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Articles 1 - 30 of 64
Full-Text Articles in Physical Sciences and Mathematics
Exploring Quaternion Neural Network Loss Surfaces, Jeremiah Bill, Bruce A. Cox
Exploring Quaternion Neural Network Loss Surfaces, Jeremiah Bill, Bruce A. Cox
Faculty Publications
This paper explores the superior performance of quaternion multi-layer perceptron (QMLP) neural networks over real-valued multi-layer perceptron (MLP) neural networks, a phenomenon that has been empirically observed but not thoroughly investigated. The study utilizes loss surface visualization and projection techniques to examine quaternion-based optimization loss surfaces for the first time. The primary contribution of this research is the statistical evidence that QMLP models yield smoother loss surfaces than real-valued neural networks, which are measured and compared using a robust quantitative measure of loss surface “goodness” based on estimates of surface curvature. Extensive computational testing validates the effectiveness of these surface …
Complete Solution Of The Lady In The Lake Scenario, Alexander Von Moll, Meir Pachter
Complete Solution Of The Lady In The Lake Scenario, Alexander Von Moll, Meir Pachter
Faculty Publications
In the Lady in the Lake scenario, a mobile agent, L, is pitted against an agent, M, who is constrained to move along the perimeter of a circle. L is assumed to begin inside the circle and wishes to escape to the perimeter with some finite angular separation from M at the perimeter. This scenario has, in the past, been formulated as a zero-sum differential game wherein L seeks to maximize terminal separation and M seeks to minimize it. Its solution is well-known. However, there is a large portion of the state space for which the canonical solution does not …
Legendre Pairs Of Lengths ℓ ≡ 0 (Mod 5), Ilias S. Kotsireas, Christopher Koutschan, Dursun Bulutoglu, David M. Arquette, Jonathan S. Turner, Kenneth J. Ryan
Legendre Pairs Of Lengths ℓ ≡ 0 (Mod 5), Ilias S. Kotsireas, Christopher Koutschan, Dursun Bulutoglu, David M. Arquette, Jonathan S. Turner, Kenneth J. Ryan
Faculty Publications
By assuming a type of balance for length ℓ = 87 and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for length ℓ = 85 , we find LPs of these lengths. We then study the power spectral density (PSD) values of m compressions of LPs of length 5 m . We also formulate a conjecture for LPs of lengths ℓ ≡ 0 (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range ≤ 200 for …
Anomaly Detection In The Molecular Structure Of Gallium Arsenide Using Convolutional Neural Networks, Timothy Roche *, Aihua W. Wood, Philip Cho *, Chancellor Johnstone
Anomaly Detection In The Molecular Structure Of Gallium Arsenide Using Convolutional Neural Networks, Timothy Roche *, Aihua W. Wood, Philip Cho *, Chancellor Johnstone
Faculty Publications
This paper concerns the development of a machine learning tool to detect anomalies in the molecular structure of Gallium Arsenide. We employ a combination of a CNN and a PCA reconstruction to create the model, using real images taken with an electron microscope in training and testing. The methodology developed allows for the creation of a defect detection model, without any labeled images of defects being required for training. The model performed well on all tests under the established assumptions, allowing for reliable anomaly detection. To the best of our knowledge, such methods are not currently available in the open …
Numerical Simulation Of The Korteweg–De Vries Equation With Machine Learning, Kristina O. F. Williams *, Benjamin F. Akers
Numerical Simulation Of The Korteweg–De Vries Equation With Machine Learning, Kristina O. F. Williams *, Benjamin F. Akers
Faculty Publications
A machine learning procedure is proposed to create numerical schemes for solutions of nonlinear wave equations on coarse grids. This method trains stencil weights of a discretization of the equation, with the truncation error of the scheme as the objective function for training. The method uses centered finite differences to initialize the optimization routine and a second-order implicit-explicit time solver as a framework. Symmetry conditions are enforced on the learned operator to ensure a stable method. The procedure is applied to the Korteweg–de Vries equation. It is observed to be more accurate than finite difference or spectral methods on coarse …
