Sentence examples for for any odd integer from inspiring English sources

Using this pair of designs, we prove there is a cocyclic Hadamard matrix of order 2ts for any odd integer s>1 and any t⩾⌊8 log2 s⌋.

Since t e t - 1 - 1 + t 2 is an even function (i.e., invariant under x ↦ - x), we see that B k = 0 for any odd integer k not smaller than 3.

Thus, F i w = z = T w, that is, z is a coincidence point of F i, T for any odd integer i ∈ N. Further, since the Cauchy sequence { T x 2 n k } converges to z ∈ C and z = F i w, z ∈ F i C ∩ C ⊆ S C, there exists v ∈ C such that S v = z.

We also show that for any prime power q and any odd integer n there exists a resolvable 3- qn+1,q+1,1) design.

For an odd integer k≥1 and prime power q, Lazebnik et al. present explicit construction for q-regular bipartite graphs with girth at least k+5 and number of edges q k−1 [15].

In detail, and it is easy to show that for an odd integer n, Since T n, 0 ( z ) = T n ( z ), we may apply results about T n, ϵ ( z ) to those about T n ( z ).

From (1.4) and (1.6), one easily derives the following identities: for any odd positive integer, (1.8).

If is even, then for some odd integer and is odd.

If is odd, then is even and for some odd integer.

Since for some odd integer and is odd, by (5.32), we have (5.33).

If d n = 0 for some odd integer n, then g has a fixed point.