498 research outputs found
Stationary Pattern of a Ratio-Dependent Food Chain Model with Diffusion
No full textThe main purpose of this erratum is to correct an error in the proof of Theorem 4.4 in [R. Peng, J. Shi, and M. Wang, SIAM J. Appl. Math., 67 (2007), pp. 1479-1503].Physic
Potential Stability of Matrix Sign Patterns
No full textThe topic of matrix stability is very important for determining the stability of solutions to systems of differential equations. We examine several problems in the field of matrix stability, including minimal conditions for a matrix sign pattern to be potentially stable, and applications of sign patterns to the study of Turing instability in the case. Furthermore, some of our work serves as a model for a new method of approaching similar problems in the future.MathematicsBachelors of Science (BS
Graph packing with constraints on edges
No full textA graph consists of a set of vertices (nodes) and a set of edges (line connecting vertices). Two graphs pack when they have the same number of vertices and we can put them in the same vertex set without overlapping edges. Studies such as Sauer and Spencer, Bollobas and Eldridge, Kostochka and Yu, have shown sufficient conditions, specifically relations between number of edges in the two graphs, for two graphs to pack, but only a few addressed packing with constraints. Kostochka and Yu proved that if e_1e_2 < (1 - \eps)n^2, then and pack with exceptions. We extend this finding by using the language of list packing introduced by Gyori, Kostochka, McConvey, and Yager, and we show that the triple with e_1e_2 + \frac{n-1}{2}\cdot e_3 < (2 - \eps)\binom{n}{2} pack with well-defined exceptions.MathematicsBachelors of Science (BS
Mathematical Studies of Optimal Economic Growth Model with Monetary Policy
No full textIn this paper, efforts will be made to study an extended Neoclassic economic growth model derived from Solow-Swan Model and Ramsey-Cass-Koopsman Model. Some growth models (e.g. Solow-Swan Model) attempt to explain long-run economic growth by looking at capital accumulation, labor or population growth, and in- creases in productivity, while our derived model tends to look at growth from individual household and how their choice of saving, consumption and money holdings would affect the overall economic capital accumulation over a long period of time. First an optimal control model is set up, and a system of differential equations and algebraic equations is derived from the optimal solutions which is an extension of the existing model. Secondly, the equilibrium points and their stability of both models are studied by calculating the determinant of their respective Jacobian matrix. Last but not least, model numerical simulations are performed for both models using Matlab, in hopes that it will give us a better understanding of the system and a clearer pattern for the behaviors of each model.Interdisciplinary StudiesBachelors of Science (BS
Meson Condensation and Holographic QCD
No full textQuantum Chromodynamics (QCD), which describes the strong nuclear force, is difficult to solve both analytically and numerically. We use a five-dimensional model of QCD motivated by the Anti-de Sitter/Conformal Field Theory (AdS/CFT) duality originally proposed by Maldacena. We discuss how this model and other variations in the literature represent chiral symmetry breaking, and test whether these models correctly reproduce chiral symmetry beyond leading order. We compare the predictions of two models, one which is correct to leading order, and another which is correct beyond leading order. The model correct to leading order does not properly predict the pion condensation phase transition, whereas the model with the correct beyond leading order qualitatively agrees with chiral perturbation theory in its description of pion condensation. Using these two models, we calculate certain observables and find agreement with experiment to within 15%
A stage-structured oyster population model for reef restoration
No full textOysters have experienced drastic declines in their population because of environmental factors and harvesting pressures, making them a focal species for restoration efforts [1, 22]. Oyster shell has become a limited resource and alternative substrates are not as suitable for larval recruitment and shell accumulation [1, 2, 7]. For this reason, restoration efforts are restricted, despite the attempts in the private and public sector. To increase the effectiveness of restoration, the dynamics of oyster reef systems must be further analyzed and understood. This thesis proposes a stage-structured ordinary differential equation model to investigate the dynamics of deterministic and stochastic oyster reef systems. Our self-replenishing oyster model expands upon the JARS stage-structured ODE model of oyster populations and allows for larval production from the natal reef [31]. There are four compartments to the model - juvenile oysters, adult oysters, oyster shell volume, and sediment volume. In this research, we evaluate the bifurcation structures of a deterministic model with different larval sources (internal and external). The deterministic stability analyses provide intuition into the overall structural dynamics of the equilibria of the system in response to changes in external (P0) and internal (P1) larval production. Parameter sensitivity analysis is conducted with this deterministic model to evaluate the impact of the parameters important to our predictions for reef restoration. Based on the results and to ensure careful estimation, the parameters for instantaneous growth rate (φ) and natural mortality (µ) are estimated using field data. Both of these parameters directly influence the change in adult oyster volume over time. The results of this section inform our understanding of the system dynamics in response to changes in parameters related to adult oyster growth and their ecosystem services. To evaluate this system in the context of environmental changes, we introduce stochastic larval availability to the model. Tests of normality and variance are used to analyze the stochastic data. Additionally, we use autocorrelation function analysis to quantify ii the time delay we observed in the data across the four model compartments. Analysis of this stochastic model informs our understanding of the distribution of sample data in each compartment as a response to annual changes in larval availability. The main implication of this work is to inform future oyster reef restoration efforts. Critical reef height (Rc) is defined as the minimum height the reef must initially be for the population to persist over time and is integral to restoration reef success. With the stochastic and deterministic models, Rc is determined. We vary the initial conditions of the Rc simulations to reflect common restoration strategies, in turn, producing biologically applicable results for reef restoration. Through this Rc analysis, and our understanding of the oyster reef system, predictions are made that can be experimentally tested in future work.Interdisciplinary StudiesBachelors of Science (BS
Non-linear Modifications of Black-Scholes Pricing Model with Diminishing Marginal Transaction Cost
No full textIn the field of quantitative financial analysis, the Black-Scholes Model has exerted significant influence on the booming of options trading strategies. Publishing in their Nobel Prize Work in 1973, the model was generated by Black and Scholes. Using Ito’s Lemma and portfolio management methodology, they employed partial differential equation to provide a theoretical estimate of the price of European-style options. This paper is interested in deriving non-linear modifications of the Black-Scholes model with diminishing marginal transaction cost.MathematicsBachelors of Science (BS
A Mathematical Study of Competition and Adoption of Two Consumer Products
No full textA mathematical model describing the competition between two consumer products in the market is constructed based on the Bass Diffusion Model and the competitive Lotka-Volterra model. Using this proposed model, the long-term behaviors of the two competing products can be forecasted. The model is analyzed and categorized into eight different cases with different settings of parameters, and under any of those cases, the two products are proved to co-exist in the long term.MathematicsBachelors of Science (BS
A New Upper Bound for the Diameter of the Cayley Graph of a Symmetric Group
No full textGiven a finite symmetric group S_n and a set S of generators, we can represent the group as a Cayley graph. The diameter of the Cayley graph is the largest distance from the identity to any other elements. We work on the conjecture that the diameter of the Cayley graph of a finite symmetric group S_n with S ={(12),(12...n)} is at most $ C(n,2). Our main result is to show that the diameter of the graph of S_n is at most (3n^2-4n)/2.MathematicsBachelors of Science (BS
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