scholarly journals Coloring the $n$-Smooth Numbers with $n$ Colors

10.37236/8492 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Andrés Eduardo Caicedo ◽  
Thomas A. C. Chartier ◽  
Péter Pál Pach
Keyword(s):  
High School ◽  
Abelian Group ◽  
General Version ◽  
Real Numbers ◽  
Smooth Numbers ◽  
The Third ◽  
School Journal ◽  
Natural Density ◽  
Positive Integers

For which values of $n$ is it possible to color the positive integers using precisely $n$ colors in such a way that for any $a$, the numbers $a,2a,\dots,na$ all receive different colors? The third-named author posed the question around 2008-2009. Particular cases appeared in the Hungarian high school journal KöMaL in April 2010, and the general version appeared in May 2010 on MathOverflow, posted by D. Pálvölgyi. The question remains open. We discuss the known partial results and investigate a series of related matters attempting to understand the structure of these $n$-satisfactory colorings. Specifically, we show that there is an $n$-satisfactory coloring whenever there is an abelian group operation $\oplus$ on the set $\{1,2,\dots,n\}$ that is compatible with multiplication in the sense that whenever $i$, $j$ and $ij$ are in $\{1,\dots,n\}$, then $ij=i\oplus j$. This includes in particular the cases where $n+1$ is prime, or $2n+1$ is prime, or $n=p^2-p$ for some prime $p$, or there is  a $k$ such that $q=nk+1$ is prime and $1^k,\dots,n^k$ are all distinct modulo $q$ (in which case we call $q$ a strong representative of order $n$). The colorings obtained by this process we call multiplicative. We also show that nonmultiplicative colorings exist for some values of $n$. There is an $n$-satisfactory coloring of $\mathbb Z^+$ if and only if there is such a coloring of the set $K_n$ of $n$-smooth numbers. We identify all $n$-satisfactory colorings for $n\leqslant 5$ and all multiplicative colorings for $n\leqslant 8$, and show that there are as many nonmultiplicative colorings of $K_n$ as there are real numbers for $n=6$ and 8. We show that if $n$ admits a strong representative $q$ then it admits infinitely many and in fact the set of such $q$ has positive natural density in the set of all primes. We also show that the question of whether there is an $n$-satisfactory coloring is equivalent to a problem about tilings, and use this to give a geometric characterization of multiplicative colorings.

scholarly journals Universality probability of a prefix-free machine

2012 ◽  
Vol 370 (1971) ◽  
pp. 3488-3511 ◽  
Author(s):  
George Barmpalias ◽  
David L. Dowe
Keyword(s):  
Turing Degree ◽  
Real Numbers ◽  
Arithmetical Hierarchy ◽  
Halting Problem ◽  
The Real ◽  
The Third ◽  
Free Machine

We study the notion of universality probability of a universal prefix-free machine, as introduced by C. S. Wallace. We show that it is random relative to the third iterate of the halting problem and determine its Turing degree and its place in the arithmetical hierarchy of complexity. Furthermore, we give a computational characterization of the real numbers that are universality probabilities of universal prefix-free machines.

scholarly journals How to solve third degree equations without moving to complex numbers

2020 ◽  
Vol 12 ◽  
pp. 123-131
Author(s):  
Antoni Leon Dawidowicz
Keyword(s):  
High School ◽  
16Th Century ◽  
Complex Numbers ◽  
Algebraic Equations ◽  
Current Version ◽  
Elementary Functions ◽  
Real Numbers ◽  
Fourth Degree ◽  
The Third ◽  
Algebraic Operations

During the Renaissance, the theory of algebraic equations developed in Europe. It is about finding a solution to the equation of the formanxn + . . . + a1x + a0 = 0,represented by coefficients subject to algebraic operations and roots of any degree. In the 16th century, algorithms for the third and fourth-degree equations appeared. Only in the nineteenth century, a similar algorithm for thehigher degree was proved impossible. In (Cardano, 1545) described an algorithm for solving third-degree equations. In the current version of this algorithm, one has to take roots of complex numbers that even Cardano didnot know.This work proposes an algorithm for solving third-degree algebraic equations using only algebraic operations on real numbers and elementary functions taught at High School.

