Searcharxiv⌕ Search

arXiv subjects

Tomoyuki Takenawa

Publications and source records attributed to Tomoyuki Takenawa.

At least 19 recordsLinked to original sources
arXiv2026

Degree growth, orbit graphs, and functoriality for birational dynamical systems

The purpose of this paper is to give a natural divisor-theoretic formulation of the counting method introduced by Halburd for computing degree growth, in a form applicable to birational dynamical systems on varieties of arbitrary dimension. Instead of counting only preimages of special values, we follow time-indexed divisorial conditions through singularity patterns. These conditions are recorded on normalized finite-window orbit graphs, where the relevant multiplicities are realized as divisorial valuations of pullbacks of time-indexed divisors. This construction explains how the elementary computations appearing in singularity patterns can be interpreted as degree relations on a single normal variety. We then show that further relations arise from the failure of functoriality of pullbacks: when the center of a divisor enters the relevant indeterminacy locus, a degree-drop divisor appears. Under suitable finite-type assumptions, the two kinds of relations lead to closed linear difference systems governing degree sequences. Several examples, including higher-dimensional ones, demonstrate that the two mechanisms are complementary and that their combination determines the degree growth in cases where either mechanism alone is insufficient.

math.DS↗
arXiv2026

Improvements to the post-processing of weather forecasts using machine learning and feature selection

This study aims to develop and improve machine learning-based post-processing models for precipitation, temperature, and wind speed predictions using the Mesoscale Model (MSM) dataset provided by the Japan Meteorological Agency (JMA) for 18 locations across Japan, including plains, mountainous regions, and islands. By incorporating meteorological variables from grid points surrounding the target locations as input features and applying feature selection based on correlation analysis, we found that, in our experimental setting, the LightGBM-based models achieved lower RMSE than the specific neural-network baselines tested in this study, including a reproduced CNN baseline, and also generally achieved lower RMSE than both the raw MSM forecasts and the JMA post-processing product, MSM Guidance (MSMG), across many locations and forecast lead times. Because precipitation has a highly skewed distribution with many zero cases, we additionally examined Tweedie-based loss functions and event-weighted training strategies for precipitation forecasting. These improved event-oriented performance relative to the original LightGBM model, especially at higher rainfall thresholds, although the gains were site dependent and overall performance remained slightly below MSMG.

physics.ao-ph↗
arXiv2025

On the geometry of a 4-dimensional extension of a $q$-Painlevé I equation with symmetry type $A_1^{(1)}$

We present a geometric study of a four-dimensional integrable discrete dynamical system which extends the autonomous form of a $q$-Painlevé I equation with symmetry of type $A_1^{(1)}$. By resolution of singularities it is lifted to a pseudo-automorphism of a rational variety obtained from $({\mathbb P}^1)^{\times 4}$ by blowing up along 28 subvarieties and we use this to establish its integrability in terms of conserved quantities and degree growth. We embed this rational variety into a family which admits an action of the extended affine Weyl group $\widetilde{W}(A_1^{(1)})\times \widetilde{W}(A_1^{(1)})$ by pseudo-isomorphisms. We use this to construct two 4-dimensional analogues of $q$-Painlevé equations, one of which is a deautonomisation of the original autonomous integrable map.

nlin.SI↗
arXiv2024

Space of initial conditions for the four-dimensional Garnier system revisited

A geometric study is given for the 4-dimensional Garnier system. By the resolution of indeterminacy, the group of its Bäklund transformations is lifted to a group of pseudo-isomorphisms between rational varieties obtained from ${\mathbb P}^2 \times {\mathbb P}^2$ by 10 or 21 blow-ups. The root basis is discussed in the Néron-Severi bilattices for the space with 10 blow-ups.

math.AG↗
arXiv2024

Interpolation of mountain weather forecasts by machine learning

Recent advances in numerical simulation methods based on physical models and their combination with machine learning have improved the accuracy of weather forecasts. However, the accuracy decreases in complex terrains such as mountainous regions because these methods usually use grids of several kilometers square and simple machine learning models. While deep learning has also made significant progress in recent years, its direct application is difficult to utilize the physical knowledge used in the simulation. This paper proposes a method that uses machine learning to interpolate future weather in mountainous regions using forecast data from surrounding plains and past observed data to improve weather forecasts in mountainous regions. We focus on mountainous regions in Japan and predict temperature and precipitation mainly using LightGBM as a machine learning model. Despite the use of a small dataset, through feature engineering and model tuning, our method partially achieves improvements in the RMSE with significantly less training time.

physics.ao-ph↗
arXiv2022

Solving the Quispel-Roberts-Thompson maps using Kajiwara-Noumi-Yamada's representation of elliptic curves

It is well known that the dynamical system determined by a Quispel-Roberts-Thompson map (a QRT map) preserves a pencil of biquadratic polynomial curves on ${\mathbb{CP}}^1 \times {\mathbb{CP}}^1$. In most cases this pencil is elliptic, i.e. its generic member is a smooth algebraic curve of genus one, and the system can be solved as a translation on the elliptic fiber to which the initial point belongs. However, this procedure is rather complicated to handle, especially in the normalization process. In this paper, for a given initial point on an invariant elliptic curve, we present a method to construct the solution directly in terms of the Weierstrass sigma function, using Kajiwara-Noumi-Yamada's parametric representation of elliptic curves.

