A universal identity for theta functions of degree eight and applications

International audience Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.

Upgrading Die Attach Epoxy Dispensing Mechanism

This research paper addressed how to improve the Die Attach Pneumatic Time-Pressure Dispensing Valve. Different assessment and statistical validation comparing the efficiency of the Pneumatic Time-Pressure Dispensing Valve and Dispensing Volume Reduction. The Musashi Super Ʃ CMIII Dispenser with Sigma function controller provides more choices for regulating the volume of dispensing, since the residual material in the syringe decreases, and can be a more suitable method for dispensing glue. The consistency of the glue volume will be more consistent after the implementation of this update. This controller has the option to compensate for pressure and vacuum as the remaining fluid in the syringe decreases. The Musashi dispenser controller is equipped with an empty syringe detection system. As the remaining adhesive, this will avoid variations in dispensed volume.

A Hirota bilinear equation for Painlevé transcendents PIV, PII and PI

We present some observations on the tau-function for the fourth Painlevé equation. By considering a Hirota bilinear equation of order four for this tau-function, we describe the general form of the Taylor expansion around an arbitrary movable zero. The corresponding Taylor series for the tau-functions of the first and second Painlevé equations, as well as that for the Weierstrass sigma function, arise naturally as special cases, by setting certain parameters to zero.

ON DEGENERATE SIGMA-FUNCTIONS IN GENUS 2

We obtain explicit expressions for genus 2 degenerate sigma-function in terms of genus 1 sigma-function and elementary functions as solutions of a system of linear partial differential equations satisfied by the sigma-function. By way of application, we derive a solution for a class of generalized Jacobi inversion problems on elliptic curves, a family of Schrödinger-type operators on a line with common spectrum consisting of a point and two segments, explicit construction of a field of three-periodic meromorphic functions. Generators of rank 3 lattice in ℂ2are given explicitly.

Arithmetical Power Series Expansion of the Sigma Function for a Plane Curve

The Weierstrass function σ(u) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an (n,s)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 curve ((3, 4)-curve)y3+ (μ1x+μ4)y2+ (μ2x2+μ5x+μ8)y=x4+μ3x3+μ6x2+μ9x+μ12(whereμjare constants) are given.