Mixed functions are analytic functions in variables z₁, ..., z_n and their conjugates $\bar z_1$, ..., $\bar z_n$. We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed hypersurface singularities. We show the existence of a canonical resolution of the singularity, and the existence of the Milnor fibration under the strong non-degeneracy condition.

In this paper, we consider a nonsingular curve C on a nonsingular compact toric surface S and intersection points of C and T-invariant divisors on S. We provide a sufficient condition for a positive integer to be a gap value of C at such points. Under a suitable assumption, it becomes the necessary and sufficient condition. We determine several Weierstrass gap sequences at infinitely near points of a point on a plane curve by using this method.

We characterize the dual spaces of restrictions of a dual pair of reproducing kernel Hilbert spaces in a discrete set. Consequently, we give a canonical dense subset to the restriction spaces. As applications, we reprove a variational principle in a dual pair of reproducing kernel Hilbert spaces. Also we give a geometric representation for the existence and ergodicity condition of equilibrium Glauber and Kawasaki dynamics for some determinantal point processes.

An afforested surface W := <P, (T_n)_n$\in$N, (σ_n)_n$\in$N>, N being the set of positive integers, is an open Riemann surface consisting of three ingredients: a hyperbolic Riemann surface P called a plantation, a sequence (T_n)_n$\in$N of hyperbolic Riemann surfaces T_n each of which is called a tree, and a sequence (σ_n)_n$\in$N of slits σ_n called the roots of T_n contained commonly in P and T_n which are mutually disjoint and not accumulating in P. Then the surface W is formed by foresting trees T_n on the plantation P at the roots for all n $\in$ N, or more precisely, by pasting surfaces T_n to P crosswise along slits σ_n for all n $\in$ N. Let ${\mathscr O}_s$ be the family of hyperbolic Riemann surfaces on which there are no nonzero singular harmonic functions. One might feel that any afforested surface W := <P, (T_n)_n$\in$N, (σ_n)_n$\in$N> belongs to the family ${\mathscr O}_s$ as far as its plantation P and all its trees T_n belong to ${\mathscr O}_s$. The aim of this paper is, contrary to this feeling, to maintain that this is not the case.

Let K be an n-dimensional knot (n ≥ 1), Q(K) the knot quandle of K, Z_q[t^±1]/J an Alexander quandle, and C_∞(K) the infinite cyclic covering space of Sⁿ⁺²$\backslash$K. We show that the set consisting of homomorphisms Q(K) → Z_q[t^±1]/J is isomorphic to Z_q[t^±1]/J ⊕ Hom_Z[t^±1] (H₁(C_∞(K)), Z_q[t^±1]/J) as Z[t^±1]-modules. Here, Hom_Z[t^±1](H₁(C_∞(K)), Z_q[t^±1]/J) denotes the set consisting of Z[t^±1]-homomorphisms H₁(C_∞(K)) → Z_q[t^±1]/J.

Real hypersurfaces in non-flat complex space forms with integrable holomorphic distribution and symmetric φ-Ricci tensor which are φ-Einstein are ruled real hypersurfaces.

We are interested in biaccessible points in the Julia sets of rational functions. D. Schleicher and S. Zakeri studied which points can be biaccessible in the Julia sets of quadratic polynomials with irrationally indifferent fixed points [SZ, Za]. In this paper, we consider the two polynomial families f_c(z) = z^d + c, g_θ(z) = e^2πiθz + z^d and the cubic rational family h_θ,a(z) = e^2πiθz²$\frac{z-a}{1-\bar{a}z}$.

We obtain several results of quotient curves of smooth plane curves with automorphisms. Such automorphisms can be divided into two types (type I and type II). The quotient curves of smooth plane curves with automorphisms of type I are extremal curves in the sense of Castelnuovo's bound. We also show some partial result on automorphisms of type II and give examples.