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Our goal in this article is to prove a form of p-adic Birch and Swinnerton-Dyer formula for the second derivative of the p-adic L-function associated to a newform f which is non-crystalline semistable at p at its central critical point, by expressing this quantity in terms of a p-adic (cyclotomic) regulator defined on an extended trianguline Selmer group. We also prove a two-variable version of this result for height pairings we construct by considering infinitesimal deformations afforded by a Coleman family passing through f. This, among other things, leads us to a proof of an appropriate version of Perrin-Riou's conjecture in this set up. [ABSTRACT FROM AUTHOR]

Let WD be a killed Brownian motion in a domain D ⊂ Rd and S an independent subordinator with Laplace exponent f. The process Y D defined by Y D t = WD St is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator -f(-Δ|D), where Δ|D is the Dirichlet Laplacian. In this paper we study the potential theory of Y D under a weak scaling condition on the derivative of f. We first show that non-negative harmonic functions of Y D satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale invariant boundary Harnack principles with explicit decay rates for non-negative harmonic functions of Y D. The first boundary Harnack principle deals with a C1,1 domain D and non-negative functions which are harmonic near the boundary of D, while the second one is for a more general domain D and non-negative functions which are harmonic near the boundary of an interior open subset of D. The obtained decay rates are not the same, reflecting different boundary and interior behaviors of Y D. [ABSTRACT FROM AUTHOR]

We prove a Julia-Wolff-Carathédory theorem on angular derivatives of infinitesimal generators of one-parameter semigroups of holomorphic self-maps of the unit ball Bn ⊂ Cn, starting from results recently obtained by Bracci and Shoikhet. [ABSTRACT FROM AUTHOR]

A set of multivariate polynomials, over a field of zero or large characteristic, can be tested for algebraic independence by the well-known Jacobian criterion. For fields of other characteristic p > 0, no analogous characterization is known. In this paper we give the first such criterion. Essentially, it boils down to a non-degeneracy condition on a lift of the Jacobian polynomial over (an unramified extension of) the ring of p-adic integers. Our proof builds on the functorial de Rham-Witt complex, which was invented by Illusie (1979) for crystalline cohomology computations, and we deduce a natural explicit generalization of the Jacobian. We call this new avatar the Witt-Jacobian. In essence, we show how to faithfully differentiate polynomials over Fp (i.e., somehow avoid ∂xp/∂x = 0) and thus capture algebraic independence. We give two applications of this criterion in algebraic complexity theory. [ABSTRACT FROM AUTHOR]

We prove a Julia-Wolff-Carathéodory type theorem for infinitesimal generators on the unit ball in ℂn. Moreover, we study jets expansions at the boundary and give necessary and sufficient conditions on such jets for an infinitesimal generator to generate a group of automorphisms of the ball. [ABSTRACT FROM AUTHOR]

We construct a Taylor tower for functors from pointed categories to abelian categories via cotriples associated to cross effect functors. The tower was inspired by Goodwillie's Taylor tower for functors of spaces, and is related to Dold and Puppe's stable derived functors and Mac Lane's $Q$-construction. We study the layers, $D_{n}F=\text{\rm fiber}(P_{n}F\rightarrow P_{n-1}F)$, and the limit of the tower. For the latter we determine a condition on the cross effects that guarantees convergence. We define differentials for functors, and establish chain and product rules for them. We conclude by studying exponential functors in this setting and describing their Taylor towers. [ABSTRACT FROM AUTHOR]