The matrix convexity and the matrix monotony of a real C¹ function f on (0,∞) are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with f, tf(t), and t²f(t). Similar characterizations are also obtained for matrix monotone functions on a finite interval (a,b).

In this note, we consider rational cuspidal plane curves having exactly one cusp whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal embedded resolution of the cusp have the maximal self-intersection number. We show that the curves given by the classification coincide with those constructed by Orevkov.

We study the notion of a strongly normal sequence in the dual E^* of a Banach space E. In particular, we prove that the following three conditions are equivalent:

(1) E^* has a strongly normal sequence,

(2) (E^*, σ(E^*,E)) has a Schauder basic sequence,

(3) E has an infinite-dimensional separable quotient.

We establish bilinear estimates in dyadic BMO as an extension of Kozono and Taniuchi's result on the usual BMO. To establish the bilinear estimates we use sharp maximal functions, while they used the boundedness of pseudo-differential operators by Coifman and Meyer. By this extension we prove that the dyadic BMO norm of the velocity controls the blow-up phenomena of smooth solutions to the Navier-Stokes equations. Moreover, we give an odd function with type II singularity which clarifies the difference between BMO and dyadic BMO.

We give a formulation of Yang-Mills gauge theory for open smooth four-dimensional manifolds whose ends are homeomorphic to S³ × [0,∞). We apply this formulation to the study of conformal structures of such manifolds. We introduce the notion of asymptotically quasiconformal homeomorphic manifolds and show that there exist manifolds which are mutually homeomorphic but not asymptotically quasiconformal homeomorphic.

We extend the work of Abe in [1], to show that the strong partition relation $C \rightarrow (n+2)^{n+1}_{<-reg}$, for every $C \in \mathsf{WNS}^{*}_{\kappa,\lambda}$, is a consequence of the existence of an n-subtle cardinal. We then build on Kanamori's result in [10], that the existence of an $n$-subtle cardinal is equivalent to the existence of a set of ordinals containing a homogeneous subset of size $n$+2 for each regressive coloring of $n$+1-tuples from the set. We use this result to show that a seemingly weaker relation, in the context of $P_{\kappa}\lambda$ is also equivalent. This relation is a new type of regressive partition relation, which we then attempt to characterize.

$\def\mathbi#1{\textbf{\em#1}}$ Let $G$ be a nonabelian $p$ group of order $p^3$ (i.e., extraspecial $p$-group), and $BG$ its classifying space. Then $CH^{*}(BG) \cong H^{2*}(BG)$ where $CH^{*}(-)$ is the Chow ring over the field $k = \textbf{C}$. We also compute mod(2) motivic cohomology and motivic cobordism of $BQ_{8}$ and $BD_{8}$.

We study the instability of standing waves for nonlinear Schrödinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a Strichartz type estimate for the propagator generated by the linearized operator around standing wave.

We deal with the property of weak normality (for non-principal ultrafilters). We characterize the situation of $|\prod_{i\lt \kappa} \lambda_i / D|=\lambda$. We have an application for a question of Depth in Boolean algebras.

In this paper, we proved the generalized resolvent estimate and the maximal $L_p$-$L_q$ regularity of the Stokes equation with first order boundary condition in the half-space, which arises in the mathematical study of the motion of a viscous incompressible one phase fluid flow with free surface. The core of our approach is to prove the $\mathcal{R}$ boundedness of solution operators defined in a sector $\Sigma_{\epsilon,\gamma_0} = \{\lambda \in C \: \backslash \: \{0\} \: | \: \left|\mathrm{arg}\lambda\right| \leq \pi - \epsilon, \left|\lambda\right| \geq \gamma_0 \}$ with $0 < \epsilon < \pi / 2$ and $\gamma_0 \geq 0$. This $\mathcal{R}$ boundedness implies the resolvent estimate of the Stokes operator and the combination of this $\mathcal{R}$ boundedness with the operator valued Fourier multiplier theorem of L. Weis implies the maximal $L_p$-$L_q$ regularity of the non-stationary Stokes. For a densely defined closed operator $A$, we know that what $A$ has maximal $L_p$ regularity implies that the resolvent estimate of $A$ in $\lambda \in \Sigma_{\epsilon,\gamma_0}$, but the opposite direction is not true in general (cf. Kalton and Lancien [19]). However, in this paper using the $\mathcal{R}$ boundedness of the operator family in the sector $\Sigma_{\epsilon,\gamma_0}$, we derive a systematic way to prove the resolvent estimate and the maximal $L_p$ regularity at the same time.

Let $H^2(D^2)$ be the Hardy space over the bidisk. Let $\{\varphi_n(z)\}_{n \geq 0}$ and $\{\psi_n(w)\}_{n \geq 0}$ be sequences of one variable inner functions satisfying some additinal conditions. Associated with them, we have a Rudin type invariant subspace $\mathcal{M}$ of $H^2(D^2)$. We study the Beurling type theorem for the fringe operator $F_w$ on $\mathcal{M} \ominus z \mathcal{M}$.

The goal of the article is to classify foliations of S³ by regular canal surfaces, that is envelopes of one-parameter families of spheres which are immersed surfaces. We will add some extra information when the leaves are “surfaces of revolution” in a conformal sense.