scholarly journals Bound Graph Polysemy

10.37236/1521 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Paul J. Tanenbaum
Keyword(s):  
Lower Bound ◽  
Partial Order ◽  
Upper Bound ◽  
Special Cases ◽  
Vertex Set

Bound polysemy is the property of any pair $(G_1, G_2)$ of graphs on a shared vertex set $V$ for which there exists a partial order on $V$ such that any pair of vertices has an upper bound precisely when the pair is an edge in $G_1$ and a lower bound precisely when it is an edge in $G_2$. We examine several special cases and prove a characterization of the bound polysemic pairs that illuminates a connection with the squared graphs.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (03) ◽  
pp. 611-629
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

scholarly journals On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1778
Author(s):  
Fangyun Tao ◽  
Ting Jin ◽  
Yiyou Tu
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Equitable Partition ◽  
Special Cases ◽  
Vertex Set ◽  
Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

NONDETERMINISTIC STATE COMPLEXITY OF PROPORTIONAL REMOVALS

2014 ◽  
Vol 25 (07) ◽  
pp. 823-835 ◽  
Author(s):  
DANIEL GOČ ◽  
ALEXANDROS PALIOUDAKIS ◽  
KAI SALOMAA
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
State Complexity ◽  
Nondeterministic State

The language [Formula: see text] consists of first halfs of strings in L. Many other variants of a proportional removal operation have been considered in the literature and a characterization of removal operations that preserve regularity is known. We consider the nondeterministic state complexity of the operation [Formula: see text] and, more generally, of polynomial removals as defined by Domaratzki (J. Automata, Languages and Combinatorics 7(4), 2002). We give an O(n2) upper bound for the nondeterministic state complexity of polynomial removals and a matching lower bound in cases where the polynomial is a sum of a monomial and a constant, or when the polynomial has rational roots.

scholarly journals A New Characterization of Tree Medians with Applications to Distributed Algorithms

1991 ◽  
Vol 20 (364) ◽  
Author(s):  
O. Gerstel ◽  
Shmuel Zaks
Keyword(s):  
Lower Bound ◽  
Sorting Algorithm ◽  
Message Complexity ◽  
Ranking Problem ◽  
Worst Case ◽  
Sorting Problem ◽  
Vertex Set ◽  
Pass Through ◽  
Case Number

A new characterization of tree medians is presented: we show that a vertex <em>m</em> is a median of a tree <em>T</em> with <em>n</em> vertices iff there exists a partition of the vertex set into [<em>n</em>/2] disjoint pairs (excluding m when <em>n</em> is odd), such that all the paths connecting the two vertices in any of the pairs pass through <em>m</em>. We show that in this case this sum is the largest possible among all such partitions, and we use this fact to discuss lower bounds on the message complexity of the distributed sorting problem. This lower bound implies that, given a network of a tree topology, choosing a median and then route all the information through it is the best possible strategy, in terms of worst-case number of messages sent during any execution of any distributed sorting algorithm. We also discuss the implications for networks of a general topology and for the distributed ranking problem.

scholarly journals Off-Diagonal Ordered Ramsey Numbers of Matchings

10.37236/8085 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Dhruv Rohatgi
Keyword(s):  
Lower Bound ◽  
Chromatic Number ◽  
Upper Bound ◽  
Edge Coloring ◽  
Ramsey Number ◽  
Relative Order ◽  
Ramsey Numbers ◽  
Asymptotic Bounds ◽  
Special Cases ◽  
Ordered Graphs

For ordered graphs $G$ and $H$, the ordered Ramsey number $r_<(G,H)$ is the smallest $n$ such that every red/blue edge coloring of the complete ordered graph on vertices $\{1,\dots,n\}$ contains either a blue copy of $G$ or a red copy of $H$, where the embedding must preserve the relative order of vertices. One number of interest, first studied by Conlon, Fox, Lee, and Sudakov, is the off-diagonal ordered Ramsey number $r_<(M, K_3)$, where $M$ is an ordered matching on $n$ vertices. In particular, Conlon et al. asked what asymptotic bounds (in $n$) can be obtained for $\max r_<(M, K_3)$, where the maximum is over all ordered matchings $M$ on $n$ vertices. The best-known upper bound is $O(n^2/\log n)$, whereas the best-known lower bound is $\Omega((n/\log n)^{4/3})$, and Conlon et al. hypothesize that there is some fixed $\epsilon > 0$ such that $r_<(M, K_3) = O(n^{2-\epsilon})$ for every ordered matching $M$. We resolve two special cases of this conjecture. We show that the off-diagonal ordered Ramsey numbers for ordered matchings in which edges do not cross are nearly linear. We also prove a truly sub-quadratic upper bound for random ordered matchings with interval chromatic number $2$.

