scholarly journals On the convergence of bumping routes to their limit shapes in the RSK algorithm: numerical experiments

2021 ◽  
pp. 2-9
Author(s):  
Nikolay Vassiliev ◽  
Vasilii Duzhin ◽  
Artem Kuzmin
Keyword(s):  
Numerical Experiments ◽  
Least Squares Method ◽  
Computer Experiments ◽  
Young Tableaux ◽  
Limit Shape ◽  
Mean Values ◽  
Limit Shapes ◽  
Rsk Algorithm ◽  
Fourth Root ◽  
Special Rules

Introduction: The Robinson — Schensted — Knuth (RSK) algorithm transforms a sequence of elements of a linearly ordered set into a pair of Young tableaux P, Q of the same shape. This transformation is based on the process of bumping and inserting elements in tableau P according to special rules. The trajectory formed by all the boxes of the tableau P shifted in the RSK algorithm is called the bumping route. D. Romik and P. Śniady in 2016 obtained an explicit formula for the limit shape of the bumping route, which is determined by its first element. However, the rate of convergence of the bumping routes to the limit shape has not been previously investigated either theoretically or by numerical experiments. Purpose: Carrying out computer experiments to study the dynamics of the bumping routes produced by the RSK algorithm on Young tableaux as their sizes increase. Calculation of statistical means and variances of deviations of bumping routes from their limit shapes in the L2 metric for various values fed to the input of the RSK algorithm. Results: A series of computer experiments have been carried out on Young tableaux, consisting of up to 10 million boxes. We used 300 tableaux of each size. Different input values (0.1, 0.3, 0.5, 0.7, 0.9) were added to each such tableau using the RSK algorithm, and the deviations of the bumping routes built from these values from the corresponding limit shapes were calculated. The graphs of the statistical mean values and variances of these deviations were produced. It is noticed that the deviations decrease in proportion to the fourth root of the tableau size n. An approximation of the dependence of the mean values of deviations on n was obtained using the least squares method.

scholarly journals Investigation of insertion tableau evolution in the Robinson-Schensted-Knuth correspondence

2019 ◽  
Vol 27 (4) ◽  
pp. 316-324
Author(s):  
Vasilii S. Duzhin
Keyword(s):  
Software Package ◽  
Ordered Set ◽  
Input Sequence ◽  
Computer Experiments ◽  
Young Tableaux ◽  
Young Diagrams ◽  
Rsk Algorithm ◽  
2D And 3D ◽  
Linearly Ordered Set

Robinson-Schensted-Knuth (RSK) correspondence occurs in different contexts of algebra and combinatorics. Recently, this topic has been actively investigated by many researchers. At the same time, many investigations require conducting the computer experiments involving very large Young tableaux. The article is devoted to such experiments. RSK algorithm establishes a bijection between sequences of elements of linearly ordered set and the pairs of Young tableaux of the same shape called insertion tableau and recording tableau . In this paper we study the dynamics of tableau and the dynamics of different concrete values in tableau during the iterations of RSK algorithm. Particularly, we examine the paths within tableaux called bumping routes along which the elements of an input sequence pass. The results of computer experiments with Young tableaux of sizes up to 108 were presented. These experiments were made using the software package for dealing with 2D and 3D Young diagrams and tableaux.

scholarly journals On the peeling procedure applied to a Poisson point process

2010 ◽  
Vol 42 (3) ◽  
pp. 620-630
Author(s):  
Y. Davydov ◽  
A. Nagaev ◽  
A. Philippe
Keyword(s):  
Convex Hull ◽  
Point Process ◽  
Numerical Experiments ◽  
Poisson Point Process ◽  
Point Processes ◽  
Asymptotic Properties ◽  
Convex Hulls ◽  
Limit Shape ◽  
Empirical Point ◽  
Stable Random Vectors

In this paper we focus on the asymptotic properties of the sequence of convex hulls which arise as a result of a peeling procedure applied to the convex hull generated by a Poisson point process. Processes of the considered type are tightly connected with empirical point processes and stable random vectors. Results are given about the limit shape of the convex hulls in the case of a discrete spectral measure. We give some numerical experiments to illustrate the peeling procedure for a larger class of Poisson point processes.

