Open Problems in Pattern Avoidance

1993 ◽  
Vol 100 (8) ◽  
pp. 790-793 ◽  
Author(s):  
James Currie
Keyword(s):  
Pattern Avoidance ◽  
Open Problems

Open Problems in Pattern Avoidance

1993 ◽  
Vol 100 (8) ◽  
pp. 790 ◽  
Author(s):  
James Currie
Keyword(s):  
Pattern Avoidance ◽  
Open Problems

ALGORITHMIC COMBINATORICS ON PARTIAL WORDS

2012 ◽  
Vol 23 (06) ◽  
pp. 1189-1206 ◽  
Author(s):  
F. BLANCHET-SADRI
Keyword(s):  
Molecular Biology ◽  
Data Compression ◽  
Pattern Avoidance ◽  
Finite Alphabet ◽  
Open Problems ◽  
The Past ◽  
Partial Word ◽  
De Bruijn ◽  
Subword Complexity ◽  
Partial Words

Algorithmic combinatorics on partial words, or sequences of symbols over a finite alphabet that may have some do-not-know symbols or holes, has been developing in the past few years. Applications can be found, for instance, in molecular biology for the sequencing and analysis of DNA, in bio-inspired computing where partial words have been considered for identifying good encodings for DNA computations, and in data compression. In this paper, we focus on two areas of algorithmic combinatorics on partial words, namely, pattern avoidance and subword complexity. We discuss recent contributions as well as a number of open problems. In relation to pattern avoidance, we classify all binary patterns with respect to partial word avoidability, we classify all unary patterns with respect to hole sparsity, and we discuss avoiding abelian powers in partial words. In relation to subword complexity, we generate and count minimal Sturmian partial words, we construct de Bruijn partial words, and we construct partial words with subword complexities not achievable by full words (those without holes).

scholarly journals Pattern-avoiding Dyck paths

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Antonio Bernini ◽  
Luca Ferrari ◽  
Renzo Pinzani ◽  
Julian West
Keyword(s):  
Asymptotic Behavior ◽  
Pattern Avoidance ◽  
Lattice Paths ◽  
Dyck Path ◽  
Typical Pattern ◽  
Open Problems ◽  
Dyck Paths ◽  
International Audience ◽  
Pattern Containment

International audience We introduce the notion of $\textit{pattern}$ in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the $\textit{Dyck pattern poset}$. Given a Dyck path $P$, we determine a formula for the number of Dyck paths covered by $P$, as well as for the number of Dyck paths covering $P$. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. Finally, we offer a conjecture concerning the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern and we pose a series of open problems regarding the structure of the Dyck pattern poset.

Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

Continuous Glucose Monitoring Time Series and Hypo/Hyperglycemia Prevention: Requirements, Methods, Open Problems

2008 ◽  
Vol 4 (3) ◽  
pp. 181-192 ◽  
Author(s):  
Giovanni Sparacino ◽  
Andrea Facchinetti ◽  
Alberto Maran ◽  
Claudio Cobelli
Keyword(s):  
Time Series ◽  
Continuous Glucose Monitoring ◽  
Glucose Monitoring ◽  
Open Problems ◽  
Monitoring Time

Secure Cloud Quantum Computing with Verification Based on Quantum Interactive Proof

Impact ◽  
2019 ◽  
Vol 2019 (10) ◽  
pp. 30-32
Author(s):  
Tomoyuki Morimae
Keyword(s):  
Quantum Computing ◽  
Quantum Information ◽  
Theoretical Physics ◽  
View Point ◽  
Research Subjects ◽  
Interactive Proof ◽  
Open Problems ◽  
Main Research ◽  
Proof Systems ◽  
Information Group

In cloud quantum computing, a classical client delegate quantum computing to a remote quantum server. An important property of cloud quantum computing is the verifiability: the client can check the integrity of the server. Whether such a classical verification of quantum computing is possible or not is one of the most important open problems in quantum computing. We tackle this problem from the view point of quantum interactive proof systems. Dr Tomoyuki Morimae is part of the Quantum Information Group at the Yukawa Institute for Theoretical Physics at Kyoto University, Japan. He leads a team which is concerned with two main research subjects: quantum supremacy and the verification of quantum computing.

Classification of Cloud Systems Cyber-security Threats and Solutions Directives

2017 ◽  
Vol 2 (3) ◽  
pp. 1
Author(s):  
Hanane Bennasar ◽  
Mohammad Essaaidi ◽  
Ahmed Bendahmane ◽  
Jalel Benothmane
Keyword(s):  
Cloud Computing ◽  
Cyber Security ◽  
Data Security ◽  
Denial Of Service ◽  
Security Threats ◽  
Huge Number ◽  
Open Problems ◽  
Long Period ◽  
Near Future

Cloud computing cyber security is a subject that has been in top flight for a long period and even in near future. However, cloud computing permit to stock up a huge number of data in the cloud stockage, and allow the user to pay per utilization from anywhere via any terminal equipment. Among the major issues related to Cloud Computing security, we can mention data security, denial of service attacks, confidentiality, availability, and data integrity. This paper is dedicated to a taxonomic classification study of cloud computing cyber-security. With the main objective to identify the main challenges and issues in this field, the different approaches and solutions proposed to address them and the open problems that need to be addressed.

scholarly journals The undirected repetition threshold and undirected pattern avoidance

2021 ◽  
Vol 866 ◽  
pp. 56-69
Author(s):  
James D. Currie ◽  
Lucas Mol
Keyword(s):  
Pattern Avoidance

Dynamics of a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination and nonlinear incidence under regime switching and Lévy jumps

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Junna Hu ◽  
Buyu Wen ◽  
Ting Zeng ◽  
Zhidong Teng
Keyword(s):  
Epidemic Model ◽  
Regime Switching ◽  
Markov Switching ◽  
Threshold Value ◽  
Open Problems ◽  
Nonlinear Incidence ◽  
Sirs Epidemic Model ◽  
Stochastic Epidemic ◽  
Lévy Jumps ◽  
White Noises

Abstract In this paper, a stochastic susceptible-infective-recovered (SIRS) epidemic model with vaccination, nonlinear incidence and white noises under regime switching and Lévy jumps is investigated. A new threshold value is determined. Some basic assumptions with regard to nonlinear incidence, white noises, Markov switching and Lévy jumps are introduced. The threshold conditions to guarantee the extinction and permanence in the mean of the disease with probability one and the existence of unique ergodic stationary distribution for the model are established. Some new techniques to deal with the Markov switching, Lévy jumps, nonlinear incidence and vaccination for the stochastic epidemic models are proposed. Lastly, the numerical simulations not only illustrate the main results given in this paper, but also suggest some interesting open problems.

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