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A linear operator on a Hilbert space *S* is shown to be densely defined and closed if and only if

In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

A linear operator on a Hilbert space *S* is shown to be densely defined and closed if and only if

In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

Let *X* be a topological space. For any positive integer *n*, we consider the *n*-fold symmetric product of *X*, ℱ* _{n}*(

*X*), consisting of all nonempty subsets of

*X*with at most

*n*points; and for a given function

*ƒ*:

*X*→

*X*, we consider the induced functions ℱ

*(*

_{n}*ƒ*): ℱ

*(*

_{n}*X*) → ℱ

*(*

_{n}*X*). Let

*M*be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ

_{+}-transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal,

*I N, T T*

_{++}, semi-open and irreducible. In this paper we study the relationship between the following statements:

*ƒ*∈

*M*and ℱ

*(*

_{n}*ƒ*) ∈

*M*.

Let *X* be a topological space. For any positive integer *n*, we consider the *n*-fold symmetric product of *X*, ℱ* _{n}*(

*X*), consisting of all nonempty subsets of

*X*with at most

*n*points; and for a given function

*ƒ*:

*X*→

*X*, we consider the induced functions ℱ

*(*

_{n}*ƒ*): ℱ

*(*

_{n}*X*) → ℱ

*(*

_{n}*X*). Let

*M*be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ

_{+}-transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal,

*I N, T T*

_{++}, semi-open and irreducible. In this paper we study the relationship between the following statements:

*ƒ*∈

*M*and ℱ

*(*

_{n}*ƒ*) ∈

*M*.

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

Infinite matroids have been defined by Reinhard Diestel and coauthors in such a way that this class is (together with the finite matroids) closed under dualization and taking minors. On the other hand, Andreas Dress introduced a theory of matroids with coefficients in a fuzzy ring which is – from a combinatorial point of view – less general, because within this theory every circuit has a finite intersection with every cocircuit. Within the present paper, we extend the theory of matroids with coefficients to more general classes of matroids, if the underlying fuzzy ring has certain properties to be specified.

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

In many clique search algorithms well coloring of the nodes is employed to find an upper bound of the clique number of the given graph. In an earlier work a non-traditional edge coloring scheme was proposed to get upper bounds that are typically better than the one provided by the well coloring of the nodes. In this paper we will show that the same scheme for well coloring of the edges can be used to find lower bounds for the clique number of the given graph. In order to assess the performance of the procedure we carried out numerical experiments.

This paper solves an enumerative problem which arises naturally in the context of Pascal’s hexagram. We prove that a general Desargues configuration in the plane is associated to *six* conical sextuples via the theorems of Pascal and Kirkman. Moreover, the Galois group associated to this problem is isomorphic to the symmetric group on six letters.

This paper solves an enumerative problem which arises naturally in the context of Pascal’s hexagram. We prove that a general Desargues configuration in the plane is associated to *six* conical sextuples via the theorems of Pascal and Kirkman. Moreover, the Galois group associated to this problem is isomorphic to the symmetric group on six letters.