Mitsunori Noguchi

Yang–Mills–Higgs theory on a compact Riemann surface

Jaffe and Taubes [Vortices and Monopoles (Birkhauser, Boston, 1980)] have shown the existence and uniqueness of n‐vortex solutions on the complex plane. In this paper, their results are generalized to an arbitrary U(1) bundle over a compact Riemann surface with a Hermitian metric. Berger’s ‘‘nonlinear analysis’’ [Nonlinearity and Functional Analysis (Academic, New York, 1977)] has provided an effective method to prove the existence part of the main theorem of this paper.

A fuzzy core equivalence theorem

Abstract We attempt to establish the equivalence of the fuzzy core, the Edgeworth core, and the set of Walrasian equilibria in economies with a measure space of agents, which is not necessarily non-atomic. The commodity spaces which we consider here include ordered separable Banach spaces whose positive cone admit an interior point, separable Banach lattices whose positive cone may lack an interior point, and L ∞ endowed with the Mackey topology. Preference relations are assumed to be convex except for the non-atomic case.

Economies with a continuum of consumers, a continuum of suppliers and an infinite dimensional commodity space

Abstract The aim of this paper is to prove the existence of a competitive equilibrium for an economy with a measure space of consumers, a measure space of suppliers, an infinite dimensional commodity space, and interdependent preferences without order and convexity.

Geometry of statistical manifolds

Abstract A statistical manifold ( M , g , ▿) is a Riemannian manifold ( M , g ) equipped with torsion-free affine connections ▿, ▿ ∗ which are dual with respect to g . A point p \te M is said to be ▿-isotropiv if the sectional curvatures have the same value k ( p ), and ( M , g , ▿) is said to be ▿-isotropic when M consists entirely of ▿-isotropic points. When the difference tensor α of ▿ and the Levi-Civita connection ▿ 0 of g is “apolar” with respect to g , Kurose has shown that α ≡ 0, and hence ▿ = ▿ ∗ = ▿ 0 , provided that k ( p ) = k (constant). His proof relies on the existence of affine immersion which may no longer hold when k ( p ) is not constant. One objective of this paper is to show that the above Kuroses result still remains valid when ( M , g , ▿) is assumed only to be ▿-isotropic. We also discuss the case where ( M , g ) is complete Riemannian.

Economies with a continuum of agents with the commodity-price pairing (l∞, l1)

Abstract Recently, the author has established an equilibrium existence theorem for economies with a measure space of consumers, a measure space of producers, an infinite-dimensional separable commodity space whose positive cone admits an interior point, and interdependent preferences without order and convexity. The aim of this paper is to establish similar results for the non-separable commodity space l ∞ .

Economies with a measure space of agents and a separable commodity space

We prove the existence of an equilibrium in an economy with a measure space of agents and a separable Banach commodity space whose positive cone admits an interior point. We follow the truncation argument given at the end of Yannelis (Yannelis, N.C., 1987. Equilibria in non- cooperative models of competition, J. Econ. Theory 41, 96-111) and the abstract economy approach as in Shafer (Shafer, W., 1976. Equilibrium in economies without ordered preferences or free disposal, J. Math. Econ. 3, 135-137) and Khan and Vohra (Khan, M.A., Vohra, R., 1984. Equilibrium in abstract economies without ordered preferences and with a measure space of agents, J. Math. Econ. 13, 133-142), which allows preferences to be interdependent. Our result may be viewed as an extension of the result in Kahn and Yannelis (Khan, M.A., Yannelis, N.C., 1991. Equilibria in markets with a continuum of agents and commodities. In: Khan, M.A., Yannelis, N.C. (Eds.), Equilibrium Theory in Infinite Dimensional Spaces, Springer-Verlag, Tokyo, pp. 233-248) employing production and allowing preferences to be interdependent. We utilize Mazurs lemma at the crucial point in the truncation argument. We assume that the preference correspondence is representable by an interdependent utility function. The method in the present paper does not rely on the usual weak openness assumption on the lower sections of the preference correspondence.

Invariant Fisher information

Abstract Let t = t(p,w) : M ×: Ω → R k be a family of R k-valued random variables parametrized by a n-dimensional manifold M, where x = (x1,…,xn) is a chart around p ∈ M, Ω a probability space and ω ∈ Ω . Assuming t possesses a density ϱp, t), the Fisher information associated with t is defined to be g(p) = ep[(∂il)(∂jl)]dxi ⊗ dxj where ep is the expectation with respect to ϱ(p, t), l(p, t) = log ϱ(p, t) and ∂ i = ∂ ∂x i . g(p) is, by definition, invariant under a change of parameters x ↦ x′ and also a change of random variables of the form t = t(t′). However, it may not be invariant under a general change of random variables t = t(p, t′). The aim of this paper is to construct information ginv(p) which is invariant under a general change of both parameters and random variables. We can, in the end, express the difference ginv(p) − g(p) in terms of two types of connections which are purely geometrical objects. If we further impose a certain “linearity” on our construction, we can express ginv(p) − g(p) in terms of a single linear connection on a vector bundle so that the vanishing of the curvature would insure the existence ofa “special” t in which ginv(p) = g(p) holds.

Mean consumption representation of consumption externalities

Abstract The author previously investigated exchange economies with a price-dependent reference coalition externality which can be represented as a mean consumption bundle of a reference coalition C [Noguchi, M., 2005. Interdependent preferences with a continuum of agents. Journal of Mathematical Economics 41, 665–686]. In the present paper, we justify, under a suitable generalization of the notion of negatively interdependent preferences [Ok, E.A., Kockesen, L., 2000. Negatively interdependent preferences. Social Choice and Welfare 17, 533–558], that consumers act as if they compute the mean consumption bundle ∫ C xdμ C of C and maximize an appropriate utility function V whose consumption externality depends only on ∫ C xdμ C . The strong law of large numbers plays a crucial role in our arguments.

Economics of information

The economics of information covers a wide range of topics such as insurance, stochastic equilibria, the theory of finance (e.g. option pricing), job search, etc. In this paper, we focus on an economic model in which traders are uncertain about the true characteristics of commodities and know only the probability distributions of those characteristics. The traders acquire information on those characteristics via the actual consumption in the past and are allowed to exchange the information among themselves prior to the forthcoming trade. Though optimal consumption at the preceding trade generally alters optimal consumption at the succeeding trade, it may happen that they both coincide. We call this particular type of optimal consumption an information stable equilibrium (ISE). At an ISE, the traders gain no additional information from consumption, which is significant enough to revise their optimal choice at the succeeding trade.