Upper bounds for the solution $l$ of the equations of the title are derived by using results concerning the equation $ax^l-by^l=c$ with $a,b,c$ non-zero integers. These solutions are also determined in some special cases.

We give numerical, complete invariants for topological orbit equivalence and Kakutani orbit equivalence in a class of substitution systems arising from primitive substitutions whose composition matrices have rational Perron-Frobenius eigenvalues.

The Schr\"odinger-type equation $-\Delta u+Vu=\lambda u$ on a noncompact Riemannian manifold $\mathcal{M}$ has no nontrivial square integrable solution $u$ for any positive constant $\lambda$, if the metric and the function $V$ satisfy certain conditions near the infinity. A set of conditions of that kind was given by the author in the case that the metric is rotationally symmetric. It contained a condition which required smallness of the curvatures of $\mathcal{M}$ in the distance. But we have had a question whether the set could remain sufficient even if we remove that condition. The present paper answers it negatively by constructing a square integrable solution for a metric which satisfies all the conditions except the one in question.

We give necessary and sufficient conditions for the existence of spacelike maximal surfaces in 4-dimensional space forms of index 2. We also discuss spacelike maximal surfaces with constant Gaussian curvature or constant normal curvature, and a rigidity type problem.

We consider the steady Navier-Stokes equations $$ \left\{ \begin{array}{@{}l@{\hspace{2pt}}ll} (\mathbf{u}\cdot\nabla)\mathbf{u} & =\nu \Delta\mathbf{u} -\nabla p & \text{in}~\Omega\,,\\ \text{div}\,\mathbf{u} & =0 & \text{in} ~\Omega\,, \end{array} \right. $$ in a 2-dimensional unbounded multiply-connected domain $\Omega$ contained in an infinite straight channel $\mathbf{R}\times(-1,1)$, under general outflow condition. We look for a solution which tends to a Poiseuille flow at infinity. In this note, we shall show the existence of solution to this problem under the assumption of symmetry with respect to the axis for the domain and the boundary value, and for small Poiseuille flow. We do not assume that the boundary value is small. The regularity and the asymptotic behavior of the solution are also discussed.

In this note Hurwitzian numbers are defined for the nearest integer, and backward continued fraction expansions, and Nakada's $\alpha$-expansions. It is shown that the set of Hurwitzian numbers for these continued fractions coincides with the classical set of such numbers.

We consider Fourier ultra-hyperfunctions and characterize them as boundary values of smooth solutions of the heat equation. Namely we show that the convolution of the heat kernel and a Fourier ultra-hyperfunction is a smooth solution of the heat equation with some exponential growth condition and, conversely that such smooth solution can be represented by the convolution of the heat kernel and a Fourier ultra-hyperfunction.

In [7], the second author classified configurations of the singularities on tame sextics of torus type. In this paper, we give a complete classification of the singularities on irreducible sextic of torus type, without assuming the tameness of the sextics. We show that there exist 121 configurations and there are 5 pairs and a triple of configurations for which the corresponding moduli spaces coincide, ignoring the respective torus decomposition.

Mean-variance hedging is well-known as one of hedging methods for incomplete markets. Our end is leading to mean-variance hedging strategy for incomplete market models whose asset price process is given by a discontinuous semimartingale and whose mean-variance trade-off process is not deterministic. In this paper, on account, we focus on this problem under the following assumptions: (1) the local martingale part of the stock price process is a process with independent increments; (2) a certain condition restricting the number and the size of jumps of the asset price process is satisfied; (3) the mean-variance trade-off process is uniformly bounded; (4) the minimal martingale measure coincides with the variance-optimal martingale measure.

Our aim is to check that the notions of positive entropy, chaos in the sense of Devaney and $\omega$-chaos are equivalent for the circle maps.

We define $q$-linear arithmetical functions and $-q$-linear ones and show the algebraic independence over $\mathbb{C}(z)$ of their generating functions.

It is known that there are no Einstein real hypersurfaces in complex space forms equipped with the K\"ahler metric. In the present paper we classified the $*$-Einstein real hypersurfaces $M$ in complex space forms $M_{n}(c)$ and such that the structure vector is a principal curvature vector.

We present a new form of the fiber surface $F$ for links arising from isolated complex plane curve singularities, in particular, the torus knot $T(p, q)$, where $F$ is a smoothing of a long thin band which has as many clasp-singularities as the unknotting number of $T(p, q)$. As an application, we give a visual proof that the Lissajous curve $(\cos{p\theta}, \cos{q\theta})$ regarded as a divide corresponds to the torus link $T(p, q)$.