We obtain a finite set of generators for the level 2 mapping class group of a closed nonorientable surface of genus g ≥ 3. This set consists of isotopy classes of Lickorish's Y-homeomorphisms also called crosscap slides.

The Banach space of bounded Dirichlet finite harmonic functions on an open Riemann surface will be seen to be reflexive and also separable if and only if the underlying Riemann surface does not carry any unbounded Dirichlet finite harmonic function.

We define the Toeplitz operators on the parabolic Bergman spaces by using a positive bilinear form. In this setting we characterize finite rank Toeplitz operators. A relation with the Carleson inclusion is also discussed.

In this paper, we prove that an Hermitian modular form with small weight over the quadratic field with class number one is a linear combination of theta series associated with Hermitian quadratic forms.

We show that if two meromorphic functions sharing a one-point set and three two-point sets CM, then one of them is a Möbius transform of the other.

Let $\cal F$ be a family of meromorphic functions defined in a domain D $subset$ C, let ψ₁, ψ₂ and ψ₃ be three meromorphic functions such that ψ_i(z) $\not\equiv$ ψ_j(z) (i ≠ j) in D, one of which may be ∞ identically, and let l₁, l₂ and l₃ be positive integers or ∞ with 1/l₁ + 1/l₂ + 1/l₃ < 1. Suppose that, for each f $in$ $\cal F$ and z $in$ D, (1) all zeros of f – ψ_i have multiplicity at least l_i for i = 1,2,3; (2) f(z₀) ≠ ψ_i(z₀) if there exist i, j $in$ {1,2,3} (i ≠ j) and z₀ $in$ D such that ψ_i(z₀) = ψ_j(z₀). Then $\cal F$ is normal in D. This improves and generalizes Montel's criterion.

Let F = (F₁, F₂, ..., F_m): Cⁿ → C^m be a polynomial dominant mapping with n > m. In this paper we give the relations between the bifurcation set of F and the set of values where F is not M-tame as well as the set of generalized critical values of F. We also construct explicitly a proper subset of C^m in terms of the Newton polyhedrons of F₁, F₂, ..., F_m and show that it contains the bifurcation set of F. In the case m = n – 1 we show that F is a locally C^∞-trivial fibration if and only if it is a locally C⁰-trivial fibration.

The leaf-wise intersection on a coisotropic submanifold of a symplectic manifold is a generalization of the Lagrangian intersection investigated by Weinstein. In a similar way as Weinstein's argument, we replace the leaf-wise intersections by zero points of some closed 1-form, and show the same result as Moser's on the existence of leaf-wise intersections under different conditions.

We improve an estimate of A. Granville (1987) on the number of vanishing Fermat quotients q_p (ℓ) modulo a prime p when ℓ runs through primes ℓ ≤ N. We use this bound to obtain an unconditional improvement of the conditional (under the Generalised Riemann Hypothesis) estimate of Y. Ihara (2006) on a certain sum, related to vanishing Fermat quotients. In turn this sum appears in the study of the index of certain subfields of cyclotomic fields Q(exp(2πi/p²)).

We investigate addition relations for multiple sine functions from the view point of formal group laws. We find that the functions which appear in the coefficients are related to classical Eisenstein serires. As application we obtain a limit formula for automorphic forms.

The aim of this paper is to study the semilocal convergence of the Super-Halley's method used for solving nonlinear equations in Banach spaces by using the recurrence relations. This convergence is established under the assumption that the second Frëchet derivative of the involved operator satisfies the Hölder continuity condition which is milder than the Lipschitz continuity condition. A new family of recurrence relations are defined based on two constants which depend on the operator. An existence-uniqueness theorem and a proori error estimates are provided for the solution x*. The R-order of the method equals to (2 + p) for p $in$ (0,1] is also established. Three numerical examples are worked out to demonstrate the efficacy of our approach. On comparison with the results obtained for the Super-Halley's method in [3] by using majorizing sequence, we observed improved existence and uniqueness regions for the solution x* by our approach.

A notion of "delta-genus" for ample vector bundles $\mathcal{E}$ of rank two on a smooth projective threefold X is defined as a couple of integers (δ₁, δ₂). This extends the classical definition holding for ample line bundles. Then pairs (X, $\mathcal{E}$) with low δ₁ and δ₂ are classified under suitable additional assumptions on $\mathcal{E}$.

In this paper, we construct a new family of contact metric structures on the unit sphere S³. Especially, the above family has the property that ∇_ξτ = 2ατφ.

We give an easy description of the barycentric extension of a map of the unit circle to the closed unit disk using some ideas from dynamical systems. We then prove that every circle endomorphism of the unit circle of degree d ≥ 2 (with a topological expansion condition) has a conformally natural extension to the closed unit disk which is real analytic on the open unit disk. If the endomorphism is uniformly quasisymmetric, then the extension is quasiconformal.

We compute the extensions of the Euler-Satake characteristic of a closed, effective, orientable 3-orbifold corresponding to free and free abelian groups in terms of the number and type of point singularities of the orbifold. Using these computations, we show that the free Euler-Satake characteristics determine the number and type of point singularities, and that it takes an infinite collection of free Euler-Satake characteristics to do so. Additionally, we show that the stringy orbifold Euler characteristic determines all of the free abelian Euler-Satake characteristics for an orbifold in this class.