scholarly journals THE PROBABILITY THAT RANDOM POSITIVE INTEGERS ARE k-WISE RELATIVELY PRIME

2013 ◽  
Vol 09 (05) ◽  
pp. 1263-1271 ◽  
Author(s):  
JERRY HU
Keyword(s):  
Exact Formula ◽  
Asymptotic Estimates ◽  
Positive Integers

The positive integers a1, a2, …, as are k-wise relatively prime if any k of them are relatively prime. For a (k-1)-tuple of positive integers u = (u1, …, uk-1), let [Formula: see text] denote the number of s-tuples of positive integers (a1, a2, …, as) with 1 ≤ a1, a2, …, as ≤ n such that a1, a2, …, as are k-wise relatively prime and are i-wise relatively prime to ui for i = 1, 2, …, k-1 when s ≥ k ≥ 2, and a1, a2, …, as are i-wise relatively prime to ui for i = 1, 2, …, s when k > s ≥ 1. Asymptotic estimates are obtained for these functions. As a corollary, exact formula is also obtained for the probability that s positive integers are k-wise relatively prime.

Generalized allocation scheme with cell occupancies from a fixed finite set

2020 ◽  
Vol 30 (5) ◽  
pp. 347-352
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Explicit Form ◽  
Asymptotic Estimates ◽  
Allocation Scheme ◽  
Number Of Components ◽  
Finite Set ◽  
Probabilistic Character ◽  
Normal Law ◽  
Generalized Allocation Scheme ◽  
Positive Integers ◽  
Number Of Particles

AbstractWe consider a generalized scheme of allocation of n particles (elements) over unordered cells (components) under the condition that the number of particles in each cell belongs to a fixed finite set A of positive integers. A new asymptotic estimates for the total number In(A) of variants of allocations of n particles are obtained under some conditions on the set A; these estimates have an explicit form (up to equivalence). Some examples of combinatorial-probabilistic character are given to illustrate by particular cases the notions introduced and results obtained. For previously known theorems on the convergence to the normal law of the total number of components and numbers of components with given cardinalities the norming parameters are obtained in the explicit form without using roots of algebraic or transcendent equations.

scholarly journals Separation of the maxima in samples of geometric random variables

2011 ◽  
Vol 5 (2) ◽  
pp. 271-282 ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher ◽  
Toufik Mansour ◽  
Stephan Wagner
Keyword(s):  
Generating Function ◽  
Probability Generating Function ◽  
Random Variables ◽  
Exact Formula ◽  
Asymptotic Estimates ◽  
Fixed Integer ◽  
Minimum Separation ◽  
Double Sum

We consider samples of n geometric random variables W1 W2 ... Wn where P{W) = i} = pqi-l, for 1 ? j ? n, with p + q = 1. For each fixed integer d > 0, we study the probability that the distance between the consecutive maxima in these samples is at least d. We derive a probability generating function for such samples and from it we obtain an exact formula for the probability as a double sum. Using Rice's method we obtain asymptotic estimates for these probabilities. As a consequence of these results, we determine the average minimum separation of the maxima, in a sample of n geometric random variables with at least two maxima.

scholarly journals Collections of sequences having the Ramsey property only for few colours

1997 ◽  
Vol 55 (1) ◽  
pp. 19-28
Author(s):  
Bruce M. Landman ◽  
Beata Wysocka
Keyword(s):  
Positive Integer ◽  
Arithmetic Progression ◽  
Exact Formula ◽  
Ramsey Property ◽  
Order Of Magnitude ◽  
The Difference ◽  
Positive Integers

