Minimal Impact One-Dimensional Arrays

: In this contribution, we consider the problem of finding the minimal Euclidean distance between a given converging decreasing one-dimensional array X in ( R + ) ∞ and arrays of the form 𝐴 (cid:3028) = (cid:4678)𝑎, 𝑎, … , 𝑎 (cid:4579)(cid:4583)(cid:4583)(cid:4580)(cid:4583)(cid:4583)(cid:4581) , 0,0, … (cid:4679) , with a being a natural number. We find a complete, if not always unique, solution. Our contribution illustrates how a formalism derived in the context of research evaluation and informetrics can be used to solve a purely mathematical problem.


Introduction
Let (R + ) ∞ be the positive cone of all infinite sequences with non-negative real values. Elements in this cone will be referred to as one-dimensional arrays, in short, arrays. We recall that any finite sequence with non-negative values can be considered as an element in (R + ) ∞ by adding infinitely many zeros. Let X = ( ) , … and Y = ( ) , … be elements of (R + ) ∞ , then X ≤ Y if for all r = 1, 2, …, xr ≤ yr. Equality only occurs if for all r, xr = yr. In this way, (R + ) ∞ becomes a cone with a (natural) partial order ≤. An array X = ( ) , … in (R + ) ∞ is said to be decreasing if for all r = 1, 2, …, xr ≥ xr+1.
We recall the definition of the h-index as introduced by Hirsch [1]. Consider, ( ) ,…, , the list of received citations of the articles (co-) authored by scientist S, ranked according to the number of citations each of these articles has received. Articles with the same number of citations are given different rankings. Then, the h-index of scientist S is h if the first h articles each received at least h citations, while the article ranked h + 1 received strictly less than h + 1 citations. Stated otherwise, scientist S' h-index is h if h is the largest natural number such that the first h publications each received at least h citations.
This index, although having many disadvantages in practical use ( [2,3]), has received a lot of attention. At this moment [1], it has received already more than 4300 citations in the Web of Science. Because of these disadvantages, many alternatives have been proposed, among which the most popular is the g-index, introduced and studied by Egghe [4]. This g-index is defined as follows: as with the calculation of the h-index, articles are ranked in decreasing order of the number of citations received; then, the g-index of this set of articles is defined as the highest rank, g, such that the first g articles together received at least citations. This can be reformulated as follows: the g-index of a set of articles is the highest-rank g such that the first g (>0) articles have an average number of citations equal to or higher than g. Indeed, ∑ ≥ ⇔ ∑ ≥ . For more information on the h-index and related indices, we refer to [5][6][7].
In [8], we defined the h-and the g-index for infinite sequences as follows:

Definition 1. The h-index for infinite sequences:
Let X = ( ) , … be a decreasing array in (R + ) ∞ . The h-index of X, denoted h(X), is the largest natural number h such that the first h coordinates each have at least a value h. If all components of a decreasing array X are strictly smaller than 1, then h(X) = 0. We will further consider only arrays X with at least one component larger than or equal to 1, hence with h(X) ≥ 1.
Note that an h-index is defined here only for decreasing arrays (although a generalization exists, see [9]). The same remark is valid for the other indices used in this article.
Similarly, a g-index has been defined in [8] as follows: The g-index for infinite sequences: Let X = ( ) , … be a decreasing array in (R + ) ∞ . The g-index of X, denoted gX, is defined as the highest natural number g such that the sum of the first g coordinates is at least equal to g 2 or, equivalently, if the average of the first g coordinates is at least equal to g.
Notation. We denote by [[a,b]] for a, b natural numbers such that a ≤ b, the intersection of the real-valued interval [a,b] and N, the set of natural numbers.

Introducing the Research Problem
Definition 3. For each natural number a > 0, we define the minimal impact array of level a, denoted as Aa, as follows: = , , … , , 0,0, … It is easy to see that Aa is the smallest array X (for the partial order ≤) for which h(X) = g(X) = a. We note that the sequence (An)n is increasing for ≤.
We say that an array X is l 2 -converging if ∑ is finite. As we only use this form of convergence, we will further on omit the specification "l 2 " and simply say converging.
Next, we formulate the research problem of this contribution.

Research Problem
Given a converging decreasing array X in (R + ) ∞ , find the largest natural number a such that the Euclidean distance d(X,Aa) is minimal.
We note that the analogous problem for differentiable functions Z(r) and a real number a has already been studied and solved in [10]. We further note that the requirements to be decreasing and convergent are independent. Indeed, if a decreasing array is convergent and we add its sum (or a larger number) to any term, except the first, then the resulting array is still convergent but not decreasing anymore. Further, the array with terms √ is decreasing but not convergent.
Minimizing d(X,Aa) is the same as finding a minimal value for the function Equation (1) shows why we need convergent arrays. Note also that a minimal value a depends on X. Hence, we write it as aX. It is trivial to see that if X = Ab for some natural value b, then b = aX (for this X) and f(b) = 0. It is clear that arrays X of the form Ab are the only ones for which the corresponding function fX becomes zero.
This leads us to the following questions: 1. Does aX exist for each X, converging and decreasing in (R + ) ∞ ? 2. Given X, converging and decreasing in (R + ) ∞ , how do we find aX (if it exists)? 3. If aX exists, is it unique?

