scholarly journals Longest Alternating Subsequences in Pattern-Restricted Permutations

10.37236/952 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Ghassan Firro ◽  
Toufik Mansour ◽  
Mark C. Wilson
Keyword(s):  
Generating Functions ◽  
Complex Analysis ◽  
Limiting Distribution ◽  
Future Research ◽  
Random Permutations ◽  
Restricted Permutations ◽  
Longest Increasing Subsequence

Inspired by the results of Stanley and Widom concerning the limiting distribution of the lengths of longest alternating subsequences in random permutations, and results of Deutsch, Hildebrand and Wilf on the limiting distribution of the longest increasing subsequence for pattern-restricted permutations, we find the limiting distribution of the longest alternating subsequence for pattern-restricted permutations in which the pattern is any one of the six patterns of length three. Our methodology uses recurrences, generating functions, and complex analysis, and also yields more detailed information. Several ideas for future research are listed.

scholarly journals Longest Increasing Subsequences in Pattern-Restricted Permutations

10.37236/1684 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Emeric Deutsch ◽  
A. J. Hildebrand ◽  
Herbert S. Wilf
Keyword(s):  
Limiting Distribution ◽  
Random Permutations ◽  
Limiting Distributions ◽  
Ordered Trees ◽  
Restricted Permutations

Inspired by the results of Baik, Deift and Johansson on the limiting distribution of the lengths of the longest increasing subsequences in random permutations, we find those limiting distributions for pattern-restricted permutations in which the pattern is any one of the six patterns of length 3. We show that the (132)-avoiding case is identical to the distribution of heights of ordered trees, and that the (321)-avoiding case has interesting connections with a well known theorem of Erdős-Szekeres.

Spectral analysis of M/G/1 and G/M/1 type Markov chains

1996 ◽  
Vol 28 (01) ◽  
pp. 114-165 ◽  
Author(s):  
H. R. Gail ◽  
S. L. Hantler ◽  
B. A. Taylor
Keyword(s):  
Markov Chains ◽  
Generating Functions ◽  
Complex Analysis ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equilibrium Analysis ◽  
Transform Method ◽  
Technical Problems ◽  
The Matrix ◽  
Transient Case

When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.

scholarly journals A Note on the Asymptotic Behavior of the Heights in $b$-Tries for $b$ Large

10.37236/1517 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Charles Knessl ◽  
Wojciech Szpankowski
Keyword(s):  
Asymptotic Behavior ◽  
Saddle Point ◽  
Generating Functions ◽  
Single Point ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Point Method ◽  
Saddle Point Method ◽  
Numerical Verification ◽  
Height Distribution

We study the limiting distribution of the height in a generalized trie in which external nodes are capable to store up to $b$ items (the so called $b$-tries). We assume that such a tree is built from $n$ random strings (items) generated by an unbiased memoryless source. In this paper, we discuss the case when $b$ and $n$ are both large. We shall identify five regions of the height distribution that should be compared to three regions obtained for fixed $b$. We prove that for most $n$, the limiting distribution is concentrated at the single point $k_1=\lfloor \log_2 (n/b)\rfloor +1$ as $n,b\to \infty$. We observe that this is quite different than the height distribution for fixed $b$, in which case the limiting distribution is of an extreme value type concentrated around $(1+1/b)\log_2 n$. We derive our results by analytic methods, namely generating functions and the saddle point method. We also present some numerical verification of our results.

scholarly journals Random Permutations, Non-Decreasing Subsequences and Statistical Independence

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1415
Author(s):  
Jesús E. García ◽  
Verónica A. González-López
Keyword(s):  
Random Variables ◽  
Continuous Data ◽  
Random Permutations ◽  
Statistical Independence ◽  
Independence Test ◽  
Independence Tests ◽  
Present Procedure ◽  
Longest Increasing Subsequence

In this paper, we show how the longest non-decreasing subsequence, identified in the graph of the paired marginal ranks of the observations, allows the construction of a statistic for the development of an independence test in bivariate vectors. The test works in the case of discrete and continuous data. Since the present procedure does not require the continuity of the variables, it expands the proposal introduced in Independence tests for continuous random variables based on the longest increasing subsequence (2014). We show the efficiency of the procedure in detecting dependence in real cases and through simulations.

A Brief Review on Numerical Studies on Film Cooling Effectiveness

2016 ◽  
Vol 852 ◽  
pp. 699-706
Author(s):  
Prakhar Jindal ◽  
Shubham Agarwal ◽  
R.P. Sharma
Keyword(s):  
Film Cooling ◽  
Complex Analysis ◽  
Numerical Models ◽  
The Body ◽  
Coolant Fluid ◽  
Future Research ◽  
Cooling Systems ◽  
Cooling Effectiveness ◽  
Film Cooling Effectiveness ◽  
Trade Offs

Film cooling is employed for effective cooling in nozzles and combustion chambers using a spray of coolant fluid in the mainstream flow to cool the body. Experimental analysis was performed elaborately in the past years to get an exact analysis of film cooling effectiveness with different parameters. This approach was however discouraged due to its high cost and time consumption. Recently, researchers switched to the use of numerical platforms for investigation of complex film cooling systems. The paper discusses in detail the numerical analysis of the film cooling systems done so far and the numerical models employed for such a complex analysis along with the advantages and trade-offs of such a numerical approach. This study was carried out to extend database knowledge about the numerical film cooling for its various applications. Therefore, an appropriate cooling technique should be designed to protect these parts. Film cooling is one of the most effective external cooling methods. Various numerical film cooling techniques presented in the literature have been investigated. Moreover, challenges and future directions of numerical film cooling techniques have been reviewed and presented in this paper. The aim of this review is to summarize recent development in research on film cooling techniques and attempt to identify some challenging issues that need to be solved for future research.

