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A method of the Riemann–Hilbert problem is applied for Zhang’s conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang’s conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang’s conjecture 1 (Main Theorem). The present work not only proves the Zhang’s conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry.

We work on special divisor classes on blow-ups F p , r of Hirzebruch surfaces over the field of complex numbers, and extend fundamental properties of special divisor classes on del Pezzo surfaces parallel to analogous ones on surfaces F p , r . We also consider special divisor classes on surfaces F p , r with respect to monoidal transformations and explain the tie-ups among them contrast to the special divisor classes on del Pezzo surfaces. In particular, the fundamental properties of quartic rational divisor classes on surfaces F p , r are studied, and we obtain interwinded relationships among rulings, exceptional systems and quartic rational divisor classes along with monoidal transformations. We also obtain the effectiveness for the rational divisor classes on F p , r with positivity condition.

Abstract: One of the feature of the integrable systems is that all forward things are determined by its initial conditions. It is also well known that the Toda lattice has poles in finite time. Thus the behavior of the blowing up of the Toda lattice have to be governed by its initial conditions. Through the Painlevé analysis it is revealed that the blowing up themselves are characterized by the Weyl group [Flaschka and Haine]. In this paper we study how the behavior at the blowing up point is governed by its initial condition. For this purpose we realize the Painlevé divisor as the analytic variety. By virtue of this description we can compactify level set by using monoidal transformation by Painlevé divisor. The study of Painlevé divisor and compactification in this paper bring us the more precise informations on the blowing up of the Toda lattice than which has ever been obtained. [Copyright &y& Elsevier]

A method of the Riemann–Hilbert problem is applied for Zhang's conjecture 1 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in the zero external field and the solution to the Zhang's conjecture 1 is constructed by use of the monoidal transform. At first, the knot structure of the ferromagnetic 3D Ising model in the zero external field is determined and the non-local behavior of the ferromagnetic 3D Ising model can be described by the non-trivial knot structure. A representation from the knot space to the Clifford algebra of exponential type is constructed, and the partition function of the ferromagnetic 3D Ising model in the zero external field can be obtained by this representation (Theorem I). After a realization of the knots on a Riemann surface of hyperelliptic type, the monodromy representation is realized from the representation. The Riemann–Hilbert problem is formulated and the solution is obtained (Theorem II). Finally, the monoidal transformation is introduced for the solution and the trivialization of the representation is constructed (Theorem III). By this, we can obtain the desired solution to the Zhang's conjecture 1 (Main Theorem). The present work not only proves the Zhang's conjecture 1, but also shows that the 3D Ising model is a good platform for studying in deep the mathematical structure of a physical many-body interacting spin system and the connections between algebra, topology, and geometry. [ABSTRACT FROM AUTHOR]

We work on special divisor classes on blow-ups F p , r of Hirzebruch surfaces over the field of complex numbers, and extend fundamental properties of special divisor classes on del Pezzo surfaces parallel to analogous ones on surfaces F p , r . We also consider special divisor classes on surfaces F p , r with respect to monoidal transformations and explain the tie-ups among them contrast to the special divisor classes on del Pezzo surfaces. In particular, the fundamental properties of quartic rational divisor classes on surfaces F p , r are studied, and we obtain interwinded relationships among rulings, exceptional systems and quartic rational divisor classes along with monoidal transformations. We also obtain the effectiveness for the rational divisor classes on F p , r with positivity condition. [ABSTRACT FROM AUTHOR]