scholarly journals New applications of Diophantine approximations to Diophantine equations.

1976 ◽  
Vol 39 ◽  
pp. 5 ◽  
Author(s):  
T. N. Shorey ◽  
R. Tijdeman
Keyword(s):  
Diophantine Equations ◽  
Diophantine Approximations ◽  
New Applications

Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken

1989 ◽  
Vol 54 (1) ◽  
pp. 234-263 ◽  
Author(s):  
H. Luckhardt
Keyword(s):  
Continued Fraction ◽  
Diophantine Equations ◽  
Growth Conditions ◽  
Number Of Solutions ◽  
Exponential Bound ◽  
Low Degree ◽  
Diophantine Approximations ◽  
Finiteness Theorems ◽  
Fourth Root ◽  
Simple Growth

AbstractA previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs ofΣ2-finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order exp(70ε−2d2) in place of exp(285ε−2d2) given by Davenport and Roth in 1955, whereαis (real) algebraic of degreed≥ 2 and ∣α−pq−1∣ <q−2−ε. (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof arepolynomial of low degree(inε−1and logd). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), log logqν, <Cα, εν5/6+ε, whereqνare the denominators of the convergents to the continued fraction ofα.

The Gross-Zagier Formula on Shimura Curves

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-wu Zhang ◽  
Wei Zhang
Keyword(s):  
Diophantine Equations ◽  
Abelian Varieties ◽  
Shimura Curves ◽  
Heegner Points ◽  
Quaternion Algebras ◽  
Complete Proof ◽  
Weil Representations ◽  
Arakelov Theory ◽  
Generating Series ◽  
New Applications

This comprehensive account of the Gross–Zagier formula on Shimura curves over totally real fields relates the heights of Heegner points on abelian varieties to the derivatives of L-series. The formula will have new applications for the Birch and Swinnerton-Dyer conjecture and Diophantine equations. The book begins with a conceptual formulation of the Gross–Zagier formula in terms of incoherent quaternion algebras and incoherent automorphic representations with rational coefficients attached naturally to abelian varieties parametrized by Shimura curves. This is followed by a complete proof of its coherent analogue: the Waldspurger formula, which relates the periods of integrals and the special values of L-series by means of Weil representations. The Gross–Zagier formula is then reformulated in terms of incoherent Weil representations and Kudla's generating series. Using Arakelov theory and the modularity of Kudla's generating series, the proof of the Gross–Zagier formula is reduced to local formulas. This book will be of great use to students wishing to enter this area and to those already working in it.

1250-kilovolt scanning transmission electron microscope

1984 ◽  
Vol 42 ◽  
pp. 624-625
Author(s):  
T. Imura ◽  
S. Maruse ◽  
K. Mihama ◽  
M. Iseki ◽  
M. Hibino ◽  
...  
Keyword(s):  
High Voltage ◽  
Electron Probe ◽  
Scanning Transmission Electron Microscope ◽  
Pressure Vessels ◽  
Practical Applications ◽  
Voltage Generator ◽  
Transmission Electron ◽  
Scanning Transmission ◽  
New Applications ◽  
High Voltage Generator

Ultra high voltage STEM has many inherent technical advantages over CTEM. These advantages include better signal detectability and signal processing capability. It is hoped that it will explore some new applications which were previously not possible. Conventional STEM (including CTEM with STEM attachment), however, has been unable to provide these inherent advantages due to insufficient performance and engineering problems. Recently we have developed a new 1250 kV STEM and completed installation at Nagoya University in Japan. It has been designed to break through conventional engineering limitations and bring about theoretical advantage in practical applications.In the design of this instrument, we exercised maximum care in providing a stable electron probe. A high voltage generator and an accelerator are housed in two separate pressure vessels and they are connected with a high voltage resistor cable.(Fig. 1) This design minimized induction generated from the high voltage generator, which is a high frequency Cockcroft-Walton type, being transmitted to the electron probe.

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