scholarly journals Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary

2001 ◽  
pp. 39-110 ◽  
Author(s):  
Bruce C. Berndt ◽  
Ken Ono
Author(s):  
Jens Meierhenrich

This chapter turns to the gestation of the first, German-language manuscript of The Dual State, known as the Urdoppelstaat of 1938. I then chart the transformation of this unpublished manuscript into the 1941 book. To lay the foundation for this detailed reconstruction, I trace in some depth the gradual destruction of the German Rechtsstaat, presenting in an accessible manner several decades worth of material culled from the historiography of Nazi law. This illustrates the enormity—and danger—of the task that Fraenkel set himself: to serve as a participant observer in the courts of the “Third Reich.” Drawing on a series of primary documents, I piece together the incredible and untold story of the gestation of The Dual State, a tale of rare courage, acumen, and insight. I pay detailed attention to similarities and differences in recently discovered manuscript drafts.


2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Boris Dubrovin ◽  
Di Yang ◽  
Don Zagier
Keyword(s):  

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.


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