Tropical variants of some complex analysis results

Tropical Nevanlinna theory studies value distribution of continuous piecewise linear functions of a real variable. In this paper, we use the reasoning from tropical Nevanlinna theory to present tropical counterparts of some classical complex results related to Fermat type equations, Hayman conjecture and Bruck conjecture.Comment: 24 pages, 6 figure

Hadamard's formula and couplings of SLEs with free field

The relation between level lines of Gaussian free fields (GFF) and SLE(4)-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field, and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard's formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann-Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE(4) is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.Comment: 26 page

Computing parametric rational generating functions with a primal Barvinok algorithm

Computations with Barvinok's short rational generating functions are traditionally being performed in the dual space, to avoid the combinatorial complexity of inclusion--exclusion formulas for the intersecting proper faces of cones. We prove that, on the level of indicator functions of polyhedra, there is no need for using inclusion--exclusion formulas to account for boundary effects: All linear identities in the space of indicator functions can be purely expressed using half-open variants of the full-dimensional polyhedra in the identity. This gives rise to a practically efficient, parametric Barvinok algorithm in the primal space.Comment: 16 pages, 1 figure; v2: Minor corrections, new example and summary of algorithm; submitted to journa

Cone Monotonicity: Structure Theorem, Properties, and Comparisons to Other Notions of Monotonicity

In search of a meaningful 2-dimensional analog to mono- tonicity, we introduce two new definitions and give examples of and dis- cuss the relationship between these definitions and others that we found in the literature. Note: After we published the article in Abstract and Applied Analysis and after we searched multiple times for previous work, we discovered that Clarke at al. had introduced the definition of cone monotonicity and given a characterization. See the addendum at the end of this paper for full reference information