In this paper, we shall prove that if an irreducible curve *X* of genus 5 over ℂ has 24 Weierstrass points of weight 5, then it has exactly three bielliptic involutions.

The *minimal unfolded region* (or the *heart*) of a bounded subset ¶ in the Euclidean space is a subset of the convex hull of ¶ the definition of which is based on reflections in hyperplanes. It was introduced to restrict the location of the points that give extreme values of certain functions, such as potentials whose kernels are monotone functions of the distance, and solutions of differential equations to which Aleksandrov's reflection principle can be applied. We show that the minimal unfolded regions of the convex hull and parallel bodies of ¶ are both included in that of ¶.

Our aim in this paper is to deal with the boundedness of the Hardy-Littlewood maximal operator in non-homogeneous central Morrey spaces of variable exponent. Further, we give Sobolev's inequality and Trudinger's exponential integrability for generalized Riesz potentials.

Flows on a network play an important role in the theory of discrete harmonic functions. In the study of discrete bi-harmonic functions, we encounter a concept of bi-flows. In this paper, we are concerned with minimization problems for bi-flows which are analogous to those for flows.

The present paper deals with the study of the geometrical properties of generic 1 and 2-parameter families of space curves by using projections into planes. It presents directions of projection and conditions on the coefficients of these families such that the projection exhibits Morsifications of the *A*_{4}, *A*_{6} and *E*_{6} singularities and transitions between the Morsifications of the *E*_{8} singularity.

We give a formula to determine the indices of special (non-totally geodesic) minimal orbits of Hermann actions. Also, we give examples of such minimal orbits of Hermann actions and calculate their indices by using the formula.

For Hénon map of nearly classical parameter values, we search numerically for Newhouse sinks. We show how to find successively the Newhouse sinks of higher period, which is the estimation of coordinates of the sinks from power laws of properties of the sinks, and investigate numerically a sequence of sinks of period from 8 to 60 that we obtained. We also show how to verify the existences of obtained sinks by interval arithmetic. The sinks of period from 8 to 14 from among our obtained sinks was verified mathematically.

In the case that we observed, when the sink exists, most orbits converge to it, and the orbit that seems to be Hénon attractor is not an attractor but just a long chaotic transient. The narrowness of the main bands of basins of the sinks causes the long chaotic transients. We also investigate numerically the chaotic transients and their rambling time.