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On the boundedness of equivariant homeomorphism groups

Lech J., Michalik I., Rybicki T.

Opuscula Mathematica

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2018

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Vol. 38, no. 3

395--408

EN

Given a principal G-bundle [formula] let HG(M) be the identity component of the group of G-equivariant homeomorphisms on M. The problem of the uniform perfectness and boundedness of HG(M) is studied. It occurs that these properties depend on the structure of H(B), the identity component of the group of homeomorphisms of B, and of B itself. Most of the obtained results still hold in the [formula] category.

2

On Sequential Compactness and Related Notions of Compactness of Metric Spaces in ZF

Keremedis K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2016

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Vol. 64, no. 1

29--46

EN

We show that: (i) If every sequentially compact metric space is countably compact then for every infinite set X, [X]<ω is Dedekind-infinite. In particular, every infinite subset of R is Dedekind-infinite. (ii) Every sequentially compact metric space is compact iff every sequentially compact metric space is separable. In addition, if every sequentially compact metric space is compact then: every infinite set is Dedekind-infinite, the product of a countable family of compact metric spaces is compact, and every compact metric space is separable. (iii) The axiom of countable choice implies that every sequentially bounded metric space is totally bounded and separable, every sequentially compact metric space is compact, and every uncountable sequentially compact, metric space has size |R|. (iv) If every sequentially bounded metric space is totally bounded then every infinite set is Dedekind-infinite. (v) The statement: “Every sequentially bounded metric space is bounded” implies the axiom of countable choice restricted to the real line. (vi) The statement: “For every compact metric space X either |X| ≤ |R|, or |R| ≤ |X|” implies the axiom of countable choice restricted to families of finite sets. (vii) It is consistent with ZF that there exists a sequentially bounded metric space whose completion is not sequentially bounded. (viii) The notion of sequential boundedness of metric spaces is countably productive.

3

Minimum energy control of fractional positive electrical circuits

Kaczorek T.

Archives of Electrical Engineering

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2016

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Vol. 65, nr 2

191--201

EN

Minimum energy control problem for the fractional positive electrical circuits is formulated and solved. Sufficient conditions for the existence of solution to the problem are established. A procedure for solving of the problem is proposed and illustrated by an example of fractional positive electrical circuit.

4

MAC Layer Outage Probability of Bounded Ad Hoc Networks

Kaynia M., Fabbri F., Øien G. E., Verdone R.

Advances in Electronics and Telecommunications

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2011

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Vol. 2, no. 4

39--46

EN

We consider a bounded square-shaped ad hoc network scenario, within which packet arrivals are distributed randomly in space and time according to a 3-dimensional Poisson Point Process. Each packet is transmitted over a single hop to its destination, located a fixed distance away. Within this context, the outage probability of the ALOHA and CSMA protocols is derived, and we evaluate the impact of edge effects in space on the performance of MAC protocols. Our analytical expressions are verified with Monte Carlo simulations. The behavior of the network is evaluated as the system parameters, such as the node density, the physical size of the network, and the distance between each transmitter and its receiver, vary. Furthermore, the obtained results are compared to those of unbounded networks, showing that edge effects reduce the average outage probability across the network significantly, due to the lower level of interference suffered by boundary nodes.

5

Boundedness of solutions of nonlinear three-dimensional difference systems with delays

Schmeidel E.

Fasciculi Mathematici

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2010

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Nr 44

107-113

EN

In this paper three-dimensional nonlinear difference system with delays ...[wzór] is investigated. The classification of nonoscillatory solutions of the considered system are presented. Next, the sufficient conditions under which nonoscillatory solution of considered system is bounded or is unbounded are given. Key words: difference equation, nonlinear system, nonoscillatory, bounded, unbounded solution.