scholarly journals Possible Probability and Irreducibility of Balanced Nontransitive Dice

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Injo Hur ◽  
Yeansu Kim
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Main Tool ◽  
Formula One ◽  
Winning Probability

We construct irreducible balanced nontransitive sets of n -sided dice for any positive integer n . One main tool of the construction is to study so-called fair sets of dice. Furthermore, we also study the distribution of the probabilities of balanced nontransitive sets of dice. For a lower bound, we show that the winning probability can be arbitrarily close to 1 / 2 . We hypothesize that the winning probability cannot be more than 1 / 2 + 1 / 9 , and we construct a balanced nontransitive set of dice whose probability is 1 / 2 + 13 − 153 / 24 ≈ 1 / 2 + 1 / 9.12 .

scholarly journals ON -K2 FOR NORMAL SURFACE SINGULARITIES

1998 ◽  
Vol 09 (06) ◽  
pp. 653-668 ◽  
Author(s):  
HAO CHEN ◽  
SHIHOKO ISHII
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Rational Number ◽  
Accumulation Point ◽  
Normal Surface ◽  
Accumulation Points ◽  
Surface Singularities

In this paper we show the lower bound of the set of non-zero -K2 for normal surface singularities establishing that this set has no accumulation points from above. We also prove that every accumulation point from below is a rational number and every positive integer is an accumulation point. Every rational number can be an accumulation point modulo ℤ. We determine all accumulation points in [0, 1]. If we fix the value -K2, then the values of pg, pa, mult, embdim and the numerical indices are bounded, while the numbers of the exceptional curves are not bounded.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals A note on the lower bound of representation functions

2021 ◽  
pp. 1-8
Author(s):  
Xing-Wang Jiang ◽  
Csaba Sándor ◽  
Quan-Hui Yang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Number Of Solutions

For a set [Formula: see text] of nonnegative integers, let [Formula: see text] denote the number of solutions to [Formula: see text] with [Formula: see text], [Formula: see text]. Let [Formula: see text] be the Thue–Morse sequence and [Formula: see text]. Let [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] for all [Formula: see text]. Previously, the first author proved that if [Formula: see text] and [Formula: see text], then [Formula: see text] for all [Formula: see text]. In this paper, we prove that the above lower bound is nearly best possible. We also get some other results.

scholarly journals On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nurdin Hinding ◽  
Hye Kyung Kim ◽  
Nurtiti Sunusi ◽  
Riskawati Mise
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
Cluster Graph ◽  
Cluster Graphs ◽  
Irregularity Strength

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ ​ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

On the values of representation functions III

2019 ◽  
Vol 16 (03) ◽  
pp. 511-522
Author(s):  
Xing-Wang Jiang
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Binary Representation ◽  
Number Of Solutions

For a set [Formula: see text] of nonnegative integers, let [Formula: see text] denote the number of solutions to [Formula: see text] with [Formula: see text]. Let [Formula: see text] be the set of all nonnegative integers [Formula: see text] with an even number of ones in the binary representation of [Formula: see text] and let [Formula: see text]. Let [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] for all [Formula: see text]. In 2011, Chen proved that, if [Formula: see text] and [Formula: see text], then [Formula: see text] for all [Formula: see text]. In this paper, we improve the lower bound by proving that [Formula: see text] for all [Formula: see text].

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

scholarly journals Monochromatic sequences whose gaps belong to {d, 2d, …, md}

1998 ◽  
Vol 58 (1) ◽  
pp. 93-101 ◽  
Author(s):  
Bruce M. Landman
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Order Of Magnitude ◽  
Positive Integers

For m and k positive integers, define a k-term hm-progression to be a sequence of positive integers {x1,…,xk} such that for some positive integer d, xi + 1 − xi ∈ {d, 2d,…, md} for i = 1,…, k - 1. Let hm(k) denote the least positive integer n such that for every 2-colouring of {1, 2, …, n} there is a monochromatic hm-progression of length k. Thus, h1(k) = w(k), the classical van der Waerden number. We show that, for 1 ≤ r ≤ m, hm(m + r) ≤ 2c(m + r − 1) + 1, where c = ⌈m/(m − r)⌉. We also give a lower bound for hm(k) that has order of magnitude 2k2/m. A precise formula for hm(k) is obtained for all m and k such that k ≤ 3m/2.

scholarly journals Forcing $k$-Repetitions in Degree Sequences

10.37236/3503 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Yair Caro ◽  
Asaf Shapira ◽  
Raphael Yuster
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Main Tool ◽  
Geometric Approach ◽  
Degree Sequences ◽  
Integer Sequences ◽  
Sum Theorem ◽  
Zero Sum

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed $k$, every graph $G$ is "close" to a graph $G'$ with  $k$ vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we show that for any positive integer $k$, there is a constant $C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$ vertices and thus obtain a new graph $G'$ that contains at least $\min\{k,|G|-C\}$ vertices of the same degree.Our main tool is a multidimensional zero-sum theorem for integer sequences, which we prove using an old geometric approach of Alon and Berman.

scholarly journals Coloring the Cube with Rainbow Cycles

10.37236/2957 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Dhruv Mubayi ◽  
Randall Stading
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Sidon Sets ◽  
Dimensional Cube

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of the $k$-cycle $C_k$ receive $k$ distinct colors. Faudree, Gyárfás, Lesniak and Schelp proved that $f(n,4)=n$ for $n=4$ or $n>5$. We consider larger $k$ and prove that if $k \equiv 0$ (mod 4), then there are positive constants $c_1, c_2$ depending only on $k$ such that$$c_1n^{k/4} < f(n,k) < c_2 n^{k/4}.$$Our upper bound uses an old construction of Bose and Chowla of generalized Sidon sets. For $k \equiv 2$ (mod 4), the situation seems more complicated. For the smallest case $k=6$ we show that $$3n-2 \le f(n, 6) < n^{1+o(1)}$$ with the lower bound holding for $n \ge 3$. The upper bound is obtained from Behrend's construction of a subset of integers with no three term arithmetic progression.

scholarly journals Large Equiangular Sets of Lines in Euclidean Space

10.37236/1533 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
D. De Caen
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Euclidean Space ◽  
Upper Bound ◽  
Equiangular Lines

A construction is given of ${{2}\over {9}} (d+1)^2$ equiangular lines in Euclidean $d$-space, when $d = 3 \cdot 2^{2t-1}-1$ with $t$ any positive integer. This compares with the well known "absolute" upper bound of ${{1}\over {2}} d(d+1)$ lines in any equiangular set; it is the first known constructive lower bound of order $d^2$ .

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