Associated to a split, semisimple linear algebraic group G is a group of invariant quadratic forms, which we denote Q(G). Namely, Q(G) is the group of quadratic forms in characters of a maximal torus which are fixed with respect to the action of the Weyl group of G. We compute Q(G) for various examples of products of the special linear, special orthogonal, and symplectic groups as well as for quotients of those examples by central subgroups. Homomorphisms between these linear algebraic groups induce homomorphisms between their groups of invariant quadratic forms. Since the linear algebraic groups are semisimple, Q(G) is isomorphic to Z^n for some n, and so the induced maps can be described by a set of integers called Rost multipliers. We consider various cases of the Kronecker tensor product map between copies of the special linear, special orthogonal, and symplectic groups. We compute the Rost multipliers of the induced map in these examples, ultimately concluding that the Rost multipliers depend only on the dimensions of the underlying vector spaces.

Let A be a von Neumann algebra of linear operators on the Hilbert space H . A linear operator T (resp. a linear bounded. functional ϕ ) on A is said to be normal if for any increasing net [formula omitted] of positive elements in A with least upper bound B , T(B) is the least upper bound of [formula omitted]. Two linear positive functionals ψ1 and ψ2 on A are said to be equivalent if ψ1 (B) = 0 <=> ψ2 (B) = 0 for any positive element B in A. Let ϕ0 be a positive normal linear functional on A . Let S be a semigroup and, {T(s) : s ε S} an antirepresentation of S as normal positive linear contraction operators on A . We find in this thesis equivalent conditions for the existence of a positive normal linear functional ϕ on A which is equivalent to ϕ0 and invariant under the semigroup {T(s) : s ε S} (i.e. ϕ(T(s)B) = ϕ(B) for all B in A and s ε S ). We also extend the concept of weakly-wandering sets, which was first introduced by Hajian-Kakutani, to weakly-wandering projections in A. We give a relation between the non-existence of weakly-wandering projections in A and the existence of positive normal linear functionals on A, invariant with respect to an antirepresentation {T(s) : s ε S} of normal *-homomorphisms on A . Finally we investigate the existence of a complete set of positive normal linear functionals on A which are invariant under the semigroup {T(s) : s ε S}. / Science, Faculty of / Mathematics, Department of / Graduate

Euclid’s Method of finding Pythagorean triples is a commonly accepted and applied technique. This study focuses on a myriad of other methods behind finding such Pythagorean triples. Specifically, we discover whether or not other ways of finding triples are special cases of Euclid’s Method.

We propose a new approach to proving results regarding channel coding schemes based on construction-A lattices for the Additive White Gaussian Noise (AWGN) channel that yields new characterizations of the code construction parameters, i.e., the primes and dimensions of the codes, as functions of the block-length. The approach we take introduces an averaging argument that explicitly involves the considered parameters. This averaging argument is applied to a generalized Loeliger ensemble [1] to provide a more practical proof of the existence of AWGN-good lattices, and to characterize suitable parameters for the lattice Gaussian coding scheme proposed by Ling and Belfiore [3].

This paper defines tropical projective space, TP^n, and the tropical general linear group TPGL(n). After discussing some simple examples of tropical polynomials and their hypersurfaces, a strategy is given for finding all conics in the tropical projective plane. The classification of conics and an analysis of the coefficient space corresponding to such conics is given.

