The Strahler number of a parity game

The Strahler number of a rooted tree is the largest height of a perfect binary tree that is its minor. The Strahler number of a parity game is proposed to be defined as the smallest Strahler number of the tree of any of its attractor decompositions. It is proved that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices~n and linear in (d/2k)k, where d is the number of priorities and k is the Strahler number. This complexity is quasi-polynomial because the Strahler number is at most logarithmic in the number of vertices. The proof is based on a new construction of small Strahler-universal trees. It is shown that the Strahler number of a parity game is a robust parameter: it coincides with its alternative version based on trees of progress measures and with the register number defined by Lehtinen~(2018). It follows that parity games can be solved in quasi-linear space and in time that is polynomial in the number of vertices and linear in (d/2k)k, where k is the register number. This significantly improves the running times and space achieved for parity games of bounded register number by Lehtinen (2018) and by Parys (2020). The running time of the algorithm based on small Strahler-universal trees yields a novel trade-off k⋅lg(d/k)=O(logn) between the two natural parameters that measure the structural complexity of a parity game, which allows solving parity games in polynomial time. This includes as special cases the asymptotic settings of those parameters covered by the results of Calude, Jain Khoussainov, Li, and Stephan (2017), of Jurdziński and Lazić (2017), and of Lehtinen (2018), and it significantly extends the range of such settings, for example to d=2O(lgn√) and k=O(lgn−√)

Drainage Basin Morphometry and River Network Analysis: A Review and Synthesis

This discussion paper is a review of drainage basin morphometry and river network analysis. It is essentially an assemblage of techniques which have become both an important part of geomorphology ahd a useful set of morphometric inputs to hydrologic models of river basins. The paper also provides a discussion of the ideas which collectively have become known as the theory of dynamic equilibrium. It is interesting that many of the concepts and techniques which grew out of this theory have been incorporated into the conventional discipline while the original conceptual framework has become somewhat suspect

Mathematical Modelling of a Brain Tumour Initiation and Early Development: A Coupled Model of Glioblastoma Growth, Pre-Existing Vessel Co-Option, Angiogenesis and Blood Perfusion

We propose a coupled mathematical modelling system to investigate glioblastoma growth in response to dynamic changes in chemical and haemodynamic microenvironments caused by pre-existing vessel co-option, remodelling, collapse and angiogenesis. A typical tree-like architecture network with different orders for vessel diameter is designed to model pre-existing vasculature in host tissue. The chemical substances including oxygen, vascular endothelial growth factor, extra-cellular matrix and matrix degradation enzymes are alculated based on the haemodynamic environment which is obtained by coupled modelling of intravascular blood flow with interstitial fluid flow. The haemodynamic changes, including vessel diameter and permeability, are introduced to reflect a series of pathological characteristics of abnormal tumour vessels including vessel dilation, leakage, angiogenesis, regression and collapse. Migrating cells are included as a new phenotype to describe the migration behaviour of malignant tumour cells. The simulation focuses on the avascular phase of tumour development and stops at an early phase of angiogenesis. The model is able to demonstrate the main features of glioblastoma growth in this phase such as the formation of pseudopalisades, cell migration along the host vessels, the pre-existing vasculature co-option, angiogenesis and remodelling. The model also enables us to examine the influence of initial conditions and local environment on the early phase of glioblastoma growth.The authors like to thank Mr. Justin Halls for his kind help on manuscript preparation. This research is supported by the National Basic Research Program of China (973 Program) (No. 2013CB733800), the National Nature Science Foundation of China (No. 11302050, No. 11272091), the Nature Science Foundation of Jiangsu Province (No. BK20130593)

Morphometric Analysis of Didessa River Catchment in Blue Nile Basin, Western Ethiopia

In the present paper an attempt has been made to study the morphometric characteristics of Didessa catchment which is part of Blue Nile River basin located in South Western part of Ethiopia. The study focused on evaluating the effect of morphometric parameters on the hydrology and morphology of the basin. The primary data for this study was an elevation data in raster format (Digital Elevation Model or DEM) also known as ASTER DEM of 30m spatial resolution. Stream networks and watersheds were delineated in ArcGIS 10.1 software environment by utilizing ArcHydro and Spatial Analyst Tools. Based on the results of the software analysis, linear, areal, and relief aspects of the basin were computed. The result indicates that Geo-spatial techniques and DEMs data are efficient for extracting stream networks and characterizing the morphology of watersheds as well as computing the various indices of drainage basin morphometry. Hence, the results of this study provide useful inputs for watershed level planning and management.Keywords: ASTER DEM, ArcHydro, Hydrology, Watershe

Digital data from shuttle photography: The effects of platform variables

Two major criticisms of using Shuttle hand held photography as an Earth science sensor are that it is nondigital, nonquantitative and that it has inconsistent platform characteristics, e.g., variable look angles, especially as compared to remote sensing satellites such as LANDSAT and SPOT. However, these criticisms are assumptions and have not been systematically investigated. The spectral effects of off-nadir views of hand held photography from the Shuttle and their role in interpretation of lava flow morphology on the island of Hawaii are studied. Digitization of photography at JSC and use of LIPS image analysis software in obtaining data is discussed. Preliminary interpretative results of one flow are given. Most of the time was spent in developing procedures and overcoming equipment problems. Preliminary data are satisfactory for detailed analysis

A Travel Network for Prehistoric Land and Water Navigation

The Hopewell culture created numerous earthworks across the landscape of Southern Ohio and Illinois between 200 BC and 400 AD. Archaeologists believe these earthworks to have been important fixtures in the Hopewell social landscape. Unfortunately, 19th and 20th century agricultural activity destroyed a large majority of the structures. Archaeologists possess little contextual data from these activities except for brief descriptions and locations provided by early settlers. This project leverages the spatial data that remain with a set of Network Analyst tools to model prehistoric human temporal costs for both land and water navigation. This project permits archaeologists to test theories about social interaction between Hopewell earthwork centers and helps researchers gain insight into the purpose of the earthworks and the communities they organized

Structural analysis and fluid flow modeling of the Kapuni Field, New Zealand

The Taranaki Basin is the largest hydrocarbon producing basin in New Zealand. A total area of over 330,000 k