Abstract:
Growths of numbers of tourists who stay long-term in tourism sites are an important policy for local governments in islands due to their locations that are far from urban areas. However, many tourists hesitate to stay long-term in islands owing to not only the expensive travel costs but also the lack of public services in islands. The purpose of this study is to examine effects of improving public services in islands for tourists’ willingness to pays (WTPs) and non-tourists’ attitudes for long-term stays. Data on tourism activity for islands, Nagasaki, Japan were used. Respondents were asked about their WTPs for long-term stays and their needs for public services of islands; reductions of costs for rent or purchasing houses for long-term stays and travel costs, easy to take a vacation, to enhance medical services, educational services and job search services. The logit model was used for estimations. Median and mean values of WTPs (per year) were calculated JPY 151,629 (USD 1184) and JPY 242,110 (USD 3008). Positive effects on five public services (without travel costs) were confirmed. For example, the median values of WTPs were increased to JPY 478,369 (USD 5943) when the medical services were improved, and JPY 1,484,704 (USD 18,446) when all public services were improved. The results showed that improvement of public services have the effect 1) to improve tourists’ benefits and 2) to change many non-tourists’ attitudes from the rejection of staying long term in islands to the acceptance. Thus, results indicate that it would be better for central and/or local governments in islands to enhance islands’ public services.

Abstract:
The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast-slow type having Bogdanov-Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow-up method. In particular, the blow-up method is effectively used for analyzing the flow near the Bogdanov-Takens type fold point in order to show that a slow manifold near the fold point is extended along the Boutroux's tritronqu\'{e}e solution of the first Painlev\'{e} equation in the blow-up space.

Abstract:
In the case of monotone independence, the transparent understanding of the mechanism to validate the central limit theorem (CLT) has been lacking, in sharp contrast to commutative, free and Boolean cases. We have succeeded in clarifying it by making use of simple combinatorial structure of peakless pair partitions.

Abstract:
In the present paper we propose two interactive directions for the integration of Non-standard Analysis and Category Theory. One direction is to develop a new framework for Non-standard Analysis by making use of an endofunctor on a topos of sets. It gives a new viewpoint for relativization and generalization of Non-standard Analysis. Another direction is to construct a new category by making use of Non-standard methods. As an attempt towards actual applications we introduce a category $\varphi Space$ for the unification of topological and coarse strucutures in terms of our endofunctor approach, giving a new viewpoint for for Coarse Geometry.

Abstract:
A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator $T$ on a Hilbert space $\mathcal{H}$ is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace $X$ of $\mathcal{H}$ such that the resolvent $(\lambda -T)^{-1}\phi$ of the operator $T$ has an analytic continuation from the lower half plane to the upper half plane as an $X'$-valued holomorphic function for any $\phi \in X$, even when $T$ has a continuous spectrum on $\mathbf{R}$, where $X'$ is a dual space of $X$. The rigged Hilbert space consists of three spaces $X \subset \mathcal{H} \subset X'$. A generalized eigenvalue and a generalized eigenfunction in $X'$ are defined by using the analytic continuation of the resolvent as an operator from $X$ into $X'$. Other basic tools of the usual spectral theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual spectral theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations.

Abstract:
We have a great interest in a framework necessary to derive various conservation laws appearing in PDEs. For usual nonlinear Schroedinger equations, some results for the framework are known. We obtain it for nonlinear Schroedinger equations with non-vanishing boundary condition. As an application for the result, we remove some of technical assumption of the nonlinearity to derive the conservation law and time global solutions.

Abstract:
The first, second and fourth Painlev\'{e} equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of the equations, a simple proof of the Painlev\'{e} property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of $\C P^3(p,q,r,s)$. In particular, for the first Painlev\'{e} equation, a well known Painlev\'{e}'s transformation is geometrically derived, which proves to be the Darboux coordinates of a certain algebraic surface with a holomorphic symplectic form. The affine Weyl group, Dynkin diagram and the Boutroux coordinates are also studied from a view point of the weighted projective space.

Abstract:
Formal series solutions and the Kovalevskaya exponents of a quasi-homogeneous polynomial system of differential equations are studied by means of a weighted projective space and dynamical systems theory. A necessary and sufficient condition for the series solution to be a convergent Laurent series is given, which improve the well known Painlev\'{e} test. In particular, if a given system has the Painlev\'{e} property, an algorithm to construct Okamoto's space of initial conditions is given. The space of initial conditions is obtained by weighted blow-ups of the weighted projective space, where the weights for the blow-ups are determined by the Kovalevskaya exponents. The results are applied to the first Painlev\'{e} hierarchy ($2m$-th order first Painlev\'{e} equation).

Abstract:
We show algorithmic randomness versions of the two classical theorems on subsequences of normal numbers. One is Kamae-Weiss theorem (Kamae 1973) on normal numbers, which characterize the selection function that preserves normal numbers. Another one is the Steinhaus (1922) theorem on normal numbers, which characterize the normality from their subsequences. In van Lambalgen (1987), an algorithmic analogy to Kamae-Weiss theorem is conjectured in terms of algorithmic randomness and complexity. In this paper we consider two types of algorithmic random sequence; one is ML-random sequences and the other one is the set of sequences that have maximal complexity rate. Then we show algorithmic randomness versions of corresponding theorems to the above classical results.

Abstract:
Generalization of the Lambalgen's theorem is studied with the notion of Hippocratic (blind) randomness without assuming computability of conditional probabilities. In [Bauwence 2014], a counter-example for the generalization of Lambalgen's theorem is shown when the conditional probability is not computable. In this paper, it is shown that (i) finiteness of martingale for blind randomness, (ii) classification of two blind randomness by likelihood ratio test, (iii) sufficient conditions for the generalization of the Lambalgen's theorem, and (iv) an example that satisfies the Lambalgen's theorem but the conditional probabilities are not computable for all random parameters.