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A Latin square, the unbordered multiplication table for a quasigroup whose 10 elements are the digits 0–9. The multiplication table of a finite quasigroup is a Latin square : an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column.
The end-loop marker specifies the name of the index variable, which must correspond to the name of the index variable at the start of the for-loop. Some languages (PL/I, Fortran 95, and later) allow a statement label at the start of a for-loop that can be matched by the compiler against the same text on the corresponding end-loop statement.
A loop is universally flexible if every one of its loop isotopes is flexible, that is, satisfies (xy)x = x(yx). A loop is middle Bol if every one of its loop isotopes has the antiautomorphic inverse property, that is, satisfies (xy) −1 = y −1 x −1. Is there a finite, universally flexible loop that is not middle Bol?
It requires memorization of the multiplication table for single digits. This is the usual algorithm for multiplying larger numbers by hand in base 10. A person doing long multiplication on paper will write down all the products and then add them together; an abacus-user will sum the products as soon as each one is computed.
In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144 ...
An example of a primitive recursive programming language is one that contains basic arithmetic operators (e.g. + and −, or ADD and SUBTRACT), conditionals and comparison (IF-THEN, EQUALS, LESS-THAN), and bounded loops, such as the basic for loop, where there is a known or calculable upper bound to all loops (FOR i FROM 1 TO n, with neither i ...
In its most general form a loop group is a group of continuous mappings from a manifold M to a topological group G.. More specifically, [1] let M = S 1, the circle in the complex plane, and let LG denote the space of continuous maps S 1 → G, i.e.
Graphs of functions commonly used in the analysis of algorithms, showing the number of operations versus input size for each function. The following tables list the computational complexity of various algorithms for common mathematical operations.