Showing papers in "Hokkaido Mathematical Journal in 2014"
TL;DR: In this article, fold maps from C∞ closed manifolds into Euclidean spaces whose singular value sets are disjoint unions of spheres embedded concentrically are studied.
Abstract: In this paper, we study fold maps from C∞ closed manifolds into Euclidean spaces whose singular value sets are disjoint unions of spheres embedded concentrically. We mainly study homology and homotopy groups of manifolds admitting such maps.
65 citations
TL;DR: In this paper, the congruency of minimal ruled real hypersurfaces in a nonflat complex space form with respect to the action of its isometry group is studied, and it is shown that those in a complex hyperbolic space are congruent to each other.
Abstract: In this paper we study congruency of minimal ruled real hypersurfaces in a nonflat complex space form with respect to the action of its isometry group. We show that those in a complex hyperbolic space are classified into 3 classes and show that those in a complex projective space are congruent to each other hence form just one class.
20 citations
TL;DR: In this article, it is shown that Ulam's cellular automata contain a linear chaotic elementary cellular automaton (Rule 150) as a subsystem, and that the partial differential equation obtained by the inverse ultradiscretization preserves the self-organizing pattern of the Ulam automata.
Abstract: In this paper we study Ulam's cellular automaton, a nonlinear almost equicontinuous two-dimensional cell-model of crystalline growths. We prove that Ulam's automaton contains a linear chaotic elementary cellular automaton (Rule 150) as a subsystem. We also study the application of the inverse ultradiscretization, a method for deriving partial differential equations from a given cellular automaton, to Ulam's automaton. It is shown that the partial differential equation obtained by the inverse ultradiscretization preserves the self-organizing pattern of Ulam's automaton.
6 citations
TL;DR: In this article, the propagation of coherent states for a general class of self interacting bosonic quantum field theories with spatial cutoffs was studied using Hepp's method, using a non-polynomial Wick quantization and a specific hypercontractive estimate.
Abstract: We study, using Hepp's method, the propagation of coherent states for a general class of self interacting bosonic quantum field theories with spatial cutoffs. This includes models with non-polynomial interactions in the field variables. We show indeed that the time evolution of coherent states, in the classical limit, is well approximated by time-dependent affine Bogoliubov unitary transformations. Our analysis relies on a non-polynomial Wick quantization and a specific hypercontractive estimate.
6 citations
TL;DR: In this article, the authors give an algebraic independence result for the two infinite products involving the Lucas sequences of the first and second kind and derive that ∆k = 1∞(1+1/F2k) and ∆m = 1 ∞( 1+ 1/L 2k) are algebraically independent over ℚ, where {Fn}n≥0 and {Ln n ≥ 0 are the Fibonacci sequence and its Lucas companion.
Abstract: The aim of this paper is to give an algebraic independence result for the two infinite products involving the Lucas sequences of the first and second kind. As a consequence, we derive that the two infinite products ∏k=1∞(1+1/F2k) and ∏k=1∞(1+1/L2k) are algebraically independent over ℚ, where {Fn}n≥0 and {Ln}n≥0 are the Fibonacci sequence and its Lucas companion, respectively.
6 citations
5 citations
TL;DR: In this article, the authors studied the question whether the indirectly prescribed radiation condition at infinity is satisfied uniformly along the asymptotic process, or whether the conveniently rescaled solution to the considered equation goes to the natural limiting Helmholtz equation.
