We study the structure of the group of equivariant Lipschitz homeomorphisms of a smooth $G$-manifold $M$ which are isotopic to the identity through equivariant Lipschitz homeomorphisms with compact support. First we show that the group is perfect when $M$ is a smooth free $G$-manifold. Secondly in the case of $C^n$with the canonical $U\left(n\right)$-action, we show that the first homology group admits continuous moduli. Thirdly we apply the result to the case of the group $L(C,0)$ of Lipschitz homeomorphisms of $C$ fixing the origin.

It is known that each 3-symmetric space admits an invariant almost complex structure $J$, so-called a canonical almost complex structure. By making use of simple graded Lie algebras and an affine Lie algebra, we classify half dimensional, totally real (with respect to $J$) and totally geodesic submanifolds of compact 3-symmetric spaces.

The structure of the ring of Hodge cycles on a certain family of abelian varieties of CM-type is investigated. This leads to an interesting combinatorial problem related to posets based on complete $p$-ary trees. A complete solution to the problem is given for the case $p=2$.

Let $b_{\alpha }^p$,, be the parabolic Bergman space, the Banach space of solutions of parabolic equations $(\partial /\partial t+(-\mathrm{\Delta }{)}^{\alpha })u=0$ on the upper half space $R_{+}^{n+1}$ which have finite $L^p$ norms. We study Carleson type measures on $b_{\alpha }^p$, and give a necessary and sufficient condition for a measure $\mu$ on $R_{+}^{n+1}$ to be of Carleson type on $b_{\alpha }^p$. As an application, we characterize bounded Toeplitz operators in the space $b_{\alpha }^2$.

We determine the Calabi-Yau dimension of the stable module categories of all symmetric algebras of tame representation type over an algebraically closed field, and derive some consequences.

In [3], Kotani proved analytically that expectations for additive functionals of Brownian motion $\{B_t,t\ge 0\}$ of the form $$E_0\left[f\left(B_t\right)g\left({\int }_0^t\phi \left(B_s\right)ds\right)\right]$$ have the asymptotics $t^{-3/2}$ as $t\to \mathrm{\infty }$ for some suitable non-negative functions $\phi$, $f$ and $g$. This generalizes, in the asymptotic form, Yor's explicit formula [10] for exponential Brownian functionals.

In the present paper, we discuss this generalization probabilistically, by using a time-change argument. We may easily see from our argument that this asymptotics $t^{-3/2}$ comes from the transition probability of 3-dimensional Bessel process.

We study combinatorics on $P_{\kappa }\left(\lambda \right)$ under the assumption that .

In 1879, Thomae discussed the relations between two generic hypergeometric ${}_3{F}_2$-series with argument 1. It is well-known since then that, in combination with the trivial ones which come from permutations of the parameters of the hypergeometric series, Thomae had found a set of 120 relations. More recently, Rhin and Viola asked the following question (in a different, but equivalent language of integrals): If there exists a linear dependence relation over $Q$ between two convergent ${}_3{F}_2$-series with argument 1, with integral parameters, and whose values are irrational numbers, is this relation a specialisation of one of the 120 Thomae relations? A few years later, Sato answered this question in the negative, by giving six examples of relations which cannot be explained by Thomae's relations. We show that Sato's counter-examples can be naturally embedded into two families of infinitely many ${}_3{F}_2$-relations, both parametrised by three independent parameters. Moreover, we find two more infinite families of the same nature. The families, which do not seem to have been recorded before, come from certain ${}_3{F}_2$-transformation formulae and contiguous relations. We also explain in detail the relationship between the integrals of Rhin and Viola and ${}_3{F}_2$-series.

Let $Y$ be a smooth curve embedded in a complex projective manifold $X$ of dimension $n\ge 2$ with ample normal bundle $N_{Y|X}$ . For every $p\ge 0$ let ${\alpha }_p$ denote the natural restriction maps $\text{Pic}\left(X\right)\to \text{Pic}\left(Y\right(p\left)\right)$, where $Y\left(p\right)$ is the $p$-th infinitesimal neighbourhood of $Y$ in $X$. First one proves that for every $p\ge 1$ there is an isomorphism of abelian groups $\text{Coker}\left({\alpha }_p\right)\cong \text{Coker}\left({\alpha }_0\right)\oplus K_p(Y,X)$, where $K_p(Y,X)$ is a quotient of the $C$-vector space $L_p(Y,X):=\underset{i=1}{\overset{p}{⨁}}{H}^1(Y,{\mathbf{S}}^i(N_{Y|X}{)}^{*})$ by a free subgroup of $L_p(Y,X)$ of rank strictly less than the Picard number of $X$. Then one shows that $L_1(Y,X)=0$ if and only if $Y\cong {\mathbf{P}}^{\mathbf{1}}$ and $N_{Y|X}\cong {𝒪}_{{\mathbf{P}}^{\mathbf{1}}}(1{)}^{\oplus n-1}$ (i.e. $Y$ is a quasi-line in the terminology of [4]). The special curves in question are by definition those for which ${\dim }_{\mathbf{C}}{L}_1(Y,X)=1$ . This equality is closely related with a beautiful classical result of B. Segre [25]. It turns out that $Y$ is special if and only if either $Y\cong {\mathbf{P}}^{\mathbf{1}}$ and $N_{Y|X}\cong {𝒪}_{{\mathbf{P}}^{\mathbf{1}}}\left(2\right)\oplus {𝒪}_{{\mathbf{P}}^{\mathbf{1}}}(1{)}^{\oplus n-2}$ , or $Y$ is elliptic and $\mathrm{deg}\left(N_{Y|X}\right)=1$ . After proving some general results on manifolds of dimension $n\ge 2$ carrying special rational curves (e.g. they form a subclass of the class of rationally connected manifolds which is stable under small projective deformations), a complete birational classification of pairs $(X,Y)$ with $X$ surface and $Y$ special is given. Finally, one gives several examples of special rational curves in dimension $n\ge 3$.

We consider the Dirichlet and the Neumann eigenvalue problem for the Laplace operator on a variable nonsmooth domain, and we prove that the elementary symmetric functions of the eigenvalues splitting from a given eigenvalue upon domain deformation have a critical point at a domain with the shape of a ball. Correspondingly, we formulate overdetermined boundary value problems of the type of the Schiffer conjecture.

We show that a John domain has finitely many minimal Martin boundary points at each Euclidean boundary point. The number of minimal Martin boundary points is estimated in terms of the John constant. In particular, if the John constant is bigger than $\sqrt{3}/2$ , then there are at most two minimal Martin boundary points at each Euclidean boundary point. For a class of John domains represented as the union of convex sets we give a sufficient condition for the Martin boundary and the Euclidean boundary to coincide.

The set $ℛ\left(B\right)$ of submanifolds conjugate to a given reflective submanifold $B$ in a Riemannian symmetric space $M$ has a structure of symmetric space. Using this structure, for a submanifold $N$ in $M$ we establish integral formulae which represent the integrals of the functions $C\mapsto \text{vol}(N\cap C)$ on $ℛ\left(B\right)$ by some extrinsic geometric amounts of $N$.