scholarly journals The Current Research Landscape of the Application of Artificial Intelligence in Managing Cerebrovascular and Heart Diseases: A Bibliometric and Content Analysis

2019 ◽  
Vol 16 (15) ◽  
pp. 2699 ◽  
Author(s):  
Bach Xuan Tran ◽  
Carl A. Latkin ◽  
Giang Thu Vu ◽  
Huong Lan Thi Nguyen ◽  
Son Nghiem ◽  
...  
Keyword(s):  
Artificial Intelligence ◽  
Latent Dirichlet Allocation ◽  
Complex Analysis ◽  
Clinical Decision Making ◽  
Heart Diseases ◽  
Medical Diagnostics ◽  
Medical Community ◽  
Future Research ◽  
Research Attention ◽  
Research And Policy

The applications of artificial intelligence (AI) in aiding clinical decision-making and management of stroke and heart diseases have become increasingly common in recent years, thanks in part to technological advancements and the heightened interest of the research and medical community. This study aims to provide a comprehensive picture of global trends and developments of AI applications relating to stroke and heart diseases, identifying research gaps and suggesting future directions for research and policy-making. A novel analysis approach that combined bibliometrics analysis with a more complex analysis of abstract content using exploratory factor analysis and Latent Dirichlet allocation, which uncovered emerging research domains and topics, was adopted. Data were extracted from the Web of Science database. Results showed topics with the most compelling growth to be AI for big data analysis, robotic prosthesis, robotics-assisted stroke rehabilitation, and minimally invasive surgery. The study also found an emerging landscape of research that was centered on population-specific and early detection of stroke and heart disease. Application of AI in health behavior tracking and improvement as well as the use of robotics in medical diagnostics and prognostication have also been found to attract significant research attention. In light of these findings, it is suggested that the currently under-researched issues of data management, AI model reliability, as well as validation of its clinical utility, need to be further explored in future research and policy decisions to maximize the benefits of AI applications in stroke and heart diseases.

scholarly journals Analysis of Statistics for Generalized Stirling Permutations

2011 ◽  
Vol 20 (6) ◽  
pp. 875-910 ◽  
Author(s):  
MARKUS KUBA ◽  
ALOIS PANHOLZER
Keyword(s):  
Generating Functions ◽  
Stein’S Method ◽  
Limiting Distribution ◽  
Permutation Statistics ◽  
Stein's Method ◽  
Urn Models ◽  
Restricted Class

In this work we give a study of generalizations of Stirling permutations, a restricted class of permutations of multisets introduced by Gessel and Stanley [15]. First we give several bijections between such generalized Stirling permutations and various families of increasing trees extending the known correspondences of [20, 21]. Then we consider several permutation statistics of interest for generalized Stirling permutations as the number of left-to-right minima, the number of left-to-right maxima, the number of blocks of specified sizes, the distance between occurrences of elements, and the number of inversions. For all these quantities we give a distributional study, where the established connections to increasing trees turn out to be very useful. To obtain the exact and limiting distribution results we use several techniques ranging from generating functions, connections to urn models, martingales and Stein's method.

scholarly journals Lindelöf Representations and (Non-)Holonomic Sequences

10.37236/275 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Philippe Flajolet ◽  
Stefan Gerhold ◽  
Bruno Salvy
Keyword(s):  
Finite Difference ◽  
Generating Functions ◽  
Complex Analysis ◽  
Integral Representations ◽  
Asymptotic Estimates ◽  
Linear Recurrences ◽  
Polynomial Coefficients ◽  
Difference Sequences ◽  
Analytic Expressions ◽  
Linear Differential

Various sequences that possess explicit analytic expressions can be analysed asymptotically through integral representations due to Lindelöf, which belong to an attractive but somewhat neglected chapter of complex analysis. One of the outcomes of such analyses concerns the non-existence of linear recurrences with polynomial coefficients annihilating these sequences, and, accordingly, the non-existence of linear differential equations with polynomial coefficients annihilating their generating functions. In particular, the corresponding generating functions are transcendental. Asymptotic estimates of certain finite difference sequences come out as a byproduct of the Lindelöf approach.

scholarly journals Analysis of an Asymmetric Leader Election Algorithm

10.37236/1302 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
Svante Janson ◽  
Wojciech Szpankowski
Keyword(s):  
Complex Analysis ◽  
Analysis Of Algorithms ◽  
Asymptotic Expression ◽  
Analytical Techniques ◽  
Leader Election ◽  
Limiting Distribution ◽  
Election Process ◽  
Precise Analysis ◽  
Election Algorithm ◽  
Leader Election Algorithm

We consider a leader election algorithm in which a set of distributed objects (people, computers, etc.) try to identify one object as their leader. The election process is randomized, that is, at every stage of the algorithm those objects that survived so far flip a biased coin, and those who received, say a tail, survive for the next round. The process continues until only one objects remains. Our interest is in evaluating the limiting distribution and the first two moments of the number of rounds needed to select a leader. We establish precise asymptotics for the first two moments, and show that the asymptotic expression for the duration of the algorithm exhibits some periodic fluctuations and consequently no limiting distribution exists. These results are proved by analytical techniques of the precise analysis of algorithms such as: analytical poissonization and depoissonization, Mellin transform, and complex analysis.

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