Number Theory and Fourier Analysis Applications in Physics, Acoustics and Computer Science

1990 ◽  
pp. 101-129
Author(s):  
M. R. Schroeder
Keyword(s):  
Number Theory ◽  
Fourier Analysis ◽  
Computer Science

scholarly journals Coleman integration for even-degree models of hyperelliptic curves

2015 ◽  
Vol 18 (1) ◽  
pp. 258-265 ◽  
Author(s):  
Jennifer S. Balakrishnan
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Hyperelliptic Curves ◽  
Open Book ◽  
Line Integral ◽  
Book Series ◽  
Numerical Examples ◽  
Lecture Notes ◽  
Integration Algorithms ◽  
Mathematical Sciences

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

scholarly journals Idempotent Factorizations of Square-free Integers

2019 ◽  
Author(s):  
Barry Fagin
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Infinite Number ◽  
Preliminary Results ◽  
Analytical Results ◽  
Hard Problems ◽  
Positive Integers

We explore the class of positive integers n that admit idempotent factorizations n=pq such that lambda(n) divides (p-1)(q-1), where lambda(n) is the Carmichael lambda function. Idempotent factorizations with p and q prime have received the most attention due to their cryptographic advantages, but there are an infinite number of n with idempotent factorizations containing composite p and/or q. Idempotent factorizations are exactly those p and q that generate correctly functioning keys in the RSA 2-prime protocol with n as the modulus. While the resulting p and q have no cryptographic utility and therefore should never be employed in that capacity, idempotent factorizations warrant study in their own right as they live at the intersection of multiple hard problems in computer science and number theory. We present some analytical results here. We also demonstrate the existence of maximally idempotent integers, those n for which all bipartite factorizations are idempotent. We show how to construct them, and present preliminary results on their distribution.

scholarly journals Some Results in the Fourier Analysis

1966 ◽  
Vol 27 (1) ◽  
pp. 55-59
Author(s):  
Tikao Tatuzawa
Keyword(s):  
Number Theory ◽  
Euclidean Space ◽  
Fourier Analysis ◽  
Analytic Number Theory ◽  
Dimensional Euclidean Space ◽  
Mathematical Methods ◽  
Analytic Number ◽  
Fundamental Theorems

There are many uses of Fourier analysis in the analytic number theory. In this paper we shall derive two fundamental theorems using Cramer’s method (Mathematical methods of statistics, 1946). Let E, E* be unit cubes in the whole n-dimensional Euclidean space X such that

Relations in the semigroup of 2 × 2 upper-triangular matrices

2020 ◽  
Vol 30 (03) ◽  
pp. 567-584
Author(s):  
Henri-Alex Esbelin ◽  
Marin Gutan
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Semigroup Theory ◽  
Theoretical Computer Science ◽  
Multiplicative Semigroup ◽  
Triangular Matrices ◽  
Theoretical Computer ◽  
Upper Triangular Matrices ◽  
Matrix Semigroups

Let [Formula: see text] with [Formula: see text] be [Formula: see text] upper-triangular matrices with rational entries. In the multiplicative semigroup generated by these matrices, we check if there are relations of the form [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] We give algorithms to find relations of the previous form. Our results are extensions of some theorems obtained by Charlier and Honkala in [The freeness problem over matrix semigroups and bounded languages, Inf. Comput. 237 (2014) 243–256]. Our paper is at the interface between algebra, number theory and theoretical computer science. While the main results concern decidability and semigroup theory, the methods for obtaining them come from number theory.

Solving 0-1 Knapsack Problems by Greedy Method and Dynamic Programming Method

2011 ◽  
Vol 282-283 ◽  
pp. 570-573
Author(s):  
Lu Liu
Keyword(s):  
Dynamic Programming ◽  
Number Theory ◽  
Computer Science ◽  
Knapsack Problem ◽  
Hot Spot ◽  
Knapsack Problems ◽  
Dynamic Programming Method ◽  
Typical Problem ◽  
Greedy Method ◽  
Simulation Results

The 0-1 knapsack problem is typical problem in computer science and its solution is a hot spot in algorithms design and verification. Because it is very hard to solve, it is very important in the research on cryptosystem and number theory. In this paper, the 0-1 knapsack problem and its algorithm is analyzed firstly. And then this paper presents two kinds of expand form, and proposes two efficient algorithms based on dynamic programming and greedy algorithm to solve the proposed problems. Simulation results show it is effective.

scholarly journals On a restricted linear congruence

2016 ◽  
Vol 12 (08) ◽  
pp. 2167-2171 ◽  
Author(s):  
Khodakhast Bibak ◽  
Bruce M. Kapron ◽  
Venkatesh Srinivasan
Keyword(s):  
Number Theory ◽  
Fourier Transform ◽  
Computer Science ◽  
Short Proof ◽  
Arithmetic Functions ◽  
Number Of Solutions ◽  
Finite Fourier Transform ◽  
Linear Congruence ◽  
Special Cases ◽  
Ramanujan Sums

Let [Formula: see text], [Formula: see text], and [Formula: see text] be all positive divisors of [Formula: see text]. For [Formula: see text], define [Formula: see text]. In this paper, by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result: the number of solutions of the linear congruence [Formula: see text], with [Formula: see text], [Formula: see text], is [Formula: see text] where [Formula: see text] is a Ramanujan sum. Some special cases and other forms of this problem have been already studied by several authors. The problem has recently found very interesting applications in number theory, combinatorics, computer science, and cryptography. The above explicit formula generalizes the main results of several papers, for example, the main result of the paper by Sander and Sander [J. Number Theory 133 (2013) 705–718], one of the main results of the paper by Sander [J. Number Theory 129 (2009) 2260–2266], and also gives an equivalent formula for the main result of the paper by Sun and Yang [Int. J. Number Theory 10 (2014) 1355–1363].

scholarly journals An orthogonality relation for (with an appendix by Bingrong Huang)

2021 ◽  
Vol 9 ◽  
Author(s):  
Dorian Goldfeld ◽  
Eric Stade ◽  
Michael Woodbury
Keyword(s):  
Number Theory ◽  
Representation Theory ◽  
Fourier Analysis ◽  
Error Term ◽  
Orthogonality Relation ◽  
Finite Abelian Groups ◽  
Real Group ◽  
Primes In Arithmetic Progressions ◽  
Orthogonality Relations ◽  
First Time

Abstract Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm {GL}(1)$ ) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Orthogonality relations for $\mathrm {GL}(2)$ and $\mathrm {GL}(3)$ have been worked on by many researchers with a broad range of applications to number theory. We present here, for the first time, very explicit orthogonality relations for the real group $\mathrm {GL}(4, \mathbb R)$ with a power savings error term. The proof requires novel techniques in the computation of the geometric side of the Kuznetsov trace formula.

scholarly journals A theorem on numbers of the form 10^x

2019 ◽  
Author(s):  
Ravin Kumar
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Natural Numbers ◽  
The Core ◽  
Pure Mathematics ◽  
Major Application

Number theory is one of the core branches of pure mathematics. It has played an important role in the study of natural numbers. In this paper, we are presenting a theorem on the numbers of form 10^x , where x ∊ Z+ . The proposed theorem have a major application in computer science. It can be used to predict ‘n’ bits which will always represent more than 10^x total numbers. We proved that the nature of the ‘n’ bits is always one of the forms 10i, 10i + 4, or 10i + 7, where i ∊ Z+ .

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