Volume 20, Issue 6

We present several new inequalities for the gamma function. One of our results states that the functional inequality holds for all x , y > 0 if and only if 1 ≤ a ≤ a 0 , where This extends the well-known result that Γ is log-convex on (0, ∞).

Let a be an integer and q a prime number. In this paper we find an asymptotic formula for the number of positive integers n ≤ x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q .

The Schwarz-Pick lemma readily implies that the derivative of any Blaschke product belongs to all the Bergman spaces A p with 0 < p < 1. It is also well known that this result is sharp: there exist a Blaschke product whose derivative does not belong to A 1 . However, the question of whether there exists an interpolating Blaschke product B with B ′ ∉ A 1 remained open. In this paper we give an explicit construction of such an interpolating Blaschke product B . A result of W. S. Cohn asserts that if and B is an interpolating Blaschke product with sequence of zeros of , then B ′ ∈ H p if and only if (1 – | a k |) 1– p < ∞. We prove that Cohn's result is no longer true for . Indeed, we construct: (a) an interpolating Blaschke product B whose sequence of zeros of satisfies (1 – | a k |) 1/2 < ∞ but B ′ ∉ H 1/2 , and (b) an interpolating Blaschke products B whose sequence of zeros of satisfies (1 – | a k |) 1– p < ∞, for all p ∈ (0, 1/2), whose derivative B ′ does not belong to the Nevanlinna class.

We refine Osserman's argument on the exceptional values of the Gauss map of algebraic minimal surfaces. This gives an effective estimate for the number of exceptional values and the totally ramified value number for a wider class of complete minimal surfaces that includes algebraic minimal surfaces. It also provides a new proof of Fujimoto's theorem for this class, which not only simplifies the proof but also reveals the geometric meaning behind it.

For any (possibly, non-unital) AF C*-algebra A with comparability of projections, let D ( A ) be the Elliott partial monoid of A , and G ( A ) the dimension group of A with scale D ( A ). For D ⊆ D ( A ) a generating set of G ( A ) let 𝒫 be the set of all formal inequalities a 1 + ⋯ + a k ≤ b 1 + ⋯ + b l satisfied by G ( A ), for any a i , b j ∈ D . By Elliott's classification, 𝒫 together with the list of all sums a 1 + ⋯ + a k ∈ D ( A ) uniquely determines A . Can 𝒫 be Gödel incomplete, i.e., effectively enumerable but undecidable? We give a negative answer in case D is finite, and a positive answer in the infinite case. We also show that the range of the map A ↦ D ( A ) precisely consists of all countable partial abelian monoids satisfying the following three conditions: (i) a + b = a + c ⇒ b = c , (ii) a + b = 0 ⇒ a = b = 0 and (iii) ∀ a , b ∈ E ∃ c ∈ E such that either a + c = b or b + c = a .

A large class of local Dirichlet forms are constructed on path spaces over noncompact Riemannian manifolds. Under reasonable conditions on curvatures and diffusion coeffcients, these Dirichlet forms are quasi-regular and thus, the corresponding diffusion processes are well-constructed by the theory of Dirichlet forms. Extensions to pinned path (loop) spaces are also derived.

By combining divided differences with symmetric function relations, we establish two expansion formulae which generalize the classical Taylor theorem and the Maclauren expansion formula. The two q -expansion formulae due to Carlitz (1973) and Liu (2002) are also contained as special cases.

We show that any weak solution to the p -Yang-Mills equations of Sobolev class W 1, p defined on an m -dimensional manifold is W 2, p -gauge equivalent to a C 1, δ -connection for some 0 < δ < 1 provided p ≥ m /2.