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Tomohiro Uchiyama

Publications and source records attributed to Tomohiro Uchiyama.

13 recordsLinked to original sources
arXiv2024

Keynesian chaos revisited: odd period cycles and ergodic properties

In this paper, we study two standard (Keynesian) dynamic macroeconomic models (one is piecewise linear and the other is nonlinear). Our purpose is twofold: (1)~For each model, we give a complete characterisation for the existence of a topological chaos (of the GDP levels), (2)~Even if a chaos exists, using ergodic theory, we show that it is possible to predict the future GDP levels "on average". This paper gives a new application of a celebrated result in ergodic theory by A. Avila (2014 fields medalist). We believe that our method/strategy in this paper is generic enough to be used to analyse many other (seemingly untractable) chaotic economic models.

econ.GN↗
arXiv2024

Odd period cycles and ergodic properties in price dynamics for an exchange economy

In the first part of this paper (Sections 1-4), we study a standard exchange economy model with Cobb-Douglas type consumers and give a necessary and sufficient condition for the existence of an odd period cycle in the Walras-Samuelson (tatonnement) price adjustment process. We also give a sufficient condition for a price to be eventually attracted to a chaotic region. In the second part (Sections 5 and 6), we investigate ergodic properties of the price dynamics showing that the existence of chaos is not necessarily bad. (The future is still predictable on average.) Moreover, supported by a celebrated work of Avila et al. (Invent. Math., 2003), we conduct a sensitivity analysis to investigate a relationship between the ergodic sum (of prices) and the speed of price adjustment. We believe that our methods in this paper can be used to analyse many other chaotic economic models.

econ.GN↗
arXiv2023

A necessary and sufficient condition for the existence of chaotic dynamics in a neoclassical growth model with a pollution effect

In this paper, we study a neoclassical growth model with a (productivity inhibiting) pollution effect. In particular, we obtain a necessary and sufficient condition for the existence of a topological chaos. We investigate how the condition changes as the strength of the pollution effect changes. This is a new application of a recent result characterising the existence of a topological chaos for a unimodal interval map by Deng, Khan, Mitra (2022).

econ.GN↗
arXiv2023

A necessary and sufficient condition for the existence of chaotic dynamics in an overlapping generations model

In this paper, we study economic dynamics in a standard overlapping generations model without production. In particular, using numerical methods, we obtain a necessary and sufficient condition for the existence of a topological chaos. This is a new application of a recent result characterising the existence of a topological chaos for a unimodal interval map by Deng, Khan, Mitra (2022).

econ.GN↗
arXiv2021

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields IV: An $F_4$ example

Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In particular, we present the first example of a connected nonabelian $k$-subgroup $H$ of $G$ that is $G$-completely reducible but not $G$-completely reducible over $k$, and the first example of a connected nonabelian $k$-subgroup $H'$ of $G$ that is $G$-completely reducible over $k$ but not $G$-completely reducible. This is new: all previously known such examples are for finite (or non-connected) $H$ and $H'$ only.

math.GR↗
arXiv2019

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 3

Let $k$ be a nonperfect separably closed field. Let $G$ be a (possibly non-connected) reductive group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In our previous work, we constructed examples of subgroups $H$ of $G$ that are $G$-completely reducible but not $G$-completely reducible over $k$ (and vice versa). In this paper, we give a theoretical underpinning of those constructions. To illustrate our result, we present a new such example in a non-connected reductive group of type $D_4$ in characteristic $2$. Then using Geometric Invariant Theory, we generalize the theoretical result above obtaining a new result on the structure of $G(k)$-(and $G$-) orbits in an arbitrary affine $G$-variety. We translate our result into the language of spherical buildings to give a topological viewpoint. A problem on centralizers of completely reducible subgroups and a problem concerning the number of conjugacy classes are also considered.

