Let G1 and G2 be simple graphs and let n1 = |V (G1)|, m1 = |E(G1)|, n2 = |V (G2)| and m2 = |E(G2)|. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G1 G2 of G1 and G2. We show that: τ(G1 G2) ≥ 2(n1−1)(n2−1) n1n2 (τ(G1)n1) n2+1 2 (τ(G2)n2) n1+1 2 and τ(G1 G2) ≤ τ(G1)τ(G2) [ 2m1 n1 − 1 + 2m2 n2 − 1 ](n1−1)(n2−1) . We also characteri...

introduction: vitamin deficiency and iron deficiency anemia are common nutritional problems, at least in children under 5. these materials shortage, especially in the first two years of life, impair physical and brain growth, reduces the child's learning ability, reduces body resistance against infections, behavioral changes, apathy and finally social and economic adverse consequences woul...

Let F (m; n1, n2) denote the number of lattice walks from (0, 0) to (n1, n2), always staying in the first quadrant {(n1, n2); n1 ≥ 0, n2 ≥ 0} and having exactly m steps, each of which belongs to the set {E = (1, 0),W = (−1, 0), NE = (1, 1), SW = (−1,−1)}. Ira Gessel conjectured that F (2n; 0, 0) = 16 (1/2)n(5/6)n (2)n(5/3)n . We pose similar conjectures for some other values of (n1, n2), and gi...

We extend PGD and its recovery guarantee [1] from one-dimensional spectrally sparse signal recovery to the multi-dimensional case. Assume the underlying multi-dimensional spectrally sparse signal is of model order r and total dimension N . We show that O(r log(N)) measurements are sufficient for PGD to achieve successful recovery with high probability provided the underlying signal satisfies so...

For all integers k with k≥2, if G is a balanced k-partite graph on n≥3 vertices minimum degree at least⌈n2⌉+⌊n+22⌈k+12⌉⌋−nk={⌈n2⌉+⌊n+2k+1⌋−nk:k odd n2+⌊n+2k+2⌋−nk:k even , then has Hamiltonian cycle unless 4 divides n and k∈{2,n2}. In the case where k∈{2,n2}, we can characterize graphs which do not have see that ⌈n2⌉+⌊n+22⌈k+12⌉⌋−nk+1 suffices. This result tight for k≥2 divisible by k.