scholarly journals Solving equations in dense Sidon sets

2021 ◽  
pp. 1-10
Author(s):  
SEAN PRENDIVILLE
Keyword(s):  
Linear Equations ◽  
Alternative Proof ◽  
Translation Invariant ◽  
Sidon Sets ◽  
Solving Equations

Abstract We offer an alternative proof of a result of Conlon, Fox, Sudakov and Zhao [CFSZ20] on solving translation-invariant linear equations in dense Sidon sets. Our proof generalises to equations in more than five variables and yields effective bounds.

scholarly journals On Systems of Complexity One in the Primes

2016 ◽  
Vol 60 (1) ◽  
pp. 133-163 ◽  
Author(s):  
Kevin Henriot
Keyword(s):  
Linear Equations ◽  
Quantitative Result ◽  
Arithmetic Progressions ◽  
Invariant System ◽  
Qualitative Result ◽  
System Of Linear Equations ◽  
Translation Invariant

AbstractConsider a translation-invariant system of linear equationsVx= 0 of complexity one, whereVis an integerr×tmatrix. We show that ifAis a subset of the primes up toNof density at leastC(log logN)–1/25t, there exists a solutionx∈ AttoVx= 0 with distinct coordinates. This extends a quantitative result of Helfgott and de Roton for three-term arithmetic progressions, while the qualitative result is known to hold for all translation-invariant systems of finite complexity by the work of Green and Tao.

Solving Equations: Exploring Instructional Exchanges as Lenses to Understand Teaching and Its Resistance to Reform

2019 ◽  
Vol 50 (1) ◽  
pp. 51-83 ◽  
Author(s):  
Orly Buchbinder ◽  
Daniel I. Chazan ◽  
Michelle Capozzoli
Keyword(s):  
Teacher Beliefs ◽  
Mathematics Teaching ◽  
Explanatory Variable ◽  
Linear Equations ◽  
Classroom Interaction ◽  
Media Representations ◽  
Theoretical Construct ◽  
Mathematics Classrooms ◽  
Rich Media ◽  
Solving Equations

Many research studies have sought to explain why NCTM's vision for mathematics classrooms has not had greater impact on everyday instruction, with teacher beliefs often identified as an explanatory variable. Using instructional exchanges as a theoretical construct, this study explores the influence of teachers' institutional positions on the solving of equations in algebra classrooms. The experimental design uses surveys with embedded rich-media representations of classroom interaction to surface how teachers appraise correct solutions to linear equations where some solutions follow suggested textbook procedures for solving linear equations and others do not. This paper illustrates the feasibility of studying teaching with rich-media surveys and suggests new ways to support changes in everyday mathematics teaching.

Representing, Solving, and Using Algebraic Equations

1989 ◽  
Vol 82 (8) ◽  
pp. 608-612
Author(s):  
Joe Dan Austin ◽  
H. J. Vollrath
Keyword(s):  
Linear Equations ◽  
Algebraic Equations ◽  
Beginning Algebra ◽  
Physical Problems ◽  
Solving Equations ◽  
Physical Objects

Students of beginning algebra are quickly expected to solve linear equations. The solution procedures are generally abstract, involving the manipulation of numbers and algebraic symbols. Many students, even after completing a year of algebra, do not understand variables, equations, and solving equations (cf. Carpenter et al. [1982]). One way to help students learn to solve equations is to use physical objects, diagrams, and then symbols to represent equations. (Bruner [1964, 1967] calls such representations enactive (concrete), iconic (pictorial), and symbolic.) Although solving equations symbolically is essential, many students can benefit from working with physical problems that can also be symbolized mathematically. This article describes one way for students to learn to solve certain linear equations using pan balances, diagrams, and then symbols.

scholarly journals Ergodic theorem involving additive and multiplicative groups of a field and patterns

2015 ◽  
Vol 37 (3) ◽  
pp. 673-692 ◽  
Author(s):  
VITALY BERGELSON ◽  
JOEL MOREIRA
Keyword(s):  
Finite Field ◽  
Ergodic Theorem ◽  
Finite Fields ◽  
Correspondence Principle ◽  
Combinatorial Problems ◽  
Alternative Proof ◽  
Large Set ◽  
Sidon Sets ◽  
New Variant ◽  
Multiplicative Groups

We establish a ‘diagonal’ ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg’s correspondence principle, prove that any ‘large’ set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular, we obtain an alternative proof for a result obtained by Cilleruelo [Combinatorial problems in finite fields and Sidon sets. Combinatorica32(5) (2012), 497–511], showing that for any finite field $F$ and any subsets $E_{1},E_{2}\subset F$ with $|E_{1}|\,|E_{2}|>6|F|$, there exist $u,v\in F$ such that $u+v\in E_{1}$ and $uv\in E_{2}$.