A family 𝑐 of sequences has the r-Ramsey property if for every positive integer k, there exists a least positive integer g(r)(k) such that for every r-colouring of {1, 2, …, g(r)(k)} there is a monochromatic k-term member of 𝑐. For fixed integers m > 1 and 0 ≤ a < m, define a k-term a (mod m)-sequence to be an increasing sequence of positive integers {x1, …, xk} such that xi − xi−1 ≡ a (mod m) for i = 2, …, k. Define an m-a.p. to be an arithmetic progression where the difference between successive terms is m. Let be the collection of sequences that are either a(mod m)-sequences or m-a.p.'s. Landman and Long showed that for all m ≥ 2 and 1 ≤ a < m, has the 2-Ramsey property, and that the 2-Ramsey function , corresponding to k-term a(mod m)-sequences or n-term m-a.p.'s, has order of magnitude mkn. We show that does not have the 4-Ramsey property and that, unless m/a = 2, it does not have the 3-Ramsey property. In the case where m/a = 2, we give an exact formula for . We show that if a ≠ 0, there exist 4-colourings or 6-colourings (depending on m and a) of the positive integers which avoid 2-term monochromatic members of , but that there never exist such 3-colourings. We also give an exact formula for .

scholarly journals Integer Programming Formulations For The Frobenius Problem

2021 ◽  
Vol 8 ◽  
pp. 60-65
Author(s):  
Imdat Kara ◽  
Halil Ibrahim Karakas
Keyword(s):  
Integer Programming ◽  
Exact Formula ◽  
Frobenius Number ◽  
Programming Approach ◽  
Postage Stamp ◽  
Frobenius Problem ◽  
The Third ◽  
Special Cases ◽  
The Given ◽  
Positive Integers

The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α2< …< αn, is the largest integer that can not be written as a nonnegative integer linear combination of the given set. Finding the Frobenius number is known as the Frobenius problem, which is also named as the coin exchange problem or the postage stamp problem. This problem is closely related with the equality constrained integer knapsack problem. It is known that this problem is NP-hard. Extensive research has been conducted for finding the Frobenius number of a given set of positive integers. An exact formula exists for the case n=2 and various formulas have been derived for all special cases of n = 3. Many algorithms have been proposed for n≥4. As far as we are aware, there does not exist any integer programming approach for this problem which is the main motivation of this paper. We present four integer linear programming formulations about the Frobenius number of a given set of positive integers. Our first formulation is used to check if a given positive integer is the Frobenius number of a given set of positive integers. The second formulation aims at finding the Frobenius number directly. The third formulation involves the residue classes with respect to the least member of the given set of positive integers, where a residue table is computed comprising all values modulo that least member, and the Frobenius number is obtained from there. Based on the same approach underlying the third formulation, we propose our fourth formulation which produces the Frobenius number directly. We demonstrate how to use our formulations with several examples. For illustrative purposes, some computa-tional analysis is also presented.

scholarly journals Minimal Automaton for Multiplying and Translating the Thue-Morse Set

10.37236/9068 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Émilie Charlier ◽  
Célia Cisternino ◽  
Adeline Massuir
Keyword(s):  
Fixed Point ◽  
Exact Formula ◽  
The State ◽  
Characteristic Sequence ◽  
State Complexity ◽  
Positive Integers

The Thue-Morse set $\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$'s. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word $${\tt 0110100110010110\cdots},$$ which is the fixed point starting with ${\tt 0}$ of the word morphism ${\tt 0\mapsto 01}$, ${\tt 1\mapsto 10}$. The numbers in $\mathcal{T}$ are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set $m\mathcal{T}+r$ (i.e. the number of states of its minimal automaton) with respect to any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\mathcal{T}+r$ for any positive integers $p$ and $m$ and any remainder $r\in\{0,\ldots,m{-}1\}$. The proposed method is general for any $b$-recognizable set of integers.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

A GENERALIZED EXACT FORMULA FOR THE SWIMMING COST OF UPSTREAM FISH MIGRATION

2018 ◽  
Vol 74 (4) ◽  
pp. I_391-I_396
Author(s):  
Hidekazu YOSHIOKA ◽  
Takeshi WATANABE ◽  
Kentaro TSUGIHASHI
Keyword(s):  
Exact Formula ◽  
Fish Migration
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