scholarly journals Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2779
Author(s):  
Petr Karlovsky
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Natural Numbers ◽  
Parametric Solutions ◽  
Special Cases ◽  
Positive Integers

Diophantine equations ∏i=1nxi=F∑i=1nxi with xi,F∈ℤ+ associate the products and sums of n natural numbers. Only special cases have been studied so far. Here, we provide new parametric solutions depending on F and the divisors of F or F2. One of these solutions shows that the equation of any degree with any F is solvable. For n = 2, exactly two solutions exist if and only if F is a prime. These solutions are (2F,2F) and (F + 1, F(F + 1)). We generalize an upper bound for the sum of solution terms from n = 3 established by Crilly and Fletcher in 2015 to any n to be F+1F+n−1 and determine a lower bound to be nnFn−1. Confining the solutions to n-tuples consisting of distinct terms, equations of the 4th degree with any F are solvable but equations of the 5th to 9th degree are not. An upper bound for the sum of terms of distinct-term solutions is conjectured to be F+1F+n−2n−1!/2+1/n−2!. The conjecture is supported by computation, which also indicates that the upper bound equals the largest sum of solution terms if and only if F=n+k−2n−2!−1, k∈ℤ+. Computation provides further insights into the relationships between F and the sum of terms of distinct-term solutions.

scholarly journals Probability that an autocommutator element of a finite group equals to a fixed element

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6241-6246 ◽  
Author(s):  
Zohreh Sepehrizadeh ◽  
Mohammad Rismanchian
Keyword(s):  
Finite Group ◽  
Lower Bound ◽  
Upper Bound ◽  
Fixed Element ◽  
Special Cases ◽  
A Finite Group

In this article we introduce a formula for the probability which an autocommutator element of a finite group G, equals to a fixed element 1 of G and derive some properties of this formula. Moreover, we obtain a lower bound and an upper bound for this probability in the special cases. This generalizes some results of Das et al. in 2010 and Moghaddam et al. in 2011.

scholarly journals The Algebraic Degree of Phase-Type Distributions

2010 ◽  
Vol 47 (3) ◽  
pp. 611-629 ◽  
Author(s):  
Mark Fackrell ◽  
Qi-Ming He ◽  
Peter Taylor ◽  
Hanqin Zhang
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Polynomial Algorithm ◽  
Algebraic Degree ◽  
Nonnegative Matrices ◽  
Stieltjes Transform ◽  
Special Cases ◽  
Phase Type ◽  
Phase Type Distributions ◽  
The Matrix

This paper is concerned with properties of the algebraic degree of the Laplace-Stieltjes transform of phase-type (PH) distributions. The main problem of interest is: given a PH generator, how do we find the maximum and the minimum algebraic degrees of all irreducible PH representations with that PH generator? Based on the matrix exponential (ME) order of ME distributions and the spectral polynomial algorithm, a method for computing the algebraic degree of a PH distribution is developed. The maximum algebraic degree is identified explicitly. Using Perron-Frobenius theory of nonnegative matrices, a lower bound and an upper bound on the minimum algebraic degree are found, subject to some conditions. Explicit results are obtained for special cases.

scholarly journals Discrepancy of Cartesian Products of Arithmetic Progressions

10.37236/1758 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Benjamin Doerr ◽  
Anand Srivastav ◽  
Petra Wehr
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Arithmetic Progressions ◽  
Cartesian Products ◽  
Fixed Integer ◽  
Higher Dimensional ◽  
Special Cases ◽  
Dimensional Version ◽  
Compact Abelian Groups ◽  
Combinatorial Discrepancy

We determine the combinatorial discrepancy of the hypergraph ${\cal H}$ of cartesian products of $d$ arithmetic progressions in the $[N]^d$–lattice ($[N] = \{0,1,\ldots,N-1\}$). The study of such higher dimensional arithmetic progressions is motivated by a multi-dimensional version of van der Waerden's theorem, namely the Gallai-theorem (1933). We solve the discrepancy problem for $d$–dimensional arithmetic progressions by proving ${\rm disc}({\cal H}) = \Theta(N^{d/4})$ for every fixed integer $d \ge 1$. This extends the famous lower bound of $\Omega(N^{1/4})$ of Roth (1964) and the matching upper bound $O(N^{1/4})$ of Matoušek and Spencer (1996) from $d=1$ to arbitrary, fixed $d$. To establish the lower bound we use harmonic analysis on locally compact abelian groups. For the upper bound a product coloring arising from the theorem of Matoušek and Spencer is sufficient. We also regard some special cases, e.g., symmetric arithmetic progressions and infinite arithmetic progressions.

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