Jump numbers, hyperrectangles and Carlitz compositions

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 1998.A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg 1998Let A = (aij) be an m x n matrix. There is a natural way to associate a poset PA with A. A jump in a linear extension of PA is a pair of consecutive elements which are incomparable in Pa. The jump number of A is the minimum number of jumps in any linear extension of PA. The maximum jump number over a class of n x n matrices of zeros and ones with constant row and column sum k, M (n, k), has been investigated in Chapter 2 and 3. Chapter 2 deals with extremization problems concerning M (n ,k). In Chapter 3, we obtain the exact values for M (11,k). M(n,Q), M (n,n-3) and M(n,n-4). The concept of frequency hyperrectangle generalizes the concept of latin square. In Chapter 4 we derive a bound for the maximum number of mutually orthogonal frequency hyperrectangles. Chapter 5 gives two algorithms to construct mutually orthogonal frequency hyperrectangles. Chapter 6 is devoted to some enumerative results about Carlitz compositions (compositions with different adjacent parts)

Local Explanations via Necessity and Sufficiency: Unifying Theory and Practice

Necessity and sufficiency are the building blocks of all successful explanations. Yet despite their importance, these notions have been conceptually underdeveloped and inconsistently applied in explainable artificial intelligence (XAI), a fast-growing research area that is so far lacking in firm theoretical foundations. In this article, an expanded version of a paper originally presented at the 37th Conference on Uncertainty in Artificial Intelligence (Watson et al., 2021), we attempt to fill this gap. Building on work in logic, probability, and causality, we establish the central role of necessity and sufficiency in XAI, unifying seemingly disparate methods in a single formal framework. We propose a novel formulation of these concepts, and demonstrate its advantages over leading alternatives. We present a sound and complete algorithm for computing explanatory factors with respect to a given context and set of agentive preferences, allowing users to identify necessary and sufficient conditions for desired outcomes at minimal cost. Experiments on real and simulated data confirm our method’s competitive performance against state of the art XAI tools on a diverse array of tasks

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Goddard trajectory determination subsystem: Mathematical specifications

The mathematical specifications of the Goddard trajectory determination subsystem of the flight dynamics system are presented. These specifications include the mathematical description of the coordinate systems, dynamic and measurement model, numerical integration techniques, and statistical estimation concepts

Mathematical theory of the Goddard trajectory determination system

Basic mathematical formulations depict coordinate and time systems, perturbation models, orbital estimation techniques, observation models, and numerical integration methods

Advances in Spacecraft Attitude Control

Spacecraft attitude maneuvers comply with Euler's moment equations, a set of three nonlinear, coupled differential equations. Nonlinearities complicate the mathematical treatment of the seemingly simple action of rotating, and these complications lead to a robust lineage of research. This book is meant for basic scientifically inclined readers, and commences with a chapter on the basics of spaceflight and leverages this remediation to reveal very advanced topics to new spaceflight enthusiasts. The topics learned from reading this text will prepare students and faculties to investigate interesting spaceflight problems in an era where cube satellites have made such investigations attainable by even small universities. It is the fondest hope of the editor and authors that readers enjoy this book

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Synthesis, optimisation and control of crystallization systems

Process systems engineering has provided with a range of powerful tools to chemical engineers for synthesis, optimisation and control using thorough understanding of the processes enhanced with the aid of sophisticated and accurate multi-faceted mathematical models. Crystallization processes have rarely benefited from these new techniques, for they lack in models that could be used to bridge the gaps in their perception before utilising the resulting insight for the three above mentioned tasks. In the present work, first a consistent and sufficiently complex models for unit operations including MSMPR crystallizer, hydrocyclone and fines dissolver are developed to enhance the understanding of systems comprising these units. This insight is then utilised for devising innovative techniques to synthesise, optimise and control such processes. A constructive targeting approach is developed for innovative synthesis of stage-wise crystallization processes. The resulting solution surpasses the performance obtained from conventional design procedure not only because optimal temperature profiles are used along the crystallizers but also the distribution of feed and product removal is optimally determined through non-linear programming. The revised Machine Learning methodology presented here for continual process improvement by analysing process data and representing the findings as zone of best average performance, has directly utilised the models to generate the data in the absence of real plant data. The methodology which is demonstrated through KNO₃ crystallization process flowsheet quickly identifies three opportunities each representing an increase of 12% on nominal operation. An optimal multi-variable controller has been designed for a one litre continuous recycle crystallizer to indirectly control total number and average size of crystals from secondary process measurements. The system identification is solely based on experimental findings. Linear Quadratic Gaussian method based design procedure is developed to design the controller which not only shows excellent set-point tracking capabilities but also effectively rejects disturbance in the simulated closed loop runs

Advances in Spacecraft Attitude Control

Spacecraft attitude maneuvers comply with Euler's moment equations, a set of three nonlinear, coupled differential equations. Nonlinearities complicate the mathematical treatment of the seemingly simple action of rotating, and these complications lead to a robust lineage of research. This book is meant for basic scientifically inclined readers, and commences with a chapter on the basics of spaceflight and leverages this remediation to reveal very advanced topics to new spaceflight enthusiasts. The topics learned from reading this text will prepare students and faculties to investigate interesting spaceflight problems in an era where cube satellites have made such investigations attainable by even small universities. It is the fondest hope of the editor and authors that readers enjoy this book