scholarly journals Profiles of Permutations

10.37236/188 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Michael Lugo
Keyword(s):  
Random Element ◽  
Asymptotic Density ◽  
Random Permutations ◽  
Expected Number ◽  
Cycle Structure ◽  
Large N ◽  
Cycle Lengths ◽  
Number Of Cycles ◽  
The One ◽  
Infinite Set

This paper develops an analogy between the cycle structure of, on the one hand, random permutations with cycle lengths restricted to lie in an infinite set $S$ with asymptotic density $\sigma$ and, on the other hand, permutations selected according to the Ewens distribution with parameter $\sigma$. In particular we show that the asymptotic expected number of cycles of random permutations of $[n]$ with all cycles even, with all cycles odd, and chosen from the Ewens distribution with parameter $1/2$ are all ${1 \over 2} \log n + O(1)$, and the variance is of the same order. Furthermore, we show that in permutations of $[n]$ chosen from the Ewens distribution with parameter $\sigma$, the probability of a random element being in a cycle longer than $\gamma n$ approaches $(1-\gamma)^\sigma$ for large $n$. The same limit law holds for permutations with cycles carrying multiplicative weights with average $\sigma$. We draw parallels between the Ewens distribution and the asymptotic-density case and explain why these parallels should exist using permutations drawn from weighted Boltzmann distributions.

A Generalization of the Erdős–Turán Law for the Order of Random Permutation

2012 ◽  
Vol 21 (5) ◽  
pp. 715-733 ◽  
Author(s):  
ALEXANDER GNEDIN ◽  
ALEXANDER IKSANOV ◽  
ALEXANDER MARYNYCH
Keyword(s):  
Markov Chain ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Walks ◽  
Random Permutation ◽  
Unit Interval ◽  
Random Permutations ◽  
Cycle Structure ◽  
Limit Laws ◽  
Cycle Lengths

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on n integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős–Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM(θ)-distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

On the probability of coincidence of cycle lengths for independent random permutations with given number of cycles

2015 ◽  
Vol 25 (6) ◽  
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Asymptotic Estimates ◽  
Random Permutations ◽  
Cycle Lengths ◽  
Number Of Cycles ◽  
Coincidence Probability

AbstractFrom the set of all permutations of the degree n with a given number N ≤ n of cycles two permutations are choosed randomly, uniformly and independently. The cycles of each permutation are numbered in some of N! possible ways. We study the coincidence probability of the cycle lengths of permutations for a given numbering. This probability up to a suitably selected renumbering of cycles of the first permutation equals to the probability of similarity of these permutations. The asymptotic estimates of the coincidence probability of the cycle lengths are obtained for five types of relations between N, n → ∞.

OnPoint: A package for online experiments in motor control and motor learning

2020 ◽  
Author(s):  
Jonathan Sanching Tsay ◽  
Alan S. Lee ◽  
Guy Avraham ◽  
Darius E. Parvin ◽  
Jeremy Ho ◽  
...  
Keyword(s):  
Motor Control ◽  
Motor Learning ◽  
Open Source Software ◽  
Research Progress ◽  
Global Pandemic ◽  
Large N ◽  
Open Source Software Package ◽  
Potential Applications ◽  
Temporal And Spatial ◽  
The One

Motor learning experiments are typically run in-person, exploiting finely calibrated setups (digitizing tablets, robotic manipulandum, full VR displays) that provide high temporal and spatial resolution. However, these experiments come at a cost, not limited to the one-time expense of purchasing equipment but also the substantial time devoted to recruiting participants and administering the experiment. Moreover, exceptional circumstances that limit in-person testing, such as a global pandemic, may halt research progress. These limitations of in-person motor learning research have motivated the design of OnPoint, an open-source software package for motor control and motor learning researchers. As with all online studies, OnPoint offers an opportunity to conduct large-N motor learning studies, with potential applications to do faster pilot testing, replicate previous findings, and conduct longitudinal studies (GitHub repository: https://github.com/alan-s-lee/OnPoint).

Density of sets with missing differences and applications

2021 ◽  
Vol 71 (3) ◽  
pp. 595-614
Author(s):  
Ram Krishna Pandey ◽  
Neha Rai
Keyword(s):  
Number Theory ◽  
Asymptotic Density ◽  
Theory Problem ◽  
Distance Graphs ◽  
Positive Integers ◽  
Infinite Set

Abstract For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M-sets, denoted by μ(M), where an M-set is a set of non-negative integers in which no two elements differ by an element in M. In 1973, Cantor and Gordon find μ(M) for |M| ≤ 2. Partial results are known in the case |M| ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ(M) for two families namely, M = {a, b,a + b, n(a + b)} and M = {a, b, b − a, n(b − a)}. For both of these families, we find some exact values and some bounds on μ(M). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M. So, as an application, we also study these coloring parameters associated with these families.

scholarly journals A watershed model of individual differences in fluid intelligence

2016 ◽  
Author(s):  
Rogier A. Kievit ◽  
Simon W. Davis ◽  
John Griffiths ◽  
Marta Correia ◽  
Cam-CAN ◽  
...  
Keyword(s):  
White Matter ◽  
Processing Speed ◽  
Fluid Intelligence ◽  
White Matter Integrity ◽  
Life Outcomes ◽  
Watershed Model ◽  
Adult Lifespan ◽  
Large N ◽  
Competing Models ◽  
The One

