We systematically construct isoparametric functions on compact symmetric spaces using vector bundles and sections of the bundles. We establish a relation between invariants of vector bundles and invariants of hypersurfaces which are the level sets of the isoparametric functions induced by sections of the bundles. We hope that this approach provides a new method for computing invariants of hypersurfaces. The Radon transform is performed to derive isoparametric functions on spheres from our functions.

We study the strong instability of standing waves $e^{i\omega t}\phi_\omega(x)$ for nonlinear Schrödinger equations with an $L^2$-supercritical nonlinearity and an attractive inverse power potential, where $\omega\in\mathbb{R}$ is a frequency, and $\phi_\omega\in H^1(\mathbb{R}^N)$ is a ground state of the corresponding stationary equation. Recently, for nonlinear Schrödinger equations with a harmonic potential, Ohta~(2018) proved that if $\partial_\lambda^2S_\omega(\phi_\omega^\lambda)|_{\lambda=1}\le0$, then the standing wave is strongly unstable, where $S_\omega$ is the action, and $\phi_\omega^\lambda(x)\mathrel{\mathop:}=\lambda^{N/2}\phi_\omega(\lambda x)$ is the scaling, which does not change the $L^2$-norm. In this paper, we prove the strong instability under the same assumption as the above-mentioned in inverse power potential case. Our proof is applicable to nonlinear Schrödinger equations with other potentials such as an attractive Dirac delta potential.

We will give a necessary condition for local connectedness of the space of Kleinian punctured torus group using Bromgerg's local coordinate system and provide a sufficient condition for local connectedness on a dense subset of the necessary condition. That is, the collection of the points where the boundary of the space of punctured torus group is not locally connected is a dense subset of the points satisfying the necessary condition.

In this paper, we prove two mutually independent theorems on the family of Fock-Bargmann-Hartogs domains. Let $D_1$ and $D_2$ be two Fock-Bargmann-Hartogs domains in $\mathbb{C}^{N_1}$ and $\mathbb{C}^{N_2}$, respectively. In Theorem 1, we obtain a complete description of an arbitrarily given proper holomorphic mapping between $D_1$ and $D_2$ in the case where $N_1 = N_2$. Also, we shall give a geometric interpretation of Theorem 1. And, in Theorem 2, we determine the structure of $\text{Aut}(D_1\times D_2)$ using the data of $\text{Aut}(D_1)$ and $\text{Aut}(D_2)$ for arbitrary $N_1$ and $N_2$.

Let $M$ be an even-dimensional closed oriented manifold and $g$ a periodic automorphism of $M$ of order $p$. In this paper, under the assumption that the fixed points of $g^k\;(1\leq k\leq p-1)$ are isolated, a calculation formula is provided for the homomorphism $I_D:{\Bbb Z}_p\to{\Bbb R}/{\Bbb Z}$ defined in [6] for equivariant twisted signature operators $D$ over $M$. The formula gives a new method to study the periodic automorphisms of oriented manifolds. As examples of the application of the formula, results about the existence of the cyclic group action for 2,4,6-dimensional closed oriented manifolds are obtained.

We give an algorithm for computing the Teichmüller polynomial for a certain class of fibered alternating links associated to trees. Furthermore, we exhibit a mutant pair of such links distinguished by the Teichmüller polynomial.

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms in ${\mathbb R}^{n}$. As is pointed out by [10], in this combination, the frictional damping term is dominant for the viscoelastic one for the global dynamics of the linear equation. In this note we observe that if the initial data is small, the frictional damping term is again dominant even in the nonlinear equation case. In other words, our main result is diffusion phenomena: the solution is approximated by the heat kernel with a suitable constant. Especially, the result obtained for the $n = 3$ case is essentially new. Our proof is based on several estimates for the corresponding linear equations.

We show that the solution of Schrödinger's functional equation is measurable in space, kernel and marginals. As an application, we show that the drift vector of the h-path process with given two end point marginals is a measurable function of space, time and marginal at each time. In particular, we show that the coefficients of mean field PDE systems which the marginals satisfy are measurable functions of space, time and marginal.

$\zeta(\cdot)$ being the Riemann zeta function, $\zeta_{\sigma}(t) := \frac{\zeta(\sigma + i t)}{\zeta(\sigma)}$ is, for $\sigma > 1$, a characteristic function of some infinitely divisible distribution $\mu_{\sigma}$. A process with time parameter $\sigma$ having $\mu_{\sigma}$ as its marginal at time $\sigma$ is called a Riemann zeta process. Ehm [2] has found a functional limit theorem on this process being a backwards Lévy process. In this paper, we replace $\zeta(\cdot)$ with a Dirichlet series $\eta(\cdot;a)$ generated by a nonnegative, completely multiplicative arithmetical function $a(\cdot)$ satisfying (3), (4) and (5) below, and derive the same type of functional limit theorem as Ehm on the process corresponding to $\eta(\cdot;a)$ and being a backwards Lévy process.

We study the asymptotic behavior of global solutions to the initial value problem for the generalized KdV-Burgers equation. One can expect that the solution to this equation converges to a self-similar solution to the Burgers equation, due to earlier works related to this problem. Actually, we obtain the optimal asymptotic rate similar to those results and the second asymptotic profile for the generalized KdV-Burgers equation.