl + 1 + . . . +nl.

mP

Using oP-, = m,-,((p - 1) log k - C~!“=P;‘-’ l/i) and

y0 = n r log k, we find the double inequality:

which is in fact a condition for mp. Using the fact that:

~L=logm+E i=l i

(E = number of Euler = 0.5772. . .) for m large, we see

that:

log k f +- (log mp + E)

so:

kP mp = -

eE

We note that this approach is already good for mP not too high. Consider the following table.


” log n ,i,$ --E A

1 0 0.42 0.42 2 0.69 0.92 0.23 3 1.1 1.25 0.15 4 1.39 1.50 0.11 5 1.61 1.70 0.09 6 1.79 1.87 0.08 7 1.95 2.01 0.06 8 2.08 2.14 0.06 9 2.2 2.25 0.05

10 2.3 2.35 0.05

Already from the second or third last group in the Brad- ford division one has mp > 10, so that the approximation is quickly obtained.

We may conclude that this “mixed” vision of the Bradford law (i.e., a frequency approach in a ranked ap-

proach) produces some new findings. We may say that the maximal productivity of a journal in each Bradford group follows an approximate Bradford law, with the

same multiplicator as in the original Bradford law, but with universal constant eeE 5: 0.56.

III. Application

A. A Theoretical Formula for k

This newly established law:

kP mp = -

eE

yields already an application-a theoretical explication of the possible values of the Bradford multiplier k. We

are indeed using the phrase “possible values of k” in- stead of “value of k” which is used in [5] and [6]. Indeed, in [5] it is remarked that k = A/J where:

A = the total number of articles

J = the total number of journals

From this it follows [6], using Lotka’s law, that:

d&g ym + E) a2

Yii 6

= 3 (log ym + E).

We note that Yablonsky in [2] ends up with k = 6/7r2 log

y,,,, due to a less accurate estimation. A private discus- sion with Professor A. Bookstein of the University of Chi- cago revealed, however, the following: The above formulas for k cannot be good from a theoretical point of view.

Indeed, since the core rO is, in principle, free to choose [ 1,4] and, since in relation to this, k is variable, we need a formula where an additional parameter appears. We remark that y,,, is fixed from the available data and hence is

not a parameter. Another drawback of the above formulas is the fact

that they are based upon the relation k = A/J which was

only experimentally explained in [41. For these reasons we started an investigation for a

mathematically proved formula for k which can also be

accepted theoretically and experimentally. In fact, this is not so difficult anymore, now that we

can use the formula:

kP mp=----.

eE

The point is that only one mp is known in advance (from a bunch of data).

Indeed, let pm be such that rokPm is the number of journals in the last Bradford group. So there are pm i- 1

Bradford groups. Then we have:

by definition of ym and mp. Hence:

k&n+’ Yf?? = mPn,+i = -

eE

or:

k = (eEy,,,)"Pm+'. (4)

Now, since the formula for mp is not so good for p small, we must assume in the above formula for k that y, is not too small. (small y,, leads to a small pm obviously).

Inspired by the formula k = A/J in [4] (which we do

not use here), we started also investigating the value A/J. We find:

A (Pm + UYO -= J ro

Since y, = y; log k duction), we have:

+ r,k + * * * -I rokPm ’

A -= J

Using once more:

and r. = (k - l)y, (see the intro-

(p, + UY, log k kPm+'-1 '

k&n+’ Yn2 = mPm+i = -

eE


we find: From a mathematical point of view the co-occurrence of

the universal constants e. 7r and E is remarkable.

A -= k~rn+~,, + 1) log k

J eE(kPm+l - 1)

_ I (pm + 1) log k A

J eE

From this, another formula for k follows:

$ (log Y, + El = (pm + 1) log k

eE

so: 6eE(logy,+E)

k =e **(P”,+l) (5)

We see now that in both formulas (4) and (5) an additional parameter pm appears as was required. We can

also express the values of r0 in terms of this parameter

(using the first formula for k):

r. = ((eEym)l’Pm+l - l)y,.

Let us compare the two formulas (4) and (5) for k, where we take pn, = 4 (so we take 5 Bradford groups, which is

often the case). In this case we also obtain a good resem- blance with the formula:

k = $(log y, + E).

It is our guess that the above formula of Yablonsky can be used, but only if we have 5 Bradford groups. This will also be investigated further on in the next section.

YO, Wy,,,YS 6(log y, + E) 6eqlogy”,+E)/sr2

d e

30 2.21 2.42 2.52

40 2.35 2.59 2.65

50 2.45 2.73 2.77

70 2.62 2.93 2.95

90 2.76 3.09 3.09

110 2.87 3.21 3.20

130 2.97 3.31 3.31

200 3.24 3.57 3.59

300 3.51 3.82 3.88

400 3.72 3.99 4.10

500 3.89 4.13 4.28

1000 4.47 4.55 4.91

Hence, we have now established a theoretical foundation of the value of the Bradford multiplicator k, based on our formula:

k” mp = -.

eE

B. Evaluations of the Expressions for k

In this section it is our purpose to apply the formulas for k to the case we have a perfect Bradford distribution; i.e., a Lotka law (see the introduction):

f(y)= Y, 2 ( > Y

Indeed, the validity of the expression for k must be checked on correct examples. We work out y, = 30 and