scholarly journals Limit Shapes via Bijections

2018 ◽  
Vol 28 (2) ◽  
pp. 187-240 ◽  
Author(s):  
STEPHEN DeSALVO ◽  
IGOR PAK
Keyword(s):  
Scaling Functions ◽  
Integer Partitions ◽  
Limit Shape ◽  
Limit Shapes

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined in [43], and extend them to include limit shapes with different scaling functions.

scholarly journals Study of Factors Influencing the Risk and their Relation to Credibility Theory

1977 ◽  
Vol 9 (1-2) ◽  
pp. 84-104 ◽  
Author(s):  
Maria Amélia Cabral ◽  
Jorge Afonso Garcia
Keyword(s):  
Least Squares Method ◽  
A Priori ◽  
Mean Value ◽  
The Other ◽  
Credibility Theory ◽  
Mean Values ◽  
Factors Influencing ◽  
The Mean ◽  
The One ◽  
Different Characteristics

The study and analysis of the various factors influencing insurance risks constitutes an intricate and usually quite extensive problem. We have to consider on the one hand the nature and heterogeneity of the elements we have been able to measure, and on the other the problem of deciding—without knowing exactly what results to expect—on the types of analysis to carry out and the form in which to present the results.These difficulties, essentially stemming from the fact that we cannot easily define “a priori” a measure of influence, can be overcome only by using highly sophisticated mathematical models. The researcher must define his objectives clearly if he is to avoid spending too much of his time in exploring such models.Either for these reasons or for lack of our experience in this field we were led to the study of three models, presenting entirely different characteristics though based on the analysis and behaviour of mean value fluctuations, measured by their variances or by the least-squares method.Our first model, described in II. 1, associates the notion of influence with the notion of variance. It analyses in detail the alteration of the mean values variance, when what we refer to as a “margination” is executed in the parameter space, taking each of the parameters in turn. We start off by having n distinct parameters, reducing them by one with each step.

The structural features of a tetrahedral 1,3-monothiolate complex: The crystal and molecular structure of Bis(ethyl 3-mercaptobut-2-enoato)zinc(II)

1977 ◽  
Vol 30 (5) ◽  
pp. 993 ◽  
Author(s):  
BF Hoskins ◽  
CD Pannan
Keyword(s):  
Molecular Structure ◽  
Least Squares Method ◽  
Crystal And Molecular Structure ◽  
Lone Pair ◽  
Structural Features ◽  
The Other ◽  
Zinc Atom ◽  
Monoclinic Space Group ◽  
Hydrogen Atoms ◽  
Mean Values

The crystal and molecular structure of bis(ethy 3-mercaptobut-2- enoato)zinc(II) has been determined by single-crystal X-ray diffraction techniques. Solved by conventional Patterson and Fourier methods the structure was refined by a least-squares method employing anisotropic thermal parameters to all non-hydrogen atoms to R and Rw values of 0.040 and 0.046 respectively. The complex crystallizes in the monoclinic space group P21/c with four molecules in a unit cell of dimensions a 11.107(1), b 18.239(2) and c 8.438(1) Ǻ and β 107.7(1)�. The intensities of 2274 independent and statistically significant [I ≥ 3σ(I)] reflections with θ values ≤ 70� were measured by counter methods using nickel- filtered Cu Kα radiation. The crystals comprise discrete monomeric molecules with the zinc atom bonded to two sulphur atoms and two oxygen atoms giving a coordination arrangement which is substantially distorted from an ideal tetrahedron. The mean values for the Zn-S and Zn-O bond distances are 2.247(1) and 2.007(3) Ǻ respectively and the average S-Zn-O intraligand bond angle is 99.25(8)�. The geometries of the ligands differ in two ways. Firstly, the two ethyl groups adopt differing conformations and secondly, while one ligand moiety is essentially planar with the zinc atom displaced about 0.1 Ǻ from that plane, the displaced atom in the other ligand is the carbon bonded to the sulphur atom and not the metal which is, in this instance, coplanar with the other members of the ring. Bond distances in each chelate ring indicate aromatic character with a lone pair of electrons on the ethoxy-oxygen participating in the delocalization.

