scholarly journals Counting Lyndon Factors

10.37236/6915 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Amy Glen ◽  
Jamie Simpson ◽  
W. F. Smyth
Keyword(s):  
Lower Bound ◽  
Golden Ratio ◽  
Fibonacci Number ◽  
Expected Number ◽  
Open Problems ◽  
Sturmian Word ◽  
Minimum Number ◽  
Lyndon Word ◽  
Lyndon Words

In this paper, we determine the maximum number of distinct Lyndon factors that a word of length $n$ can contain. We also derive formulas for the expected total number of Lyndon factors in a word of length $n$ on an alphabet of size $\sigma$, as well as the expected number of distinct Lyndon factors in such a word. The minimum number of distinct Lyndon factors in a word of length $n$ is $1$ and the minimum total number is $n$, with both bounds being achieved by $x^n$ where $x$ is a letter. A more interesting question to ask is what is the minimum number of distinct Lyndon factors in a Lyndon word of length $n$? In this direction, it is known (Saari, 2014) that a lower bound for the number of distinct Lyndon factors in a Lyndon word of length $n$ is $\lceil\log_{\phi}(n) + 1\rceil$, where $\phi$ denotes the golden ratio $(1 + \sqrt{5})/2$. Moreover, this lower bound is sharp when $n$ is a Fibonacci number and is attained by the so-called finite Fibonacci Lyndon words, which are precisely the Lyndon factors of the well-known infinite Fibonacci word $\boldsymbol{f}$ (a special example of an infinite Sturmian word). Saari (2014) conjectured that if $w$ is Lyndon word of length $n$, $n\ne 6$, containing the least number of distinct Lyndon factors over all Lyndon words of the same length, then $w$ is a Christoffel word (i.e., a Lyndon factor of an infinite Sturmian word). We give a counterexample to this conjecture. Furthermore, we generalise Saari's result on the number of distinct Lyndon factors of a Fibonacci Lyndon word by determining the number of distinct Lyndon factors of a given Christoffel word. We end with two open problems.

SIGACT News Online Algorithms Column 37

2021 ◽  
Vol 52 (2) ◽  
pp. 71-71
Author(s):  
Rob van Stee
Keyword(s):  
Lower Bound ◽  
Online Algorithms ◽  
Packet Scheduling ◽  
Open Problems ◽  
Competitive Algorithm ◽  
Long Time ◽  
News Online

For this issue, Pavel Vesely has contributed a wonderful overview of the ideas that were used in his SODA paper on packet scheduling with Marek Chrobak, Lukasz Jez and Jiri Sgall. This is a problem for which a 2-competitive algorithm as well as a lower bound of ϕ ≈ 1:618 was known already twenty years ago, but which resisted resolution for a long time. It is great that this problem has nally been resolved and that Pavel was willing to explain more of the ideas behind it for this column. He also provides an overview of open problems in this area.

scholarly journals Generalizations of the Poincaré Birkhoff fixed point theorem

1977 ◽  
Vol 17 (3) ◽  
pp. 375-389 ◽  
Author(s):  
Walter D. Neumann
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Periodic Points ◽  
Open Problems ◽  
Number Of Components ◽  
Birkhoff Theorem

It is shown how George D. Birkhoff's proof of the Poincaré Birkhoff theorem can be modified using ideas of H. Poincaré to give a rather precise lower bound on the number of components of the set of periodic points of the annulus. Some open problems related to this theorem are discussed.

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

scholarly journals Minimum Number of Monotone Subsequences of Length 4 in Permutations

2014 ◽  
Vol 24 (4) ◽  
pp. 658-679 ◽  
Author(s):  
JÓZSEF BALOGH ◽  
PING HU ◽  
BERNARD LIDICKÝ ◽  
OLEG PIKHURKO ◽  
BALÁZS UDVARI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Complete Graphs ◽  
Formula Group ◽  
Minimum Number ◽  
Additional Stability ◽  
Edge Colourings ◽  
Image Position

We show that for every sufficiently largen, the number of monotone subsequences of length four in a permutation onnpoints is at least\begin{equation*} \binom{\lfloor{n/3}\rfloor}{4} + \binom{\lfloor{(n+1)/3}\rfloor}{4} + \binom{\lfloor{(n+2)/3}\rfloor}{4}. \end{equation*}Furthermore, we characterize all permutations on [n] that attain this lower bound. The proof uses the flag algebra framework together with some additional stability arguments. This problem is equivalent to some specific type of edge colourings of complete graphs with two colours, where the number of monochromaticK4is minimized. We show that all the extremal colourings must contain monochromaticK4only in one of the two colours. This translates back to permutations, where all the monotone subsequences of length four are all either increasing, or decreasing only.

scholarly journals Lower Bounds for the Size of Random Maximal $H$-Free Graphs

10.37236/93 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Guy Wolfovitz
Keyword(s):  
Lower Bound ◽  
Random Process ◽  
Lower Bounds ◽  
Random Permutation ◽  
Bipartite Graphs ◽  
Expected Number ◽  
Free Graph ◽  
Vertex Graph