math.GR↗
arXiv2017

Complete reducibility, Kulshammer's question, conjugacy classes: a D_4 example

Let $k$ be a nonperfect separably closed field. Let $G$ be a connected reductive algebraic group defined over $k$. We study rationality problems for Serre's notion of complete reducibility of subgroups of $G$. In particular, we present a new example of subgroup $H$ of $G$ of type $D_4$ in characteristic $2$ such that $H$ is $G$-completely reducible but not $G$-completely reducible over $k$ (or vice versa). This is new: all known such examples are for $G$ of exceptional type. We also find a new counterexample for Külshammer's question on representations of finite groups for $G$ of type $D_4$. A problem concerning the number of conjugacy classes is also considered. The notion of nonseparable subgroups plays a crucial role in all our constructions.

math.GR↗
arXiv2017

Non-separability and complete reducibility: $E_n$ examples with an application to a question of Külshammer

Let $G$ be a simple algebraic group of type $E_n (n=6,7,8)$ defined over an algebraically closed field $k$ of characteristic $2$. We present examples of triples of closed reductive groups $H<M<G$ such that $H$ is $G$-completely reducible, but not $M$-completely reducible. As an application, we consider a question of Külshammer on representations of finite groups in reductive groups. We also consider a rationality problem for $G$-complete reducibility and a problem concerning conjugacy classes.

math.GR↗
arXiv2017

Separability and complete reducibility of subgroups of the Weyl group of a simple algebraic group of type $E_7$

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $k$. The aim of this paper is to present a method to find triples $(G,M,H)$ with the following three properties. Property 1: $G$ is simple and $k$ has characteristic 2. Property 2: $H$ and $M$ are closed reductive subgroups of $G$ such that $H<M<G$, and $(G,M)$ is a reductive pair. Property 3: $H$ is $G$-completely reducible, but not $M$-completely reducible. We exhibit our method by presenting a new example of such a triple in $G=E_7$. Then we consider a rationality problem and a problem concerning conjugacy classes as important applications of our construction.

math.GR↗
arXiv2017

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 1

Let $k$ be a nonperfect field of characteristic $2$. Let $G$ be a $k$-split simple algebraic group of type $E_6$ (or $G_2$) defined over $k$. In this paper, we present the first examples of nonabelian non-$G$-completely reducible $k$-subgroups of $G$ which are $G$-completely reducible over $k$. Our construction is based on that of subgroups of $G$ acting non-separably on the unipotent radical of a proper parabolic subgroup of $G$ in our previous work. We also present examples with the same property for a non-connected reductive group $G$. Along the way, several general results concerning complete reducibility over nonperfect fields are proved using the recently proved Tits center conjecture for spherical buildings. In particular, we show that under mild conditions a $k$-subgroup of $G$ is pseudo-reductive if it is $G$-completely reducible over $k$.

math.GR↗
arXiv2017

A geometric approach to the transfer problem for a finite number of traders

We present a complete characterization of the classical transfer problem for an exchange economy with an arbitrary finite number of traders. Our method is geometric, using an equilibrium manifold developed by Debreu, Mas-Colell, and Balasko. We show that for a regular equilibrium the transfer problem arises if and only if the index at the equilibrium is $-1$. This implies that the transfer problem does not happen if the equilibrium is Walras tatonnement stable. Our result generalizes Balasko's analogous result for an exchange economy with two traders.

econ.GN↗
arXiv2017

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields 2

Let k be a separably closed field. Let G be a reductive algebraic k-group. In this paper, we study Serre's notion of complete reducibility of subgroups of G over k. In particular, using the recently proved center conjecture of Tits, we show that the centralizer of a k-subgroup H of G is G-completely reducible over k if it is reductive and H is G-completely reducible over k. We also show that a regular reductive k-subgroup of G is G-completely reducible over k. Various open problems concerning complete reducibility are discussed. We present examples where the number of overgroups of irreducible subgroups and the number of G(k)-conjugacy classes of k-anisotropic unipotent elements are infinite.

math.GR↗