math.DS↗
arXiv2020

Space of initial conditions for the four-dimensional Fuji-Suzuki-Tsuda system

A geometric study for an integrable 4-dimensional dynamical system so called the Fuji-Suzuki-Tsuda system is given. By the resolution of indeterminacy, the group of its Bäklund transformations is lifted to a group of pseudo-isomorphisms between rational varieties obtained from $(P^1)^4$ by blowing-up along eight 2-dimensional subvarieties and four 1-dimensional subvarieties. The root basis is realised in the Néron-Severi bilattices. A discrete Painlevé system with quadratic degree growth is also realised as its translational element.

nlin.SI↗
arXiv2019

An algebraically stable variety for a four-dimensional dynamical system reduced from the lattice super-KdV equation

In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by the traveling wave reduction from the lattice super-KdV equation in a case of finitely generated Grassmann algebra. The system is a coupling of a Quispel-Roberts-Thompson map and a linear map but does not satisfy the singularity confinement criterion. It was conjectured that the dynamical degree of this system grows quadratically. In this paper, constructing a rational variety where the system is lifted to an algebraically stable map and using the action of the map on the Picard lattice, we prove this conjecture. We also show that invariants can be found through the same technique.

math.DS↗
arXiv2018

Space of initial conditions and geometry of two 4-dimensional discrete Painlevé equations

A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they are lifted to pseudo-automorphisms of rational varieties obtained from $({\mathbb P}^1)^4$ by blowing-up along sixteen 2-dimensional subvarieties. The symmetry groups, the invariants and the degree growth rates are computed from the linearisation on the corresponding Néron-Severi bilattices. It turns out that the deautonomised version of one of the mappings is a Bäcklund transformation of a direct product of the fourth Painlevé equation which has $A_2^{(1)}+A_2^{(1)}$ type affine Weyl group symmetry, while that of the other mapping is of Noumi-Yamada's $A_5^{(1)}$ Painlevé equation.

math.DS↗
arXiv2018

On Some Applications of Sakai's Geometric Theory of Discrete Painlevé Equations

Although the theory of discrete Painlevé (dP) equations is rather young, more and more examples of such equations appear in interesting and important applications. Thus, it is essential to be able to recognize these equations, to be able to identify their type, and to see where they belong in the classification scheme. The definite classification scheme for dP equations was proposed by H. Sakai, who used geometric ideas to identify 22 different classes of these equations. However, in a major contrast with the theory of ordinary differential Painlevé equations, there are infinitely many non-equivalent discrete equations in each class. Thus, there is no general form for a dP equation in each class, although some nice canonical examples in each equation class are known. The main objective of this paper is to illustrate that, in addition to providing the classification scheme, the geometric ideas of Sakai give us a powerful tool to study dP equations. We consider a very complicated example of a dP equation that describes a simple Schlesinger transformation of a Fuchsian system and we show how this equation can be identified with a much simpler canonical example of the dP equation of the same type and moreover, we give an explicit change of coordinates transforming one equation into the other. Among our main tools are the birational representation of the affine Weyl symmetry group of the equation and the period map. Even though we focus on a concrete example, the techniques that we use are general and can be easily adapted to other examples.

math-ph↗
arXiv2017

Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painlevé equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. %By choosing a particular Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painlevé equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painlevé equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painlevé equations, including examples whose symmetry groups do not appear explicitly in Sakai's classification.

math.DS↗
arXiv2014

Geometric Analysis of Reductions from Schlesinger Transformations to Difference Painlevé Equations

We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painlevé equations: difference Painlevé equation d-$P\left({A}_{2}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painlevé equation d-$P\left({A}_{1}^{(1)*}\right)$ with the symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coordinate system describing the equation, and how to compare it with other instances of equations of the same type.

math-ph↗
arXiv2014

Discrete Schlesinger Transformations, their Hamiltonian Formulation, and Difference Painlevé Equations

Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlevé equations; this is also the main theme behind our work. We derive \emph{discrete Schlesinger evolution equations} describing discrete dynamical systems generated by elementary Schlesinger transformations and give their discrete Hamiltonian description w.r.t.~the standard symplectic structure on the space of Fuchsian systems. As an application, we compute explicitly two examples of reduction from Schlesinger transformations to difference Painlevé equations. The first example, d-$P\big(D_{4}^{(1)}\big)$ (or difference Painlevé V), corresponds to Bäcklund transformations for continuous $P_{\text{VI}}$. The second example, d-$P\big(A_{2}^{(1)*}\big)$ (with the symmetry group $E_{6}^{(1)}$), is purely discrete. We also describe the role played by the geometry of the Okamoto space of initial conditions in comparing different equations of the same type.

math-ph↗
arXiv2013

A note on minimization of rational surfaces obtained from birational dynamical systems

In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable mappings, including discrete versions of reduced Nahm equations.

math.DS↗
arXiv2009

Real ideal and the duality of semidefinite programming for polynomial optimization

We study the ideal generated by polynomials vanishing on a semialgebraic set and propose an algorithm to calculate the generators, which is based on some techniques of the cylindrical algebraic decomposition. By applying these, polynomial optimization problems with polynomial equality constraints can be modified equivalently so that the associated semidefinite programming relaxation problems have no duality gap. Elementary proofs for some criteria on reality of ideals are also given.

math.OC↗
arXiv2008

Tropical representation of Weyl groups associated with certain rational varieties

Starting from certain rational varieties blown-up from (P^1)^N, we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. Furthermore, we develop an algebro-geometric framework of tau-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields a class of (higher order) q-difference Painleve equations and its algebraic degree grows quadratically.

math.AG↗