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Best approximation in quotient probabilistic normed space

Sen M., Nath S., Tripathy B. C.

Journal of Applied Analysis

2017

Vol. 23, nr 1

53--57

In this article, we study the best approximation in quotient probabilistic normed space. We define the notion of quotient space of a probabilistic normed space, then prove some theorems of approximation in quotient space are extended to quotient probabilistic normed space.

A New Algorithm for Optimal Path Finding in Complex Networks Based on the Quotient Space

Zhang L., He F-g, Zhang Y-p, Zhao S.

Fundamenta Informaticae

2009

Vol. 93, nr 4

459-469

The optimal path finding problem in weighted edge networks is an old and interesting one in many fields. There were many well-known algorithms to deal with that issue. But they were confronted with the high computational complexity while the network becoming larger. We present a hierarchical quotient space model based algorithm that reduces the computational complexity. The basic idea is the following. The nodes of a given network are partitioned with respect to the weights of their adjacent edges. We construct a variety of coarser versions of the given network with new nodes corresponding to the blocks of partitions at various levels of granularity. They are called the quotient spaces (networks) of the original network. The construction of the (sub-) optimal path is then done incrementally, throughout the hierarchy of quotient networks. Since each version of the network is much simpler than the original one, especially of the coarsest spaces, the computational complexity is reduced. In this paper, we present the basic principles of the algorithm and its experimental comparison to other well-known algorithms.

The Quotient Space Theory of Problem Solving

Zhang L., Zhang B.

Fundamenta Informaticae

2004

Vol. 59, nr 2,3

287--298

The paper introduces a framework of quotient space theory of problem solving. In the theory, a problem (or problem space) is represented as a triplet, including the universe, its structure and attributes. The problem spaces with different grain sizes can be represented by a set of quotient spaces. Given a problem, the construction of its quotient spaces is discussed. Based on the model, the computational complexity of hierarchical problem solving and the information combination are also dealt with. The model can also be extended to the fuzzy granular world.

The separable quotient problem for symmetric function spaces

Śliwa W.

Bulletin of the Polish Academy of Sciences. Mathematics

2000

Vol. 48, no 1

13-27

We study the separable quotient problem for some classes of symmetric function spaces on a measure space (Omega,Sigma,mi). In particular we prove that an infinite dimensional Orlicz space L[sup fi](Omega,Sigma,mi) has an infinite dimensional separable quotient if and only if the dual of L[sup fi](Omega,Sigma,mi) is total or there exists a measurable subset A of[Omega] such that the space L[sup fi] (Alpha,Sigma[sub Alpha],mi[sub Alpha)) is infinite dimensional and separable.