scholarly journals Geometric Construction of Some Lehmer Means

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 251 ◽  
Author(s):  
Ralph Høibakk ◽  
Dag Lukkassen ◽  
Annette Meidell ◽  
Lars-Erik Persson
Keyword(s):  
Positive Integer ◽  
Geometric Construction ◽  
Geometric Constructions ◽  
Line Segments ◽  
Integer Power

The main aim of this paper is to contribute to the recently initiated research concerning geometric constructions of means, where the variables are appearing as line segments. The present study shows that all Lehmer means of two variables for integer power k and for k = m 2 , where m is an integer, can be geometrically constructed, that Lehmer means for power k = 0 , 1 and 2 can be geometrically constructed for any number of variables and that Lehmer means for power k = 1 / 2 and - 1 can be geometrically constructed, where the number of variables is n = 2 m and m is a positive integer.

scholarly journals On normal numbers

1982 ◽  
Vol 32 (1) ◽  
pp. 79-87 ◽  
Author(s):  
C. E. M. Pearce ◽  
M. S. Keane
Keyword(s):  
Positive Integer ◽  
Simple Proof ◽  
Normal Numbers ◽  
Integer Power ◽  
Positive Integers

AbstractSchmidt has shown that if r and s are positive integers and there is no positive integer power of r which is also a positive integer power of s, then there exists an uncountable set of reals which are normal to base r but not even simply normal to base s. We give a structurally simple proof of this result

scholarly journals Trace of Positive Integer Power of Squared Special Matrix

CAUCHY ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 200-211
Author(s):  
Rahmawati Rahmawati ◽  
Aryati Citra ◽  
Fitri Aryani ◽  
Corry Corazon Marzuki ◽  
Yuslenita Muda
Keyword(s):  
Positive Integer ◽  
Direct Proof ◽  
Integer Power ◽  
The Matrix ◽  
Special Matrix ◽  
Matrix Trace

The rectangle matrix to be discussed in this research is a special matrix where each entry in each line has the same value which is notated by An. The main aim of this paper is to find the general form of the matrix trace An powered positive integer m. To prove whether the general form of the matrix trace of An powered positive integer can be confirmed, mathematics induction and direct proof are used.  

scholarly journals The Adjacency Matrix of One Type of Directed Graph and the Jacobsthal Numbers and Their Determinantal Representation

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Fatih Yılmaz ◽  
Durmuş Bozkurt
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Intensive Study ◽  
Integer Power ◽  
Determinantal Representation ◽  
Adjacency Matrices

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we consider one type of directed graph. Then we obtain a general form of the adjacency matrices of the graph. By using the well-known property which states the(i,j)entry ofAm(Ais adjacency matrix) is equal to the number of walks of lengthmfrom vertexito vertexj, we show that elements ofmth positive integer power of the adjacency matrix correspond to well-known Jacobsthal numbers. As a consequence, we give a Cassini-like formula for Jacobsthal numbers. We also give a matrix whose permanents are Jacobsthal numbers.

scholarly journals Trace of Positive Integer Power of Real 2 × 2 Matrices

2015 ◽  
Vol 05 (04) ◽  
pp. 150-155 ◽  
Author(s):  
Jagdish Pahade ◽  
Manoj Jha
Keyword(s):  
Positive Integer ◽  
Integer Power

CUTTING AND PASTING OF RIEMANN SURFACES WITH ABELIAN DIFFERENTIALS I

1999 ◽  
Vol 10 (05) ◽  
pp. 587-617
Author(s):  
YOSHITAKE HASHIMOTO ◽  
KIYOSHI OHBA
Keyword(s):  
Riemann Surface ◽  
Positive Integer ◽  
Elliptic Curves ◽  
Complex Plane ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Complex Structures ◽  
Line Segments ◽  
Abelian Differentials ◽  
The Family

We introduce a method of constructing once punctured Riemann surfaces by cutting the complex plane along "line segments" and pasting by "parallel transformations". The advantage of this construction is to give a good visualization of the deformation of complex structures of Riemann surfaces. In fact, given a positive integer g, there appears a family of once punctured Riemann surfaces of genus g which is complete and effectively parametrized at any point. Our construction naturally gives each of the resulting surfaces what we call a Lagrangian lattice Λ, a certain subgroup of the first homology. Furthermore Λ and the puncture determine an Abelian differential ωΛ of the second kind on the Riemann surface. Using Λ and ωΛ we consider the Kodaira–Spencer maps and some extension of the family to obtain any once punctured Riemann surface with a Lagrangian lattice. In particular we describe the moduli space of once punctured elliptic curves with Lagrangian lattices.

