The $n$ dimensional hypergeometric integrals associated with a hypersphere arrangement $S$ are formulated by the pairing of $n$ dimensional twisted cohomology $H_{\nabla}^{n} (X, \Omega^{\cdot} (*S))$ and its dual. Under the condition of general position there are stated some results and conjectures which concern a representation of the standard form by a special basis of the twisted cohomology, the variational formula of the corresponding integral in terms of special invariant 1-forms using Calyley-Menger minor determinants, a connection relation of the unique twisted $n$-cycle identified with the unbounded chamber to a special basis of twisted $n$-cycles identified with bounded chambers. General conjectures are presented under a much weaker assumption.

We give Sturm bounds for Siegel modular forms of degree 2 with Fourier coefficients in an arbitrary algebraic number field $K$ and for any prime ideal $\mathfrak{p}$ in $K$.

In this paper, we prove an analog of Younis’s theorem 5.2 in~[4] for the generalized Fourier-Bessel transform on the Half line for functions satisfying the $(\beta, \gamma)$-generalized Bessel Lipschitz condition in the space $\mathrm{L}^{2}_{\alpha,n}$.

We study Fano manifolds with nef tangent bundle and large Picard number.