scispace - formally typeset
Search or ask a question
Topic

Symmetric relation

About: Symmetric relation is a research topic. Over the lifetime, 166 publications have been published within this topic receiving 2601 citations. The topic is also known as: symmetrical relationship.


Papers
More filters
Journal ArticleDOI
TL;DR: Some fundamental properties of generalized rough sets induced by binary relations on algebras are considered and it is shown that any reflexive binary relation determines a topology.

289 citations

Journal ArticleDOI
TL;DR: In this paper, various classes of extensions of symmetric operators with equal (finite or infinite) defect numbers are described in terms of abstract boundary conditions, and the dual problem of describing extensions of a symmetric binary relation is also considered.
Abstract: Various classes of extensions of symmetric operators with equal (finite or infinite) defect numbers are described in terms of abstract boundary conditions. The dual problem of the description of extensions of a symmetric binary relation is also considered.

200 citations

Journal ArticleDOI
TL;DR: The main result establishes the NP-completeness of a problem of the best approximation of a symmetric relation on a finite set by an equivalence relation, thus answering in the negative a question proposed implicitly by C.T. Zahn.
Abstract: We consider a class of optimization problems of hierarchical-tree clustering and prove that these problems are NP-hard. The sequence of polynomial reductions and/or transformations used in our proof is based on relatively laborious graph-theoretical constructions and starts in the NP-complete problem of 3-dimensional matching. Using our main result we establish the NP-completeness of a problem of the best approximation of a symmetric relation on a finite set by an equivalence relation, thus answering in the negative a question proposed implicitly by C.T. Zahn.

190 citations

Journal ArticleDOI
TL;DR: In this paper, the concepts of boundary relations and the corresponding Weyl families are introduced, and fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : (H 2, J(H)) -> (H-2, J (H)), where Gamma need not be surjective and is even allowed to be multivalued.
Abstract: The concepts of boundary relations and the corresponding Weyl families are introduced. Let S be a closed symmetric linear operator or, more generally, a closed symmetric relation in a Hilbert space h, let H be an auxiliary Hilbert space, let [GRAPHICS] and let JH be defined analogously. A unitary relation G from the Krein space (h(2), J(h)) to the Kre. in space (H-2, J(H)) is called a boundary relation for the adjoint S* if ker Gamma = S. The corresponding Weyl family M(lambda) is de. ned as the family of images of the defect subspaces (n) over cap (lambda), lambda is an element of C \ R under Gamma. Here Gamma need not be surjective and is even allowed to be multi-valued. While this leads to fruitful connections between certain classes of holomorphic families of linear relations on the complex Hilbert space H and the class of unitary relations Gamma : ( H-2, J(H)) -> (H-2, J(H)), it also generalizes the notion of so-called boundary value space and essentially extends the applicability of abstract boundary mappings in the connection of boundary value problems. Moreover, these new notions yield, for instance, the following realization theorem: every H-valued maximal dissipative (for lambda is an element of C+) holomorphic family of linear relations is the Weyl family of a boundary relation, which is unique up to unitary equivalence if certain minimality conditions are satisfied. Further connections between analytic and spectral theoretical properties of Weyl families and geometric properties of boundary relations are investigated, and some applications are given.

172 citations

Journal ArticleDOI
TL;DR: In this paper, a new and novel variant of classical Banach contraction principle on a complete metric space with a binary relation is presented, which, under universal relation, reduces to Banach's contraction principle.
Abstract: In this paper, we present yet another new and novel variant of classical Banach contraction principle on a complete metric space en- dowed with a binary relation which, under universal relation, reduces to Banach contraction principle. In process, we observe that various kinds of binary relations, such as partial order, preorder, transitive relation, tolerance, strict order, symmetric closure, etc., utilized by earlier authors in several well-known metrical fixed point theorems can be weakened to the extent of an arbitrary binary relation.

122 citations


Network Information
Related Topics (5)
Categorical variable
13.1K papers, 665.5K citations
76% related
Generalization
23.5K papers, 483.5K citations
75% related
Axiom
11K papers, 257.9K citations
74% related
Mathematical proof
13.8K papers, 374.4K citations
74% related
Type (model theory)
38.9K papers, 670.5K citations
74% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20215
20208
20194
20183
20176
20165