We show that for all integers m > 4 there exists a 2m × 2m ×m latin cuboid that cannot be completed to a 2m × 2m × 2m latin cube. We also show that for all even m / ∈ {2, 6} there exists a (2m−1) × (2m−1) × (m−1) latin cuboid that cannot be extended to any (2m−1)× (2m−1)×m latin cuboid.

Abstract For the $p$ -localized sphere $\mathbb {S}^{2m-1}_{(p)}$ with $p >3$ a prime, we prove that homotopy nilpotency satisfies $\mbox {nil}\ \mathbb {S}^{2m-1}_{(p)}<\infty$ , respect to any associative $H$ -structure on . We also {S}^{2m-1}_{(p)}= 1$ for all but finite number of primes Then, loop space associated -projective {S}^{2m-1}_{(p)}P(n-1)$ $m,n\ge 2$ and $m\mid p-1$ derive \...

We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K 2m,2m , a result connected to a conjecture of Graham and Häggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m 2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a ...

A  $ mu $-way  $ G $-trade ($ mu geq 2) $  consists of $ mu $ disjoint decompositions of some simple (underlying) graph $ H $ into copies of a graph $ G. $  The   number of copies of the  graph $ G $ in  each of the decompositions is the volume of the $ G $-trade and   denoted by $ s. $ In this paper, we determine all values  $ s $ for which there exists a $  mu $-way   $ K_{1,m} $-trade of  vo...

beta 2-Microglobulin (beta 2M) is a major constituent of amyloid fibrils in hemodialysis-associated amyloidosis, a complication of long-term hemodialysis patients. Amyloid fibril proteins were isolated from connective tissues forming carpal tunnels in hemodialysis patients with carpal tunnel syndrome. Two-dimensional polyacrylamide gel electrophoresis and Western blotting demonstrated that most...

First note that for any ρ ∈ (0, 1) and x ∈ R the function f (x) = x − 1 2 log ρ 2m 2 2x + 2πe(1 − ρ 2m) (1) is an monotonically increasing function with respect to x, because f ′ (x) = 2πe(1 − ρ 2m) ρ 2m 2 2x + 2πe(1 − ρ 2m) > 0. (2) By applying Shannon's EPI we have. h(s b |f a) ≥ 1 2 log ρ 2m 2 2h(sa|fa) + 2πe(1 − ρ 2m) (3) and thus, h(s a |f a) − h(s b |f a) ≤ h(s a |f a) − 1 2 log ρ 2m 2 2h...