alternating

1. 1
   verb

   Present participle and gerund of alternate.

   Bands of heavy rain were alternating with brief spells of sunshine all afternoon.

   The farmer has been alternating corn and soybeans in that field for years.

   Note

   The -ing form does two jobs: after be it builds progressive tenses (the lights were alternating), and on its own it names the action as a noun (she tried alternating hot and cold packs). When two things take turns, the verb usually brings in the second with with: rain alternating with sunshine. Placed before a noun, as in alternating rows, the same spelling is the adjective.

2. 2
   adjective

   Happening or acting by turns, or changing repeatedly from one state to another and back.

   The hallway floor is laid in alternating black and white tiles.

   The infection brought alternating bouts of fever and chills.

3. 3
   adjective

   Of a mathematical series, having terms whose signs switch between positive and negative from one term to the next.

   The series 1 − 1/2 + 1/3 − 1/4 + … is alternating, and it converges even though its terms made positive would not.

   She applied the alternating series test to show that the sum settles on a limit.

   Note

   Signs that flip at every step give such series a convergence check of their own, the alternating series test: when the terms shrink steadily toward zero, the sum settles on a limit, even if the same terms with every sign made positive would grow without bound.

4. 4
   adjective

   Of a multilinear map whose arguments all lie in the same space, taking the value zero whenever any two adjacent arguments are equal.

   The cross product is alternating: the cross product of any vector with itself is zero.

   Every alternating bilinear form is also skew-symmetric.

   Note

   Requiring zero only for equal neighbors turns out to buy much more. Such a map also gives zero whenever any two of its arguments coincide, wherever they sit, and exchanging any two arguments reverses its sign. The determinant, taken as a function of a matrix's rows, is the textbook case.

5. 5
   adjective

   Of a knot, having a flat diagram in which the crossings switch between over and under, one after another, as you follow the strand.

   The trefoil is an alternating knot: trace its strand and you pass over, under, over, under.

   Every knot with seven or fewer crossings is alternating, but some eight-crossing knots are not.

   Note

   Knot theorists attach the label to the knot rather than to a single drawing: a knot counts as alternating if any one of its diagrams alternates, even when other drawings of it do not. That makes alternating knots unusually tractable, since a reduced alternating diagram already shows the fewest crossings the knot can have.

Showing meanings only. Show all 5