Let $\mathcal X$ be the space of type-preserving $\mathrm{SL}(2,\mathbb{C})$ characters of the punctured torus $T$. The Bowditch space $\mathcal{X}_{\mathrm{BQ}}$ is the largest open subset of $\mathcal{X}$ on which the mapping class group acts properly discontinuously, this is characterized by two simple conditions called the BQ-conditions. In this note, we show that $[\rho] \in \operatorname{int}(\mathcal{X} \setminus \mathcal{X}_{\mathrm{BQ}})$ if there exists an essential simple closed curve $X$ on $T$ such that $|\tr\rho(X)|<0.5$.

The first and the third authors have proved the global existence of holomorphic solutions to partial differential equations with several Fuchsian variables in the sense of N.S. Madi, under some assumptions on some coefficients and on the Fuchsian characteristic polynomial. This article shows a similar global existence of solutions which are holomorphic with respect to Fuchsian variables and of projective Gevrey class with respect to non-Fuchsian variables. The proof is based on the concept of formal norms of Leray-Waelbroeck and the application of Gevrey's operator of C. Wagschal.

Given a log smooth log scheme $X$ over $\operatorname{Spec} \mathbb{C}$, in this article we analyze and compare different filtrations defined on the log de Rham complex $\omega^{\bullet}_X$ associated to $X$. We mainly refer to the articles of Ogus ([23]), Danilov ([1]), Ishida ([16]). In this context, we analyze two filtrations on $\omega^{\bullet}_X$: the decreasing Ogus filtration $\tilde{L}^{\bullet}$, which is a sort of extension of the Deligne weight filtration $W_{\bullet}$ to log smooth log schemes over $\Spec \mathbb{C}$, and an increasing filtration, which we call the Ishida filtration and denote by $I_{\bullet}$, defined by using the Ishida complex $\tilde{\Omega}^{\bullet}_X$ of $X$. Moreover, we have the Danilov de Rham complex $\Omega^{\bullet}_X(\log D)$ with logarithmic poles along $D= X - X_{\mathrm{triv}}$ ($X_{\mathrm{triv}}$ being the trivial locus for the log structure on $X$), endowed with an increasing weight filtration (the Danilov weight filtration $\mathcal{W}_{\bullet}$). Then we prove that the Danilov de Rham complex $\Omega^{\bullet}_X(\log D)$ coincides with the log de Rham complex $\omega^{\bullet}_X$ and the Ishida filtration $I_{\bullet}$ (which is a globalization of the Danilov weight filtration $\mathcal W_{\bullet}$) coincides with the opposite Ogus filtration $\tilde{L}^{-\bullet}$.

Special Lagrangian cones in complex Euclidean spaces are obtained as cones over compact minimal Legendrian submanifolds in the odd dimenisonal standard hypersphere. The notion of the stability, the Legendrian stability and the rigidity of special Lagrangian cones were recently introduced and investigated by D. Joyce, M. Haskins etc. In this paper we determine explicitly the stability-index, the Legendrian-index, and the rigidity of special Lagrangian cones over compact irreducible symmeric spaces of type $A$ obtained as minimal Legendrian orbits and over a minimal Legendrian $\mathit{SU}(2)$-orbit. We obtain the examples of stable and rigid special Lagrangian cones in higher dimensions. Moreover we discuss a relationship of these properties with the Hamiltonian stability of minimal Lagrangian submanifolds in complex projective spaces.

We consider finite-dimensional interacting diffusions which are defined by adding a linear drift term to independent one dimensional diffusions. For these processes we prove that the distribution of the occupation time at the first quadrant converges to a generalized arc-sine law.

We give a Turaev-Viro type construction for the LMO invariant. More precisely, we construct an invariant of closed oriented 3-manifolds from data of their spines or their simplicial decompositions and the values of Kontsevich invariant of the unknotted tetrahedron and the Hopf link by using Bar-Natan and Thurston's operations.

In this paper we shall deal with hyperbolic systems of first order with constant coefficient characteristic polynomials and give a necessary and sufficient condition for the Cauchy problem to be $C^{\infty}$ well-posed under the maximal rank condition (see the condition (R) below).

We provide in this paper a counterexample to the Benson-Ratcliff conjecture about a cohomology class invariant on coadjoint orbits on nilpotent Lie groups. We prove that this invariant never vanishes on generic coadjoint orbits for some restrictive classes. As such, it does separate up to invariant factor, unitary representations associated to generic orbits in some cases.

We consider half-line media which consist of many kinds of substances. We assume that the waves through this media are described by the one-dimensional wave equation. We can directly observe the data near the boundary point of the half-line, but we cannot directly observe the data of things away from the boundary point. In this situation, we try to identify these unknown things by creating an artificial explosion and observing on the boundary point the waves generated by the explosion. In the previous works related to this problem, only the speeds of the waves were treated, but we also take into account the impedances of the media in our setting.

Virtual knot theory, introduced by L. Kauffman, is a generalization of classical knot theory. It naturally yields the notion of a virtual braid, which is closely related to the notion of a welded braid due to R. Fenn, R. Rimányi and C. Rourke. In this paper we prove that any virtual link or welded link can be described as the closure of a virtual braid or welded braid, respectively, which is unique up to certain basic moves. This is analogous to the Alexander and Markov theorems for classical braids and links.

In this paper, we shall investigate some potential theory for time change of Markov processes. Under weak duality, it is proved that the jumping measure and Feller measure are actually independent of time change, and the jumping measure of a time changed process induced by a PCAF supported on $V$ coincides with the sum of the Feller measure on $V$ and the trace of the original jumping measure on $V$.

Let $G$ be a subgroup of rank two of the Möbius group $\textit{PSL}(2, \mathbb{C})$. The Jørgensen number $J(G)$ of $G$ is defined by \[ J(G) = \inf \{|\tr^{2} A - 4| + \mathopen|\tr[A, B]-2|\colon \langle A, B\rangle= G\}. \] We describ e all subgroups $G$ of the Picard group $\textit{PSL}(2, \mathbb{Z} + i\mathbb{Z})$ with $J(G) = 1$.

Given a homogeneous almost Kähler manifold $(M,J,g)$ with nonpositive curvature operator, we prove that if $g$ is an Einstein metric having negative sectional curvature, then the almost complex structure $J$ must be integrable. Furthermore, such $(M,J,g)$ eventually has constant negative holomorphic sectional curvature and hence is holomorphically isometric to a complex hyperbolic space.