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We present an algorithm to compute the primary decomposition of a submodule *N* of the free module ℤ[*x*
_{1},...,*x*
_{n}]^{m}. For this purpose we use algorithms for primary decomposition of ideals in the polynomial ring over the integers. The idea is to compute first the minimal associated primes of *N*, i.e. the minimal associated primes of the ideal Ann (ℤ[*x*
_{1},...,*x*
_{n}]^{m}/*N*) in ℤ[*x*
_{1},...,*x*
_{n}] and then compute the primary components using pseudo-primary decomposition and extraction, following the ideas of Shimoyama-Yokoyama. The algorithms are implemented in Singular.

Let *A*
_{1},...,*A*
_{N} and *B*
_{1},...,*B*
_{M} be two sequences of events and let *ν*
_{N}(*A*) and *ν*
_{M}(*B*) be the number of those *A*
_{i} and *B*
_{j}, respectively, that occur. Based on multivariate Lagrange interpolation, we give a method that yields linear bounds in terms of *S*
_{k,t}, *k*+*t* ≤ *m* on the distribution of the vector (*ν*
_{N}(*A*), *ν*
_{M}(*B*)). For the same value of *m*, several inequalities can be generated and all of them are best bounds for some values of *S*
_{k,t}. Known bivariate Bonferroni-type inequalities are reconstructed and new inequalities are generated, too.

M. Giusti’s classification of the simple complete intersection singularities is characterized in terms of invariants. This is a basis for the implementation of a classifier in the computer algebra system Singular.

Let *S* be a set of *n* points distributed uniformly and independently in a convex, bounded set in the plane. A four-gon is called empty if it contains no points of *S* in its interior. We show that the expected number of empty non-convex four-gons with vertices from *S* is 12*n*
^{2}log*n* + *o*(*n*
^{2}log*n*) and the expected number of empty convex four-gons with vertices from *S* is Θ(*n*
^{2}).

This paper attempts an exposition of the connection between valuation theory and hyperstructure theory. In this regards, by considering the notion of totally ordered canonical hypergroup we define a hypervaluation of a hyperfield onto a totally ordered canonical hypergroup and obtain some related basic results.

We provide sufficient conditions for a mapping acting between two Banach spaces to be a diffeomorphism. We get local diffeomorhism by standard method while in making it global we employ a critical point theory and a duality mapping. We provide application to integro-differential initial value problem for which we get differentiable dependence on parameters.

We obtain new lower and upper bounds for probabilities of unions of events. These bounds are sharp. They are stronger than earlier ones. General bounds may be applied in arbitrary measurable spaces. We have improved the method that has been introduced in previous papers. We derive new generalizations of the first and second parts of the Borel-Cantelli lemma.

It is proved that there exists an *NI* ring *R* over which the polynomial ring *R*[*x*] is not an *NLI* ring. This answers an open question of Qu and Wei (*Stud. Sci. Math. Hung.*, **51(2)**, 2014) in the negative. Moreover a sufficient condition of *R*[*x*] to be an *NLI* ring is included for an *NLI* ring *R*.

A space *X* is *almost star countable (weakly star countable)* if for each open cover *U* of *X* there exists a countable subset *F* of *X* such that

*G*be a nilpotent group with finite abelian ranks (e.g. let

*G*be a finitely generated nilpotent group) and suppose

*φ*is an automorphism of

*G*of finite order

*m*. If

*γ*and

*ψ*denote the associated maps of

*G*given by

*Gγ*· ker

*γ*and

*Gψ*· ker

*ψ*are both very large in that they contain subgroups of finite index in

*G*.