Given a graph G and a non-negative integer h, the h-restricted connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whos...Given a graph G and a non-negative integer h, the h-restricted connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whose deletion disconnects G and every remaining component has the minimum degree of vertex at least h; and the h-extra connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, if any, whose deletion disconnects G and every remaining component has order more than h. This paper shows that for the hypercube Qn and the folded hypercube FQn, κ1（Qn）=κ（1）（Qn）=2n-2 for n≥3, κ2（Qn）=3n-5 for n≥4, κ1（FQn）=κ（1）（FQn）=2n for n≥4 and κ（2）（FQn）=4n-4 for n≥8.展开更多
A model of the growth curve of microorganisms was proposed,which reveals a relation-ship with the number of a‘golden section’,1.618…,for main parameters of the growth curves.The treatment mainly concerns the ratio ...A model of the growth curve of microorganisms was proposed,which reveals a relation-ship with the number of a‘golden section’,1.618…,for main parameters of the growth curves.The treatment mainly concerns the ratio of the maximum asymptotic value of biomass in the phase of slow growth to the real value of biomass accumulation at the end of exponential growth,which is equal to thc square of the'golden section',i.e.,2.618.There are a few relevant theorems to explain these facts.New,yet simpler,methods were considered for deterrmining the model parameters based on hyperbolic functions.A comparison was made with one of the alternative models to demonstrate the advantage of the proposed model.The proposed model should be useful to apply at various stages of fermentation in scientific and industrial units.Further,the model could give a new impetus to the development of new mathematical knowledge regarding the algebra of the‘golden section'as a whole,as well as in connection with the introduction of a new equation at decomposing of any roots with any degrees for differences between constants and/or variables.展开更多
文摘Given a graph G and a non-negative integer h, the h-restricted connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whose deletion disconnects G and every remaining component has the minimum degree of vertex at least h; and the h-extra connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, if any, whose deletion disconnects G and every remaining component has order more than h. This paper shows that for the hypercube Qn and the folded hypercube FQn, κ1（Qn）=κ（1）（Qn）=2n-2 for n≥3, κ2（Qn）=3n-5 for n≥4, κ1（FQn）=κ（1）（FQn）=2n for n≥4 and κ（2）（FQn）=4n-4 for n≥8.
文摘A model of the growth curve of microorganisms was proposed,which reveals a relation-ship with the number of a‘golden section’,1.618…,for main parameters of the growth curves.The treatment mainly concerns the ratio of the maximum asymptotic value of biomass in the phase of slow growth to the real value of biomass accumulation at the end of exponential growth,which is equal to thc square of the'golden section',i.e.,2.618.There are a few relevant theorems to explain these facts.New,yet simpler,methods were considered for deterrmining the model parameters based on hyperbolic functions.A comparison was made with one of the alternative models to demonstrate the advantage of the proposed model.The proposed model should be useful to apply at various stages of fermentation in scientific and industrial units.Further,the model could give a new impetus to the development of new mathematical knowledge regarding the algebra of the‘golden section'as a whole,as well as in connection with the introduction of a new equation at decomposing of any roots with any degrees for differences between constants and/or variables.