Yl?Z = 50. We always use the formula:

yo =~t log k

and

r, = (k - l)y,

and use for k the 3 different values from the preceding table. First Example f(y) = (30/y)*. We approximate f (y) in the usual way to obtain a whole number. We have the following table:

y(# Articles) # Journals

30 1

29 1

28 1

27 1

26 1

25 1

24 2

23 2

22 2

21 2

20 2

19 2

18 3

17 3

16 4

15 4

14 5

13 5

12 6

11 7

10 9

9 11

8 14

7 18

6 25

5 36

4 56

3 100

2 225

1 900


(a) k = (eE 3O)1’5 = 2.21

Here y, = 714 and F-~ = 36. So, as always, we start for the first zone with the first 36 journals. This yields 708 articles. Then, for the other zones, we approximate as

close as possible the average of 708 and 714. We find:

Zone # Articles #Journals

1 708 36

2 715 81

3 713 180

4 714 435

5 717 717

k

-

2.25

2.22

2.42

(b) k = (6/n*)(log 30 + E) = 2.42

Here y, = 795 and r0 = 43. Proceeding as in (a) we find:

Zone # Articles # Journals k

1 799 43 -

2 794 105 2.44

3 794 261 2.49

4 795 655 2.51 5 385 385 -

(c) k = e6eE(~og30+~)/~~2 = 2 52

Here y0 = 832 and r0 = 46.

Zone # Articles #Journals

1 835 46

2 833 117

3 833 303

4 832 749

5 234 234

k

-

2.54

2.59

2.47

Second Example f(y) = (SO/y)*. Now we use the following table:

y(# Articles) #Journals

50 1

49 1

48 1

47 1

46 1 45 1

44 1

43 1

42 1 41 1

40 2

39 2

38 2

37 2

36 2

35 2

34 2

33 2

32 2

31 3

30 3

29 3

28 3

27 3

26 4

2.5 4

24 4

23 5

22 5

21 6

20 6

19 7

18 8

17 9

16 10

15 11

14 13

13 15

12 17

11 21

10 25

9 31

8 39

7 51

6 69

5 100

4 156

3 278

2 625

1 2500

(a) k = (eE 50)“’ = 2.45

Here y0 = 2240 and r, = 73. We find:


1 2229 73 -

2 2240 179 2.45

3 2233 437 2.44

4 2244 1123 2.51 5 2250 2250 -

(b) k = (6/?r2)(log 50 + E) = 2.73

Here y,, = 511 and r, = 87. We find:


1 2496 87

2 2508 237 2.72 3 2512 648 2.73 4 2510 1920 2.96 5 1170 1170 -


tc) k = e6eEb3g50+EvSn2 = 2.77

Here y, = 2547 and r,, = 89. we now have:

Zone # Articles # Journals A

1 2532 89 -

2 2542 245 2.75

3 2546 690 2.82

4 2546 2008 2.91 5 1030 1030 -

IV. mp In Function of y,,,

The two previous paragraphs yield the following theoretically established formula of mp in function of ym:

mp = e 6eEp(logy,+E)/*2(p,+I)--E

an intricate, but easily calculated formula. So, for the values of y, in the previous table,

mp = (e6eE(~wy~+W*2)p~ 0.56

where we can use the last column in the table in Section

III, Part A. From this, we can derive some practical ta- bles such as the following:

Ym = 110

( pth La: Group)

Number of Articles in the

nrp Journals in This Group

1 1.79 1, 2 2 5.73 2, 3, 435, 6 3 18.35 6 . ., 19

4 58.72 15:. . .,59

5 187.90 59,. . 110 .(

Yl?l = 300

( pth Las’I Group)


mP Journals in This Group

1 2.17 1, 2, 3

2 8.43 3, 4, 5, 6, 7, 8, 9 3 32.71 9, . . .,33 4 126.92 33, . . ., 127

5 492.43 127, . . .,300

Ym = 500

( pth La: Group)


tn,, Journals in This Group

1 2.40 1, 2, 3 2 10.26 3, 4, 5, 6, 7, 8, 9, 10, 11 3 43.91 11, . . ., 44

4 187.92 44, . . ., 188

5 804.28 188, . . .,500

Remark: The above theory can also be worked out if we

do not have the Lotka law:

f(Y) = 5

but the more general Lotka law:

f(y) = -5 Y”

with o # 2. In practice, we mostly have o E (1, S, 3). In fact, the same methods apply, without any addi-

tional difficulty. Now we obtain (with the same notations

as before):

ayP = mg-l

(

p$y - y’ 1

i=l iCi-1

>

yielding the condition (for m,):

so:

However, the condition for mp in function of y, is not so clear, since there is not an easy approximation of the se- ries C,C=, l/i”-’ (being the classical zeta-function) for a f 2.

Of course, it must also be remarked that for (Y f 2 we do not have the classical Bradford law, so that also from a

theoretical point of view we cannot expect to construct exact Bradford groups.

References

1. Bradford, S. C. “Sources of information on specific subjects”, En-

gineering, 137: 85-88; 1934.

2. Yablonsky, A. I. “On fundamental regularities of the distribution

of scientific productivity”, Scientometrics 2, 1:3-34; 1980.

3. Egghe, L. “Consequences of Lotka’s law for the law of Bradford”,

Jourd of Documentation, 41(3):173-189; 1985.

4. Goffman, W. and Warren, K. S. “Dispersion of papers among

journals based on a mathematical analysis of two diverse medical

literatures.” Nature, 221:1205-1207; March 1969.

5. Bookstein, A. “Robustness Properties of the Bibliometric Distribu-

tions.” Preprint 1984.

6. Drott M. C. “Bradford’s law: theory, empiricism and the gaps be-

tween”, Library Trends, 30(1):41-52; 1981.


Documents

The dual of Bradford's law