In this paper, we explore the 2-extra connectivity and 2-extra-edge-connectivity of the folded hypercube FQn. We show that j2(FQn) = 3n 2 for n P 8; and k2(FQn) = 3n 1 for n P 5. That is, for n P 8 (resp. n P 5), at least 3n 2 vertices (resp. 3n 1 edges) of FQn are removed to get a disconnected graph that contains no isolated vertices (resp. edges). When the folded hypercube is used to model th...

Solution. (a) Since φ(3n) = 2 · 3n−1, the problem amounts to showing that 3n 23 − 1 and 3n 22·3 − 1 (when n ≥ 2). The first claim follows from reduction mod 3, and the second claim follows from the exponent lifting trick, as 3 ‖ 22−1, so that 3n−1 ‖ 22·3−1. (b) Since φ(pn) = φ(2pn) = (p − 1)pn−1, it suffices to show pn g(p−1)p − 1 and pn gdp − 1 for any divisor d of p − 1 with d < p − 1. The fi...

Drisko proved that 2n−1 matchings of size n in a bipartite graph have rainbow matching n. For general graphs it is conjectured 2n suffice for this purpose (and when even). The known showing sharpness conjecture even are called badges. We improve the previously best bound from 3n−2 to 3n−3, using new line proof involves analysis appearance also prove “cooperative” generalization: t>0 and n≥3, an...

A loop of order $n$ possesses at least $3n^2-3n+1$ associative triples. However, no $n>1$ that achieves this bound seems to be known. If the is involutory, then it $3n^2-2n$ Involutory loops with triples can obtained by prolongation certain maximally nonassociative quasigroups whenever $n-1$ a prime greater than or equal $13$ $n-1=p^{2k}$, $p$ an odd prime. For orders $n\le 9$ minimum number re...

In the present study, fifteen growing pigs were used to determine the whole-body oxidation, retention efficiency (RE) and apparent conversion (AC) of α-linolenic acid (18 : 3n-3) to n-3 highly unsaturated fatty acids (HUFA), including EPA (20 : 5n-3) and DHA (22 : 6n-3). The pigs were fed a diet containing 10% flaxseed for 30 d. Whole-body fatty acid composition was determined at initial (27.7 ...

In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence Sn = ( 3n)( 3n 2n) 2( n )(2n+1) , and the binomial coefficients ( 3n n )

The meson-nucleon dynamics that generates the hard core of the RuhrPot two-nucleon interaction is shown to vanish in the irreducible 3N force. This result indicates a small 3N force dominated by conventional light meson-exchange dynamics and holds for an arbitrary meson-theoretic Lagrangian. The resulting RuhrPot 3N force is defined in the appendix. A completely different result is expected whe...

Intron prediction is an important problem of the constantly updated genome annotation. Using two model plant (rice and Arabidopsis) genomes, we compared two well-known intron prediction tools: the Blast-Like Alignment Tool (BLAT) and Sim4cc. The results showed that each of the tools had its own advantages and disadvantages. BLAT predicted more than 99% introns of whole genomic introns with a sm...

A matchstick graph is a crossing-free unit-distance in the plane. Harborth conjectured 1981 that maximum number of edges with $n$ vertices $\lfloor 3n-\sqrt{12n-3}\rfloor$. Using Euler formula and isoperimetric inequality, it can be shown has no more than $3n-\sqrt{2\pi\sqrt{3}\cdot n}+O(1)$ edges. We improve this upper bound to $3n-c\sqrt{n-1/4}$ edges, where $c=\frac12(\sqrt{12} + \sqrt{2\pi\...

The number Nn of non-crossing trees of size n satis/es Nn+1 = Tn where Tn enumerates ternary trees of size n. We construct a new bijection to establish that fact. Since Tn =(1=(2n+ 1))( 3n n ), it follows that 3(3n− 1)(3n− 2)Tn−1 = 2n(2n+ 1)Tn. We construct two bijections “explaining” this recursion; one of them easily extends to the case of t-ary trees. c © 2002 Elsevier Science B.V. All right...