scholarly journals On the Number of Indecomposable Permutations with a Given Number of Cycles

10.37236/2071 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Robert Cori ◽  
Claire Mathieu ◽  
John Michael Robson
Keyword(s):  
Fixed Ratio ◽  
Threshold Phenomenon ◽  
Asymptotic Probability ◽  
Number Of Cycles

A permutation $a_1a_2\ldots a_n$ is indecomposable if there does not exist $p<n$ such that $a_1a_2\ldots a_p$ is a permutation of $\{ 1,2,\ldots,p\}$. We consider the  probability that a permutation of ${\mathbb S}_n$  with $m$ cycles is indecomposable and prove that  this probability is monotone non-increasing in $n$.We compute  also the asymptotic probability when  $n$ goes to infinity with $m/n$ tending to a fixed ratio.  The asymptotic probability is monotone in $m/n$, and there is no threshold phenomenon: it degrades gracefully from 1 to 0. When $n=2m$, a slight majority ($51.117\ldots$ percent) of the permutations are indecomposable.

scholarly journals Indecomposable permutations with a given number of cycles

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Robert Cori ◽  
Claire Mathieu
Keyword(s):  
Fixed Point ◽  
Error Term ◽  
Nous Obtenons ◽  
Threshold Phenomenon ◽  
Arbitrary Genus ◽  
International Audience ◽  
Asymptotic Probability ◽  
Number Of Cycles ◽  
Pour Cent ◽  
Orientable Surfaces

International audience A permutation $a_1a_2 \ldots a_n$ is $\textit{indecomposable}$ if there does not exist $p \lt n$ such that $a_1a_2 \ldots a_p$ is a permutation of $\{ 1,2, \ldots ,p\}$. We compute the asymptotic probability that a permutation of $\mathbb{S}_n$ with $m$ cycles is indecomposable as $n$ goes to infinity with $m/n$ fixed. The error term is $O(\frac{\log(n-m)}{ n-m})$. The asymptotic probability is monotone in $m/n$, and there is no threshold phenomenon: it degrades gracefully from $1$ to $0$. When $n=2m$, a slight majority ($51.1 \ldots$ percent) of the permutations are indecomposable. We also consider indecomposable fixed point free involutions which are in bijection with maps of arbitrary genus on orientable surfaces, for these involutions with $m$ left-to-right maxima we obtain a lower bound for the probability of being indecomposable. Une permutation $a_1a_2 \ldots a_n$ est $\textit{indécomposable}$, s’il n’existe pas de $p \lt n$ tel que $a_1a_2 \ldots a_p$ est une permutation de $\{ 1,2, \ldots ,p\}$. Nous calculons la probabilité pour qu’une permutation de $\mathbb{S}_n$ ayant $m$ cycles soit indécomposable et plus particulièrement son comportement asymptotique lorsque $n$ tend vers l’infini et que $m=n$ est fixé. Cette valeur décroît régulièrement de $1$ à $0$ lorsque $m=n$ croît, et il n’y a pas de phénomène de seuil. Lorsque $n=2m$, une faible majorité ($51.1 \ldots$ pour cent) des permutations sont indécomposables. Nous considérons aussi les involutions sans point fixe indécomposables qui sont en bijection avec les cartes de genre quelconque plongées dans une surface orientable, pour ces involutions ayant $m$ maxima partiels (ou records) nous obtenons une borne inférieure pour leur probabilité d’êtres indécomposables.

scholarly journals Short Cycles in Random Regular Graphs

10.37236/1819 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Brendan D. McKay ◽  
Nicholas C. Wormald ◽  
Beata Wysocka
Keyword(s):  
Bipartite Graphs ◽  
Regular Graphs ◽  
Asymptotic Probability ◽  
Number Of Cycles

Consider random regular graphs of order $n$ and degree $d=d(n)\ge 3$. Let $g=g(n)\ge 3$ satisfy $(d-1)^{2g-1}=o(n)$. Then the number of cycles of lengths up to $g$ have a distribution similar to that of independent Poisson variables. In particular, we find the asymptotic probability that there are no cycles with sizes in a given set, including the probability that the girth is greater than $g$. A corresponding result is given for random regular bipartite graphs.

The generalized regularities of damageability and integrity in estimations of the endurance in conditions of variability of loading regimes

2019 ◽  
Vol 85 (9) ◽  
pp. 61-65
Author(s):  
N. A. Makhutov
Keyword(s):  
Life Time ◽  
Cyclic Strength ◽  
Laboratory Diagnostics ◽  
Methodological Issues ◽  
Break Points ◽  
Number Of Cycles ◽  
Cyclic Damage ◽  
Damage Summation ◽  
Temperature Factors

We consider and analyze general methodological issues regarding the strength and endurance (life-time) of the materials and structure elements under a combined effect of various force, deformation and temperature factors. The Journal "Zavodskaya laboratoriya. Diagnostika materialov" (Industrial laboratory. Diagnostics of materials) has launched systematic publications on this problematic since 2018. For many decades, domestic and foreign laboratory studies have gleaned to a traditional methodology for obtaining initial curves of the long-term and cyclic strength that related the breaking stresses with time or number of cycles. These curves, with the characteristic sections and break points, separating the areas of elastic and inelastic (plastic strain or creep strain) strain, are used in analysis of long-term and cyclic damage. Using the elementary linear law of damage summation, it is possible to calculate at a first approximation the strength and endurance under varying conditions of loading. Stepping up the requirements to the accuracy of calculations necessitates a transition from force fracture criteria (at stresses a) to deformation criteria (in elastic and inelastic deformations e). Thus, it becomes possible to construct and use a unified expression for the curve of the long-term cyclic fracture (taking into account the temporal x and cyclic N factors) and a long-term cyclic damage. With such approach it is possible to remain the linear law of damage summation though those damages are obviously nonlinear. The goal of the study is to continue and support the discussion of the most complex problems of a comprehensive assessment of the strength, resource, survivability and safety of high-risk engineering equipment within the journal pages.

Assessment of thermal behavior of a cooling system to reduce thermal fatigue cracks in aluminum injection molds

2020 ◽  
pp. 75-86
Author(s):  
Sergio Antonio Camargo ◽  
Lauro Correa Romeiro ◽  
Carlos Alberto Mendes Moraes
Keyword(s):  
Thermal Fatigue ◽  
Cooling System ◽  
Fatigue Cracks ◽  
Cooling Water ◽  
Thermal Conditions ◽  
Thermal Gradients ◽  
Ansys Software ◽  
Cooling Systems ◽  
Thermal Fatigue Cracks ◽  
Number Of Cycles

The present article aimed to test changes in cooling water temperatures of males, present in aluminum injection molds, to reduce failures due to thermal fatigue. In order to carry out this work, cooling systems were studied, including their geometries, thermal gradients and the expected theoretical durability in relation to fatigue failure. The cooling system tests were developed with the aid of simulations in the ANSYS software and with fatigue calculations, using the method of Goodman. The study of the cooling system included its geometries, flow and temperature of this fluid. The results pointed to a significant increase in fatigue life of the mold component for the thermal conditions that were proposed, with a significant increase in the number of cycles, to happen failures due to thermal fatigue.

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