A Bit-Parallel Tabu Search Algorithm For Finding Es2 -Optimal And Minimax-Optimal Supersaturated Designs, Luis B. Morales, Dursun A. Bulotuglu
A Bit-Parallel Tabu Search Algorithm For Finding Es2 -Optimal And Minimax-Optimal Supersaturated Designs, Luis B. Morales, Dursun A. Bulotuglu
Faculty Publications
We prove the equivalence of two-symbol supersaturated designs (SSDs) with N (even) rows, m columns, smax=4t+i, where i ∈ {0,2}, t ∈ Z≥0 and resolvable incomplete block designs (RIBDs) whose any two blocks intersect in at most (N+4t+i)/4 points. Using this equivalence, we formulate the search for two-symbol E(s2)-optimal and minimax-optimal SSDs with smax ∈ {2,4,6} as a search for RIBDs whose blocks intersect accordingly. This allows developing a bit-parallel tabu search (TS) algorithm. The TS algorithm found E(s2)-optimal and minimax-optimal SSDs achieving the sharpest known E(s2) lower bound with …
A Comparison Of Quaternion Neural Network Backpropagation Algorithms, Jeremiah Bill, Bruce A. Cox, Lance Champaign
A Comparison Of Quaternion Neural Network Backpropagation Algorithms, Jeremiah Bill, Bruce A. Cox, Lance Champaign
Faculty Publications
This research paper focuses on quaternion neural networks (QNNs) - a type of neural network wherein the weights, biases, and input values are all represented as quaternion numbers. Previous studies have shown that QNNs outperform real-valued neural networks in basic tasks and have potential in high-dimensional problem spaces. However, research on QNNs has been fragmented, with contributions from different mathematical and engineering domains leading to unintentional overlap in QNN literature. This work aims to unify existing research by evaluating four distinct QNN backpropagation algorithms, including the novel GHR-calculus backpropagation algorithm, and providing concise, scalable implementations of each algorithm using a …
Node Generation For Rbf-Fd Methods By Qr Factorization, Tony Liu, Rodrigo B. Platte
Node Generation For Rbf-Fd Methods By Qr Factorization, Tony Liu, Rodrigo B. Platte
Faculty Publications
Polyharmonic spline (PHS) radial basis functions (RBFs) have been used in conjunction with polynomials to create RBF finite-difference (RBF-FD) methods. In 2D, these methods are usually implemented with Cartesian nodes, hexagonal nodes, or most commonly, quasi-uniformly distributed nodes generated through fast algorithms. We explore novel strategies for computing the placement of sampling points for RBF-FD methods in both 1D and 2D while investigating the benefits of using these points. The optimality of sampling points is determined by a novel piecewise-defined Lebesgue constant. Points are then sampled by modifying a simple, robust, column-pivoting QR algorithm previously implemented to find sets of …
Estimating Turbulence Distribution Over A Heterogeneous Path Using Time‐Lapse Imagery From Dual Cameras, Benjamin Wilson, Santasri Bose-Pillai, Jack E. Mccrae, Kevin J. Keefer, Steven T. Fiorino
Estimating Turbulence Distribution Over A Heterogeneous Path Using Time‐Lapse Imagery From Dual Cameras, Benjamin Wilson, Santasri Bose-Pillai, Jack E. Mccrae, Kevin J. Keefer, Steven T. Fiorino
Faculty Publications
Knowledge of turbulence distribution along an experimental path can help in effective turbulence compensation and mitigation. Although scintillometers are traditionally used to measure the strength of turbulence, they provide a path-integrated measurement and have limited operational ranges. A technique to profile turbulence using time-lapse imagery of a distant target from spatially separated cameras is presented here. The method uses the turbulence induced differential motion between pairs of point features on a target, sensed at a single camera and between cameras to extract turbulence distribution along the path. The method is successfully demonstrated on a 511 m almost horizontal path going …
Defect Detection In Atomic Resolution Transmission Electron Microscopy Images Using Machine Learning, Philip Cho, Aihua W. Wood, Krishnamurthy Mahalingam, Kurt Eyink
Defect Detection In Atomic Resolution Transmission Electron Microscopy Images Using Machine Learning, Philip Cho, Aihua W. Wood, Krishnamurthy Mahalingam, Kurt Eyink
Faculty Publications