Investigation of properties of equivalence classes of permutations by inverse Robinson — Schensted — Knuth transformation

2019 ◽  
pp. 11-22
Author(s):  
N. N. Vassiliev ◽  
V. S. Duzhin ◽  
A. D. Kuzmin
Keyword(s):  
Software Package ◽  
Numerical Experiments ◽  
Young Tableau ◽  
Infinite Sequence ◽  
Equivalence Classes ◽  
Young Tableaux ◽  
Young Diagrams ◽  
C Programming ◽  
Set Of Permutations ◽  
Knuth Equivalence

Introduction:All information about a permutation, i.e. about an element of a symmetric groupS(n), is contained in a pair of Young tableaux mapped to it by RSK transformation. However, when considering an infinite sequence of natural or real numbers instead of a permutation, all information about it is contained only in an insertion infinite Young tableau. The connection between the first element of an infinite sequence of uniformly distributed random values and the limit angle of the recording tableau nerve was found in a recent work by D. Romik and P. Śniady. However, so far there were no massive numerical experiments devoted to the reconstruction of the beginning of such a sequence by the beginning of an insertion Young tableau. The reconstruction accuracy is very important, because even the value of the first element of a sequence can be determined only by an infinite tableau.Purpose:Developing a software package for operations on Young diagrams and Young tableaux, and its application for numerical experiments with large Young tableaux. Studying the properties of Knuth equivalence classes and dual Knuth equivalence classes on a set of permutations by numerical experiments using direct and inverse RSK transformation.Results:A software package is developed using the C ++ programming language. It includes functions for dealing with Young diagrams and tableaux. The dependence of values of the first element of a permutation obtained by inverse RSK transformation on the recording tableau nerve end coordinates was investigated by conducting massive numerical experiments. Standard deviations of these values were calculated for permutations of different sizes. We determined possible positions of 1 in permutations of the same Knuth equivalence class. It has been found out that the number of these positions does not exceed the number of corner boxes of the corresponding Young diagram. Experiments showed that for a fixed insertion tableau, the value of the first element of a permutation depends only on the recording tableau nerve end coordinates.

scholarly journals Image Restoration Using a Fixed-Point Method for a TVL2 Regularization Problem

Algorithms ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 1
Author(s):  
Kyoum Sun Kim ◽  
Jae Heon Yun
Keyword(s):  
Fixed Point ◽  
Image Restoration ◽  
Iterative Methods ◽  
Numerical Experiments ◽  
Least Squares Method ◽  
Fixed Point Method ◽  
Point Method ◽  
Test Problems ◽  
Regularization Problem ◽  
Methods Numerical

In this paper, we first propose a new TVL2 regularization model for image restoration, and then we propose two iterative methods, which are fixed-point and fixed-point-like methods, using CGLS (Conjugate Gradient Least Squares method) for solving the new proposed TVL2 problem. We also provide convergence analysis for the fixed-point method. Lastly, numerical experiments for several test problems are provided to evaluate the effectiveness of the proposed two iterative methods. Numerical results show that the new proposed TVL2 model is preferred over an existing TVL2 model and the proposed fixed-point-like method is well suited for the new TVL2 model.

Method for automatic evaluation of the voice signals quality with low bit rate voice coding

2021 ◽  
Author(s):  
V.A. Aladinskiy ◽  
S.V. Kuzminskiy
Keyword(s):  
Initial Data ◽  
Covariance Matrix ◽  
Least Squares Method ◽  
Reference Signal ◽  
Reference Sample ◽  
Subjective Assessment ◽  
Training Sample ◽  
Pulse Code Modulation ◽  
Reference Pattern ◽  
Mean Values

