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This is a continuation of the recent work [36] by one of the authors. According to [36], there are four 4-dimensional Painlevé-type equations derived from isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painlevé system. In this paper we degenerate these four source equations, and systematically obtain other 4-dimensional Painlevé-type equations. If we only consider Painlevé-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painlevé-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equations. Some well-known equations such as Noumi-Yamada systems are included in this list. They are written as Hamiltonian systems, and their Hamiltonians are simply written by using Hamiltonians of the classical Painlevé equations.We denote the field of formal Laurent series in z by C((z)), and the field of Puiseux series ∪ p>0 C((z 1 p )) by K z .Theorem 1.2 (Hukuhara, Turrittin, Levelt). For anysuch that the transformation of dependent variable Y = P (z)Z brings the system into the following canonical form:• Θ is a (not necessarily diagonal) Jordan matrix which commutes with all D j 's.Here l 0 , . . . , l s are uniquely determined only by the original system (1.5).If the equationis another canonical form corresponding to the same system, there exist a constant matrix g ∈ GL m (C) and a natural number k ∈ Z ≥1 such thatWe call the number l 0 − 1 the Poincaré rank of the singular point. When there is a rational number l j that is not an integer, the singular point is called a ramified irregular singular point. A linear differential equation is said to be of ramified type if it has a ramified irregular singular point. In the present paper, as we have mentioned, we consider only linear equations of unramified type, namely, we treat linear equations which do not have ramified irregular singular points.At an unramified (irregular) singular point, the canonical form can be written in the formis a fundamental solution matrix of (1.9). The degree of the polynomial in the exponential function with respect to z −1 is the Poincaré rank, which we have defined above.In the rest of this subsection, we briefly outline how the canonical form at an irregular singular point is computed, and how the canonical form is described by the refining sequence of partitions.We first consider the case r = 0 in (1.4), namely when the order of the pole is 1. The case of the simple poleWe start with the following Lemma concerning so-called the Sylvester equation. We will use this Lemma very effectively to solve differential equations formally. Lemma 1.3. Let A ∈ M m (C), B ∈ M n (C), and C ∈ M m,n (C). The matrix equation (1.12) AX − XB = C with respect to m × n matrix X has a solution for any C if and only if {eigenvalue of A} ∩ {eigenvalue of B} = ∅. Proof. First, we prove the following: AX − XB = O has a non-trivial solution ⇐⇒ {eigenvalue of A}...

We present a special solutions of the discrete Painlevé equations associated with A 0 -surface discrete Painlevé equation is the most generic difference equation, as all discrete Painlevé equations can be obtained by its degeneration limit. These special solutions exist when the parameters of the discrete Painlevé equation satisfy a particular constraint. We consider that these special functions belong to the hypergeometric family although they seems to go beyond the known discrete and q-discrete hypergeometric functions. We also discuss the degeneration scheme of these solutions.

In order to decipher the origin and tectonothermal history of the Kuncha nappe, we undertook a geological investigation in the Taplejung window in eastern Nepal, and carried out multichronological analyses of zircon, apatite, and mica of the Kuncha Formation and Taplejung granites. Three granite bodies that intrude into the Kuncha Formation show fission-track (FT) ages of 6.2 to 4.8 Ma for zircon and 2.9 to 2.

A q-di¤erence analog of the Garnier system is presented. It arises as the condition for preserving the connection matrix of linear q-di¤erence equations, in close analogy with the monodromy preserving deformation of linear di¤erential equations. The continuous limit is also studied.

The Capitanian (Late Guadalupian) Maokou Formation at Chaotian in northern Sichuan, South China, is composed mainly of shallow marine shelf carbonates deposited on the Tethyan side of South China. By detailed field mapping and scientific drilling, we newly found out unique fossil assemblages and a sharp lithologic change in the upper part of the Maokou Formation. The main part of the Maokou Formation (over 130 m thick) is composed of algal packstone with Wordian-Capitanian large-tested fusulines, rugose corals and other sessile benthos, whereas the Uppermost Member (13 m thick) is composed of black limy mudstone/chert with Capitanian offshore biota (ammonoids, radiolarians, and conodonts). The topmost Capitanian conodont zones are missing; however, the Maokou Formation is disconformably overlain by 260 AE 4 Ma volcanic ash (Wangpo bed) and the Early Lopingian Wujiaping Formation with plant-bearing coaly mudstone and shallow marine carbonates (packstone). The newly identified facies change indicates that northern Sichuan has experienced rapid sea-level changes in the late Guadalupian, i.e., first a transgression in the mid-Capitanian and then a regression across the Guadalupian-Lopingian boundary. As the end-Guadalupian is characterized by a global regression, such a volatile sea-level fluctuation, in particular the sea-level rise, is unique to the Tethyan side of South China. The newly recognized relatively deep-water late Guadalupian sequence adds new paleo-environmental information and further provides a paleotectonic interpretation of the low-latitude eastern Tethyan margin immediately before the end-Guadalupian mass extinction.