scholarly journals A Pseudoline Counterexample to the Strong Dirac Conjecture

10.37236/4015 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ben Lund ◽  
George B. Purdy ◽  
Justin W. Smith
Keyword(s):  
Infinite Family ◽  
Open Problems ◽  
Dirac Conjecture ◽  
Pseudoline Arrangements

We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of $n$ pseudolines has no member incident to more than $4n/9$ points of intersection. This shows the "Strong Dirac" conjecture to be false for pseudolines.We also raise a number of open problems relating to possible differences between the structure of incidences between points and lines versus the structure of incidences between points and pseudolines.

From the Outside In

2015 ◽  
Author(s):  
Derek Smith
Keyword(s):  
Positive Integer ◽  
Infinite Family ◽  
Open Problems ◽  
The Family

This chapter discusses Slothouber–Graatsma–Conway puzzle, which asks one to assemble six 1 × 2 × 2 pieces and three 1 × 1 × 1 pieces into the shape of a 3 × 3 × 3 cube. The puzzle has been generalized to larger cubes, and there is an infinite family of such puzzles. The chapter's primary argument is that, for any odd positive integer n = 2k + 1, there is exactly one way, up to symmetry, to make an n × n × n cube out of n tiny 1 × 1 × 1 cubes and six of each of a set of rectangular blocks. The chapter describes a way to solve each puzzle in the family and explains why there are no other solutions. It then presents several related open problems.

scholarly journals Hirzebruch-Type Inequalities Viewed as Tools in Combinatorics

10.37236/8335 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Piotr Pokora
Keyword(s):  
Projective Plane ◽  
Plane Curve ◽  
Combinatorial Problems ◽  
Complex Projective Plane ◽  
Open Problems ◽  
Line Arrangements ◽  
Dirac Conjecture ◽  
Complex Projective

The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.

scholarly journals Residual $q$-Fano Planes and Related Structures

10.37236/7107 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Tuvi Etzion ◽  
Niv Hooker
Keyword(s):  
Block Design ◽  
Infinite Family ◽  
Steiner System ◽  
Interesting Case ◽  
Common Belief ◽  
Steiner Systems ◽  
Fano Plane ◽  
Open Problems ◽  
One Step ◽  
Existence Question

One of the most intriguing problems for $q$-analogs of designs, is the existence question of an infinite family of $q$-Steiner systems that are not spreads. In particular the most interesting case is the existence question for the $q$-analog of the Fano plane, known also as the $q$-Fano plane. These questions are in the front line of open problems in block design. There was a common belief and a conjecture that such structures do not exist. Only recently, $q$-Steiner systems were found for one set of parameters. In this paper, a definition for the $q$-analog of the residual design is presented. This new definition is different from previous known definition, but its properties reflect better the $q$-analog properties. The existence of a design with the parameters of the residual $q$-Steiner system in general and the residual $q$-Fano plane in particular are examined. We construct different residual $q$-Fano planes for all $q$, where $q$ is a prime power. The constructed structure is just one step from a construction of a $q$-Fano plane.

scholarly journals The Frobenius and Factor Universality Problems of the Kleene Star of a Finite Set of Words

2021 ◽  
Vol 68 (3) ◽  
pp. 1-22
Author(s):  
Maksymilian Mika ◽  
Marek Szykuła
Keyword(s):  
Infinite Family ◽  
Upper Bounds ◽  
Open Problems ◽  
Rewriting Systems ◽  
Finite Set ◽  
General Tool

We solve open problems concerning the Kleene star of a finite set of words over an alphabet . The Frobenius monoid problem is the question for a given finite set of words , whether the language is cofinite. We show that it is PSPACE-complete. We also exhibit an infinite family of sets such that the length of the longest words not in (when is cofinite) is exponential in the length of the longest words in and subexponential in the sum of the lengths of words in . The factor universality problem is the question for a given finite set of words , whether every word over is a factor (substring) of some word from . We show that it is also PSPACE-complete. Besides that, we exhibit an infinite family of sets such that the length of the shortest words not being a factor of any word in is exponential in the length of the longest words in and subexponential in the sum of the lengths of words in . This essentially settles in the negative the longstanding Restivo’s conjecture (1981) and its weak variations. All our solutions are based on one shared construction, and as an auxiliary general tool, we introduce the concept of set rewriting systems . Finally, we complement the results with upper bounds.

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.

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