Consider a rigid body moving with a prescribed constant non-zero velocity and rotating with a prescribed constant non-zero angular velocity in a three-dimensional Navier-Stokes liquid. The asymptotic structure of a steady-state solution to the corresponding equations of motion is analyzed. In particular, an asymptotic expansion of the corresponding velocity field is obtained.

Let $F$ be a non-Archimedean local field of characteristic zero. Jacquet, Piatetski-Shapiro and Shalika introduced the notion of newforms for irreducible generic representations of $GL_n(F)$. In this paper, we give an explicit formula for Whittaker functions associated to newforms on the diagonal matrices in $GL_n(F)$.

The enhancement to the Milnor number is an invariant of the homotopy classes of fibered links in the sphere $S^{2n-1}$ and belongs to $\mathbb{Z}/r\mathbb{Z}$, where $r=0$ if $n=2$ and $r=2$ if $n=2$. Mixed polynomials are polynomials in complex variables $z_1,\dots,z_n$ and their conjugates $\bar{z}_1,\dots,\bar{z}_n$. M. Oka showed that mixed polynomials have Milnor fibrations under the strongly non-degeneracy condition. In this present paper, we study fibered links which are defined by a certain class of mixed polynomials which admit Milnor fibrations and show that any element of $\mathbb{Z}/r\mathbb{Z}$ is realized by the enhancement to the Milnor number of such a fibered link.

The main purpose of this paper is to classify the real forms $M$ of simple irreducible pseudo-Hermitian symmetric spaces $G/R$ with $G$ non-compact. That provides an extension of Jaffee's results (Bull. Amer. Math. Soc. '75; J. Differential Geom. '78), Leung's result (J. Differential Geom. '79) and Takeuchi's result (Tohoku Math. J. '84) concerning the classification of real forms of irreducible Hermitian symmetric spaces of the non-compact type. Moreover, that enables us to classify the pairs of simple para-Hermitian symmetric Lie algebras and their para-holomorphic involutions, which includes Kaneyuki-Kozai's result (Tokyo J. Math. '85) of the classification of simple para-Hermitian symmetric Lie algebras.

In this paper we investigate Wulff shapes in $\mathbb{R}^{n+1}$ $(n\ge 0)$ from the topological viewpoint. A topological characterization of the limit of Wulff shapes and the dual Wulff shape of the given Wulff shape are provided. Moreover, we show that the given Wulff shape is never a polytope if its support function is of class $C^1$. Several characterizations of the given Wulff shape from the viewpoint of pedals are also provided. One of such characterizations may be regarded as a bridge between the mathematical aspect of crystals at equilibrium and the mathematical aspect of perspective projections.

This is a continuation of our previous paper. We consider a certain order-like relation for positive operators on a Hilbert space. This relation is defined by using the Jensen inequality with respect to the square-root function. We show that this relation is antisymmetric if the operators are invertible.

We study the geometry of transversality of holomorphic foliations of codimension one in ${\mathbb C}^n$ with spheres, from a viewpoint of dynamics of anti-holomorphic maps in the projective space. A point of non-degenerate contact of a leaf with a sphere is a hyperbolic fixed point of the corresponding dynamics. Around a point of degenerate contact, the intersection of branches of the variety of contacts is described as a bifurcation diagram of a neutral fixed point of dynamics. The Morse index for the distance function from the origin is computed as the complex dimension of an unstable manifold.

Exact critical values of symmetric fourth $L$ function of the Ramanujan Delta function $\Delta$ were conjectured by Don Zagier in 1977. They are given as products of explicit rational numbers, powers of $\pi$, and the cube of the inner product of $\Delta$. In this paper, we prove that the ratio of these critical values are as conjectured by showing that the critical values are products of the same explicit rational numbers, powers of $\pi$, and the inner product of some vector valued Siegel modular form of degree two. Our method is based on the Kim-Ramakrishnan-Shahidi lifting, the pullback formulas, and differential operators which preserve automorphy under restriction of domains. We also show a congruence between a lift and a non-lift. Furthermore, we show the algebraicity of the critical values of the symmetric fourth $L$ function of any elliptic modular form and give some conjectures in general case.

We investigate how Viro's integral calculus applies for the study of the topology of stable maps. We also discuss several applications to Morin maps and complex maps.

W. Zhang showed a higher dimensional version of Rochlin congruence for $8k+4$-dimensional manifolds. We give an equivariant version of Zhang's theorem for $8k+4$-dimensional compact Spin$^c$-$G$-manifolds with spin boundary, where we define equivariant indices with values in $R(G)/RSp(G)$. We also give a similar congruence relation for $8k$-dimensional compact Spin$^c$-$G$-manifolds with spin boundary, where we define equivariant indices with values in $R(G)/RO(G)$.

Takeda ([30]) solved the Feynman-Kac penalization problem for positive continuous additive functionals. We extend his result to additive functionals with jumps. We further give concrete examples of jumping functions.

It is known that if a classical link group is a free abelian group, then its rank is at most two. It is also known that a $k$-component 2-link group ($k$ > 1) is not free abelian. In this paper, we give examples of $T^2$-links each of whose link groups is a free abelian group of rank three or four. Concerning the $T^2$-links of rank three, we determine the triple point numbers and we see that their link types are infinitely many.

We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to present an alternative formulation of some known characterizations of equivariantizations for fusion categories, and to extend these characterizations to equivariantizations of finite tensor categories under finite group scheme actions. In particular, we obtain a simple characterization of equivariantizations under actions of finite abelian groups. As an application, we show that if ${\mathcal C}$ is a fusion category and $F: {\mathcal C} \to {\mathcal D}$ is a dominant tensor functor of Frobenius-Perron index $p$, then $F$ is an equivariantization if $p=2$, or if ${\mathcal C}$ is weakly integral and $p$ is the smallest prime factor of FPdim ${\mathcal C}$.

We give a simple proof of the symmetry of a minimal diffusion $X^0$ on a one-dimensional open interval $I$ with respect to the attached canonical measure $m$ along with the identification of the Dirichlet form of $X^0$ on $L^2(I; m)$ in terms of the triplet $(s,m,k)$ attached to $X^0$. The $L^2$-generators of $X^0$ and its reflecting extension $X^r$ are then readily described. We next use the associated reproducing kernels in connecting the $L^2$-setting to the traditional $C_b$-setting and thereby deduce characterizations of the domains of $C_b$-generators of $X^0$ and $X^r$ by means of boundary conditions. We finally identify the $C_b$-generators for all other possible symmetric diffusion extensions of $X^0$ and construct by that means all diffusion extensions of $X^0$ in [IM2].

We introduce a natural definition of Riesz measures and Wishart laws associated to an $\Omega$-positive (virtual) quadratic map, where $\Omega \subset$ R$^n$ is a regular open convex cone. In this context we prove new general formulas for moments of the Wishart laws on non-symmetric cones. For homogeneous cases, all the quadratic maps are characterized and the associated Riesz measure and Wishart law with its moments are described explicitly. We apply the theory of relatively invariant distributions and a matrix realization of homogeneous cones obtained recently by the second author.