We are interested in computing alternate sums of Euler characteristics of some particular semialgebraic sets, intersections of an algebraic one, smooth or with finitely many singularities, with sets given by just one polynomial inequality. We state theorems relating these alternate sums of characteristics to some topological degrees at infinity of polynomial mappings.

In the present paper, we study the behavior of adic sheaves over a weighted projective line of genus one or an elliptic curve, and describe the relationship between adic sheaves and generic sheaves. Moreover, we construct the generic sheaf through the coherent sheaf and adic sheaves in the sense of extension.

Let $E_C$ be the hypergeometric system of differential equations satisfied by Lauricella's hypergeometric series $F_C$ of $m$ variables. This system is irreducible in the sense of $D$-modules if and only if $2^{m+1}$ non-integral conditions for parameters are satisfied. We find a linear transformation of the classically known $2^m$ solutions so that the transformed ones always form a fundamental system of solutions under the irreducibility conditions. By using this fundamental system, we give an elementary proof of the irreducibility of the monodromy representation of $E_C$. When one of the conditions is not satisfied, we specify a non-trivial invariant subspace, which implies that the monodromy representation is reducible in this case.

Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. They are time-dependent dynamical systems with hyperbolic behavior. In addition to presenting several properties and examples of Anosov families, in this paper we build local stable and local manifolds for such families.

We prove that the sign of the Gaussian curvature, which is closely related to the diagonalizability of the shape operator, of any timelike minimal surface in the 3-dimensional Lorentz-Minkowski space is determined by the degeneracy and the signs of the two null regular curves that generate the surface. We also investigate the behavior of the Gaussian curvature near singular points of a timelike minimal surface with some kinds of singular points, which is called a minface. In particular we determine the sign of the Gaussian curvature near any non-degenerate singular point of a minface.

In this paper, we introduce the truncated Euler-Carlitz numbers as analogues of hypergeometric Euler numbers. In a special case, Euler-Carlitz numbers are defined, which is an analogue of the classical Euler numbers. We give several interesting properties for these numbers. We also show some determinant expressions of Euler-Carlitz numbers.

The purpose of this paper is to give geometric characterizations of minimal ruled real hypersurfaces $M$ in a non-flat complex space form by paying particular attention to integral curves of the characteristic vector field on $M$.

The aim of this paper is to investigate the decompositional structure of generalized fractional integral operators whose kernels are the generalized hypergeometric functions of certain type. By using the Mellin transform theory proposed by Butzer and Jansche [J. Fourier Anal. 3 (1997), 325-376], we prove that these operators can be decomposed in terms of Laplace and inverse Laplace transforms. As applications, we derive two very general results involving the $H$-function. We also show that these fractional integral operators when being understood as integral equations possess the $\mathcal{L}$ and $\mathcal{L}^{-1}$ solutions. We also consider the applications of the decompositional structures of the fractional integral operators to some specific integral equations and one of such integral equations is shown to possess a solution in terms of an Aleph $(\aleph)$-function.