Diagonal forms, linear algebraic methods and Ramsey-type problems

This thesis focuses mainly on linear algebraic aspects of combinatorics. Let N_t(H) be an incidence matrix with edges versus all subhypergraphs of a complete hypergraph that are isomorphic to H. Richard M. Wilson and the author find the general formula for the Smith normal form or diagonal form of N_t(H) for all simple graphs H and for a very general class of t-uniform hypergraphs H.

As a continuation, the author determines the formula for diagonal forms of integer matrices obtained from other combinatorial structures, including incidence matrices for subgraphs of a complete bipartite graph and inclusion matrices for multisets.

One major application of diagonal forms is in zero-sum Ramsey theory. For instance, Caro's results in zero-sum Ramsey numbers for graphs and Caro and Yuster's results in zero-sum bipartite Ramsey numbers can be reproduced. These results are further generalized to t-uniform hypergraphs. Other applications include signed bipartite graph designs.

Research results on some other problems are also included in this thesis, such as a Ramsey-type problem on equipartitions, Hartman's conjecture on large sets of designs and a matroid theory problem proposed by Welsh.

Algebraic Techniques in Coding Theory: Entropy Vectors, Frames, and Constrained Coding

The study of codes, classically motivated by the need to communicate information reliably in the presence of error, has found new life in fields as diverse as network communication, distributed storage of data, and even has connections to the design of linear measurements used in compressive sensing. But in all contexts, a code typically involves exploiting the algebraic or geometric structure underlying an application. In this thesis, we examine several problems in coding theory, and try to gain some insight into the algebraic structure behind them.

The first is the study of the entropy region - the space of all possible vectors of joint entropies which can arise from a set of discrete random variables. Understanding this region is essentially the key to optimizing network codes for a given network. To this end, we employ a group-theoretic method of constructing random variables producing so-called "group-characterizable" entropy vectors, which are capable of approximating any point in the entropy region. We show how small groups can be used to produce entropy vectors which violate the Ingleton inequality, a fundamental bound on entropy vectors arising from the random variables involved in linear network codes. We discuss the suitability of these groups to design codes for networks which could potentially outperform linear coding.

The second topic we discuss is the design of frames with low coherence, closely related to finding spherical codes in which the codewords are unit vectors spaced out around the unit sphere so as to minimize the magnitudes of their mutual inner products. We show how to build frames by selecting a cleverly chosen set of representations of a finite group to produce a "group code" as described by Slepian decades ago. We go on to reinterpret our method as selecting a subset of rows of a group Fourier matrix, allowing us to study and bound our frames' coherences using character theory. We discuss the usefulness of our frames in sparse signal recovery using linear measurements.

The final problem we investigate is that of coding with constraints, most recently motivated by the demand for ways to encode large amounts of data using error-correcting codes so that any small loss can be recovered from a small set of surviving data. Most often, this involves using a systematic linear error-correcting code in which each parity symbol is constrained to be a function of some subset of the message symbols. We derive bounds on the minimum distance of such a code based on its constraints, and characterize when these bounds can be achieved using subcodes of Reed-Solomon codes.

Mass transfer from a cylinder to an air stream in axisymmetrical flow (an experimental study)

Mass transfer from wetted surfaces on one-inch cylinders with unwetted approach sections was studied experimentally by means of the evaporation of n-octane and n-heptane into an air stream in axisymmetrical flow, for Reynolds numbers from 5,000 to 310,000. A transition from the laminar to the turbulent boundary layer was observed to occur at Reynolds numbers from 10,000 to 15,000. The results were expressed in terms of the Sherwood number as a function of the Reynolds number, the Schmidt number, and the ratio of the unwetted approach length to the total length. Empirical formulas were obtained for both laminar and turbulent regimes. The rates of mass transfer obtained were higher than theoretical and experimental results obtained by previous investigators for mass and heat transfer from flat plates.

Part I: Latent heat of vaporization of n-decane. Part II: Heat and mass transfer from a cylinder placed in a turbulent air stream - Sherwood and Frössling numbers as a function of free stream turbulence level and Reynolds number

Part I

The latent heat of vaporization of n-decane is measured calorimetrically at temperatures between 160° and 340°F. The internal energy change upon vaporization, and the specific volume of the vapor at its dew point are calculated from these data and are included in this work. The measurements are in excellent agreement with available data at 77° and also at 345°F, and are presented in graphical and tabular form.

Part II

Simultaneous material and energy transport from a one-inch adiabatic porous cylinder is studied as a function of free stream Reynolds Number and turbulence level. Experimental data is presented for Reynolds Numbers between 1600 and 15,000 based on the cylinder diameter, and for apparent turbulence levels between 1.3 and 25.0 per cent. n-heptane and n-octane are the evaporating fluids used in this investigation.

Gross Sherwood Numbers are calculated from the data and are in substantial agreement with existing correlations of the results of other workers. The Sherwood Numbers, characterizing mass transfer rates, increase approximately as the 0.55 power of the Reynolds Number. At a free stream Reynolds Number of 3700 the Sherwood Number showed a 40% increase as the apparent turbulence level of the free stream was raised from 1.3 to 25 per cent.

Within the uncertainties involved in the diffusion coefficients used for n-heptane and n-octane, the Sherwood Numbers are comparable for both materials. A dimensionless Frössling Number is computed which characterizes either heat or mass transfer rates for cylinders on a comparable basis. The calculated Frössling Numbers based on mass transfer measurements are in substantial agreement with Frössling Numbers calculated from the data of other workers in heat transfer.