We consider the next random process for generating a maximal $H$-free graph: Given a fixed graph $H$ and an integer $n$, start by taking a uniformly random permutation of the edges of the complete $n$-vertex graph $K_n$. Then, traverse the edges of $K_n$ according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving $n$-vertex graph - unless its addition creates a copy of $H$. The result of this process is a maximal $H$-free graph ${\Bbb M}_n(H)$. Our main result is a new lower bound on the expected number of edges in ${\Bbb M}_n(H)$, for $H$ that is regular, strictly $2$-balanced. As a corollary, we obtain new lower bounds for Turán numbers of complete, balanced bipartite graphs. Namely, for fixed $r \ge 5$, we show that ex$(n, K_{r,r}) = \Omega(n^{2-2/(r+1)}(\ln\ln n)^{1/(r^2-1)})$. This improves an old lower bound of Erdős and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in ${\Bbb M}_n(H)$, conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.

scholarly journals Finding the Number of Clusters in Data and Better Initial Centers for K-means Algorithm

2020 ◽  
Vol 12 (6) ◽  
pp. 1-20
Author(s):  
Ahmed Fahim ◽  
Keyword(s):  
Data Clustering ◽  
Linear Time ◽  
Original Data ◽  
Local Minima ◽  
Expected Number ◽  
Open Problems ◽  
Number Of Clusters ◽  
Benchmark Datasets ◽  
Selection Of

The k-means is the most well-known algorithm for data clustering in data mining. Its simplicity and speed of convergence to local minima are the most important advantages of it, in addition to its linear time complexity. The most important open problems in this algorithm are the selection of initial centers and the determination of the exact number of clusters in advance. This paper proposes a solution for these two problems together; by adding a preprocess step to get the expected number of clusters in data and better initial centers. There are many researches to solve each of these problems separately, but there is no research to solve both problems together. The preprocess step requires o(n log n); where n is size of the dataset. This preprocess step aims to get initial portioning of data without determining the number of clusters in advance, then computes the means of initial clusters. After that we apply k-means on original data using the resulting information from the preprocess step to get the final clusters. We use many benchmark datasets to test the proposed method. The experimental results show the efficiency of the proposed method.

scholarly journals On Reay's Relaxed Tverberg Conjecture and Generalizations of Conway's Thrackle Conjecture

10.37236/6516 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Megumi Asada ◽  
Ryan Chen ◽  
Florian Frick ◽  
Frederick Huang ◽  
Maxwell Polevy ◽  
...  
Keyword(s):  
Geometric Property ◽  
Convex Sets ◽  
Convex Hulls ◽  
Open Problems ◽  
Higher Dimensional ◽  
Special Cases ◽  
Minimum Number ◽  
Point Set ◽  
Colored Version ◽  
Special Case

Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean $d$-space that guarantees any such point set admits a partition into $r$ parts, any $k$ of whose convex hulls intersect. Here we give new and improved lower bounds for this number, which Reay conjectured to be independent of $k$. We prove a colored version of Reay's conjecture for $k$ sufficiently large, but nevertheless $k$ independent of dimension $d$. Pairwise intersecting convex hulls have severely restricted combinatorics. This is a higher-dimensional analogue of Conway's thrackle conjecture or its linear special case. We thus study convex-geometric and higher-dimensional analogues of the thrackle conjecture alongside Reay's problem and conjecture (and prove in two special cases) that the number of convex sets in the plane is bounded by the total number of vertices they involve whenever there exists a transversal set for their pairwise intersections. We thus isolate a geometric property that leads to bounds as in the thrackle conjecture. We also establish tight bounds for the number of facets of higher-dimensional analogues of linear thrackles and conjecture their continuous generalizations.

The multi-region index of a knot

2020 ◽  
Vol 29 (04) ◽  
pp. 2050022
Author(s):  
Sarah Goodhill ◽  
Adam M. Lowrance ◽  
Valeria Munoz Gonzales ◽  
Jessica Rattray ◽  
Amelia Zeh
Keyword(s):  
Lower Bound ◽  
Crossing Number ◽  
Number Of Generators ◽  
Branched Cover ◽  
Minimum Number

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover of [Formula: see text] over the knot is strictly less than the multi-region index. Our proof of this lower bound uses Goeritz matrices.

scholarly journals Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 615
Author(s):  
Hongzhuan Wang ◽  
Piaoyang Yin
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Extremal Graphs ◽  
Graph Invariant ◽  
Resistance Distance ◽  
Harary Index ◽  
H Index ◽  
Electronic Networks ◽  
Minimum Number ◽  
Graph Parameters

Resistance distance is a concept developed from electronic networks. The calculation of resistance distance in various circuits has attracted the attention of many engineers. This report considers the resistance-based graph invariant, the Resistance–Harary index, which represents the sum of the reciprocal resistances of any vertex pair in the figure G, denoted by R H ( G ) . Vertex bipartiteness in a graph G is the minimum number of vertices removed that makes the graph G become a bipartite graph. In this study, we give the upper bound and lower bound of the R H index, and describe the corresponding extremal graphs in the bipartite graph of a given order. We also describe the graphs with maximum R H index in terms of graph parameters such as vertex bipartiteness, cut edges, and matching numbers.

scholarly journals Crossing Numbers and Hard Erdős Problems in Discrete Geometry

1997 ◽  
Vol 6 (3) ◽  
pp. 353-358 ◽  
Author(s):  
LÁSZLÓ A. SZÉKELY
Keyword(s):  
Lower Bound ◽  
Discrete Geometry ◽  
Crossing Number ◽  
Plane Geometry ◽  
Crossing Numbers ◽  
Minimum Number

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

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