scholarly journals Soliton Decomposition of the Box-Ball System

2021 ◽  
Vol 9 ◽  
Author(s):  
Pablo A. Ferrari ◽  
Chi Nguyen ◽  
Leonardo T. Rolla ◽  
Minmin Wang
Keyword(s):  
Invariant Measures ◽  
Vries Equation ◽  
Linear Equations ◽  
Large Family ◽  
Ergodic Measure ◽  
System Of Linear Equations ◽  
Translation Invariant ◽  
Discrete Counterpart ◽  
Product Measures ◽  
Effective Speed

Abstract The box-ball system (BBS) was introduced by Takahashi and Satsuma as a discrete counterpart of the Korteweg-de Vries equation. Both systems exhibit solitons whose shape and speed are conserved after collision with other solitons. We introduce a slot decomposition of ball configurations, each component being an infinite vector describing the number of size k solitons in each k-slot. The dynamics of the components is linear: the kth component moves rigidly at speed k. Let $\zeta $ be a translation-invariant family of independent random vectors under a summability condition and $\eta $ be the ball configuration with components $\zeta $ . We show that the law of $\eta $ is translation invariant and invariant for the BBS. This recipe allows us to construct a large family of invariant measures, including product measures and stationary Markov chains with ball density less than $\frac {1}{2}$ . We also show that starting BBS with an ergodic measure, the position of a tagged k-soliton at time t, divided by t converges as $t\to \infty $ to an effective speed $v_k$ . The vector of speeds satisfies a system of linear equations related with the generalised Gibbs ensemble of conservative laws.

scholarly journals A Gyrogeometric Mean in the Einstein Gyrogroup

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1333 ◽  
Author(s):  
Takuro Honma ◽  
Osamu Hatori
Keyword(s):  
Alternative Proof ◽  
Left Translation ◽  
Elementary Calculation ◽  
Translation Invariant

In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation.

scholarly journals PPROBLEMS OF PHYSICAL CONTENT, REDUCED TO A SYSTEM OF LINEAR EQUATIONS WITH TWO VARIABLES

2020 ◽  
Vol 72 (4) ◽  
pp. 48-55
Author(s):  
Zh.А. Nurmaganbetova ◽  
◽  
N.К. Аshirbayev ◽  
А.M. Polatbek ◽  
А.О. Bаidibekova ◽  
...  
Keyword(s):  
Teaching Methods ◽  
Graphical Method ◽  
Linear Equations ◽  
Linear Functions ◽  
Teaching Mathematics ◽  
System Of Linear Equations ◽  
Physical Content ◽  
Concept Of Function ◽  
Solving Equations ◽  
Mathematics Teaching Methods

Functional and graphic lines are one of the foundations of mathematics teaching methods. The advantage of this line is that the study of other important lines of mathematics is carried out through the prism of the concept of function. Based on the experience of teaching mathematics, we know that the concept of function is abstract and very difficult for students to understand, so in order to enhance the visualization of the researching objects and concepts when implementing functional and graphic lines, students need to increase the system of physical content tasks for studying and understanding functions. In school course of algebra, the functional-graphical method is rarely used for solving a system of equations with two unknowns, as well as for solving equations with two unknowns. The article deals with the problems of solving problems of physical content when studying a system of linear equations with two variables in school course of algebra. The emphasis is on the fact that the considered problems with physical content are interconnected with functionalgraphic lines in algebra and allow deepening the topic, revealing the practical content. The problems with physical content presented in the article are intended for studying linear functions of algebra and their graphs, studying functions, constructing and solving equations and a system of linear equations associated with these functions.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

The influence of extrinsic motivation and work experience on employee performance (Case study in the Serang Baru Sub-district, Bekasi District)

2020 ◽  
Vol 5 (02) ◽  
pp. 167
Author(s):  
Nur’enny Nur’enny ◽  
Rahmat Hidayat
Keyword(s):  
Work Experience ◽  
Extrinsic Motivation ◽  
Employee Performance ◽  
Linear Equations ◽  
Test Reliability ◽  
Coefficient Of Determination ◽  
District Office ◽  
Normality Test ◽  
Observation Methods ◽  
Saturated Sample

This study aims to obtain information about extrinsic motivation and work experience and its effect on employee performance in the Serang Baru District Office. This study uses a saturated sample so that the population is the same as the sample of 80 employees, at the Serang Baru District Office. The method used is validation test, reliability test, then classical assumption test, which includes normality test and multicollinearity, as well as heteroscedasticity test, multiple linear analysis test, multiple linear equations, F test, coefficient of determination, and t test. The data of this research used observation methods and questionnaires distributed to 80 samples which were addressed to employees of the Serang Baru District Office. Based on the results of research and discussion, it can be concluded: 1) Extrinsic motivation does not affect employee performance because employees are willing to work more than expected regardless of extrinsic motivation or not. 2) Employee performance is strongly influenced by work experience. The more experience, they get while working, the more knowledge they will get. 3) Employee performance will be better with the support of experienced employees so as to increase the level of output produced.             Keywords: Employee Performance, Extrinsic Motivation, Work Experience

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