<p>Studying properties of random polynomials has marked a shift in algebraic geometry. Instead of worst-case analysis, which often leads to overly pessimistic perspectives, randomness helps perform average-case analysis, and thus obtain a more realistic view. Also, via Erdos' astonishing 'probabilistic method', one can potentially obtain deterministic results by introducing randomness into a question that apriori had nothing to do with randomness. </p> <p><br></p> <p>In this thesis, we study topological questions in real algebraic geometry, o-minimal geometry and random algebraic geometry, with motivation from incidence combinatorics. Specifically, we prove results along two different threads:</p> <p><br></p> <p>1. Topology of semi-algebraic and definable (over any o-minimal structure over R) sets, in both deterministic and random settings.</p><p>2. Topology of random hypersurface arrangements. In this case, we also prove a result that could be of independent interest in random graph theory.</p> <p><br></p> <p>Towards the first thread, motivated by applications in o-minimal incidence combinatorics, we prove bounds (both deterministic and random) on the topological complexity (as measured by the Betti numbers) of general definable hypersurfaces restricted to algebraic sets. Given any sequence of hypersurfaces, we show that there exists a definable hypersurface G, and a sequence of polynomials, such that each manifold in the sequence of hypersurfaces appears as a component of G restricted to the zero set of some polynomial in the sequence of polynomials. This shows that the topology of the intersection of a definable hypersurface and an algebraic set can be made <i>arbitrarily pathological</i>. On the other hand, we show that for random polynomials, the Betti numbers of the restriction of the zero set of a random polynomial to any definable set deviates from a Bezout-type bound with <i>bounded probability</i>.</p> <p><br></p> <p>Progress in o-minimal incidence combinatorics has lagged behind the developments in incidence combinatorics in the algebraic case due to the absence of an o-minimal version of the Guth-Katz <i>polynomial partitioning</i> theorem, and the first part of our work explains why this is so difficult. However, our average result shows that if we can prove that the measure of the set of polynomials which satisfy a certain property necessary for polynomial partitioning is suitably bounded from below, by the <i>probabilistic method</i>, we get an o-minimal polynomial partitioning theorem. This would be a tremendous breakthrough and would enable progress on multiple fronts in model theoretic combinatorics. </p> <p><br></p> <p>Along the second thread, we have studied the average Betti numbers of <i>random hypersurface arrangements</i>. Specifically, we study how the average Betti numbers of a finite arrangement of random hypersurfaces grows in terms of the degrees of the polynomials in the arrangement, as well as the number of polynomials. This is proved using a random Mayer-Vietoris spectral sequence argument. We supplement this result with a better bound on the average Betti numbers when one considers an <i>arrangement of quadrics</i>. This question turns out to be equivalent to studying the expected number of connected components of a certain <i>random graph model</i>, which has not been studied before, and thus could be of independent interest. While our motivation once again was incidence combinatorics, we obtained the first bounds on the topology of arrangements of random hypersurfaces, with an unexpected bonus of a result in random graphs.</p>

<p>Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is compact because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces.</p>

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<p> To provide a logical framework for the comparison of various methods of kinematic flood routing a general method of kinematic flood routing is developed.</p> <p> After presenting the general framework, the properties of the numerical model are investigated by: l. Algebraic examination of the finite difference scheme. 2.Numerical experiments using a high speed digital computer. 3. Comparison of the kinematic flood routing results with results of simulations using the complete one dimensional dynamic representation.</p> <p> Particular facets of the numerical kinematic model that were studied included: 1. The stability of the numeric schematizations. 2. The degree of approximation with the finite difference system. 3. The applicability of kinematic methods to unsteady flow systems. 4. Methods of extending the kinematic solutions to predict attenuation as well as translation of the flood wave through the channel systems. </p> <p> The results indicate that kinematic flood routing methods differ primarily in the point about which the finite difference equation is formulated, hereafter termed the nucleus, and that the general framework is capable of emulating such methods as the Muskinghum Method, other non-linear kinematic methods and reservoir routing. By varying the location of the nucleus the stability and degree of approximation is significantly altered. This results in the outflow hydrograph being sensitive to the location of the nucleus and the size of the finite difference steps.</p> <p> To facilitate further research and application of the methods outlined in the thesis, a computer program was developed to enable kinematic flood routing to be performed in a natural channel with arbitrary geometry. Furthermore, the data is compatible with a program that is capable of performing a flood routing analysis using a numerical solution of the complete Saint-Venant equations. Documentation of the computer program for kinematic analysis is included with this thesis.</p> / Thesis / Master of Engineering (MEngr)