In this article we investigate the rate of the approximation by Cesàro means of the quadratical partial sums of double Walsh-Kaczmarz-Fourier series of a function in the Lebesgue space over the Walsh group. The approximation properties of Cesàro means of negative order of one- and two-dimensional Walsh-Fourier series was discussed earlier by Goginava.

Let $\Delta$ be a spherical building each of whose irreducible components is infinite, has rank at least 2 and satisfies the Moufang condition. We show that $\Delta$ can be given the structure of a topological building that is compact and totally disconnected precisely when $\Delta$ is the building at infinity of a locally finite affine building.

Skoruppa and Zagier established a bijective correspondence from the space of Jacobi forms $\phi$ of index $m$ to that of elliptic modular forms $f$ of level $m$. Gross, Kohnen and Zagier formulated this correspondence by means of kernel functions. Moreover, they proved that the squares of Fourier coefficients of $\phi$ are essentially equal to the critical values of the zeta functions $L(s,f,\chi)$ of $f$ twisted by a quadratic character $\chi$.

The purpose of this paper is to prove a generalization of such results concerning liftings and Fourier coefficients of Jacobi forms to the case of Jacobi forms of index $N$ over totally real number fields $F$. Using kernel functions associated with the space of quadratic forms, we shall establish the existence of a lifting from the space of Jacobi forms $\phi$ of index $N$ over $F$ to that of Hilbert modular forms $f$ of level $N$ over $F$. Moreover, we determine explicitly the Fourier coefficients of $f$ from those of $\phi$. We prove that an analogue of Waldspurger's theorem in the case of Jacobi forms of index $N$ over $F$ holds.

We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and 3-dimensional $D_4^\pm$ singularities. We give criteria for these singularity types in terms of differential geometry (Theorems 3.4 and 3.5).

A topological invariant of reduced sextic curves with simple singularities is given. Twelve new Zariski pairs of sextic curves are determined. Each pair consists of two curves with non-isomorphic fundamental groups.

In this paper, we prove the algebraic independence over ${\boldsymbol C}(z)$ of the generating functions of pattern sequences defined in distinct $\langle q,r \rangle$-numeration systems. Our result asserts that any nontrivial linear combination over ${\boldsymbol C}$ of pattern sequences chosen from distinct $\langle q,r \rangle$-numeration systems can not be a linear recurrence sequence. As an application, we give a linear independence over ${\boldsymbol C}$ of the pattern sequences.

We study the heat kernel transform on a nilmanifold $ \Gamma \backslash N $ associated to an H-type group. Using a reduction technique we reduce the problem to the case of Heisenberg groups. The image of $ L^2(\Gamma \backslash N) $ under the heat kernel transform is shown to be a direct sum of weighted Bergman spaces.

In this paper, we shall prove the uniqueness of Sasaki-Einstein metrics on compact Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure. This generalizes the result of Cho, Futaki and Ono for compact toric Sasaki manifolds.