Point defects play a fundamental role in the discovery of new materials due to their strong influence on material properties and behavior. At present, imaging techniques based on transmission electron microscopy (TEM) are widely employed for characterizing point defects in materials. However, current methods for defect detection predominantly involve visual inspection of TEM images, which is laborious and poses difficulties in materials where defect related contrast is weak or ambiguous. Recent efforts to develop machine learning methods for the detection of point defects in TEM images have focused on supervised methods that require labeled training data that is generated via …
Meta-Heuristic Optimization Methods For Quaternion-Valued Neural Networks, Jeremiah Bill, Lance E. Champagne, Bruce Cox, Trevor J. Bihl
Meta-Heuristic Optimization Methods For Quaternion-Valued Neural Networks, Jeremiah Bill, Lance E. Champagne, Bruce Cox, Trevor J. Bihl
Faculty Publications
In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued …
Acceleration Of Boltzmann Collision Integral Calculation Using Machine Learning, Ian Holloway, Aihua W. Wood, Alexander Alekseenko
Acceleration Of Boltzmann Collision Integral Calculation Using Machine Learning, Ian Holloway, Aihua W. Wood, Alexander Alekseenko
Faculty Publications
The Boltzmann equation is essential to the accurate modeling of rarefied gases. Unfortunately, traditional numerical solvers for this equation are too computationally expensive for many practical applications. With modern interest in hypersonic flight and plasma flows, to which the Boltzmann equation is relevant, there would be immediate value in an efficient simulation method. The collision integral component of the equation is the main contributor of the large complexity. A plethora of new mathematical and numerical approaches have been proposed in an effort to reduce the computational cost of solving the Boltzmann collision integral, yet it still remains prohibitively expensive for …
A Radial Basis Function Finite Difference Scheme For The Benjamin–Ono Equation, Benjamin F. Akers, Tony Liu, Jonah A. Reeger
A Radial Basis Function Finite Difference Scheme For The Benjamin–Ono Equation, Benjamin F. Akers, Tony Liu, Jonah A. Reeger
Faculty Publications
A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on ℝ, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.
Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, Matthew C. Fickus, Courtney A. Schmitt
Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, Matthew C. Fickus, Courtney A. Schmitt
Faculty Publications
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin …
Legendre G-Array Pairs And The Theoretical Unification Of Several G-Array Families, K. T. Arasu, Dursun A. Bulutoglu, J. R. Hollon
Legendre G-Array Pairs And The Theoretical Unification Of Several G-Array Families, K. T. Arasu, Dursun A. Bulutoglu, J. R. Hollon
Faculty Publications
We investigate how Legendre G-array pairs are related to several different perfect binary G-array families. In particular we study the relations between Legendre G-array pairs, Sidelnikov-Lempel-Cohn-Eastman ℤq−1-arrays, Yamada-Pott G-array pairs, Ding-Helleseth-Martinsen ℤ2×ℤmp-arrays, Yamada ℤ(q−1)/2-arrays, Szekeres ℤmp-array pairs, Paley ℤmp-array pairs, and Baumert ℤm1p1×ℤm2p2-array pairs. Our work also solves one of the two open problems posed in Ding~[J. Combin. Des. 16 (2008), 164-171]. Moreover, we provide several computer search based existence and non-existence results regarding Legendre ℤn-array pairs. …
Polyphase Equiangular Tight Frames And Abelian Generalized Quadrangles, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson, Cody E. Watson
Polyphase Equiangular Tight Frames And Abelian Generalized Quadrangles, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson, Cody E. Watson
Faculty Publications
An equiangular tight frame (ETF) is a type of optimal packing of lines in a finite-dimensional Hilbert space. ETFs arise in various applications, such as waveform design for wireless communication, compressed sensing, quantum information theory and algebraic coding theory. In a recent paper, signature matrices of ETFs were constructed from abelian distance regular covers of complete graphs. We extend this work, constructing ETF synthesis operators from abelian generalized quadrangles, and vice versa. This produces a new infinite family of complex ETFs as well as a new proof of the existence of certain generalized quadrangles. This work involves designing matrices whose …
