In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. It is known that there is no other convex domain which can form a two, three or fourfold lattice tiling in the Euclidean plane, but there are centrally symmetric convex octagons and decagons which can form fivefold lattice tilings. This paper characterizes all the convex domains which can form five or sixfold lattice tilings of the Euclidean plane. They are parallelograms, centrally symmetric hexagons, three types of centrally symmetric octagons and three types of centrally symmetric decagons.

One of the important research subjects in the study of multiple zeta functions is to clarify the linear relations and functional equations among them. The Schur multiple zeta functions are a generalization of the multiple zeta functions of Euler–Zagier type. Among many relations, the duality formula and its generalization are important families for both Euler–Zagier type and Schur type multiple zeta values. In this paper, following the method of previous works for multiple zeta values of Euler–Zagier type, we give an interpolation of the sums in the generalized duality formula, called Ohno relation, for Schur multiple zeta values. Moreover, we prove that the Ohno relation for Schur multiple zeta values is valid for complex numbers.

The paper applies the JSJ decomposition and Koda–Ozawa's annulus classification to analyze the annulus configuration in a handlebody-knot exterior. We introduce the notion of the annulus diagram, to pack the configuration into a labeled graph, and classify genus two handlebody-knots in terms of their annulus diagrams. Applications to handlebody-knot symmetries are discussed; methods to produce handlebody-knots with various types of annulus diagrams are also presented.

We study Hadamard variation of eigenvalues of Laplacian with respect to general domain perturbations. We show their existence up to the second order rigorously and characterize the derivatives, using associated eigenvalue problems in finite dimensional spaces. Then smooth rearrangement of multiple eigenvalues is explicitly given. This result follows from an abstract theory, applicable to general perturbations of symmetric bilinear forms.

In this paper, we give a refinement of a generalized Dedekind's theorem. In addition, we show that all possible values of integer group determinants of any group are also possible values of integer group determinants of its any abelian subgroup. By applying the refinement of a generalized Dedekind's theorem, we determine all possible values of integer group determinants of the direct product group of the cyclic group of order 8 and the cyclic group of order 2.

In this work, we translate at the level of decorated trees some of the crucial arguments which have been used by P. Linares et al. in their recent paper to propose a diagram-free approach for the convergence of the model in regularity structures. This allows us to broaden the perspective and enlarge the scope of singular SPDEs covered by this approach. It also sheds new light on algebraic structures introduced in the foundational paper of M. Hairer on regularity structures which was used later for recursively described renormalised models.

G. Lupton conjectured that the group of self-homotopy equivalences of an $F_{0}$-space inducing the identity on the homotopy groups is finite. Thus, the aim of this paper is to establish this conjecture.

Let $k$ be a number field and $\mathbb{A}$ be its ring of adeles. Let $U$ be a unipotent group defined over $k$, and $\sigma$ a $k$-rational involution of $U$ with fixed points $U^{+}$. As a consequence of the results of Moore, the space $L^{2}(U(k) \backslash U_{\mathbb{A}})$ is multiplicity free as a representation of $U_{\mathbb{A}}$. Setting $p^+$ to be the period integral attached to $\sigma$ on the space of smooth vectors of $L^{2}(U(k) \backslash U_{\mathbb{A}})$, we prove that if $\Pi$ is a topologically irreducible subspace of $L^{2}(U(k) \backslash U_{\mathbb{A}})$, then $p^+$ is nonvanishing on the subspace of smooth vectors in $\Pi$ if and only if $\Pi^{\vee} = \Pi^{\sigma}$. This is a global analogue of local results of Benoist and the author, on which the proof relies.

Let $G$ be a finite group. If $n \leq 5$ then any $n$-dimensional homotopy sphere never admits a smooth action of $G$ with exactly one fixed point. Let $A_{n}$ and $S_{n}$ denote the alternating group and the symmetric group on some $n$ letters. If $n \geq 6$ then the $n$-dimensional sphere possesses a smooth action of $A_{5}$ with exactly one fixed point. Let $V$ be an $n$-dimensional real $G$-representation with exactly one fixed point. It is interesting to ask whether there exists a smooth $G$-action with exactly one fixed point on the $n$-dimensional sphere such that the associated tangential $G$-representation is isomorphic to $V$. In this paper, we study this problem for nonsolvable groups $G$ and real $G$-representations $V$ satisfying certain hypotheses. Applying a theory developed in this paper, we can prove that the $n$-dimensional sphere has an effective smooth action of $S_{5}$ with exactly one fixed point if and only if $n = 6$, 10, 11, 12, or $n \geq 14$ and that the $n$-dimensional sphere has an effective smooth action of $A_{5} \times Z$ with exactly one fixed point if $n$ satisfies $n \geq 6$ and $n \neq 9$, where $Z$ is a group of order 2.

