Odd periodic oscillations in Comb-drive finger actuators

In this work we consider the existence of solutions of a Dirichlet problem with a prescribed number of zeros for a large class of second order differential equations with singularities. Additionally, for the model of a well known kind of micro-electromechanical system, the transverse Comb-drive finger device, we provide sufficient conditions in terms of control parameters that lead to odd periodic oscillations with a certain number of zeros that ranges from zero to some positive integer. This work is motivated and follows the lines of R. Ortega (2016) where the existence of odd periodic oscillations was studied for second order differential equations without singularities, in particular, for the Sitnikov problem. The main results of this paper are obtained through a truncation process, the application of classical tools as the shooting method, and the Sturm comparison theory. A numerical approach based on a bisection method is employed to illustrate the main results with simulations by using literature reference parameters.     

More odd graph theory from another point of view

The Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $\left[n\right]=\{1,2,\dots ,n\}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n,k)$ is called an odd graph. Let $n>4$ and $1<k<\frac{n}{2}$. In the present paper, we show that if the Kneser graph $K(n,k)$ is of even order where $n$ is an odd integer or both of the integers $n,k$ are even, then $K(n,k)$ is a vertex-transitive non Cayley graph. Although, these are special cases of Godsil [7], unlike his proof that uses some very deep group-theoretical facts, ours uses no heavy group-theoretic facts. We obtain our results by using some rather elementary facts of number theory and group theory. We show that ‘almost all’ odd graphs are of even order, and consequently are vertex-transitive non Cayley graphs. Finally, we show that if $k>4$ is an even integer such that $k$ is not of the form $k=2^t$ for some $t>2$, then the line graph of the odd graph $O_{k+1}$ is a vertex-transitive non Cayley graph.     

积图P_n×P_m的奇优美性和奇强协调性

给出了积图P_n×P_m的奇优美标号和奇强协调标号.     

Note on robust critical graphs with large odd girth

A graph G is (k+1)-critical if it is not k-colourable but G−e is k-colourable for any edge e∈E(G). In this paper we show that for any integers k≥3 and l≥5 there exists a constant c=c(k,l)>0, such that for all , there exists a (k+1)-critical graph G on n vertices with and odd girth at least ?, which can be made (k−1)-colourable only by the omission of at least cn² edges.     

Planar graph bipartization in linear time

For each constant k, we present a linear time algorithm that, given a planar graph G, either finds a minimum odd cycle vertex transversal in G or guarantees that there is no transversal of size at most k.     

Extending Independent Sets to Bases and the Axiom of Choice

We show that the both assertions “in every vector space B over a finite element field every subspace V ? B has a complementary subspace S” and “for every family ?? of disjoint odd sized sets there exists a subfamily ?={F_j:j ?ω} with a choice function” together imply the axiom of choice AC. We also show that AC is equivalent to the statement “in every vector space over ? every generating set includes a basis”.     

Visconata: A rare flavonol having long chain fatty acid from Dodonaea viscosa which inhibits Human neutrophil elastase (HNE)

A series of flavonoids were isolated from Dodonaea Viscosa and tested for inhibition of human neutrophil elastase (HNE), enzyme involved in inflammatory disorders. Isolated compounds were identified as a novel flavonol (1) along with eight known flavonoids (2–9). Novel flavonol, visconata (1) has a very rare skeleton having odd numbered long chain (C19) fatty acid, which was completely identified by mass fragmentation and 2D NMR analysis. All compounds (1–9) inhibited HNE in dose dependent manner with IC₅₀s ranging between 2.4 and 150 μM. Visconata (1) emerged to be the most potent compound with 2.4 μM of IC₅₀. In kinetic studies, compound (1) was observed to be reversible, noncompetitive inhibitor having K_i = 1.8 μM, whereas other flavonoids (2–9) displayed mixed type inhibition.