Let $A$ be a real commutative Banach algebra with unity. Let $a_0in Asetminus{0}$. Let $mathbb Z a_0:={na_0}_{nin mathbb Z}$. Then, $mathbb Z a_0$ is a discrete subgroup of $A$. For any $nin mathbb Z$, the Frechet derivative of the mapping $$x , in ,
A mapsto x+na_0 , in , A$$ is the identity map on $A$ and, especially, an $A$-linear transformation on $A$. So, the quotient group $A/(mathbb Z a_0)$ is a $1$-dimensional $A$-manifold and the covering projection $$x , in , A mapsto x+mathbb Z a_0 , in , A/(mathbb Z a_0)$$ is an $A$-map. We call $A/(mathbb Z a_0)$ the $1$-dimensional $A$-cylinder by $a_0$. Let $T$ be a compact Hausdorff space. Suppose that there exist $t_1in T$ and $t_2in T$ such that $t_1 ot=t_2$ holds. Then, the set $C(T;mathbb R)$ of all real-valued continuous functions on $T$ is a real commutative Banach algebra with unity and $mathbb R , subsetneq , C(T;mathbb R)$ holds. In this paper, we show that there exists $a_0 , in , C(T;mathbb R)setminus mathbb R$ such that for any $k, in , mathbb N$, the $1$-dimensional $C(T;mathbb R)$-cylinder $(C(T;mathbb R))/(mathbb Z a_0)$ by $a_0$ cannot be embedded in the finite direct product space $(C(T;mathbb R))^k$ as a $C(T;mathbb R)$-submanifold.
Let $X$ and $Y$ be topological spaces. Let $C$ be a path-connected closed set of $Xtimes Y$. Suppose that $C$ is locally direct product, that is, for any $(a,b)in Xtimes Y$, there exist an open set $U$ of $X$, an open set $V$ of $Y$, a subset $I$ of
$U$ and a subset $J$ of $V$ such that $(a,b) in Utimes V$ and $$Ccap (Utimes V)=Itimes J$$ hold. Then, in this memo, we show that $C$ is globally so, that is, there exist a subset $A$ of $X$ and a subset $B$ of $Y$ such that $$C=Atimes B$$ holds. The proof is elementary. Here, we note that one might be able to think of a (perhaps, open) similar problem for a fiber product of locally trivial fiber spaces, not just for a direct product of topological spaces. In Appendix, we mentioned a simple example of a $C([0,1];mathbb R)$-manifold that cannot be embedded in the direct product $(C([0,1];mathbb R))^n$ as a $C([0,1];mathbb R)$-submanifold. In addition, we introduce the concept of topological 2-space, which is locally the direct product of topological spaces and an analog of homotopy category for topological 2-space. Finally, we raise a question on the existence of an $mathbb R^n$-Morse function and the existence of an $mathbb R^n$-immersion in a finite-dimensional $mathbb R^n$-Euclidean space. Here, we note that the problem of defining the concept of an $mathbb R^n$-handle body may also be considered.
Cartan-Thullen theorem is a basic one in the theory of analytic functions of several complex variables. It states that for any open set $U$ of ${mathbb C}^k$, the following conditions are equivalent: (a) $U$ is a domain of existence, (b) $U$ is a dom
ain of holomorphy and (c) $U$ is holomorphically convex. On the other hand, when $f , (, =(f_1,f_2,cdots,f_n), )$ is a $mathbb C^n$-valued function on an open set $U$ of $mathbb C^{k_1}timesmathbb C^{k_2}timescdotstimesmathbb C^{k_n}$, $f$ is said to be $mathbb C^n$-analytic, if $f$ is complex analytic and for any $i$ and $j$, $i ot=j$ implies $frac{partial f_i}{partial z_j}=0$. Here, $(z_1,z_2,cdots,z_n) in mathbb C^{k_1}timesmathbb C^{k_2}timescdotstimesmathbb C^{k_n}$ holds. We note that a $mathbb C^n$-analytic mapping and a $mathbb C^n$-analytic manifold can be easily defined. In this paper, we show an analogue of Cartan-Thullen theorem for a $mathbb C^n$-analytic function. For $n=1$, it gives Cartan-Thullen theorem itself. Our proof is almost the same as Cartan-Thullen theorem. Thus, our generalization seems to be natural. On the other hand, our result is partial, because we do not answer the following question. That is, does a connected open $mathbb C^n$-holomorphically convex set $U$ exist such that $U$ is not the direct product of any holomorphically convex sets $U_1, U_2, cdots, U_{n-1}$ and $U_n$ ? As a corollary of our generalization, we give the following partial result. If $U$ is convex, then $U$ is the direct product of some holomorphically convex sets. Also, $f$ is said to be $mathbb C^n$-triangular, if $f$ is complex analytic and for any $i$ and $j$, $i<j$ implies $frac{partial f_i}{partial z_j}=0$. Kasuya suggested that a $mathbb C^n$-analytic manifold and a $mathbb C^n$-triangular manifold might, for example, be related to a holomorphic web and a holomorphic foliation.
