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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?
The Erdős–Tenenbaum–Ford constant is a mathematical constant that appears in number theory. [1] Named after mathematicians Paul Erdős, Gérald Tenenbaum, and Kevin Ford, it is defined as
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.
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. Fortran also allows the EXIT and CYCLE statements to name this text; in a nest of loops, this makes clear which loop is intended.
Loop optimization acts on the statements that make up a loop, such as a for loop, for example loop-invariant code motion. Loop optimizations can have a significant impact because many programs spend a large percentage of their time inside loops. [3]: 596 Some optimization techniques primarily designed to operate on loops include:
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.
The pictures on the right show how to calculate 345 × 12 using lattice multiplication. As a more complicated example, consider the picture below displaying the computation of 23,958,233 multiplied by 5,830 (multiplier); the result is 139,676,498,390. Notice 23,958,233 is along the top of the lattice and 5,830 is along the right side.
Matrix chain multiplication (or the matrix chain ordering problem [1]) is an optimization problem concerning the most efficient way to multiply a given sequence of matrices. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved.