Browsing by Author "Braverman, Elena"
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- ItemOpen AccessA spatial generalization of the ricker model and the break of chaos with applications(2009) Haroutunian, Jeff; Braverman, ElenaThe Ricker model is a well-studied discrete model of population dynamics given through the map Xn+i = Xner(l -xn). It is known that this map is chaotic for large r, but that the addition of a constant perturbation will induce a series of period-doubling reversals until at last there is a stable 2-cycle for large r. In this thesis we propose a spatial Ricker model on a discrete lattice. The interactions are such that each cell is influenced only by itself and its nearest neighbours at the previous stage. Under the influence of a constant perturbation we find that there is no chaos for sufficiently large r, and that in place of chaos there may be two kinds of points: points which have 2-cycle dynamics, and points which exhibit nearly stable dynamics. That is, while other points are in 2-cycle, these points only ever have values very close to the lower phase of this 2-cycle. We also consider the case of a negative perturbation, as well a..c; a few others, one of which is of spatial nature and unique to this problem. We also make a few modifications to the model to simulate environmental biases. Another problem of great interest 1s: under what conditions does a discrete map with a constant perturbation exhibit this 'break of chaos' behaviour? We study this problem by focusing on unimodal functions that relate to population dynamics. For all of the models we study, we observe 2-cycle dynamics for large enough choice of growth parameter (and 3-cycle dynamics for a special case). We outline the consequences of this study to the field of population dynamics, and we mention applications to synchronization and cellular automata.
- ItemOpen AccessApplications of difference equations to some intial value problems(2008) Zhukovskiy, Sergey; Braverman, Elena
- ItemOpen AccessCharacterization of Surface-Plasmon Polaritons and Electromagnetic Waveguides With Positive, Negative and Near-Zero Permittivity and Permeability(2017) Sang-Nourpour, Nafiseh; Sanders, Barry C.; Kheradmand, Reza; Hobill, David Wesley; Potter, Mike; Braverman, Elena; Sipe, John E.This thesis reports advances in characterization of electromagnetic waveguides and surface-plasmon polaritons. I advance the applications of electromagnetic waveguides through investigations of electromagnetic duality in waveguides and the tunability of particular waveguide modes. Moreover, an accurate and precise procedure is devised for characterizing surface-plasmon polaritons at lossy planar and curved interfaces. The explanations of each step in this work are summarized as follows. A description of waveguides that respects the duality of electromagnetism, namely the symmetry of the equations arising through the inclusion of magnetic monopoles in addition to electrons, is presented in the thesis. To ensure manifest electromagnetic duality in waveguides, I employ generic electromagnetic susceptibilities that are dual in both electric charges and magnetic monopoles using the generalized Drude-Lorentz model. Our description accommodates exotic media, such as double-negative, near-zero and zero-index materials. I consider metamaterials and metamaterial waveguides, as well as metal waveguides, as examples of waveguides constructed of electromagnetic materials. In particular, in the slab and cylindrical waveguides, exchanging electric and magnetic material properties leads to the exchange of transverse magnetic and transverse electric modes and dispersion equations, which suggests a good test of the potential duality of waveguides. Our advances establish an intuitive microscopic-level understanding of the electromagnetic duality in waveguides and its applications. The properties of metamaterials are then employed to tailor the modes of metamaterial-dielectric waveguides operating at optical frequencies. I survey the effects of 3D isotropic metamaterial structural parameters on the refractive index of metamaterials and on the modes in slab metamaterial-dielectric waveguides. Hybrid modes refer to hybrid ordinary-surface-plasmon polariton modes in the waveguide structures. I investigate how robust metamaterials are to fluctuations in their structural parameters; specifically, the effects of Gaussian errors are examined on the metamaterials EM behaviour. Our survey enables us to determine the allowable fluctuation limits and from this to identify appropriate unit-cell structure for further applications of metamaterials in waveguides technologies. I also characterize surface-plasmon polaritons at lossy planar interfaces between one dispersive and one nondispersive linear isotropic homogeneous media, i.e., materials or metamaterials. Specifically, Maxwell's equations are solved to obtain strict bounds for the permittivity and permeability of these media such that satisfying these bounds implies surface-plasmon polaritons successfully propagate at the interface, and violation of the bounds signifies propagation is impeded, i.e., the field delocalizes from the surface into the bulk. Our characterization of surface-plasmon polaritons is valuable for checking viability of a proposed application, and, as an example, our method is employed to falsify a previous prediction that surface-plasmon propagation through a surface of a double-negative refractive index medium occurs for any permittivity and permeability; instead our results show that propagation can occur only for certain medium parameters. Finally, a theoretical study of surface-plasmon polaritons propagation along lossy curved interfaces is presented here. Specifically, conformal transformation is employed to map the curved interface between one lossy dispersive and one nondispersive linear isotropic homogeneous material to a planar interface between inhomogeneous materials. My characterization of surface-plasmon polaritons is valuable for checking the viability of a proposed application.
- ItemOpen AccessCompact High-order Finite Difference Schemes for Acoustic Wave Equations(2021-01-05) Li, Keran; Liao, Wenyuan; Lamoureux, Michael P.; Liao, Wenyuan; Braverman, Elena; Liang, Dong; Ware, Antony FrankThis study developed three compact high-order finite difference schemes for acoustic wave equations. Benefiting from the compactness, the new schemes require less layers of boundary conditions than conventional finite difference schemes. All the three schemes work for acoustic wave equations with variable coefficients in homogeneous media, with the third one also being applicable to the case of heterogeneous density media. The first scheme is based on Padé approximation which is formally a product of the inverse of a finite difference operator and the conventional 2nd-order finite difference operator, thus some algebraic manipulation is necessary to discuss the product of operators. The second scheme is based on so-called combined finite difference method, which needs the boundary conditions for the second spatial derivatives and the needed boundary conditions can be derived by using the wave equation and usual Dirichlet boundary conditions themselves. The third scheme is also based on combined finite difference method, and it generalizes the second scheme so that it can also work in heterogeneous density media case, i.e., the Laplacian in the wave equations being divergence form. The stability of the first two schemes are established by an energy method, while the stability of the last scheme is obtained by an analogy of von Neumann analysis. All of these new schemes are proven to be conditionally stable with given Courant-Friedrichs-Lewy (CFL) numbers. Numerical experiments are conducted to verify the efficiency, accuracy and stability of the new schemes. It is expected that these new schemes will find extensive applications in both research and engineering areas.
- ItemOpen AccessDiscrete and delay continuous population models with unimodal reproduction functions(2006) Kinzebulatov, Damir; Braverman, Elena
- ItemOpen AccessExponential Stability of Difference Equations with Several Delays: Recursive Approach(Hindawi Publishing Corporation, 2009-04-23) Berezansky, Leonid; Braverman, Elena
- ItemOpen AccessNew Stability Conditions for Linear Differential Equations with Several Delays(2011-06-05) Berezansky, Leonid; Braverman, ElenaNew explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation (...) with measurable delays and coefficients. These results are compared to known stability tests.
- ItemOpen AccessNew Stability Conditions for Linear Differential Equations with Several Delays(Hindawi Publishing Corporation, 2011-04-04) Berezansky, Leonid; Braverman, Elena
- ItemOpen AccessNonoscillation of First-Order Dynamic Equations with Several Delays(Hindawi Publishing Corporation, 2010-07-21) Braverman, Elena; Karpuz, Basak
- ItemOpen AccessNonoscillation of Second-Order Dynamic Equations with Several Delays(2011-04-14) Braverman, Elena; Karpuz, BaşakExistence of nonoscillatory solutions for the second-order dynamic equation is investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the case and for second-order nondelay difference equations. Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitrary and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced.
- ItemOpen AccessNonoscillation of Second-Order Dynamic Equations with Several Delays(Hindawi Publishing Corporation, 2011-02) Braverman, Elena; Karpuz, Basak
- ItemOpen AccessNovel Optimization Schemes for Full Waveform Inversion: Optimal Transport and Inexact Gradient Projection(2021-03-09) Li, Da; Lamoureux, Michael P.; Liao, Wenyuan; Lamoureux, Michael P.; Liao, Wenyuan; Braverman, Elena; Innanen, Kristopher A.; Fear, Elise C.; Haber, EldadFull waveform inversion (FWI) is an important seismic inversion technique that provides high-resolution estimates of underground physical parameters. However, high-accuracy inverse results are not guaranteed due to the essential non-convexity characteristics of the FWI problem. This thesis focuses on designing novel optimization schemes for the FWI problem which improve the inverse results. Applying optimal transport (OT) based distances to the FWI problem is popular because they provide additional geometric information. The OT distances are designed for positive measures with equal mass, and the unbalanced optimal transport (UOT) distance can overcome the mass equality condition. A mixed distance is constructed which can also overcome the mass equality condition, and the convex properties for the shift, dilation, and amplitude change are proved. Both UOT distance and the proposed distance are applied to the FWI problem with normalization methods transforming the signals into positive functions. Numerical examples show that the optimal transport based distances outperform the traditional L2 distance in certain cases. The gradient projection methods are often used to solve constrained optimization problems, and the closed-form projection function is necessary since the projection has to be evaluated exactly. A constraint set expanding strategy is designed for the gradient projection methods such that the projection can be evaluated inexactly, which extends the application scope of the gradient projection methods. The convergence analysis is provided with proper assumptions. A priori information of the model is important to improve the inverse result, and an optimization scheme is proposed for incorporating multiple a priori information into the FWI problem. The optimization scheme is a combination of the scaled gradient projection method and a projection onto convex sets algorithm. Also, the L-BFGS Hessian approximation and the above constraint set expanding strategy are used. Numerical examples show that the proposed optimization scheme is flexible for integrating multiple types of constraint sets such as total variation constraint, sparsity constraint, box constraint, and hyperplane constraint into the FWI problem.
- ItemOpen AccessOn Nonoscillation of Advanced Differential Equations with Several Terms(Hindawi Publishing Corporation, 2011-01-27) Berezansky, L.; Braverman, Elena
- ItemOpen AccessOn oscillation of a food-limited population model with time delay(2003-01-01) Berezansky, Leonid; Braverman, ElenaFor a scalar nonlinear delay differential equation Ṅ(t) = r(t)N(t)(K − N(h(t)))/(K + s(t)N(g(t))),r(t) ≥ 0, h(t) ≤ t, g(t) ≤ t and some generalizations of this equation, we establish explicit oscillation and nonoscillation conditions. Coefficient r(t) and delays are not assumed to be continuous.
- ItemOpen AccessOptimal impulsive sustainable harvesting(2007) Mamdani, Reneeta; Braverman, Elena
- ItemOpen AccessOptimal Policy for Blood Inventory Management Problem(2018-08-22) Grushevska, Iaryna; Braverman, Elena; Sabouri, Alireza; Zinchenko, Yuriy; Alp, OsmanBlood units that are used for transfusion can be stored for a limited amount of time. The blood that is older than 42 days must be discarded. In order not to face shortage usually the oldest blood is used. But the risk of complications after surgery is growing as the age of used blood is growing as well. In this work we find the optimal policy to use blood for transfusion for two blood types. The main goal is to find the policy that will reduce the shortage and minimize the risk of complications at the same time. For this purpose, we use two methods (Linear Programming and Approximate Dynamic Programming) and compare the results of two approaches.