Trace Matriks Berbentuk Khusus 2 X 2 Berpangkat Bilangan Bulat Positif

2019 ◽  
Vol 2 (2) ◽  
Author(s):  
Fitri Aryani ◽  
Titik Fatonah
Keyword(s):  
Positive Integer ◽  
Direct Proof ◽  
Mathematical Induction ◽  
Main Diagonal ◽  
General Shape ◽  
Integer Power ◽  
Square Matrix

Trace matriks ialah jumlah dari elemen diagonal utama dari matriks kuadrat. Penelitian ini membahas mengenai jejak kekuatan bilangan bulat positif matriks nyata 2x2. Ada dua langkah dalam membentuk bentuk umum dari trace matriks. Pertama, tentukan bentuk umum (An) dan buktikan menggunakan induksi matematika. Kedua, tentukan jejak bentuk umum (An) dan buktikan dengan bukti langsung. Hasilnya diperoleh bentuk umum jejak daya bilangan bulat positif dari matriks nyata 2x2 nyata untuk n ganjil dan n genap.   Trace matrix is ​​the sum of the main diagonal elements of the square matrix. This Paper discusses the trace of positive integer power of  real 2x2 special matrices. There are two steps in forming the general shape of the trace matrix. First, determine the general form of (An) and prove it using mathematical induction. Second, determine the general form trace (An) and prove it by direct proof. The results obtained a general shape of trace of positive integer power power of  real 2x2 special matrices for n odd and n even.

Alternative Geometric Constructions: Promoting Mathematical Reasoning

2002 ◽  
Vol 95 (1) ◽  
pp. 32-36 ◽  
Author(s):  
Eric A. Pandiscio
Keyword(s):  
Euclidean Geometry ◽  
Mathematical Reasoning ◽  
Geometric Constructions ◽  
Line Segments ◽  
Mathematical Ideas ◽  
Geometric Concepts ◽  
Circular Arcs ◽  
Multiple Tasks ◽  
History Of ◽  
History Of Use

Construction tools in most high school Euclidean geometry classes have typically been limited to a compass for drawing circular arcs and a straightedge for drawing line segments. The strengths of these tools include both mathematical precision and a long history of use. However, alternatives can provide fresh possibilities for engaging students in the mathematical reasoning that lies at the heart of traditional geometry (Gibb 1982; Robertson 1986). This article proposes that a single task completed with a variety of construction tools fosters a greater sense of mathematical contemplation than multiple tasks done with the same tool. The premises are simple: each tool fosters different mathematical ideas, and using multiple tools not only requires understanding of a greater breadth and depth of geometric concepts but also highlights the connections that exist among different ideas.

scholarly journals One Can Hear the Area of a Torus by Hearing the Eigenvalues of the Polyharmonic Operators

2014 ◽  
Vol 47 (3) ◽  
Author(s):  
Shunzi Guo ◽  
Jinyun Jin
Keyword(s):  
Positive Integer ◽  
Asymptotic Formula ◽  
Spectral Problem ◽  
Asymptotic Properties ◽  
Beltrami Operator ◽  
Dimensional Torus ◽  
Integer Power ◽  
Real Positive Number

AbstractThis paper considers the asymptotic properties for the spectrum of a positive integer power l of the Laplace-Beltrami operator acting on an n-dimensional torus T. If N(λ) is the number of eigenvalues counted with multiplicity, smaller than a real positive number, we establish a Weyl-type asymptotic formula for the spectral problem of the polyharmonic operators on T, that is, as λ → +∞N (λ) ~ ωwhere ω

Discrimination of Occluded Line Segments by Pigeons

2009 ◽  
Author(s):  
Robert G. Cook ◽  
Carl Erick Hagmann
Keyword(s):  
Line Segments
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