scholarly journals A PARTIAL ORDERING OF KNOTS AND LINKS THROUGH DIAGRAMMATIC UNKNOTTING

2009 ◽  
Vol 18 (04) ◽  
pp. 505-522 ◽  
Author(s):  
YUANAN DIAO ◽  
CLAUS ERNST ◽  
ANDRZEJ STASIAK
Keyword(s):  
Special Property ◽  
Partial Ordering ◽  
Single Crossing ◽  
Knots And Links ◽  
Minimal Diagram

In this paper we define a partial ordering of knots and links using a special property derived from their minimal diagrams. A link [Formula: see text] is called a predecessor of a link [Formula: see text] if [Formula: see text] and a diagram of [Formula: see text] can be obtained from a minimal diagram D of [Formula: see text] by a single crossing change. In such a case, we say that [Formula: see text]. We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.

scholarly journals A dichotomy theorem for minimizers of monotone recurrence relations

2013 ◽  
Vol 35 (1) ◽  
pp. 215-248 ◽  
Author(s):  
BLAŽ MRAMOR ◽  
BOB RINK
Keyword(s):  
Recurrence Relations ◽  
Nearest Neighbor ◽  
Special Property ◽  
Solid State Physics ◽  
Twist Map ◽  
Global Minimizers ◽  
Mather Theory ◽  
Single Crossing ◽  
Crystal Model ◽  
Kontorova Model

AbstractVariational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel–Kontorova model for a ferromagnetic crystal. For such problems, Aubry–Mather theory establishes the existence of ‘ground states’ or ‘global minimizers’ of arbitrary rotation number. A nearest neighbor crystal model is equivalent to a Hamiltonian twist map. In this case, the global minimizers have a special property: they can only cross once. As a non-trivial consequence, every one of them has the Birkhoff property. In crystals with a larger range of interaction and for higher order recurrence relations, the single crossing property does not hold and there can exist global minimizers that are not Birkhoff. In this paper we investigate the crossings of global minimizers. Under a strong twist condition, we prove the following dichotomy: they are either Birkhoff, and thus very regular, or extremely irregular and non-physical: they then grow exponentially and oscillate. For Birkhoff minimizers, we also prove certain strong ordering properties that are well known for twist maps.

Knots and Links

2004 ◽  
Author(s):  
Peter R. Cromwell
Keyword(s):  
Knots And Links

Babbitt’s Beguiling Surfaces, Improvised Inside; Part I: Freedoms

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Joshua Banks Mailman
Keyword(s):  
Partial Ordering ◽  
Personal Traits ◽  
Milton Babbitt ◽  
Iconic Figure ◽  
Inside Part

Milton Babbitt has been a controversial and iconic figure, which has indirectly led to fallacious assumptions about how his music is made, and therefore to fundamental misconceptions about how it might be heard and appreciated. This video (the first of a three-part video essay) reconsiders his music in light of both his personal traits and a more precise examination of the constraints and freedoms entailed by his unusual and often misunderstood compositional practices, which are based inherently on partial ordering (as well as pitch repetition), which enables a surprising amount of freedom to compose the surface details we hear. The opening of Babbitt’s Composition for Four Instruments (1948) and three recompositions (based on re-ordering of pitches) demonstrate the freedoms intrinsic to partial ordering.

Minimal distributions in a stochastic partial ordering and bounds for Gaussian processes

1983 ◽  
Vol 20 (03) ◽  
pp. 529-536
Author(s):  
W. J. R. Eplett
Keyword(s):  
Gaussian Process ◽  
Gaussian Processes ◽  
Partial Ordering ◽  
Boundary Crossing ◽  
Conditional Probabilities ◽  
Life Distribution ◽  
Life Distributions ◽  
Bounded Below ◽  
Crossing Probabilities