Let $A$ be a commutative Banach algebra. Let $M$ be a complex manifold on $A$ (an $A$-manifold). Then, we define an $A$-holomorphic vector bundle $(wedge^kT^*)(M)$ on $M$. For an open set $U$ of $M$, $omega$ is said to be an $A$-holomorphic different
ial $k$-form on $U$, if $omega$ is an $A$-holomorphic section of $(wedge^kT^*)(M)$ on $U$. So, if the set of all $A$-holomorphic differential $k$-forms on $U$ is denoted by $Omega_{M}^k(U)$, then ${Omega_{M}^k(U)}_{U}$ is a sheaf of modules on the structure sheaf $O_M$ of the $A$-manifold $M$ and the cohomology group $H^l(M,Omega_{M}^k)$ with the coefficient sheaf ${Omega_{M}^k(U)}_{U}$ is an $O_M(M)$-module and therefore, in particular, an $A$-module. There is no new thing in our definition of a holomorphic differential form. However, this is necessary to get the cohomology group $H^l(M,Omega_{M}^k)$ as an $A$-module. Furthermore, we try to define the structure sheaf of a manifold that is locally a continuous family of $mathbb C$-manifolds (and also the one of an analytic family). Directing attention to a finite family of $mathbb C$-manifolds, we mentioned the possibility that Dolbeault theorem holds for a continuous sum of $mathbb C$-manifolds. Also, we state a few related problems. One of them is the following. Let $nin mathbb N$. Then, does there exist a $mathbb C^n$-manifold $N$ such that for any $mathbb C$-manifolds $M_1, M_2, cdots, M_{n-1}$ and $M_n$, $N$ can not be embedded in the direct product $M_1times M_2 times cdots times M_{n-1} times M_n$ as a $mathbb C^n$-manifold ? So, we propose something that is likely to be a candidate for such a $mathbb C^2$-manifold $N$.
An n-dimensional complex manifold is a manifold by biholomorphic mappings between open sets of the finite direct product of the complex number field. On the other hand, when A is a commutative Banach algebra, Lorch gave a definition that an A-valued
function on an open set of A is holomorphic. The definition of a holomorphic function by Lorch can be straightforwardly generalized to an A-valued function on an open set of the finite direct product of A. Therefore, a manifold modeled on the finite direct product of A (an n-dimensional A-manifold) is easily defined. However, in my opinion, it seems that so many nontrivial examples were not known (including the case of n=1, that is, Riemann surfaces). By the way, if X is a compact Hausdorff space, then the algebra C(X) of all complex valued continuous functions on X is the most basic example of a commutative Banach algebra (furthermore, a commutative C*-algebra). In this note, we see that if the set of all continuous cross sections of a continuous family M of compact complex manifolds (a topological deformation M of compact complex analytic structures) on X is denoted by G(M), then the structure of a C(X)-manifold modeled on the C(X)-modules of all continuous cross sections of complex vector bundles on X is introduced into G(M). Therefore, especially, if X is contractible, then G(M) is a finite-dimensional C(X)-manifold.
We consider a nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure for the convolution is absolutely continuous. In order to show the main result, we modify a recursive method for abstract monotone discrete dynamical
systems by Weinberger. We note that the monotone semiflow generated by the equation does not have compactness with respect to the compact-open topology. At the end, we propose a discrete model that describes the measurement process.
We study two initial value problems of the linear diffusion equation and a nonlinear diffusion equation, when Cauchy data are bounded and oscillate mildly. The latter nonlinear heat equation is the equation of the curvature flow, when the moving curv
es are represented by graphs. By using an elementary scaling technique, we show some formulas for space-time behavior of the solution. Keywords: scaling argument, self-similar solution, nonstabilizing solution, nontrivial dynamics, nontrivial large-time behavior, irregular behavior.
We consider traveling fronts to the nonlocal bistable equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We show that there is a traveling wave solution with monotone profile. In order to
prove this result, we would develop a recursive method for abstract monotone dynamical systems and apply it to the equation.
We consider the nonlocal analogue of the Fisher-KPP equation. We do not assume that the Borel-measure is absolutely continuous with respect to the Lebesgue measure. We gives a sufficient condition for existence of traveling waves, and a necessary condition for existence of periodic traveling waves.
