We prove the vanishing of the Koszul homology group$H$_{μ}(Kos($M$)_{μ}), where μ is the minimal number of generators of M. We give a counterexample that the Koszul complex of a module is not always acyclic and show its relationship with the homology of commutative rings.

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Let$A$and$B$be positive numbers and$m$and$n$positive integers,$m<n$. Then there is for complex valued functions φ on$R$with sufficient differentiability and boundedness properties a representationwhere$v$_{$1$}and$v$_{$2$}are bounded Borel measures with$v$_{$1$}absolutely continuous, such that there exists a function φ with ∣φ^{(n)}∣ ⩽$A$and ∣φ∣ ⩽$A$on$R$and satisfying $$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$ This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If$G$is a 2-connected graph of order$n$and max{$d$($u$),$d$($v$)}≥$n$/2 for each pair of vertices$u$and$v$with distance$d$($u$,$v$)=2, then$G$is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if$G$is a 2-connected graph and |$N$($u$)∪$N$($v$)|+δ($G$)≥$n$for each pair of nonadjacent vertices$u$,$v$∈$V$($G$), then$G$is Hamiltonian. This paper generalizes the above results when$G$is 3-connected. We show that if$G$is a 3-connected graph of order$n$and max{|$N$($x$)∪$N$($y$)|+$d$($u$),|$N$($w$)∪$N$($z$)|+$d$($v$)}≥$n$for every choice of vertices$x$,$y$,$u$,$w$,$z$,$v$such that$d$($x$,$y$)=$d$($y$,$u$)=$d$($w$,$z$)=$d$($z$,$v$)=$d$($u$,$v$)=2 and where$x$,$y$and$u$are three distinct vertices and$w$,$z$and$v$are also three distinct vertices (and possibly |{$x$,$y$}∩{$w$,$z$}| is 1 or 2), then$G$is Hamiltonian.