A Quick Proof of the Order-Extension Principle

AbstractIt is shown that the boolean prime ideal theorem BPIT: every boolean algebra has a prime ideal, does not follow from the order-extension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a Fraenkel–Mostowski model, where the family of atoms is indexed by a countable universal-homogeneous boolean algebra whose boolean partial ordering has a ‘generic’ extension to a linear ordering. To illustrate the technique for proving that the order-extension principle holds in the model we also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure.

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Estimation of Fuzzy Parameters in the Linear Muskingum Model with the Aid of Particle Swarm Optimization

Sustainability ◽

10.3390/su13137152 ◽

2021 ◽

Vol 13 (13) ◽

pp. 7152

Author(s):

Mike Spiliotis ◽

Alvaro Sordo-Ward ◽

Luis Garrote

Keyword(s):

Particle Swarm Optimization ◽

Fitness Function ◽

Particle Swarm ◽

Flood Routing ◽

Performance Criteria ◽

Extension Principle ◽

Swarm Optimization ◽

Muskingum Model ◽

Muskingum Method ◽

Fuzzy Parameters

The Muskingum method is one of the widely used methods for lumped flood routing in natural rivers. Calibration of its parameters remains an active challenge for the researchers. The task has been mostly addressed by using crisp numbers, but fuzzy seems a reasonable alternative to account for parameter uncertainty. In this work, a fuzzy Muskingum model is proposed where the assessment of the outflow as a fuzzy quantity is based on the crisp linear Muskingum method but with fuzzy parameters as inputs. This calculation can be achieved based on the extension principle of the fuzzy sets and logic. The critical point is the calibration of the proposed fuzzy extension of the Muskingum method. Due to complexity of the model, the particle swarm optimization (PSO) method is used to enable the use of a simulation process for each possible solution that composes the swarm. A weighted sum of several performance criteria is used as the fitness function of the PSO. The function accounts for the inclusive constraints (the property that the data must be included within the produced fuzzy band) and for the magnitude of the fuzzy band, since large uncertainty may render the model non-functional. Four case studies from the references are used to benchmark the proposed method, including smooth, double, and non-smooth data and a complex, real case study that shows the advantages of the approach. The use of fuzzy parameters is closer to the uncertain nature of the problem. The new methodology increases the reliability of the prediction. Furthermore, the produced fuzzy band can include, to a significant degree, the observed data and the output of the existent crisp methodologies even if they include more complex assumptions.

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Constructing weak simulations from linear implications for processes with private names

Mathematical Structures in Computer Science ◽

10.1017/s0960129518000452 ◽

2019 ◽

Vol 29 (8) ◽

pp. 1275-1308 ◽

Cited By ~ 1

Author(s):

Ross Horne ◽

Alwen Tiu

Keyword(s):

Expressive Power ◽

Dual Pair ◽

Proof System ◽

Branching Time ◽

First Order ◽

Order Extension

AbstractThis paper clarifies that linear implication defines a branching-time preorder, preserved in all contexts, when used to compare embeddings of process in non-commutative logic. The logic considered is a first-order extension of the proof system BV featuring a de Morgan dual pair of nominal quantifiers, called BV1. An embedding of π-calculus processes as formulae in BV1 is defined, and the soundness of linear implication in BV1 with respect to a notion of weak simulation in the π -calculus is established. A novel contribution of this work is that we generalise the notion of a ‘left proof’ to a class of formulae sufficiently large to compare embeddings of processes, from which simulating execution steps are extracted. We illustrate the expressive power of BV1 by demonstrating that results extend to the internal π -calculus, where privacy of inputs is guaranteed. We also remark that linear implication is strictly finer than any interleaving preorder.

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On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

Canadian Mathematical Bulletin ◽

10.4153/cmb-2013-015-9 ◽

2014 ◽

Vol 57 (2) ◽

pp. 254-263 ◽

Cited By ~ 6

Author(s):

Ole Christensen ◽

Hong Oh Kim ◽

Rae Young Kim

Keyword(s):

Extension Principle ◽

Sufficient Condition ◽

Wavelet Frame ◽

Wavelet Frames ◽

Wavelet System ◽

Number Of Generators ◽

Unitary Extension

AbstractThe unitary extension principle (UEP) by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

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XXI. Differential equations of the first order. Extension of integrable forms

The London Edinburgh and Dublin Philosophical Magazine and Journal of Science ◽

10.1080/14786446508643841 ◽

1865 ◽

Vol 29 (194) ◽

pp. 121-128

Author(s):

Charles James Hargreave

Keyword(s):

Differential Equations ◽

First Order ◽

Order Extension

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Adaptive Synchronization for Uncertain Delayed Fractional-Order Hopfield Neural Networks via Fractional-Order Sliding Mode Control

Mathematical Problems in Engineering ◽

10.1155/2018/1603629 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8 ◽

Cited By ~ 6

Author(s):

Bo Meng ◽

Xiaohong Wang

Keyword(s):

Neural Networks ◽

Fractional Order ◽

Sliding Mode ◽

Small Region ◽

Hopfield Neural Networks ◽

Adaptive Synchronization ◽

Unknown Parameters ◽

External Disturbances ◽

Lyapunov Stability Criterion ◽

Order Extension

Adaptive synchronization for a class of uncertain delayed fractional-order Hopfield neural networks (FOHNNs) with external disturbances is addressed in this paper. For the unknown parameters and external disturbances of the delayed FOHNNs, some adaptive estimations are designed. Firstly, a fractional-order switched sliding surface is proposed for the delayed FOHNNs. Then, according to the fractional-order extension of the Lyapunov stability criterion, a fractional-order sliding mode controller is constructed to guarantee that the synchronization error of the two uncertain delayed FOHNNs converges to an arbitrary small region of the origin. Finally, a numerical example of two-dimensional uncertain delayed FOHNNs is given to verify the effectiveness of the proposed method.

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