Evaluation of the VSQ is an important problem. In known methods for еvaluation the quality of analog or digital voice signals with pulse-code modulation (PCM) are used. When automatic quality assessing, it is assumed that the parameters of voice signal are determined. Then the corresponding subjective assessment of VSQ is selected. Objective methods for quality assessing are obtained by comparing the reference signal and the signal transmitted through the communication channel. To eliminate the identified contradictions, it is proposed to use an objective method that provides the assessing of VSQ in the absence of a reference signal and without conversion to the PCM format, i.e. based on the analysis of the LBR voice digital stream. The analyzed digital stream is considered as a system of random variables, which is characterized by a set of mean values and a covariance matrix that make up the pattern of input implementation. The reference pattern also includes a set of mean values and a covariance matrix. The initial data for the formation of a reference pattern are digital streams formed according to a given protocol of LBR voice coding, having the highest VSQ. The proposed method is based on calculating the divergence between the reference pattern and input implementation pattern. At the training stage, several distorted samples with a known number of bit errors are formed from the reference sample. When comparing their patterns, the divergence values are calculated, which are put in compliance with the VSQ assessment. Two approaches have been identified for making the compliance: experimental and experimental-analytical. The choice of approach is determined by the presence of initial data characterizing the VSQ for particular vocoder. Then generated compliance is interpolated. At the assessment stage for each implementation is calculated the divergence between the explored pattern and the reference pattern, which is formed from the training sample. Based on the divergence, the VSQ assessment is calculated. To test the proposed method a reference sample of digital streams obtained by compressing voice signals with a LPC-10-2400 vocoder was formed. Based on the initial data, which are presented in Recommendation-R F.1112-1, the compliance between the values of sound intelligibility and divergence was compiled by experimental-analytical approach. The obtained correspondence was interpolated by the least squares method using a third degree polynomial.

scholarly journals Asymptotics for the number of standard tableaux of skew shape and for weighted lozenge tilings

2021 ◽  
pp. 1-24
Author(s):  
Alejandro H. Morales ◽  
Igor Pak ◽  
Martin Tassy
Keyword(s):  
Variational Principle ◽  
Partition Function ◽  
Stable Limit ◽  
Young Tableaux ◽  
Limit Shape ◽  
Lozenge Tilings ◽  
Standard Young Tableaux

Abstract We prove and generalise a conjecture in [MPP4] about the asymptotics of $\frac{1}{\sqrt{n!}} f^{\lambda/\mu}$ , where $f^{\lambda/\mu}$ is the number of standard Young tableaux of skew shape $\lambda/\mu$ which have stable limit shape under the $1/\sqrt{n}$ scaling. The proof is based on the variational principle on the partition function of certain weighted lozenge tilings.

scholarly journals Uncorrected Least-Squares Temporal Difference with Lambda-Return

2020 ◽  
Vol 34 (04) ◽  
pp. 5323-5330
Author(s):  
Takayuki Osogami
Keyword(s):  
Monte Carlo ◽  
Reinforcement Learning ◽  
Linear Regression ◽  
Least Squares ◽  
Numerical Experiments ◽  
Least Squares Method ◽  
Training Data ◽  
Temporal Difference

Temporal difference, TD(λ), learning is a foundation of reinforcement learning and also of interest in its own right for the tasks of prediction. Recently, true online TD(λ) has been shown to closely approximate the “forward view” at every step, while conventional TD(λ) does this only at the end of an episode. We re-examine least-squares temporal difference, LSTD(λ), which has been derived from conventional TD(λ). We design Uncorrected LSTD(λ) in such a way that, when λ = 1, Uncorrected LSTD(1) is equivalent to the least-squares method for the linear regression of Monte Carlo (MC) return at every step, while conventional LSTD(1) has this equivalence only at the end of an episode, since the MC return is corrected to be unbiased. We prove that Uncorrected LSTD(λ) can have smaller variance than conventional LSTD(λ), and this allows Uncorrected LSTD(λ) to sometimes outperform conventional LSTD(λ) in practice. When λ = 0, however, Uncorrected LSTD(0) is not equivalent to LSTD. We thus also propose Mixed LSTD(λ), which % mixes the two LSTD(λ)s in a way that it matches conventional LSTD(λ) at λ = 0 and Uncorrected LSTD(λ) at λ = 1. In numerical experiments, we study how the three LSTD(λ)s behave under limited training data.

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