Abstract:
The establishment of Intensity-Duration-Frequency (IDF) curves for precipitation remains a powerful tool in the risk analysis of natural hazards. Indeed the IDF-curves allow for the estimation of the return period of an observed rainfall event or conversely of the rainfall amount corresponding to a given return period for different aggregation times. There is a high need for IDF-curves in the tropical region of Central Africa but unfortunately the adequate long-term data sets are frequently not available. The present paper assesses IDF-curves for precipitation for three stations in Central Africa. More physically based models for the IDF-curves are proposed. The methodology used here has been advanced by Koutsoyiannis et al. (1998) and an inter-station and inter-technique comparison is being carried out. The IDF-curves for tropical Central Africa are an interesting tool to be used in sewer system design to combat the frequently occurring inundations in semi-urbanized and urbanized areas of the Kinshasa megapolis.

Abstract:
The
change in rainfall pattern and intensity is becoming a great concern for
hydrologic engineers and planners. Many parts of the world are experiencing
extreme rainfall events such as experienced on 26^{th} July 2005 in
Mumbai, India. For the appropriate design and planning of urban drainage system
in an area, Intensity Duration Frequency (IDF) curves for given rainfall
conditions are required. The aim of the present study is to derive the IDF
curves for the rainfall in the Mumbai city, Maharashtra, India. Observed
rainfall data from 1901 pertaining to Colaba and from 1951 of the Santacruz
rain gauge stations in Mumbai are used in the present study to derive the IDF
curves. Initially, the proposed IDF curves are derived using an empirical
equation (Kothyari and Garde), by using probability distribution for annual
maximum rainfall and then IDF curves are derived by modifying the equation. IDF
curves developed by the modified equation gives good results in the changing
hydrologic conditions and are compatible even with the extreme rainfall of 26^{th} July 2005 in Mumbai.

Abstract:
A
regional analysis of design storms, defined as the expected rainfall intensity
for given storm duration and return period, is conducted to determine storm Rainfall
Intensity-Duration-Frequency (IDF) relationships. The ultimate purpose was to
determine IDF curves for homogeneous regions identified in Botswana. Three
homogeneous regions were identified based on topographic and rainfall
characteristics which were constructed with the K-Means Clustering algorithm.
Using the mean annual rainfall and the 24 hr annual maximum rainfall as an
indicator of rainfall intensity for each homogeneous region, IDF curves and
maps of rainfall intensities of 1 to 24 hr and above durations were produced.
The Gamma and Lognormal probability distribution functions were able to provide
estimates of rainfall depths for low and medium return periods (up to 100
years) in any location in each homogeneous region of Botswana.

Abstract:
The analysis of maximum precipitation is usually carried out by using IDF curves (Intensity-Duration-Frequency), which in turn could be expressed as MAI curves (Maximum Average Intensities). An index “n” has been developed in this work, defined from the exponent obtained when adjusting IDF climatic curves to MAI curves. That index provides information about how maximum precipitation is achieved in a certain climatic area, according to the relative temporal distribution of maximum intensities. From the climatic analysis of index “n”, large areas could be distinguished in the Iberian Peninsula, characterized by rain maxima of a stormier origin (peninsular inland), and areas characterized by rain maxima of a more frontal origin (southwest, Atlantic coast and Mediterranean coast). Additionally, these areas could be more specifically divided according to the persistence of maximum precipitation.

Abstract:
We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.

Abstract:
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using the development of ``mechanism'' which is based on ``distortion'' values and their matrices, we discuss some aspects refereing to quadrics with respect to their dual objects. This topic includes also the induced dual subdivision of Newton Polytope and its compatible properties. Finally, a regularity of tropical curves in the duality sense is generally defined and, studied for families of tropical quadrics.

Abstract:
In this paper, we study the correspondence between tropical curves and holomorphic curves. The main subjects in this paper are superabundant tropical curves. First we give an effective combinatorial description of these curves. Based on this description, we calculate the obstructions for appropriate deformation theory, describe the Kuranishi map, and study the solution space of it. The genus one case is solved completely, and the theory works for many of the higher genus cases, too.

Abstract:
Harmonic amoebas are generalizations of amoebas of algebraic curves embedded in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of classical tropical geometry of curves. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces, and show how they can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends the fact that tropical curves in affine spaces always arise as degenerations of amoebas of algebraic curves. The flexibility of this machinery gives an alternative proof of Mikhalkin's approximation theorem for regular phase-tropical morphisms to any affine space, as stated e.g. in \cite{Mikh06}. All the approximation results presented here are obtained as corollaries of a theorem on convergence of imaginary normalized differentials on families of Riemann surfaces.

Abstract:
This is mostly* a non-technical exposition of the joint work arXiv:1212.0373 with Caporaso and Payne. Topics include: Moduli of Riemann surfaces / algebraic curves; Deligne-Mumford compactification; Dual graphs and the combinatorics of the compactification; Tropical curves and their moduli; Non-archimedean geometry and comparison. * Maybe the last section is technical.

Abstract:
We introduce the notion of families of n-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of n-marked abstract rational tropical curves.