- ItemOpen AccessOptimality and Sustainability of Delayed Impulsive Harvesting(2022-07-18) Lawson, Jennifer Lynn; Braverman, Elena; Liao, Wenyuan; Rios, Cristian; Post, John RobertOptimal and sustainable management of natural resources requires knowledge about the behaviour of mathematical models of harvesting under many different types of conditions. In this thesis, the effects of delays on the optimality and sustainability of harvesting models are studied, with a particular focus on delayed impulsive harvesting models. We begin by considering delays within a continuous harvesting model and derive sufficient conditions for stability of harvesting models with general growth and harvesting rate. We also derive maximum sustainable yields for models with both logistic and Gompertz growth, and show that they are delay dependent. Then we consider the main object of the thesis, a logistic differential equation subject to impulsive delayed harvesting, where the deduction information is a function of the population size at the time of one of the previous impulses. A close connection to the dynamics of high-order difference equations is used to conclude that while the inclusion of a delay in the impulsive condition does not impact the optimality of the yield, sustainability may be highly affected and is once again delay-dependent. Maximum and other types of yields are explored, and sharp stability tests are obtained for the model, as well as explicit sufficient conditions. It is also shown that persistence of the solution is not guaranteed for all positive initial conditions, and extinction in finite time is possible, as is illustrated in the simulations. The results of this thesis imply that delays within harvesting should be kept short to maintain the sustainability of resources.
- ItemOpen AccessOptimization of Quantum Algorithms for Applications(2022-05-04) Nerem, Robert Riley; Sanders, Barry; Hoyer, Peter; Gour, Gilad; Eberly, Wayne; Braverman, ElenaI aim to design and evaluate quantum algorithms that perform optimally with respect to metrics that make or break the applicability of these algorithms. Specifically, I analyze two applications: Bitcoin mining and estimating expectation values from a system of linear equations. For the former I develop a quantum algorithm for Bitcoin mining which optimizes the probability of successfully mining Bitcoin. For the later I give a quantum algorithm with query complexity that is optimally dependent on accuracy. I ensure that my quantum algorithms are relevant to applications by designing algorithms that are end-to-end for their applications, as opposed to algorithms that only address a subroutine. My work yields quantum algorithms that are directly comparable to their classical counterparts. By making this comparison, I develop necessary conditions for quantum algorithms to outperform classical algorithms at solving systems of linear equations and Bitcoin mining.
- ItemOpen AccessRegularity of Solutions to a Class of Infinitely Degenerate Second Order Quasilinear Equations(2013-10-02) Korobenko, Lyudmila; Rios, Cristian; Braverman, ElenaThis thesis studies regularity of weak solutions to quasilinear infinitely degenerate second order equations. We show that every weak solution to a certain class of degenerate quasilinear equations of divergence form is continuous, thus completing the result on hypoellipticity. We also study properties of subunit metric spaces associated to degenerate second order operators.
- ItemOpen AccessA Study on Cross-Frequency Coupling Effects on Intra-cranial EEG (iEEG) Data of Epileptic Patients(2020-12) Bazhan, Yanina; Braverman, Elena; Vasudevan, Kris; Zinchenko, Yuriy; Federico, PaoloHuman epilepsy is ascribed to neuronal disorder of the brain. Neuronal activity of drug-resistant epileptic patients awaiting resection surgeries is routinely studied using intracranial electroencephalogram (iEEG) recording techniques. Analysis methods applied to the measured iEEG data seek answers to the neuronal behaviour at different stages or ictal periods of a seizure. Cross-frequency coupling (CFC) that influences the phase and amplitude of the frequencies of neuronal signals is a basic brain activity striking a relationship between normal and disorder conditions. Shedding light on the CFC in measured iEEG data of epilepsy patients in vivo is important to understanding of and mediation to seizures. In this thesis, we examine a few CFC metrics, such as the generalized linear model (GLM), the modulation index (MI) and the phase-locking value (PLV). We compute the numerical values for these metrics to establish cross-frequency coupling “within” signals and also “between” brain regions of four different iEEG datasets of epileptic patients. Finally, we draw some preliminary conclusions on the results and suggest future directions.