Hadamard Equiangular Tight Frames, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson
Hadamard Equiangular Tight Frames, Matthew C. Fickus, John Jasper, Dustin G. Mixon, Jesse D. Peterson
Faculty Publications
An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. They are often represented as the columns of a short, fat matrix. In certain applications we want this matrix to be flat, that is, have the property that all of its entries have modulus one. In particular, real flat ETFs are equivalent to self-complementary binary codes that achieve the Grey-Rankin bound. Some flat ETFs are (complex) Hadamard ETFs, meaning they arise by extracting rows from a (complex) Hadamard matrix. These include harmonic ETFs, which are obtained by extracting the rows of a character table …
Periodic Traveling Interfacial Hydroelastic Waves With Or Without Mass Ii: Multiple Bifurcations And Ripples, Benjamin F. Akers, David M. Ambrose, David W. Sulon
Periodic Traveling Interfacial Hydroelastic Waves With Or Without Mass Ii: Multiple Bifurcations And Ripples, Benjamin F. Akers, David M. Ambrose, David W. Sulon
Faculty Publications
In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic traveling waves on infinite depth, and computed such traveling waves. The formulation of the traveling wave problem used both analytically and numerically allows for waves with multi-valued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and non-resonant variants. We apply an implicit function theorem argument to prove existence of traveling waves …
Cocyclic Hadamard Matrices: An Efficient Search Based Algorithm, Jonathan S. Turner
Cocyclic Hadamard Matrices: An Efficient Search Based Algorithm, Jonathan S. Turner
Theses and Dissertations
This dissertation serves as the culmination of three papers. “Counting the decimation classes of binary vectors with relatively prime fixed-density" presents the first non-exhaustive decimation class counting algorithm. “A Novel Approach to Relatively Prime Fixed Density Bracelet Generation in Constant Amortized Time" presents a novel lexicon for binary vectors based upon the Discrete Fourier Transform, and develops a bracelet generation method based upon the same. “A Novel Legendre Pair Generation Algorithm" expands upon the bracelet generation algorithm and includes additional constraints imposed by Legendre Pairs. It further presents an efficient sorting and comparison algorithm based upon symmetric functions, as well …
Finding The Symmetry Group Of An Lp With Equality Constraints And Its Application To Classifying Orthogonal Arrays, Andrew J. Geyer, Dursun A. Bulutoglu, Kenneth J. Ryan
Finding The Symmetry Group Of An Lp With Equality Constraints And Its Application To Classifying Orthogonal Arrays, Andrew J. Geyer, Dursun A. Bulutoglu, Kenneth J. Ryan
Faculty Publications
Excerpt: For a given linear program (LP) a permutation of its variables that sends feasible points to feasible points and preserves the objective function value of each of its feasible points is a symmetry of the LP. The set of all symmetries of an LP, denoted by GLP, is the symmetry group of the LP. Margot (2010) described a method for computing a subgroup of the symmetry group GLP of an LP. This method computes GLP when the LP has only non-redundant inequalities and its feasible set satisfies no equality constraints.
Atmospheric Propagation Of High Energy Lasers: Thermal Blooming Simulation, Jonathan Gustafsson, Benjamin F. Akers, Jonah A. Reeger, Sivaguru S. Sritharan
Atmospheric Propagation Of High Energy Lasers: Thermal Blooming Simulation, Jonathan Gustafsson, Benjamin F. Akers, Jonah A. Reeger, Sivaguru S. Sritharan
Faculty Publications
High Energy Laser (HEL) propagation through turbulent atmosphere is examined via numerical simulation. The beam propagation is modeled with the paraxial equation, which in turn is written as a system of equations for a quantum fluid, via the Madelung transform. A finite volume solver is applied to the quantum fluid equations, which supports sharp gradients in beam intensity. The atmosphere is modeled via a coupled advection-diffusion equation whose initial data have Kolmogorov spectrum. In this model the combined effects of thermal blooming, beam slewing, and deep turbulence are simulated.