A natural requirement to impose upon the life distribution of a component is that after inspection at some randomly chosen time to check whether it is still functioning, its life distribution from the time of checking should be bounded below by some specified distribution which may be defined by external considerations. Furthermore, the life distribution should ideally be minimal in the partial ordering obtained from the conditional probabilities. We prove that these specifications provide an apparently new characterization of the DFRA class of life distributions with a corresponding result for IFRA distributions. These results may be transferred, using Slepian's lemma, to obtain bounds for the boundary crossing probabilities of a stationary Gaussian process.

scholarly journals Inequalities in the European Union—A Partial Order Analysis of the Main Indicators

2021 ◽  
Vol 13 (11) ◽  
pp. 6278
Author(s):  
Lars Carlsen ◽  
Rainer Bruggemann
Keyword(s):  
Indicator System ◽  
Partial Ordering ◽  
Disposable Income ◽  
The European Union ◽  
Member States ◽  
Eu Member States ◽  
Per Capita ◽  
Educational Aspect ◽  
The Eu

The inequality within the 27 European member states has been studied. Six indicators proclaimed by Eurostat to be the main indicators charactere the countries: (i) the relative median at-risk-of-poverty gap, (ii) the income distribution, (iii) the income share of the bottom 40% of the population, (iv) the purchasing power adjusted GDP per capita, (v) the adjusted gross disposable income of households per capita and (vi) the asylum applications by state of procedure. The resulting multi-indicator system was analyzed applying partial ordering methodology, i.e., including all indicators simultaneously without any pretreatment. The degree of inequality was studied for the years 2010, 2015 and 2019. The EU member states were partially ordered and ranked. For all three years Luxembourg, The Netherlands, Austria, and Finland are found to be highly ranked, i.e., having rather low inequality. Bulgaria and Romania are, on the other hand, for all three years ranked low, with the highest degree of inequality. Excluding the asylum indicator, the risk-poverty-gap and the adjusted gross disposable income were found as the most important indicators. If, however, the asylum application is included, this indicator turns out as the most important for the mutual ranking of the countries. A set of additional indicators was studied disclosing the educational aspect as of major importance to achieve equality. Special partial ordering tools were applied to study the role of the single indicators, e.g., in relation to elucidate the incomparability of some countries to all other countries within the union.

scholarly journals Regular ω-semigroups

1968 ◽  
Vol 9 (1) ◽  
pp. 46-66 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Partial Ordering ◽  
Image Position

Let S be a semigroup whose set E of idempotents is non-empty. We define a partial ordering ≧ on E by the rule that e ≧ f and only if ef = f = fe. If E = {ei: i∈ N}, where N denotes the set of all non-negative integers, and if the elements of E form the chainthen S is called an ω-semigroup.

A New Category of Three-Dimensional Chaotic Flows with Identical Eigenvalues

2020 ◽  
Vol 30 (02) ◽  
pp. 2050026 ◽  
Author(s):  
Zahra Faghani ◽  
Fahimeh Nazarimehr ◽  
Sajad Jafari ◽  
Julien C. Sprott
Keyword(s):  
Chaotic Systems ◽  
Autonomous Systems ◽  
Three Dimensional ◽  
Special Property ◽  
Computer Search ◽  
Chaotic Flows ◽  
Stable Equilibria ◽  
Dynamical Properties ◽  
Simple Systems ◽  
First Time

In this paper, some new three-dimensional chaotic systems are proposed. The special property of these autonomous systems is their identical eigenvalues. The systems are designed based on the general form of quadratic jerk systems with 10 terms, and some systems with stable equilibria. Using a systematic computer search, 12 simple chaotic systems with identical eigenvalues were found. We believe that systems with identical eigenvalues are described here for the first time. These simple systems are listed in this paper, and their dynamical properties are investigated.

On some partial ordering of interest in reliability

1996 ◽  
Vol 36 (10) ◽  
pp. 1337-1346 ◽  
Author(s):  
A.N. Ahmed ◽  
A.A. Soliman ◽  
S.E. Khider
Keyword(s):  
Partial Ordering
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