Two cross caps in Euclidean 3-space are said to be infinitesimally isometric if their Taylor expansions of the first fundamental forms coincide by taking a local coordinate system. For a given $C^\infty$ cross cap $f$, we give a method to find all cross caps which are infinitesimally isomeric to $f$. More generally, we show that for a given $C^{\infty}$ metric with singularity having certain properties like as induced metrics of cross caps (called a Whitney metric), there exists locally a $C^\infty$ cross cap infinitesimally isometric to the given one. Moreover, the Taylor expansion of such a realization is uniquely determined by a given $C^{\infty}$ function with a certain property (called characteristic function). As an application, we give a countable family of intrinsic invariants of cross caps which recognizes infinitesimal isometry classes completely.

Let $f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C}^{n+1},0)$ be a corank 1 finitely determined map germ. For a generic linear form $p:(\mathbb{C}^{n+1},0)\to(\mathbb{C},0)$ we denote by $g:(\mathbb{C}^{n-1},0)\rightarrow (\mathbb{C}^{n},0)$ the transverse slice of $f$ with respect to $p$. We prove that the sum of the image Milnor numbers $\mu_I(f)+\mu_I(g)$ is equal to the number of critical points of $p|_{X_s}:X_s\to\mathbb{C}$ on all the strata of $X_s$, where $X_s$ is the disentanglement of $f$ (i.e., the image of a stabilisation $f_s$ of $f$).

We prove certain vector valued inequalities on $\Bbb R^n$ related to Littlewood-Paley theory. They can be used in proving characterization of the Hardy spaces in terms of Littlewood-Paley operators by methods of real analysis.

The aim of this paper is to discuss the Lipschitz continuity of $\alpha$-harmonic functions.

We give new viewpoints of Campanato spaces with variable growth condition for applications to the Navier-Stokes equation. Namely, we formulate a blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of stationary Euler flows and show that the properties of Campanato spaces with variable growth condition are very useful for this formulation, since variable growth condition can control the continuity and integrability of functions on the neighborhood at each point. Our criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. If geometric behavior of the velocity vector field near the maximum point has a kind of stationary Euler flow configuration up to a possible blowup time, then the solution can be extended to be the strong solution beyond the possible blowup time. As another application we also mention the Cauchy problem for the Navier-Stokes equation.

A Châtelet surface over a field is a typical geometrically rational surface. Its Chow group of zero-cycles has been studied as an important birational invariant by many researchers since the 1970s. Recently, S. Saito and K. Sato obtained a duality between the Chow and Brauer groups from the Brauer-Manin pairing. For a Châtelet surface over a local field, we combine their result with the known calculation of the Chow group to determine the structure and generators of the Brauer group of a regular proper flat model of the surface over the integer ring of the base field.

In this paper we consider the equation system describing the motion of the air and the variation of the radiation intensity and the quantity of water droplets in the air, including also the process of water phase transition. Under a suitable condition we prove the existence and uniqueness of the local solution. By eliminating the approximation by regularization of vapor density and by including the equation of radiation, this result improves previous ones.

Let $\frak{a}$ denote an ideal in a commutative Noetherian local ring $(R,\frak{m})$ and $M$ a non-zero finitely generated $R$-module of dimension $d$. Let $d:=\dim(M/\frak{a} M)$. In this paper we calculate the annihilator of the top formal local cohomology module $\mathfrak{F}_{\frak{a}}^d (M)$. In fact, we prove that ${\rm Ann}_R(\mathfrak{F}_{\frak{a}}^d (M))={\rm Ann}_R(M/U_R(\frak{a}, M))$, where $$ U_R(\frak{a}, M):=\cup\lbrace N: N\leqslant M \text{ and } \dim(N/\frak{a}N) \lt \dim(M/\frak{a}M) \rbrace. $$ We give a description of $U_R(\frak{a}, M)$ and we will show that $$ {\rm Ann}_R (\mathfrak{F}_{\frak{a}}^d(M)) = {\rm Ann}_R (M/\cap_{\frak{p}_j \in {\rm Assh}_R M \cap {\rm V}(\frak{a})} N_j), $$ where $0=\bigcap_{j=1}^{n} N_{j}$ denotes a reduced primary decomposition of the zero submodule $0$ in $M$ and $N_j$ is a $\frak{p}_j$-primary submodule of $M$, for all $j=1,\dots, n$. Also, we determine the radical of the annihilator of $\mathfrak{F}_{\frak{a}}^d (M)$. We will prove that $$ \sqrt{{\rm Ann}_R(\mathfrak{F}_{\frak{a}}^d (M))} = {\rm Ann}_R(M/G_R(\frak{a}, M)), $$ where $G_R(\frak{a}, M)$ denotes the largest submodule of $M$ such that ${\rm Assh}_R(M)\cap {\rm V}(\frak{a}) \subseteq {\rm Ass}_R(M/G_R(\frak{a}, M))$ and ${\rm Assh}_R(M)$ denotes the set $\{\frak{p} \in {\rm Ass} M:\dim R/\frak{p} = \dim M\}.$

It is shown that the cubic derivative nonlinear Schrödinger equation is locally well-posed in Besov spaces $B^{s}_{2,\infty}(\mathbb X)$, $s\ge 1/2$, where we treat the non-periodic setting $\mathbb X=\mathbb R$ and the periodic setting $\mathbb X=\mathbb T$ simultaneously. The proof is based on the strategy of Herr for initial data in $H^{s}(\mathbb T)$, $s\ge 1/2$.