We construct a system of non-linear differential equations from the uniformizing differential equations of an orbifold attached to certain Hilbert modular surface. Generic solutions of this system can be given by the logarithmic derivatives of Hilbert modular forms.

For integers $d \geq 0$ and $1 \neq k \neq n$, let $\mathrm{Hol}_{d}(\mathbb{C}P^{k},\mathbb{C}P^{n})$ denote the space consisting of all holomorphic maps $f : \mathbb{C}P^{k}\to \mathbb{C}P^{n}$ of degree $d$. We shall compute the fundamental group of $\mathrm{Hol}_{d}(\mathbb{C}P^{k},\mathbb{C}P^{n})$ and study $\mathrm{PGL}_{n+1}(\mathbb{C})$-action on it.

We decompose the most generalized Cantor set into a spectral class using weak lower (upper) local dimension. Each member of the spectral class is related to a quasi-self-similar measure, so the information of its Hausdorff (packing) dimension can be obtained. In the end, we give an example of the Cantor set having countable members composing the spectral class.

We prove that, as Wiener functionals, local times of two-parameter martingales belong to some fractional Sobolev spaces over the Wiener space and, therefore, exist $(2,\infty )$-quasi surely.

T. Ikawa obtained in [5] the following characteristic ordinary differential equation \[ \begin{array}{cc} \nabla _{X}\nabla _{X}\nabla _{X}X-K\nabla _{X}X=0,& K=k^{2}-\tau ^{2} \end{array} \] for the circular helix which corresponds to the case that the curvatures $k$ and $\tau$ of a time-like curve $\alpha$ on the Lorentzian manifold $M$ are constant.

N. Ekmekçi and H. H. Hacısalihoğlu generalized in [4] T. Ikawa’s this result, i.e., $k$ and $\tau$ are variable, but $\frac{k}{\tau}$ is constant. In [1] H. Balgetir, M. Bektaş and M. Ergüt obtained a geometric characterization of null Frenet curve with constant ratio of curvature and torsion (called null general helix).

In this paper, making use of method in [1, 4, 5] , we obtained characterizations of a curve with respect to the Frenet frame of ruled surfaces in the 3-dimensional pseudo-Galilean space $G_{3}^{1}$.

In this paper, we study Riccati solutions of Painlevé equations from a view point of geometry of Okamoto-Painlevé pairs $(S,Y)$. After establishing the correspondence between (rational) nodal curves on $S-Y$ and Riccati solutions, we give the complete classification of the configurations of nodal curves on $S-Y$ for each Okamoto-Painlevé pair $(S,Y)$. As an application of the classification, we prove the non-existence of Riccati solutions of Painlev´e equations of types $P_{I}$, $P_{III}^{\Bar{D}_8}$ and $P_{III}^{\Bar{D}_7}$. We will also give a partial answer to the conjecture in [STT] and [T1] that the dimension of the local cohomology $H_{Y_{red}}^{1}(S,\Theta _{S}(-\log Y_{red}))$ is one.

We construct a new homotopy fibration at the prime 5, involving $E_{8}$ and Harper’s rank two finite mod-5 $H$-space. We then use this to show that the 5-primary homotopy exponent of $E_{8}$ is bounded above by $5^{31}$, which is at most one power of 5 from being optimal.

Let $GL(n,R)$ be the general linear group of $n \times n$ real matrices. Definitions of $GL(n,R)$-equivalence and the centro-affine type of curves are introduced. All possible centro-affine types are founded. For every centro affine type all invariant parametrizations of a curve are described. The problem of $GL(n,R)$-equivalence of curves is reduced to that of paths. A generating system of the differential field of invariant differential rational functions of a path is described. They can be viewed as centro-affine curvatures of a path. Global conditions of $GL(n,R)$-equivalence of curves are given in terms of the centro-affine type and the generating differential invariants. Independence of elements of the generating differential invariants is proved.

The difference of the integrated densities of states (IDS) of the two components of a random Pauli Hamiltonian is shown to equal a constant given in terms of the expectation of the magnetic field. This formula is a random version of the Aharonov and Casher theory or that of the Atiyah and Singer index theorem. By this formula, the IDS is shown to jump at 0 if the expectation of the magnetic field is nonzero. For simple cases where the expectation of the magnetic field is zero, a lower estimate of the asymptotics of the IDS at 0 is given. This lower estimate shows that the IDS decays slower than known results for random Schrödinger operators whose infimum of the spectrum is 0. Moreover the strong-magnetic-field limit of the IDS is identified in a general setting.

In this paper we investigate the homotopy set $[X,U(n)]$, where $X$ is a finite CW-complex with its dimension $2n + 1$ and $U(n)$ is the unitary group. This homotopy set has the group structure and can be described as a central extension using ordinary cohomology and $K$-theory.