New oscillation criteria for third order nonlinear neutral difference equations

2011 ◽  
Vol 61 (4) ◽  
Author(s):  
S. Saker
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Nonnegative Integer ◽  
Real Numbers ◽  
Third Order ◽  
Neutral Difference Equation ◽  
Neutral Difference Equations ◽  
Oscillation Criteria ◽  
The Third ◽  
Positive Integers

AbstractIn this paper, we are concerned with oscillation of the third-order nonlinear neutral difference equation $\Delta (c_n [\Delta (d_n \Delta (x_n + p_n x_{n - \tau } ))]^\gamma ) + q_n f(x_{g(n)} ) = 0,n \geqslant n_0 ,$ where γ > 0 is the quotient of odd positive integers, c n, d n, p n and q n are positive sequences of real numbers, τ is a nonnegative integer, g(n) is a sequence of nonnegative integers and f ∈ C(ℝ,ℝ) such that uf(u) > 0 for u ≠ 0. Our results extend and improve some previously obtained ones. Some examples are considered to illustrate the main results.

THE EFFECT OF APPLYING DEMONSTRATION METHOD ON STUDENTS’ ACHIEVEMENT IN SPEAKING SKILL AT THE THIRD GRADE OF SMK PAB 3 MEDAN ESTATE

2015 ◽  
Vol 4 (2) ◽  
Author(s):  
Meryanti Napitupulu And Anni Holila Pulungan
Keyword(s):  
High School ◽  
Control Group ◽  
Vocational High School ◽  
Control Groups ◽  
Speaking Test ◽  
The Third ◽  
Speaking Skill ◽  
The Subject ◽  
And Control ◽  
Experimental Group

This study was conducted as an attempt to discover the effect of applying Demonstration Method on students’ achievement in speaking skill. It was an experimental research. The subject was students of Grade XII, Vocational High School (Sekolah Menengah Kejuruan: SMK), which consisted of 79 students. The research was divided into two groups: experimental and control groups. The instrument used to collect the data was speaking test. To obtain the reliability of the test, the writer applied Kuder Richardson 21 formula. The result of the reliability was 0.7, and it was found that the test was reliable. The data were analyzed by using t-test formula. The analysis showed that the scores of the students in the experimental group were significantly higher than the scores of the students in the control group at the level of significant m = 0.05 with the degree of freedom (df) 77, t-observed value 8.9 > t-table value 1.99. The findings indicate that using Demonstration Method significantly affected the students’ achievement in speaking skill. So, English teachers are suggested to use Demonstration Method in order to improve students’ achievement in speaking skill.

Gaunilo Parodies Anselm

2014 ◽  
Vol 17 (1) ◽  
pp. 45-71
Author(s):  
Geo Siegwart
Keyword(s):  
Detailed Comparison ◽  
The Third ◽  
Common Structure ◽  
Logical Reconstruction ◽  
Points Of View ◽  
And Function

The main objective is an interpretation of the island parody, in particular a logical reconstruction of the parodying argument that stays close to the text. The parodied reasoning is identified as the proof in the second chapter of the Proslogion, more specifically, this proof as it is represented by Gaunilo in the first chapter of his Liber pro insipiente. The second task is a detailed comparison between parodied and parodying argument as well as an account of their common structure. The third objective is a tentative characterization of the nature and function of parodies of arguments. It seems that parodying does not add new pertinent points of view to the usual criticism of an argument.

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

scholarly journals HEYDE’S CHARACTERIZATION THEOREM FOR DISCRETE ABELIAN GROUPS

2010 ◽  
Vol 88 (1) ◽  
pp. 93-102 ◽  
Author(s):  
MARGARYTA MYRONYUK
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Real Line ◽  
Linear Form ◽  
Conditional Distribution ◽  
Random Variables ◽  
Characterization Theorem ◽  
Gaussian Distributions ◽  
Discrete Abelian Group

AbstractLet X be a countable discrete abelian group with automorphism group Aut(X). Let ξ1 and ξ2 be independent X-valued random variables with distributions μ1 and μ2, respectively. Suppose that α1,α2,β1,β2∈Aut(X) and β1α−11±β2α−12∈Aut(X). Assuming that the conditional distribution of the linear form L2 given L1 is symmetric, where L2=β1ξ1+β2ξ2 and L1=α1ξ1+α2ξ2, we describe all possibilities for the μj. This is a group-theoretic analogue of Heyde’s characterization of Gaussian distributions on the real line.

Spectral synthesis and residually finite-dimensional algebras

2017 ◽  
Vol 16 (10) ◽  
pp. 1750200 ◽  
Author(s):  
László Székelyhidi ◽  
Bettina Wilkens
Keyword(s):  
Spectral Analysis ◽  
Abelian Group ◽  
Theoretical Approach ◽  
Abelian Groups ◽  
Spectral Synthesis ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Analysis And Synthesis ◽  
Spectral Analysis And Synthesis

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.

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