In this paper, we prove that if n ≥ 2 and x0 is an isolated singularity of a non-negative infinity harmonic function u, then either x0 is a removable singularity of u or u(x) = u(x0) + c|x − x0| + o(|x − x0|) near x0 for some fixed constant c = 0. In particular, if x0 is nonremovable, then u has a local maximum or a local minimum at x0. We also prove a Bernstein-type theorem, which asserts that...

For any positive integer N, we completely determine the structure of rational cuspidal divisor class group X0(N), which is conjecturally equal to torsion subgroup J0(N). More specifically, for a given prime ℓ, construct Zℓ(d) non-trivial d N. Also, compute order linear equivalence and show that ℓ-primary X0(N) isomorphic direct sum cyclic subgroups generated by classes Zℓ(d).

and Applied Analysis 3 Lemma 7. Assume that conditions (H1), (H2) hold. Suppose that x(t, t0, x0) is a solution of (1) satisfying initial value x(t + 0 ) = x0. Then −x(t + T, t0, x0) is also a solution of (1), and −x (t + T, t0, x0) = x (t, t0, −x (t0 + T, t0, x0)) , t ∈ R. (5) Proof. Let φ(t) ≡ −x(t + T, t0, x0), ψ(t) ≡ x(t, t0, −x(t0 + T, t0, x0)). Then for t ̸ = τk, k ∈ Z, we have by (H2) tha...

Given a C1,1–function f : U → R (where U ⊂ Rn open) we deal with the question of whether or not at a given point x0 ∈ U there exists a local minorant φ of f of class C2 that satisfies φ(x0) = f(x0), Dφ(x0) = Df(x0) and Dφ(x0) ∈ Hf(x0) (the generalized Hessian of f at x0). This question is motivated by the second-order viscosity theory of the PDE, since for nonsmooth functions, an analogous resu...

The papers [2], [7], [13], [3], [1], [6], [9], [4], [14], [8], [5], [15], [11], [12], and [10] provide the notation and terminology for this paper. We adopt the following rules: n denotes an element of N, h, k, x, x0, x1, x2, x3 denote real numbers, and f , g denote functions from R into R. Next we state a number of propositions: (1) If x0 > 0 and x1 > 0, then loge x0 − loge x1 = loge( x1 ). (2...

1.1 Topology Let us first recall some basic Euclidean topology. For x0 = (x0, y0, z0) ∈ R3 and r > 0 we define the open ball of radius r > 0 centered at x0 to be B(x0, r) = {x ∈ R | ‖x− x0‖ < r}. Recall that ‖x‖ = √ x2 + y2 + z2 is the Euclidean norm, or length, of the vector x. Definition 1 (Open set). We say a set D ⊂ R3 is open if for every x0 ∈ D, there exists a radius r > 0 such that B(x0,...

In this note we will obtain defining equations of modular curves X0(2). The key ingredient is a recursive formula for certain generators of the function fields on X0(2).