Mathematics Masters Theses

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Browsing Mathematics Masters Theses by browse.metadata.department "Mathematics"

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    Absolute Stability of a Class of Second Order Feedback Non-Linear, Time-Varying Systems
    (North Dakota State University, 2010) Omotoyinbo, Tayo
    In this thesis, we consider the problem of absolute stability of continuous time feedback systems with a single, time-varying nonlinearity. Necessary and sufficient conditions for absolute stability of second-order systems in terms of system parameters are developed, which are characterized by eigenvalue locations on the complex plane. More specifically, our results are presented in terms of the associated matrix-pencil {A+ bvc*, v E [11,1, /L2]}, where /Li, /J,2 E ffi., A is n x n-matrix, b and c are n-vectors. The stability conditions require that the eigenvalues of all matrices A +bvc*, p1 ~ v ~ μ2, lie in the interior of a specific region of the complex plane ( a cone to be specific). Thus, we have the following reformulation of the problem. Find the maximal cone satisfying the following condition: If all eigenvalues of corresponding linear systems belong to this cone, then system is absolutely stable. Known results show that this cone is not smaller than { z E
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    Bifurcation Portrait of a Hybrid Spiking Neuron Model
    (North Dakota State University, 2012) Goodell, Brandon Grae
    A bifurcation portrait classifies the behavior of a dynamical system and how it transitions between different behaviors. A hybrid dynamical system displays both continuous and discrete dynamics and may display nonsmooth bifurcations. Herein, we analyze a novel hybrid model of a spiking neuron proposed by E.M. Izhikevich [9] that is based on a previous hybrid model with a convex spike-activation function f(x), but modified with a conductance reversal potential term. We analyze the model proposed by Izhikevich and obtain a bifurcation portrait for the continuous dynamics for an arbitrary convex spike activation function f(x). Both subcritical and supercritical Andronov-Hopf bifurcations are possible, and we numerically confirm the presence of a Bautin bifurcation for a particular choice of spike activation function. The model is capable of simulating common cortical neuron types and presents several possibilities for generalizations that may be capable of more complicated behavior.
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    Local Risk Minimization Under Time-Varying Transaction Costs
    (North Dakota State University, 2010) Nitschke, Matthew Cody
    Closely following the results of Lamberton, Pham, and Schweizer [5] we construct a locally risk-minimizing strategy in a general incomplete market including transactiou costs. This is done in dbcrete time under the assurnptious of a bounded meanvariance tracleoff and substantial risk. Once we establbh all the required integrability conditions, a backward induction argument is implemented to obtain the desired strategy for every square-integrable contingent claim. \Ve model the trnusactiou costs as an adapted stochastic process aud provide all necessary proofs in detail.
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    A New Generalization of Cohen-Kaplansky Domains
    (North Dakota State University, 2015) Kennedy, Diana Michelle
    The goal of this thesis is to provide an new generalization of Cohen-Kaplansky domains, stemming from questions related to valuation domains. Recall that a Cohen-Kaplansky domain is an atomic integral domain that contains only a nite number of irreducible elements (up to units). In the new generalization presented in this thesis, we remove the atomic condition required in the de nition of a Cohen-Kaplansky domain and add in the extra condition that our integral domain has nitely many irreducible elements, say 1; 2; ; n, such that for every nonzero nonunit y in the domain there exists an irreducible element, say i with 1 i n, such that i j y.
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    The Square of Adjacency Matrices
    (North Dakota State University, 2011) Kranda, Daniel Joseph
    It can be shown that any symmetric (0, 1)-matrix A with tr A = 0 can be interpreted as the adjacency matrix of a simple, finite graph. The square of an adjacency matrix A2 = (si1) has the property that Sij represents the number of walks of length two from vertex i to vertex j. With this information, the motivating question behind this paper was to determine what conditions on a matrix S are needed to have S = A(G)2 for some graph G. Structural results imposed by the matrix S include detecting bipartiteness or connectedness, counting four cycles and determining plausible neighborhoods of vertices. Some characterizations will be given and the problem of when S represents several non-isomorphic graphs is also explored.