James Ax and Simon Kochen. Diophantine problems over local fields I. American journal of mathematics, vol. 87 (1965), pp. 605–630. - James Ax and Simon Kochen. Diophantine problems over local fields II. A complete set of axioms for p-adic number theory. American journal of mathematics, vol. 87 (1965), pp. 631–648. - James Ax and Simon Kochen. Diophantine problems over local fields III. Decidable fields. Annals of mathematics, vol. 83 (1966), pp. 437–456.

1971 ◽  
Vol 36 (4) ◽  
pp. 683-684
Author(s):  
Abraham Robinson
Keyword(s):  
Number Theory ◽  
Local Fields ◽  
Complete Set ◽  
Diophantine Problems

Diophantine Problems Over Local Fields I

1965 ◽  
Vol 87 (3) ◽  
pp. 605 ◽  
Author(s):  
James Ax ◽  
Simon Kochen
Keyword(s):  
Local Fields ◽  
Diophantine Problems

Diophantine Problems Over Local Fields: III. Decidable Fields

1966 ◽  
Vol 83 (3) ◽  
pp. 437 ◽  
Author(s):  
James Ax ◽  
Simon Kochen
Keyword(s):  
Local Fields ◽  
Diophantine Problems

Diophantine approximation

2021 ◽  
Vol 34 (2) ◽  
pp. 205-229
Author(s):  
Noriko Hirata-Kohno
Keyword(s):  
Number Theory ◽  
Diophantine Approximation ◽  
Recent Progress ◽  
Padé Approximation ◽  
Pade Approximation ◽  
Fundamental Properties ◽  
Subspace Theorem ◽  
Diophantine Problems

This article gives an introductory survey of recent progress on Diophantine problems, especially consequences coming from Schmidt’s subspace theorem, Baker’s transcendence method and Padé approximation. We present fundamental properties around Diophantine approximation and how it yields results in number theory.

scholarly journals Generators and Relations for Cyclotomic Units

1966 ◽  
Vol 27 (2) ◽  
pp. 401-407 ◽  
Author(s):  
Hyman Bass
Keyword(s):  
Number Theory ◽  
Roots Of Unity ◽  
Generators And Relations ◽  
Complex Roots ◽  
Complete Set ◽  
Cyclotomic Units

We prove here an unpublished conjecture of Milnor which gives a complete set of multiplicative relations between the numbers e′(ζ) = 1−ζ,where ranges over complex roots of unity. Information of this type is useful in certain areas of topology as well as in number theory.

An Introduction to Micromechanics

2016 ◽  
Vol 828 ◽  
pp. 3-24 ◽  
Author(s):  
Wenbin Yu
Keyword(s):  
Representative Volume Element ◽  
Volume Element ◽  
Local Fields ◽  
Homogenization Theory ◽  
Element Analysis ◽  
Full Field ◽  
Linear Elastic ◽  
Representative Volume ◽  
Complete Set ◽  
Periodic Materials

This article provides a brief introduction to micromechanics using linear elastic materials as an example. The fundamental micromechanics concepts including homogenization and dehomogenization, representative volume element (RVE), unit cell, average stress and strain theories, effective stiffness and compliance, Hill-Mandel macrohomogeneity condition. This chapter also describes the detailed derivations of the rules of mixtures, and three full field micromechanics theories including finite element analysis of a representative volume element (RVE analysis), mathematical homogenization theory (MHT), and mechanics of structure genome (MSG). Theoretical connections among the three full field micromechanics theories are clearly shown. Particularly, it is shown that RVE analysis, MHT and MSG are governed by the same set of equations for 3D RVEs with periodic boundary conditions. RVE analysis and MSG can also handle aperiodic or partially periodic materials for which MHT is not applicable. MSG has the unique capability to obtain the complete set of 3D properties and local fields for heterogeneous materials featuring 1D or 2D heterogeneities.

NUMBER THEORY AND QUANTUM MECHANICS

1994 ◽  
Vol 09 (20) ◽  
pp. 1845-1851 ◽  
Author(s):  
DARWIN CHANG ◽  
PALASH B. PAL
Keyword(s):  
Number Theory ◽  
Quantum Mechanics ◽  
Angular Momentum ◽  
Classical Limit ◽  
Total Spin ◽  
Complete Set

We demonstrate a simple contradiction of naive deterministic theories with quantum mechanics by showing that the spin components cannot have deterministic values for many values of the total spin of a particle. Using a theorem in number theory, we work out the complete set of spin values which display such properties. The fact that the set is infinite should prompt some modifications in the usual statement about achieving the classical limit when the value of angular momentum becomes very large.

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