Abstract: We consider the high frequency Helmholtz equation with a variable refraction index $n^2(x)$ ($x \in \R^d$), supplemented with a given high frequency source term supported near the origin $x=0$. A small absorption parameter $\alpha_{\varepsilon}>0$ is added, which somehow prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is $\varepsilon>0$. We let $\eps$ and $\a_\eps$ go to zero {\em simultaneaously}. We study the question whether the indirectly prescribed radiation condition at infinity is satisfied {\em uniformly} along the asymptotic process $\eps \to 0$, or, in other words, whether the conveniently rescaled solution to the considered equation goes to the {\em outgoing} solution to the natural limiting Helmholtz equation. This question has been previously studied by the first autor. It is proved that the radiation condition is indeed satisfied uniformly in $\eps$, provided the refraction index satisfies a specific {\em non-refocusing condition}, a condition that is first pointed out in this reference. The non-refocusing condition requires, in essence, that the rays of geometric optics naturally associated with the high-frequency Helmholtz operator, and that are sent from the origin $x=0$ at time $t=0$, should not refocus at some later time $t>0$ near the origin again. In the present text we show the {\em optimality} of the above mentionned non-refocusing condition, in the following sense. We exhibit a refraction index which {\em does} refocus the rays of geometric optics sent from the origin near the origin again, and, on the other hand, we completely compute the asymptotic behaviour of the solution to the associated Helmholtz equation: we show that the limiting solution {\em does not} satisfy the natural radiation condition at infinity. More precisely, we show that the limiting solution is a {\em perturbation} of the outgoing solution to the natural limiting Helmholtz equation, and that the perturbing term explicitly involves the contribution of the rays radiated from the origin which go back to the origin. This term is also conveniently modulated by a phase factor, which turns out to be the action along the above rays of the hamiltonian associated with the semiclassical Helmholtz equation.
4 citations
TL;DR: In this article, it was shown that holomorphic loop algebra-valued 1-forms over Riemann surfaces yield minimal Lagrangian immersions into the complex 2-dimensional projective space via the Weierstrass type representation.
Abstract: We show that certain holomorphic loop algebra-valued 1-forms over Riemann surfaces yield minimal Lagrangian immersions into the complex 2-dimensional projective space via the Weierstrass type representation, hence 3-dimensional special Lagrangian submanifolds of ℂ3. A particular family of 1-forms on ℂ corresponds to solutions of the third Painleve equation which are smooth on (0, +∞).
3 citations
TL;DR: In this article, the formulation of the biharmonic map equation in terms of the Maurer-Cartan form for all smooth maps of a compact Riemannian manifold into a compact Lie group (G,h) with the bi-invariant metric h is obtained.
Abstract: In this paper, the formulation of the biharmonic map equation in terms of the Maurer-Cartan form for all smooth maps of a compact Riemannian manifold into a compact Lie group (G,h) with the bi-invariant Riemannian metric h is obtained. Using this, all biharmonic curves into compact Lie groups are determined exactly, and all the biharmonic maps of an open domain of ℝ2 equipped with a Riemannian metric conformal to the standard Euclidean metric into (G,h) are determined.
2 citations
TL;DR: In this article, the authors give necessary and sufficient conditions for the existence of solutions of the cohomological equation f - f ∘ γ = g on the Frechet space C ∞ (G) of complex C∞ functions on G.
Abstract: This paper deals with two analytic questions on a connected compact Lie group G. i) Let a ∈ G and denote by γ the diffeomorphism of G given by γ (x) = ax (left translation by a). We give necessary and sufficient conditions for the existence of solutions of the cohomological equation f - f ∘ γ = g on the Frechet space C∞ (G) of complex C∞ functions on G. ii) When G is the torus ${\Bbb T}^n$, we compute explicitly the distributions on ${\Bbb T}^n$ invariant by an affine automorphism γ, that is, γ (x) = A (x + a) with A ∈ GL(n, ℤ) and a ∈ ${\Bbb T}^n$. iii) We apply these results to describe the infinitesimal deformations of some Lie foliations.
2 citations
TL;DR: In this paper, the description of bi-harmonic map equa- tion in terms of the Maurer-Cartan form for all smooth map of a compact Riemannian manifold into a RiemANNian symmetric space (G/K,h) induced from the bi-invariant metric h on G is obtained.
Abstract: In this paper, the description of biharmonic map equa- tion in terms of the Maurer-Cartan form for all smooth map of a compact Riemannian manifold into a Riemannian symmetric space (G/K,h) induced from the bi-invariant Riemannian metric h on G is obtained. By this formula, all biharmonic curves into symmetric spaces are determined, and all the biharmonic maps of an open domain of R 2 with the standard Riemannian metric into (G/K,h) are determined.
TL;DR: In this article, the authors give examples of certain kind of minimal orbits of Hermann actions and discuss whether each of the examples is austere or not, and show that some of them are not austere.
Abstract: We give examples of certain kind of minimal orbits of Hermann actions and discuss whether each of the examples is austere.