Concept

Autocorrelation

The sum of products of a sequence with a shifted copy of itself, one value for each shift. Small values at every non-zero shift mean the sequence looks unlike its own echoes, which is what radar pulses and flat polynomials need.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as littlewood polynomial — the same set of essays touches all of them, so they are one junction rather than several.

Three polynomials of signs around the circle. The size of the polynomial around the unit circle divided by √64 for all-plus, random and Rudin–Shapiro sign polynomials of length 64; maxima 8.00, 2.50, 1.41.

The flattest polynomials of signs

A polynomial whose coefficients are all +1 or −1 has average size √n on the unit circle. Keeping it near √n everywhere is the problem Littlewood posed: the Rudin–Shapiro polynomials never exceed √2 times it, searching every sign pattern up to length 22 finds the best are the Barker sequences, and whether the maximum can come arbitrarily close to √n is still open.

algebra · Polynomial roots

Two sign sequences laid on themselves. Length 41: random signs (seed 41) with E = 552 and merit factor 1.523; the rotated Legendre sequence of 41 with E = 176 and merit factor 4.776.

A quarter turn quiets the echoes

Lay a sequence of signs on a shifted copy of itself and add up the squares of the overlaps. Random signs leave the total near n²/2, and the squares modulo a prime, started a quarter of the way along, cut it to a twelfth. Exhaustive search finds far quieter short sequences, and no infinite family is proved to do better than 6.342.

algebra · Polynomial roots

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive search Littlewood polynomial Gray code Legendre symbol Open problem Parseval identity Probabilistic method Quadratic residue Unit circle Variance

All concepts