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A pivot table is a table of values which are aggregations of groups of individual values from a more extensive table (such as from a database, spreadsheet, or business intelligence program) within one or more discrete categories. The aggregations or summaries of the groups of the individual terms might include sums, averages, counts, or other ...
Unlike many other nimber related games, the number of spaces between the two tokens on each row are the sizes of the Nim heaps. If your opponent increases the number of spaces between two tokens, just decrease it on your next move. Else, play the game of Nim and make the Nim-sum of the number of spaces between the tokens on each row be 0. [3]
Another numerical construction of the Klein four-group is the set { 1, 3, 5, 7 }, with the operation being multiplication modulo 8. Here a is 3, b is 5, and c = ab is 3 × 5 = 15 ≡ 7 (mod 8). The Klein four-group also has a representation as 2 × 2 real matrices with the operation being matrix multiplication:
(where Z[1/p] denotes the group of all rational numbers whose denominator is a power of p, using addition of rational numbers as group operation). For each natural number n, consider the quotient group Z/p n Z and the embedding Z/p n Z → Z/p n+1 Z induced by multiplication by p. The direct limit of this system is Z(p ∞):
This system has the exact solution of x 1 = 10.00 and x 2 = 1.000, but when the elimination algorithm and backwards substitution are performed using four-digit arithmetic, the small value of a 11 causes small round-off errors to be propagated. The algorithm without pivoting yields the approximation of x 1 ≈ 9873.3 and x 2 ≈ 4.
In its simplest form, if a number had a plus sign on one side and a multiplication sign on the other side, the multiplication acts first. If we were to express this idea using symbols of grouping, the factors in a product. Example: 2+3×4 = 2 +(3×4)=2+12=14. In understanding expressions without symbols of grouping, it is useful to think of ...
The linking number is chiral: taking the mirror image of link negates the linking number. The convention for positive linking number is based on a right-hand rule. The winding number of an oriented curve in the x-y plane is equal to its linking number with the z-axis (thinking of the z-axis as a closed curve in the 3-sphere).
The set {3,19} generates the group, which means that every element of (/) is of the form 3 a × 19 b (where a is 0, 1, 2, or 3, because the element 3 has order 4, and similarly b is 0 or 1, because the element 19 has order 2). Smallest primitive root mod n are (0 if no root exists)