The isometric embeddings 2;K → p;K (m ≥ 2, p ∈ 2N) over a field K ∈ {R,C,H} are considered, and an upper bound for the minimal n is proved. In the commutative case (K = H) the bound was obtained by Delbaen, Jarchow and Pe lczyński (1998) in a different way. Let K be one of three fields R,C,H (real, complex or quaternion). Let K be the K-linear space consisting of columns x = [ξi] n 1 , ξi ∈ K, ...

We study the -server problem when the off-line algorithm has fewer than servers. We give two upper bounds of the cost WFA of the Work Function Algorithm. The first upper bound is OPT OPT , where OPT denotes the optimal cost to service by servers. The second upper bound is OPT OPT for . Both bounds imply that the Work Function Algorithm is -competitive. Perhaps more important is our technique wh...

it is well known that in operations research‎, ‎degeneracy can cause a cycle in a network‎ ‎simplex algorithm which can be prevented by maintaining strong‎ ‎feasible bases in each pivot‎. ‎also‎, ‎in a network consists of n arcs‎ ‎and m nodes‎, ‎not considering any new conditions on the entering‎ ‎variable‎, ‎the upper bound of consecutive degenerate pivots is equal‎ $left( ‎begin{array}{c}‎ ‎n...

The band structure for a crystal generally consists of connected components in energy-momentum space, known as complexes. Here, we explore fundamental aspect regarding the maximal number bands that can be accommodated single complex. We show that, principle, complex have no finite upper bound certain space groups. This means infinitely many entangle together, forming pattern stable against symm...