A Riemannian metric $g$ on a domain $\Omega$ in $ \mathbb{R}^n$ defines a map $F_g$ from $(\Omega,g)$ into the symmetric space of positive definite real symmetric $n \times n$ matrices $(\textrm{Sym}^+(n),h)$, where $h$ is the Cheng-Yau metric on $\textrm{Sym}^+(n)$. We show that the map $F_g$ is a harmonic immersion if $\Omega$ is a regular convex cone and $g$ is the Cheng-Yau metric on $\Omega$. We also prove that the map $F_g$ is totally geodesic if $\Omega$ is a homogeneous self-dual regular convex cone and $g$ is the Cheng-Yau metric on $\Omega$.

In this paper, we derive systems of ordinary differential equations (ODEs) satisfied by modular forms of level three, which are level three versions of Ramanujan's system of ODEs satisfied by the classical Eisenstein series.

We prove three conjectures of Tsukano about exponential sums stated in his Master's thesis written at Osaka University. These conjectures are variations of earlier conjectures made by Lee and Weintraub which were first proved by Ibukiyama and Saito.

In this short article, we give a necessary and sufficient condition for the form (1.3), like the one in Hamza's theorem, to be closable and prove that every regular and strongly local Dirichlet form is the closure of a form of Hamza's type, which can be represented in terms of effective intervals introduced in [7].

In this paper we prove two results about ${\rm SLF}(\bar U_q)$, the algebra of symmetric linear forms on the restricted quantum group $\bar U_q = \bar U_q\left(\mathfrak{sl}(2)\right)$. First, we express any trace on finite dimensional projective $\bar U_q$-modules as a linear combination in the basis of ${\rm SLF}(\bar U_q)$ constructed by Gainutdinov - Tipunin and also by Arike. In particular, this allows us to determine the symmetric linear form corresponding to the modified trace on projective $\bar U_q$-modules. Second, we give the explicit multiplication rules between symmetric linear forms in this basis.

We study the Cauchy problem for effectively hyperbolic operators $P$ with triple characteristics points lying on the initial plane $t= 0$. Under some conditions on the principal symbol of $P$ one proves that the Cauchy problem for $P$ in $[0, T] \times \Omega \subset {\mathbb R}^{n+1}$ is well posed for every choice of lower order terms. Our results improves those in [11] since we do not assume the condition (E) of [11] to be satisfied.

In this paper we study the smoothing of a semistable Fano variety over a perfect field $k$. In characteristic 0, the reduced semistable Fano degenerate fibers of Mori fibrations are classified. In positive characteristic, under a suitable $W_2$ lifting assumption, we prove that a semistable Fano variety always appears as a degenerate fiber in a semistable family if it has a global log structure (in the sense of Fontaine-Illusie-Kato) of semistable type. A geometric criterion for the existence of a log structure of semistable type is given.

Bowman and Bradley obtained a remarkable formula among multiple zeta values. The formula states that the sum of multiple zeta values for indices which consist of the shuffle of two kinds of the strings $\{1,3,\ldots,1,3\}$ and $\{2,\ldots,2\}$ is a rational multiple of a power of $\pi^2$. Recently, Saito and Wakabayashi proved that analogous but more general sums of finite multiple zeta values in an adelic ring $\mathcal{A}_1$ vanish. In this paper, we partially lift Saito-Wakabayashi's theorem from $\mathcal{A}_1$ to $\mathcal{A}_2$. Our result states that a Bowman-Bradley type sum of finite multiple zeta values in $\mathcal{A}_2$ is a rational multiple of a special element and this is closer to the original Bowman-Bradley theorem.

An extrinsic symmetric space is a submanifold $M\subset V = {\mathbb R}^n$ which is kept invariant by the reflection $s_x$ along every normal space $N_xM$. An extrinsic symmetric subspace is a connected component $M'$ of the intersection $M\cap V'$ for some subspace $V'\subset V$ which is $s_x$-invariant for any $x\in M'$. We give an algebraic charactrization of all such subspaces $V'$.

The Goeritz matrix of a link is obtained from the Jacobian matrix of a modified Dehn presentation associated to a diagram using Fox's free differential calculus. When the diagram is special the Seifert matrix can also be determined from the presentation.

Let $W$ be a manifold with boundary $M$ given together with a conformal class $\bar C$ which restricts to a conformal class $C$ on $M$. Then the relative Yamabe constant $Y_{\bar C}(W,M;C)$ is well-defined. We study the short-time behavior of the relative Yamabe constant $Y_{[\bar g_t]}(W,M;C)$ under the Ricci flow $\bar g_t$ on $W$ with boundary conditions that mean curvature $H_{\bar g_t}\equiv 0$ and $\bar{g}_t|_M\in C = [\bar{g}_0]$. In particular, we show that if the initial metric $\bar{g}_0$ is a Yamabe metric, then, under some natural assumptions, $\left.\frac{d}{dt}\right|_{t=0}Y_{[\bar g_t]}(W,M;C)\geq 0$ and is equal to zero if and only the metric $\bar{g}_0$ is Einstein.

Brauer tree algebras are important and fundamental blocks in the representation theory of finite dimensional algebras. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between an ordinary Brauer tree algebra and the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tree algebra to the Brauer star algebra. Our main purpose in this research is to combine these two approaches. We first find an algorithm based on centers which are all terminal edges, for which we are able to obtain a tilting complex constructed from irreducible complexes of length two [13], which is obtained from a sequence of mutations. In [1], Aihara gave an algorithm for reducing from tree to star by mutations and showed that it gave a two-term tree-to-star complex. We prove that Aihara's complex is obtained from the corresponding completely folded Rickard tree-to-star complex by a permutation of projectives.

Relatively cuspidal representations attached to a $p$-adic symmetric space $G/H$ are thought of as the building blocks for all the irreducible $H$-distinguished representations of $G$. This work provides certain new examples of relatively cuspidal representations. We study three examples of symmetric spaces; ${\rm GL}_n(E)/{\rm GL}_n(F)$, ${\rm GL}_{2m}(F)/{\rm GL}_m(E)$, and ${\rm GL}_n(F)/\bigl({\rm GL}_{n-r}(F)\times{\rm GL}_r(F)\bigr)$ where $E/F$ is a quadratic extension of $p$-adic fields. Those representations are given by induction from cuspidal distinguished representations of particular kinds of parabolic subgroups stable under the involution.

A handlebody-knot is a handlebody embedded in the $3$-sphere. We introduce an invariant for handlebody-knots derived from their Alexander polynomials. The value of the invariant is a vertex-weighted graph. As an application, we describe a sufficient condition for a handlebody-knot to be irreducible and a necessary condition for a link to be a constituent link of a handlebody-knot.