Let $f_{1} : (\mathbb{R}^{n}, \mathbf{0}_{n}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ and $f_{2} : (\mathbb{R}^{m}, \mathbf{0}_{m}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ be real analytic map germs of independent variables, where $n, m \geq 2$. Then the pair $(f_{1}, f_{2})$ of $f_{1}$ and $f_{2}$ defines a real analytic map germ from $(\mathbb{R}^{n+m}, \mathbf{0}_{n+m})$ to $(\mathbb{R}^{4}, \mathbf{0}_{4})$. We assume that $f_{1}$ and $f_{2}$ satisfy the $a_{f}$-condition at $\mathbf{0}_{2}$. Let $g$ be a strongly non-degenerate mixed polynomial of 2 complex variables which is locally tame along vanishing coordinate subspaces. A mixed polynomial $g$ defines a real analytic map germ from $(\mathbb{C}^{2}, \mathbf{0}_{4})$ to $(\mathbb{C}, \mathbf{0}_{2})$. If we identify $\mathbb{C}$ with $\mathbb{R}^{2}$, then $g$ also defines a real analytic map germ from $(\mathbb{R}^{4}, \mathbf{0}_{4})$ to $(\mathbb{R}^{2}, \mathbf{0}_{2})$. Then the real analytic map germ $f : (\mathbb{R}^{n} \times \mathbb{R}^{m}, \mathbf{0}_{n+m}) \rightarrow (\mathbb{R}^{2}, \mathbf{0}_{2})$ is defined by the composition of $g$ and $(f_{1}, f_{2})$, i.e., $f(\mathbf{x}, \mathbf{y}) = (g \circ (f_{1}, f_{2}))(\mathbf{x}, \mathbf{y}) = g(f_{1}(\mathbf{x}), f_{2}(\mathbf{y}))$, where $(\mathbf{x}, \mathbf{y})$ is a point in a neighborhood of $\mathbf{0}_{n+m}$.

In this paper, we first show the existence of the Milnor fibration of $f$. We next show a generalized join theorem for real analytic singularities. By this theorem, the homotopy type of the Milnor fiber of $f$ is determined by those of $f_{1}, f_{2}$ and $g$. For complex singularities, this theorem was proved by A. Némethi. As an application, we show that the zeta function of the monodromy of $f$ is also determined by those of $f_{1}, f_{2}$ and $g$.

Let $\tilde{G}$ be a finite group, $G$ a normal subgroup of $\tilde{G}$ and $k$ an algebraically closed field of characteristic $p > 0$. The first main result in this paper is to show that support $\tau$-tilting $k\tilde{G}$-modules with some properties are support $\tau$-tilting modules as $kG$-modules, too. As the second main result, we give equivalent conditions for support $\tau$-tilting $k\tilde{G}$-modules to satisfy the above properties, and show that the set of the support $\tau$-tilting $k\tilde{G}$-modules with the properties is isomorphic to the set of $\tilde{G}$-invariant support $\tau$-tilting $kG$-modules as posets. As an application, we show that the set of $\tilde{G}$-invariant support $\tau$-tilting $kG$-modules is isomorphic to the set of support $\tau$-tilting $k\tilde{G}$-modules in the case that the index of $G$ in $\tilde{G}$ is a $p$-power. As a further application, we give a feature of vertices of indecomposable $\tau$-rigid $k\tilde{G}$-modules. Finally, we give block versions of the above results.

The joint distribution of the first hitting time of a Brownian motion with constant drift to a sphere and the hitting place is studied. Explicit formulae by means of spherical harmonics for the joint density are presented. The result is applied to a study on the asymptotic behavior of the distribution function.