Rayleigh processes, real trees, and root growth with re-grafting

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N tends to infinity of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous--Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N tends to infinity of the Aldous--Broder chain. A key technical ingredient in this work is the use of a pointed Gromov--Hausdorff distance to metrize the space of rooted compact real trees.Comment: 48 Pages. Minor revision of version of Feb 2004. To appear in Probability Theory and Related Field

Zero-Temperature Complex Replica Zeros of the ±J\pm J Ising Spin Glass on Mean-Field Systems and Beyond

Zeros of the moment of the partition function $[Z^n]_{\bm{J}}$ with respect to complex $n$ are investigated in the zero temperature limit $\beta \to \infty$, $n\to 0$ keeping $y=\beta n \approx O(1)$. We numerically investigate the zeros of the $\pm J$ Ising spin glass models on several Cayley trees and hierarchical lattices and compare those results. In both lattices, the calculations are carried out with feasible computational costs by using recursion relations originated from the structures of those lattices. The results for Cayley trees show that a sequence of the zeros approaches the real axis of $y$ implying that a certain type of analyticity breaking actually occurs, although it is irrelevant for any known replica symmetry breaking. The result of hierarchical lattices also shows the presence of analyticity breaking, even in the two dimensional case in which there is no finite-temperature spin-glass transition, which implies the existence of the zero-temperature phase transition in the system. A notable tendency of hierarchical lattices is that the zeros spread in a wide region of the complex $y$ plane in comparison with the case of Cayley trees, which may reflect the difference between the mean-field and finite-dimensional systems.Comment: 4 pages, 4 figure

Development of an in-field tree imaging system : a thesis presented in partial fulfilment of the requirements for the degree of Master of Technology at Massey University

Quality inventory information is essential for optimal resource utilisation in the forestry industry. In-field tree imaging is a method which has been proposed to improve the preharvest inventor assessment of standing trees. It involves the application of digital imaging technology to this task. The method described generates a three dimensional model of each tree through the capture of two orthogonal images from ground level. The images are captured and analysed using the "TreeScan" in-field tree imaging system. This thesis describes the design, development, and evaluation of the TreeScan system. The thesis can also be used as a technical reference for the system and as such contains appropriate technical and design detail. The TreeScan system consists of a portable computer, a custom designed high resolution scanner with integral microcontroller, a calibration rod, and custom designed processing software. Images of trees are captured using the scanner which contains a CCD line scan camera and a precision scanning mechanism. Captured images are analysed on the portable computer using customised image processing software to estimate real world tree dimensions and shape. The TreeScan system provides quantitative estimates of five tree parameters; height, sweep, stem diameter, branch diameter, and feature separation such as internodal distance. In addition to these estimates a three dimensional model is generated which can be further processed to determine the optimal stem breakdown into logs

PQ TREES, CONSECUTIVE ONES PROBLEM AND APPLICATIONS

A PQ tree is an advanced tree–based data structure, which represents a family of permutations on a set of elements. In this research article, we considered the significance of PQ trees and the Consecutive ones Problem to Computer Science and bioinformatics and their various applications. We also went further to demonstrate the operations of the characteristics of the Consecutive ones property by simulation, using high level programming languages. Attempt was also made at developing a PQ tree–Consecutive Ones analyzer, which could be instrumental not only as an educative tool to inquisitive students, but also serve as an important tool in developing clustering software in the field of bioinformatics and other application domains, with respect to solving real life problems

Solving kk-SUM using few linear queries

The $k$-SUM problem is given $n$ input real numbers to determine whether any $k$ of them sum to zero. The problem is of tremendous importance in the emerging field of complexity theory within $P$, and it is in particular open whether it admits an algorithm of complexity $O(n^c)$ with $c<\lceil \frac{k}{2} \rceil$. Inspired by an algorithm due to Meiser (1993), we show that there exist linear decision trees and algebraic computation trees of depth $O(n^3\log^3 n)$ solving $k$-SUM. Furthermore, we show that there exists a randomized algorithm that runs in $\tilde{O}(n^{\lceil \frac{k}{2} \rceil+8})$ time, and performs $O(n^3\log^3 n)$ linear queries on the input. Thus, we show that it is possible to have an algorithm with a runtime almost identical (up to the $+8$) to the best known algorithm but for the first time also with the number of queries on the input a polynomial that is independent of $k$. The $O(n^3\log^3 n)$ bound on the number of linear queries is also a tighter bound than any known algorithm solving $k$-SUM, even allowing unlimited total time outside of the queries. By simultaneously achieving few queries to the input without significantly sacrificing runtime vis-\`{a}-vis known algorithms, we deepen the understanding of this canonical problem which is a cornerstone of complexity-within-$P$. We also consider a range of tradeoffs between the number of terms involved in the queries and the depth of the decision tree. In particular, we prove that there exist $o(n)$-linear decision trees of depth $o(n^4)$

Practical path planning and obstacle avoidance for autonomous mowing

There is a need for systems which can autonomously perform coverage tasks on large outdoor areas. Unfortunately, the state-of-the-art is to use GPS based localization, which is not suitable for precise operations near trees and other obstructions. In this paper we present a robotic platform for autonomous coverage tasks. The system architecture integrates laser based localization and mapping using the Atlas Framework with Rapidly-Exploring Random Trees path planning and Virtual Force Field obstacle avoidance. We demonstrate the performance of the system in simulation as well as with real world experiments

Assessment of the Accuracy of a Multi-Beam LED Scanner Sensor for Measuring Olive Canopies

MDPI. CC BYCanopy characterization has become important when trying to optimize any kind of agricultural operation in high-growing crops, such as olive. Many sensors and techniques have reported satisfactory results in these approaches and in this work a 2D laser scanner was explored for measuring canopy trees in real-time conditions. The sensor was tested in both laboratory and field conditions to check its accuracy, its cone width, and its ability to characterize olive canopies in situ. The sensor was mounted on a mast and tested in laboratory conditions to check: (i) its accuracy at different measurement distances; (ii) its measurement cone width with different reflectivity targets; and (iii) the influence of the target’s density on its accuracy. The field tests involved both isolated and hedgerow orchards, in which the measurements were taken manually and with the sensor. The canopy volume was estimated with a methodology consisting of revolving or extruding the canopy contour. The sensor showed high accuracy in the laboratory test, except for the measurements performed at 1.0 m distance, with 60 mm error (6%). Otherwise, error remained below 20 mm (1% relative error). The cone width depended on the target reflectivity. The accuracy decreased with the target density