Mathematics of Harmony as a New Interdisciplinary Direction and “Golden” Paradigm of Modern Science. Alexey Stakhov
be the segment of the unit length Δ = 1, that is,
Let’s now consider the second situation X C1C2 among the situations (1.25). In this situation, after the first step, we have the (n – 1) steps and one IE, which is to the left of the point X;all other IE are to the right of point X and, according to the restriction S = 1, cannot participate in the measurement process. Then we can apply to the segment C1 C2 the optimal (n – 1, 1, 1)-algorithm, which, according to the “inductive hypothesis” (1.24), divides the segment C1C2 on the F(n –1, 1) equal parts of the unit length Δ = 1, that is,
Let’s now consider the situation X
Finally, in the last situation X CkB among the situations (1.25), all the k IE are to the left of point X; this means that we can enclose to the segment CkB the optimal (n– 1, k, 1)-algorithm, which, according to the “inductive hypothesis” (1.24), divides the segment CkB on the F(n – 1, k) equal parts of the unit length Δ = 1, that is,
Taking into consideration the relation (1.26), as well as the expressions (1.27)–(1.30), we can write the following recurrent relation for the (n, k)-exactness of the optimal (n, k, 1)-algorithm:
Now, we consider the sum:
taken from the expression (1.31). According to the recurrent formula (1.31), the sum (1.32) is equal to F(n, k – 1), that is,
By using (1.33), we can simplify the recurrent relation (1.31) and write it in the following compact form:
Now let’s construct the table of the numbers of F(n, k)by using the recurrent formula (1.34). To do this, we find out the extreme values of the function F(n, k), corresponding to the values n = 0 and k = 0, that is, the values of F(0, k) and F(n, 0). Recall that F(0, k)is the (0, k)-exactness of the optimal (0, k, 1)-algorithm, and F(n, 0) is (n, 0)-exactness of the optimal (n, 0, 1)-algorithm. But, according to our definitions, (0, k, 1)-algorithm is the (n, k, 1)-algorithm, in which the number of steps is equal to n = 0, and (n, 0, 1)-algorithm is (n, k, 1)-algorithm, in which the number of IE is equal to k = 0. But, from the “physical” sense of the task, the (n, k, 1)-algorithms, in which either n = 0 or k = 0, cannot narrow the initial uncertainty interval, and therefore, for such (n, k, 1)-algorithms, the (n, k)-exactness or efficiency function is always identically equal to 1, that is,
1.10.2. Arithmetic square
By using the recurrent relation (1.34) and the initial conditions (1.35), we can construct Table 1.1 for the numerical values of the efficiency function F(n, k) of the optimal (n, k, 1)-algorithm.
Table 1.1. Arithmetic square.
By comparing Table 1.1 with the table for binomial coefficients, known as arithmetic square or Tartaglia rectangle [132], we come to the unexpected conclusion that the efficiency function of F(n, k) is expressed by using the formula for the binomial coefficients:
Niccolo Fontana Tartaglia (1499/1500–1557) (Fig. 1.10) was the famous Italian mathematician and engineer (designing fortifications) [132].
Fig. 1.10. Italian mathematician and engineer Niccoló Fontana Tartaglia.
Tartaglia published many books, including the first Italian translations of Archimedes and Euclid. He was the first one, who used mathematics for the investigation of the paths of cannonballs, known as ballistics, in his book A New Science.
1.10.3. Optimal (n, k, 1)-algorithm
The operation of the optimal (n, k, 1)-algorithm can be demonstrated by using the arithmetic square (Table 1.1). Indeed, for the given n and k, the value of the efficiency function F(n, k) is at the intersection of the nth column and the kth row of the arithmetic square (see Table 1.1).
The first step of the optimal (n, k, 1)-algorithm is as follows. The coordinates of the application points of k IE on the segment AB (the points C1, C2, …, Cj, Cj+i, …, Ck) relative to point A (Fig. 1.8) are in the nth column of the arithmetic square (Table 1.1), that is,
If after the first step of the (n, k, 1)-algorithm, the jth IE turned to be to the left of point X, and the remaining (k – j) IE turned to be to the right of point X, then the uncertainty interval relative to point X decreases to the segment Cj Cj+1. The length of this segment is equal to F(n – 1, j) This binomial coefficient is at the intersection of the (n – 1)th column and jth row of the