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 Functional Equations and Fourier Analysis

2013 ◽  
Vol 56 (1) ◽  
pp. 218-224 ◽  
Author(s):  
Dilian Yang
Keyword(s):  
Representation Theory ◽  
Fourier Analysis ◽  
Harmonic Analysis ◽  
Functional Equations ◽  
Compact Groups ◽  
Wilson Equation ◽  
D’Alembert Equation

AbstractBy exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation-on compact groups.

scholarly journals A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

1959 ◽  
Vol 11 ◽  
pp. 59-60 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Representation Theory ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Orthogonality Relations ◽  
Schur’S Lemma ◽  
Image Position ◽  
A Finite Group ◽  
Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

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

scholarly journals Studying in Lee groups and most important examples of it (Heisenberg group): دراسة في زمر لي وأهم الأمثلة عنها (زمرة هايزنبرغ)

2020 ◽  
Vol 4 (1) ◽  
Author(s):  
Soha Ali Salamah
Keyword(s):  
Number Theory ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Representation Theory ◽  
Heisenberg Group ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
General Definition ◽  
Nilpotent Lie Groups ◽  
Basic Facts

In this research, we present some basic facts about Lie algebra and Lie groups. We shall require only elementary facts about the general definition and knowledge of a few of the more basic groups, such as Euclidean groups. Then we introduce the Heisenberg group which is the most well-known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis.

On the computation of overorders

2019 ◽  
Vol 16 (04) ◽  
pp. 857-879
Author(s):  
Tommy Hofmann ◽  
Carlo Sircana
Keyword(s):  
Number Theory ◽  
Representation Theory ◽  
Maximal Order ◽  
Global Field ◽  
Semisimple Algebra ◽  
Practical Algorithms ◽  
Ring Extensions ◽  
Algorithmic Representation

The computation of a maximal order of an order in a semisimple algebra over a global field is a classical well-studied problem in algorithmic number theory. In this paper, we consider the related problems of computing all minimal overorders as well as all overorders of a given order. We use techniques from algorithmic representation theory and the theory of minimal integral ring extensions to obtain efficient and practical algorithms, whose implementation is publicly available.

On modal analysis of ship strength

1974 ◽  
Vol 341 (1624) ◽  
pp. 121-134 ◽  
Keyword(s):  
Rigid Body ◽  
Structural Dynamics ◽  
Model Test ◽  
Time History ◽  
Body Motion ◽  
Strength Analysis ◽  
Fluid Forces ◽  
History Effects ◽  
Orthogonality Relations ◽  
First Time

An approach is formulated by which structural dynamics of ships may be analysed in a linear modal form. By employing the principal modes of the ship in vacuo , simple orthogonality relations can be retained without dependence on the necessarily approximate techniques used to estimate fluid forces. It is also possible to identify modal contributions to mass, damping and stiffness for the hull and for the hydrodynamic actions separately. Those contributions of hydrodynamic origin may depend significantly on time history effects which can be measured by means of a model test; these effects can be admitted into the ship strength analysis, it is believed for the first time. It is shown how existing modal theories of ship strength and theories of seakeeping (i. e. of ‘rigid body’ motion in a seaway) fit into this more general analysis.

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