Algebro-geometric approach to the Yang-Baxter equation and related topics

Vladimir Dragovic

doi:10.2298/pim1205025d

Algebro-geometric approach to the Yang-Baxter equation and related topics

Publications de l Institut Mathematique ◽

10.2298/pim1205025d ◽

2012 ◽

Vol 91 (105) ◽

pp. 25-48 ◽

Cited By ~ 2

Author(s):

Vladimir Dragovic

Keyword(s):

Computational Experiment ◽

Geometric Approach ◽

Mathematical Physics ◽

Baxter Equation ◽

Geometric Properties ◽

Reflection Theorem ◽

Pencils Of Conics

We review the results of algebro-geometric approach to 4 ? 4 solutions of the Yang-Baxter equation. We emphasis some further geometric properties, connected with the double-reflection theorem, the Poncelet porism and the Euler-Chasles correspondence. We present a list of classifications in Mathematical Physics with a similar geometric background, related to pencils of conics. In the conclusion, we introduce a notion of discriminantly factorizable polynomials as a result of a computational experiment with elementary n-valued groups.

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New Investigation on the Geometry of the Contact in Gear Generation

Volume 4: 9th International Power Transmission and Gearing Conference, Parts A and B ◽

10.1115/detc2003/ptg-48087 ◽

2003 ◽

Author(s):

Marco Gabiccini ◽

Massimo Guiggiani ◽

Francesca Di Puccio

Keyword(s):

Reference Frame ◽

Relative Position ◽

Geometric Approach ◽

Gear Ratio ◽

Reference Systems ◽

Geometric Properties ◽

Envelope Surface ◽

Enveloping Surface ◽

Contact Matrix

Based on a recently developed geometric approach to the theory of gearing that does not make use of any reference systems [1], this paper presents some useful relations between the geometric properties of the enveloping surface and those of its envelope. Treating vectors as such, that is without expressing their components in any reference systems, it is possible to obtain compact expressions for the coefficients of the first and second fundamental forms of the envelope surface. These coefficients show to be central in the determination of the contact matrix between mating surfaces. Moreover, since this approach is coordinate free, it is valid regardless of the reference frame actually employed to perform calculations and allows a, hopefully, clearer understanding of the roles played by the intrinsic geometric properties of the enveloping surface, the relative position of the gear axes and the gear ratio.

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CALCULATION OF ANOMALIES IN GAUGE THEORIES USING A GEOMETRIC APPROACH

International Journal of Modern Physics E ◽

10.1142/s0218301304002417 ◽

2004 ◽

Vol 13 (04) ◽

pp. 801-810 ◽

Cited By ~ 1

Author(s):

PAUL BRACKEN

Keyword(s):

Gauge Theory ◽

Gauge Theories ◽

Geometric Approach ◽

Geometric Properties ◽

Elementary Derivation ◽

Geometric Methods

Gauge theory and anomalies which arise under quantization of these theories are formulated using differential geometric methods. The Wess–Zumino conditions are developed and some other relevant geometric properties of anomalies are mentioned. An elementary derivation of the non-Abelian anomaly in 2n dimensions is presented using this formalism.

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