Let H be a connected graph and G be a supergraph of H . It is trivial that for any k-flow (D, f ) of G, the restriction of (D, f ) on the edge subset E(G/H) is a k-flow of the contracted graph G/H . However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow (D′, f ′) of the contracted graph G/H , whether or not the supergraph G admits a k-flow (D,...

The technical details of RSA works on the idea that it is easy to generate modulus by multiplying two sufficiently large prime numbers together, but factorizing number back into original extremely difficult. Suppose \(N=p^r q^s\) are modulus, where \(p\) and \(q\) product unknown unbalance primes for \(2 \leq s<r\). paper proves using an approximation \(\phi(N) \approx\) \(N-N^{\frac{r+8-1}{...

We exhibit a separable commutative C*-algebra A such that A⊗K is homotopy equivalent to zero, without Mn(A) being so for any n ≥ 1.

To study the ionic basis of salt sensitivity in hypertension, 19F-, 13P-, and 23Na-nuclear magnetic resonance techniques were used to measure cytosolic free calcium (Cai), pH (pHi), free magnesium (Mgi), and sodium (Nai) in erythrocytes of essential hypertensive subjects (n = 19). Individuals were studied for 2 mo each on low- (UNaV < 50 meq/d) and high- (UNaV > 200 meq/d) salt diets, with the ...

McCuaig and Ota proved that every 3-connected graph G on at least 9 vertices admits a contractible triple, i. e. a connected subgraph H on three vertices such that G − V (H) is 2-connected. Here we show that every 3-connected graph G on at least 9 vertices has more than |V (G)|/10 many contractible triples. If, moreover, G is cubic, then there are at least |V (G)|/3 many contractible triples, w...

Contractible transformations of graphs consist of contractible gluing and deleting of vertices and edges of graphs. They partition all graphs into the family of homotopy classes. Contractible transformations do not change the Euler characteristic and the homology groups of graphs. In this paper we describe the minimal representatives of some homotopy classes and find the formula for computing t...

let $mathcal {a} $ and $mathcal {b} $ be c$^*$-algebras. assume that $mathcal {a}$ is of real rank zero and unital with unit $i$ and $k>0$ is a real number. it is shown that if $phi:mathcal{a} tomathcal{b}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $phi(|a|^k)=|phi(a)|^k $ for all normal elements $ainmathcal a$, $phi(i)$ is a projection, and there exists a posit...

Let $[a_1(x),a_2(x),\ldots, a_n(x), \ldots]$ be the continued fraction expansion of an irrational number $x\in (0, 1)$. The study growth rate product consecutive partial quotients $a_n(x)a_{n+1}(x)$ is associated with improvements to Dirichlet's theorem (1842). We establish both weak and strong laws large numbers for sums $S_n(x)= \sum_{i=1}^n a_i(x)a_{i+1}(x)$ as well as, from a multifractal a...

The template requirements for the formation of the phi 29 protein p3-dAMP initiation complex in vitro have been studied. The initiation reaction requires the parental protein p3 but not an intact DNA molecule. Protein p3-containing fragments from the left- or right-hand DNA ends were active as template for formation of the initiation complex provided they had a minimal size: a 26-base-pair-long...