Completions of epsilon-dense partial Latin squares

A classical question in combinatorics is the following: given a partial Latin square $P$, when can we complete $P$ to a Latin square $L$? In this paper, we investigate the class of textbf{$epsilon$-dense partial Latin squares}: partial Latin squares in which each symbol, row, and column contains no more than $epsilon n$-many nonblank cells. Based on a conjecture of Nash-Williams, Daykin and H"aggkvist conjectured that all $frac{1}{4}$-dense partial Latin squares are completable. In this paper, we will discuss the proof methods and results used in previous attempts to resolve this conjecture, introduce a novel technique derived from a paper by Jacobson and Matthews on generating random Latin squares, and use this novel technique to study $ epsilon$-dense partial Latin squares that contain no more than $delta n^2$ filled cells in total.

In Chapter 2, we construct completions for all $ epsilon$-dense partial Latin squares containing no more than $delta n^2$ filled cells in total, given that $epsilon < frac{1}{12}, delta < frac{ left(1-12epsilonright)^{2}}{10409}$. In particular, we show that all $9.8 cdot 10^{-5}$-dense partial Latin squares are completable. In Chapter 4, we augment these results by roughly a factor of two using some probabilistic techniques. These results improve prior work by Gustavsson, which required $epsilon = delta leq 10^{-7}$, as well as Chetwynd and H"aggkvist, which required $epsilon = delta = 10^{-5}$, $n$ even and greater than $10^7$.

If we omit the probabilistic techniques noted above, we further show that such completions can always be found in polynomial time. This contrasts a result of Colbourn, which states that completing arbitrary partial Latin squares is an NP-complete task. In Chapter 3, we strengthen Colbourn's result to the claim that completing an arbitrary $left(frac{1}{2} + epsilonright)$-dense partial Latin square is NP-complete, for any $epsilon > 0$.

Colbourn's result hinges heavily on a connection between triangulations of tripartite graphs and Latin squares. Motivated by this, we use our results on Latin squares to prove that any tripartite graph $G = (V_1, V_2, V_3)$ such that begin{itemize} item $|V_1| = |V_2| = |V_3| = n$, item For every vertex $v in V_i$, $deg_+(v) = deg_-(v) geq (1- epsilon)n,$ and item $|E(G)| > (1 - delta)cdot 3n^2$ end{itemize} admits a triangulation, if $epsilon < frac{1}{132}$, $delta < frac{(1 -132epsilon)^2 }{83272}$. In particular, this holds when $epsilon = delta=1.197 cdot 10^{-5}$.

This strengthens results of Gustavsson, which requires $epsilon = delta = 10^{-7}$.

In an unrelated vein, Chapter 6 explores the class of textbf{quasirandom graphs}, a notion first introduced by Chung, Graham and Wilson cite{chung1989quasi} in 1989. Roughly speaking, a sequence of graphs is called "quasirandom"' if it has a number of properties possessed by the random graph, all of which turn out to be equivalent. In this chapter, we study possible extensions of these results to random $k$-edge colorings, and create an analogue of Chung, Graham and Wilson's result for such colorings.

Weighted least-squares phase unwrapping algorithm based on derivative variance correlation map

Among different phase unwrapping approaches, the weighted least-squares minimization methods are gaining attention. In these algorithms, weighting coefficient is generated from a quality map. The intrinsic drawbacks of existing quality maps constrain the application of these algorithms. They often fail to handle wrapped phase data contains error sources, such as phase discontinuities, noise and undersampling. In order to deal with those intractable wrapped phase data, a new weighted least-squares phase unwrapping algorithm based on derivative variance correlation map is proposed. In the algorithm, derivative variance correlation map, a novel quality map, can truly reflect wrapped phase quality, ensuring a more reliable unwrapped result. The definition of the derivative variance correlation map and the principle of the proposed algorithm are present in detail. The performance of the new algorithm has been tested by use of a simulated spherical surface wrapped data and an experimental interferometric synthetic aperture radar (IFSAR) wrapped data. Computer simulation and experimental results have verified that the proposed algorithm can work effectively even when a wrapped phase map contains intractable error sources. (c) 2006 Elsevier GmbH. All rights reserved.

Seismic model parameter estimation using least-squares migration

We describe a method for verifying seismic modelling parameters. It is equivalent to performing several iterations of unconstrained least-squares migration (LSM). The approach allows the comparison of modelling/imaging parameter configurations with greater confidence than simply viewing the migrated images. The method is best suited to determining discrete parameters but can be used for continuous parameters albeit with greater computational expense.

Shallow water equations: Magic and limitations

D Liang from Cambridge University explains the shallow water equations and their applications to the dam-break and other steep-fronted flow modeling. They assume that the horizontal scale of the flow is much greater than the vertical scale, which means the flow is restricted within a thin layer, thus the vertical momentum is insignificant and the pressure distribution is hydrostatic. The left hand sides of the two momentum equations represent the acceleration of the fluid particle in the horizontal plane. If the fluid acceleration is ignored, then the two momentum equations are simplified into the so-called diffusion wave equations. In contrast to the SWEs approach, it is much less convenient to model floods with the Navier-Stokes equations. In conventional computational fluid dynamics (CFD), cumbersome treatments are needed to accurately capture the shape of the free surface. The SWEs are derived using the assumptions of small vertical velocity component, smooth water surface, gradual variation and hydrostatic pressure distribution.

Weighted least squares with expectation-maximization algorithm for burst detection in U.K. water distribution systems

Flow measurement data at the district meter area (DMA) level has the potential for burst detection in the water distribution systems. This work investigates using a polynomial function fitted to the historic flow measurements based on a weighted least-squares method for automatic burst detection in the U.K. water distribution networks. This approach, when used in conjunction with an expectationmaximization (EM) algorithm, can automatically select useful data from the historic flow measurements, which may contain normal and abnormal operating conditions in the distribution network, e.g., water burst. Thus, the model can estimate the normal water flow (nonburst condition), and hence the burst size on the water distribution system can be calculated from the difference between the measured flow and the estimated flow. The distinguishing feature of this method is that the burst detection is fully unsupervised, and the burst events that have occurred in the historic data do not affect the procedure and bias the burst detection algorithm. Experimental validation of the method has been carried out using a series of flushing events that simulate burst conditions to confirm that the simulated burst sizes are capable of being estimated correctly. This method was also applied to eight DMAs with known real burst events, and the results of burst detections are shown to relate to the water company's records of pipeline reparation work. © 2014 American Society of Civil Engineers.

A new beam propagation method based on least-squares expansion approximating

A new finite difference wide-angle beam propagation method is developed by introducing the least-squares expansion approximant in the propagator expansion. In this new method it is not necessary to select the reference index point because of the whole region approaching the lease-square expansion. This method avoids the problems induced by error selection of the reference index in the old methods based on Taylor or Pade expansion. Several typical structures are simulated by the new method and the results prove the validity of it.