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 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 A note on the ternary Diophantine equation x 2 − y 2 m = z n

2021 ◽  
Vol 29 (2) ◽  
pp. 93-105
Author(s):  
Attila Bérczes ◽  
Maohua Le ◽  
István Pink ◽  
Gökhan Soydan
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Special Cases ◽  
Positive Integers

Abstract Let ℕ be the set of all positive integers. In this paper, using some known results on various types of Diophantine equations, we solve a couple of special cases of the ternary equation x2 − y2m = zn , x, y, z, m, n ∈ ℕ, gcd(x, y) = 1, m ≥ 2, n ≥ 3.

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 An Explicit Edge-Coloring of $K_n$ with Six Colors on Every $K_5$

10.37236/7852 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Alex Cameron
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Edge Coloring ◽  
Minimum Number ◽  
Positive Integers

Let $p$ and $q$ be positive integers such that $1 \leq q \leq {p \choose 2}$. A $(p,q)$-coloring of the complete graph on $n$ vertices $K_n$ is an edge coloring for which every $p$-clique contains edges of at least $q$ distinct colors. We denote the minimum number of colors needed for such a $(p,q)$-coloring of $K_n$ by $f(n,p,q)$. This is known as the Erdös-Gyárfás function. In this paper we give an explicit $(5,6)$-coloring with $n^{1/2+o(1)}$ colors. This improves the best known upper bound of $f(n,5,6)=O\left(n^{3/5}\right)$ given by Erdös and Gyárfás, and comes close to matching the order of the best known lower bound, $f(n,5,6) = \Omega\left(n^{1/2}\right)$.

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 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.

Pointless metric spaces

1990 ◽  
Vol 55 (1) ◽  
pp. 207-219 ◽  
Author(s):  
Giangiacomo Gerla
Keyword(s):  
Lower Bound ◽  
Metric Space ◽  
Upper Bound ◽  
Metric Spaces ◽  
Space Theory ◽  
Research Projects ◽  
Natural Numbers ◽  
Real Numbers ◽  
Euclidean Metric

In the last years several research projects have been motivated by the problem of constructing the usual geometrical spaces by admitting “regions” and “inclusion” between regions as primitives and by defining the points as suitable sequences or classes of regions (for references see [2]).In this paper we propose and examine a system of axioms for the pointless space theory in which “regions”, “inclusion”, “distance” and “diameter” are assumed as primitives and the concept of point is derived. Such a system extends a system proposed by K. Weihrauch and U. Schreiber in [5].In the sequel R and N denote the set of real numbers and the set of natural numbers, and E is a Euclidean metric space. Moreover, if X is a subset of R, then ⋁X is the least upper bound and ⋀X the greatest lower bound of X.

scholarly journals A note on the exponential Diophantine equation (A^2n)^x+(B^2n)^y=((A^2+B^2)n)^z

2020 ◽  
Vol 55 (2) ◽  
pp. 195-201
Author(s):  
Maohua Le ◽  
◽  
Gökhan Soydan ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Exponential Diophantine Equation ◽  
Positive Integer Solution ◽  
Exponential Diophantine Equations ◽  
Integer Solutions ◽  
Positive Integers ◽  
Upper Bound For Solutions

Let A, B be positive integers such that min{A,B}>1, gcd(A,B) = 1 and 2|B. In this paper, using an upper bound for solutions of ternary purely exponential Diophantine equations due to R. Scott and R. Styer, we prove that, for any positive integer n, if A >B3/8, then the equation (A2 n)x + (B2 n)y = ((A2 + B2)n)z has no positive integer solutions (x,y,z) with x > z > y; if B>A3/6, then it has no solutions (x,y,z) with y>z>x. Thus, combining the above conclusion with some existing results, we can deduce that, for any positive integer n, if B ≡ 2 (mod 4) and A >B3/8, then this equation has only the positive integer solution (x,y,z)=(1,1,1).

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