Characterizing the Minimum of fX
Taking into account that aX is possibly not unique for some X, we want to characterize aX-if it exists and is strictly larger than 1-as the largest natural number such that Note that if the minimum of fX occurs in two (or more) natural numbers, we choose the largest one. We still have to show that inequality (2) actually characterizes the minimum we are searching for. Indeed, theoretically, it may happen that the function fX(a) decreases first to a (local) minimum b, then increases again, and then decreases to a lower minimum value than the one in b. This might, in theory, even occur infinitely many times. We will prove that this behavior does not occur. Moreover, if we want to use inequality (2), we first have to deal with the case aX = 1, as this case is not covered by inequality (2) If aX = 1, then fX(1) < fX (2). This inequality is equivalent to -2x1 + 1< -4(x1 + x2) + 8 or 2x1 + 4x2 < 7 or 2x1 + 4x2 -7 < 0.

The Generalized Discrete h-and g-Index
We next show that aX exists for each converging and decreasing X in (R + ) ∞ . For this, we recall the definitions of the generalized discrete h-and g-index [11].
We note that if a and b are natural numbers and X = (x ) , … is a decreasing array in (R + ) ∞ , then the property for r ≤ a: ∑ x ≥ θa implies that gθ(X) ≥ a; similarly, the property for r > b: We finally also define the discrete f-index, already introduced in [10], for the continuous case.
In [10], we found that in the continuous case, the solution of our problem was obtained as f(3/4)(X) (where f is the continuous analog of the discrete f-index introduced above). We will show further on that this is not the case for the discrete case studied here.

Excluding the Theoretical Case of Infinitely Many Minima
Next, we need two lemmas.

Lemma 1.
If X is decreasing, then with = ∑ is also decreasing. This decrease is strict if x1 > x2.
Proof of Lemma 1. The easy proof is left to the reader. □ As for given X and n > 0, xn = θn, for = , it is clear that ℎ ( ), > 0 = (where n = 0 is reached for θ > x1). Now, we prove a similar result for gθ(X).

Theorem 2. Given X is decreasing and convergent and a > g(0.5)(X), then fX(x) is strictly increasing for x > a.
Proof of Theorem 2. From Lemma 2, it follows that there exists θ0 < 0.5 such that = ( ).

It follows from Theorem 2 that if aX exists, it belongs to [[1, g(0.5)(X)]]
, which excludes the theoretical case of infinitely many minima.

Excluding the Case of More Than One Minimum
Next, to exclude the case of a local maximum, following a first local minimum, we continue as follows.

Theorem 3. For all X, decreasing and convergent in (R + ) ∞ and for all a ∈
, we have the following property:
We next reformulate inequality (2), leading to a refinement of the previous observation.

Proof of Theorem 4. From Equations
. □
As aX is the largest natural number with this property, this ends the proof of Theorem 5. □
We already observed that g(3/4)(X) can be smaller than, equal to, and larger than aX. We next show that aX ≤ g(3/4)(X) +1.

An Upper Bound for aX
We already know that g(3/4)(X) is not an upper bound for aX and that g(0.5)(X) is. Hence, we wonder if there a number strictly between 0.5 and 0.75 that leads to an upper bound for all X.

Theorem 6. An upper bound for aX is provided by g(7/12)(X).
Proof of Theorem 6. Take a ≥ gs(X), with s being any real number strictly smaller than 0.75.
We already observed that if aX = 1, then h(X) = 1. What about the converse? The next proposition answers this question.

Applications
First, we give a new characterization of the classical h-index [1], i.e., the case θ = 1.

Proposition 5. Given X decreasing and convergent, then h(X) = max{a ∈ N ; Aa ≤ X}.
Proof of Proposition 5. Writing h(X) simply as h, we see that Ah ≤ X because for Ah and j ≤ h, xj ≥ h, while for all j > h, xj ≥ 0. This shows that h ≤ max{a ∈ N ; Aa ≤ X}. Now, let am = max{a ∈ N ; Aa ≤ X}. Then, we see that for all j ≤ am, xj ≥ am, while for all j > am, xj ≥ 0. As h is defined as the largest number with this property, we see that h ≥ am = max{a ∈ N ; Aa ≤ X}. This proves this proposition. □ Before continuing with the next proposition, we recall the definition of the majorization partial order for finite sequences.
Definition 5. The majorization order [12]: Let X, Y ∈ (R + ) k , where k is any finite number in N0 = {1,2,3, … }. The array X is majorized by Y, or X is smaller than or equal to Y in the majorization order, denoted as X -< Y if for all i = 1,…,N:

Proposition 6. If X is finite with length N and ∑
where -< denotes the majorization partial order.

Theorem 7. If X < Y, then aX ≤ aY.
Proof of Theorem 7. We know that aX is the largest index such that As aY is the largest index with property (12), this shows that aX ≤ aY. □ Remark 2. If X < Y (strict), then it is possible that aX = aY. An example is given by X = (6,1,1) < Y = (6,2,1), for which aX = aY = 3.

Conclusions
In this article, we studied the following problem: Given a converging decreasing array X in (R + ) ∞ , find the largest natural number a such that the Euclidean distance d(X,Aa) is minimal.
We have shown that this problem has a solution, which is always situated in the interval ℎ ( ), ( ) . Yet, the solution is not necessarily unique. It was shown that a discrete and an analogous continuous problem have related but not the same solutions. Our contribution illustrates how a formalism derived in the context of research evaluation and informetrics [1] can be used to solve a purely mathematical problem.