AbstractFluid intelligence is a crucial cognitive ability that predicts key life outcomes across the lifespan. Strong empirical links exist between fluid intelligence and processing speed on the one hand, and white matter integrity and processing speed on the other. We propose a watershed model that integrates these three explanatory levels in a principled manner in a single statistical model, with processing speed and white matter figuring as intermediate endophenotypes. We fit this model in a large (N=555) adult lifespan cohort from the Cambridge Centre for Ageing and Neuroscience (Cam-CAN) using multiple measures of processing speed, white matter health and fluid intelligence. The model fit the data well, outperforming competing models and providing evidence for a many-to-one mapping between white matter integrity, processing speed and fluid intelligence. The model can be naturally extended to integrate other cognitive domains, endophenotypes and genotypes.

scholarly journals A note on primes in certain residue classes

2018 ◽  
Vol 14 (08) ◽  
pp. 2219-2223
Author(s):  
Paolo Leonetti ◽  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Residue Classes ◽  
Euler Totient Function ◽  
The One ◽  
Positive Integers

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

Generation of the symmetric group Sn2

2021 ◽  
pp. 2150107
Author(s):  
Carlos Zequeira Sánchez ◽  
Evaristo José Madarro Capó ◽  
Guillermo Sosa-Gómez
Keyword(s):  
Symmetric Group ◽  
Random Element ◽  
Matrix Representation ◽  
Random Permutation ◽  
Random Permutations ◽  
Dimension Formula ◽  
Indispensable Tool

In various scenarios today, the generation of random permutations has become an indispensable tool. Since random permutation of dimension [Formula: see text] is a random element of the symmetric group [Formula: see text], it is necessary to have algorithms capable of generating any permutation. This work demonstrates that it is possible to generate the symmetric group [Formula: see text] by shifting the components of a particular matrix representation of each permutation.

THE REFLECTIVITY OF THE MIXED CRYSTAL AB1-xCx

2010 ◽  
Vol 24 (27) ◽  
pp. 5337-5343
Author(s):  
XIAOYAN ZHANG ◽  
JINGFENG WANG ◽  
GUOLIANG FAN
Keyword(s):  
Mixed Crystal ◽  
Random Element ◽  
Mixed Crystals ◽  
Phonon Interaction ◽  
Long Wavelength ◽  
Electron Phonon Interaction ◽  
The Third ◽  
Optical Character ◽  
Mode Behavior ◽  
The One

The properties of long wavelength optical phonons in mixed crystals AB 1-x C x is discussed by a model similar to Modified Random Element Isodisplacement (MREI). Using this method we investigate the frequencies, the dielectric functions, and the reflectivity of several mixed crystals. It is found that this model can be applied to the one-mode behavior, the two-mode behavior, and that of the third category. So the model provides a possible way to understand the optical character of the ternary mixed crystal. Based on it, we can discuss other problems similar to electron–phonon interaction and so on.

TOPOLOGICAL σ MODELS AND LARGE N MATRIX INTEGRAL

1995 ◽  
Vol 10 (29) ◽  
pp. 4203-4224 ◽  
Author(s):  
TOHRU EGUCHI ◽  
KENTARO HORI ◽  
SUNG-KIL YANG
Keyword(s):  
Matrix Model ◽  
Riemann Surfaces ◽  
Holomorphic Maps ◽  
Integrable Structure ◽  
Toda Hierarchy ◽  
Matrix Integral ◽  
Large N ◽  
The Matrix ◽  
Lax Operator ◽  
The One

In this paper we describe in some detail the representation of the topological CP1 model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the CP1 model and show that it is governed by an extension of the one-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto CP1. We compute intersection numbers on the moduli space of curves using a geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the CP1 model using a superpotential eX + et0,Q e-X given by the Lax operator of the Toda hierarchy (X is the LG field and t0,Q is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between CP1 and N=2 supersymmetric sine-Gordon theory which was noted sometime ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present an LG formulation of the topological CP2 model.

scholarly journals The Influence of the Carbo-Glass Geogrid-Reinforcement on the Fatigue Life of the Asphalt Pavement Structure / Wpływ Zbrojenia Siatka Weglowo–Szklana Na Trwałosc Zmeczeniowa Asfaltowej Nawierzchni Drogowej

2012 ◽  
Vol 58 (1) ◽  
pp. 97-113 ◽  
Author(s):  
J. Górszczyk ◽  
S. Gaca
Keyword(s):  
Fatigue Life ◽  
Asphalt Pavement ◽  
Field Tests ◽  
Traffic Loading ◽  
Geogrid Reinforcement ◽  
Traffic Conditions ◽  
Pavement Structure ◽  
Number Of Cycles ◽  
The One ◽  
Stress Fatigue

Abstract This paper describes the analyses of the fatigue life of the asphalt pavement reinforced with geogrid interlayer under traffic loading. Finite Element ANSYS package with using nCode applications, as well as macros specially designed in APDL programming script and VBA were used to model the considered problem. Our analysis included computation of stress, fatigue life, damage matrix and rainflow matrix. The method applied was the one of fatigue calculation: stress - number of cycles in short S-N. On the basis of the performed high cycle fatigue analysis, the influence of the location of the used geogrid and of its bond with asphalt layers on the fatigue life and the work of the asphalt pavement structure were determined. The study was carried out for three temperature seasons i.e. spring and fall (assumed as one season), winter and summer. The variability of the traffic conditions were taken into account by assuming weekly blocks of traffic loading. The calculations were made using the real values of loading measured in field tests on the German highways by means of HS-WIM weighing system. As a result of the performed tests, it was proved that the use of geogrid-reinforcement may prolong the fatigue life of the asphalt pavement. However, it is required that: the geogrid should be located in the tension zone as low as possible in the structure of the asphalt layers. Moreover, it is necessary to provide high stiffness of the bond between the geogrid and the asphalt layers.

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