Piezoelectric Sensor Crack Detection On Airframe Systems, Kevin J. Lin
Piezoelectric Sensor Crack Detection On Airframe Systems, Kevin J. Lin
Theses and Dissertations
In 2008, the Department of Defense published a guidebook for a methodology named Condition-Based Maintenance Plus (CBM+) which capabilities include improving productivity, shortening maintenance cycles, lowering costs, and increasing availability and reliability. This push replaces existing inspection criteria, often conducted as non-destructive testing (NDT), with structural health monitoring (SHM) systems. The SHM system addressed utilizes guided Lamb waves generated by piezoelectric wafer active sensors (PWAS) to detect the existence, size, and location of damage from through-thickness cracks around a rivet hole. The SHM field lacks an experiment testing how small changes in receiver sensor distances affect damage detection. In addition, …
Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, Courtney A. Schmitt
Harmonic Equiangular Tight Frames Comprised Of Regular Simplices, Courtney A. Schmitt
Theses and Dissertations
An equiangular tight frame (ETF) is a sequence of equal-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs, which are formed by restricting the characters of a finite abelian group to a difference set. Recently, it was shown that some harmonic ETFs are themselves comprised of regular simplices. In this thesis, we continue the …
Analyzing A Method To Determine The Utility Of Adding A Classification System To A Sequence For Improved Accuracy, Kevin S. Pamilagas
Analyzing A Method To Determine The Utility Of Adding A Classification System To A Sequence For Improved Accuracy, Kevin S. Pamilagas
Theses and Dissertations
Frequently, ensembles of classification systems are combined into a sequence in order to better enhance the accuracy in classifying objects of interest. However, there is a point in which adding an additional system to a sequence no longer enhances the system as either the increase in operational costs exceeds the benefit of improvements in classification or the addition of the system does not increase accuracy at all. This research will examine a utility measure to determine the valid or invalid nature of adding a classification system to a sequence of such systems based on the ratio of the change in …
Schlieren Imaging And Flow Analysis On A Cone/Flare Model In The Afrl Mach 6 Ludwieg Tube Facility, David A. Labuda
Schlieren Imaging And Flow Analysis On A Cone/Flare Model In The Afrl Mach 6 Ludwieg Tube Facility, David A. Labuda
Theses and Dissertations
High-speed Schlieren photography was utilized to visualize flow in the Air Force Research Laboratory Mach 6 Ludwieg tube facility. A 7° half-angle cone/flare model with variable nosetip radius and flare angle options was used in the study. Testing was performed at two driver tube pressures, generating freestream Reynolds numbers of 10.0x106 and 19.8x106 per meter. The variable-angle flare portion of the model provided a method for adjusting the intensity of the adverse pressure gradient at the cone/flare junction. As expected from existing literature, boundary layer separation along the cone frustum occurred further upstream as the magnitude of the …
Wall Model Large Eddy Simulation Of A Diffusing Serpentine Inlet Duct, Ryan J. Thompson
Wall Model Large Eddy Simulation Of A Diffusing Serpentine Inlet Duct, Ryan J. Thompson
Theses and Dissertations
The modeling focus on serpentine inlet ducts (S-duct), as with any inlet, is to quantify the total pressure recovery and ow distortion after the inlet, which directly impacts the performance of a turbine engine fed by the inlet. Accurate prediction of S-duct ow has yet to be achieved amongst the computational fluid dynamics (CFD) community to improve the reliance on modeling reducing costly testing. While direct numerical simulation of the turbulent ow in an S-duct is too cost prohibitive due to grid scaling with Reynolds number, wall-modeled large eddy simulation (WM-LES) serves as a tractable alternative. US3D, a hypersonic research …
Solving The Traveling Salesman Problem Using Ordered-Lists, Petar D. Jackovich
Solving The Traveling Salesman Problem Using Ordered-Lists, Petar D. Jackovich
Theses and Dissertations
The arc-greedy heuristic is a constructive heuristic utilized to build an initial, quality tour for the Traveling Salesman Problem (TSP). There are two known sub-tour elimination methodologies utilized to ensure the resulting tours are viable. This thesis introduces a third novel methodology, the Greedy Tracker (GT), and compares it to both known methodologies. Computational results are generated across multiple TSP instances. The results demonstrate the GT is the fastest method for instances below 400 nodes while Bentley's Multi-Fragment maintains a computational advantage for larger instances. A novel concept called Ordered-Lists is also introduced which enables TSP instances to be explored …
Time Series Analysis Of Stochastic Networks With Correlated Random Arcs, Brendon T. Sands
Time Series Analysis Of Stochastic Networks With Correlated Random Arcs, Brendon T. Sands
Theses and Dissertations
While modern day weather forecasting is not perfect, there are many benefits given by the multitude and variety of predictive models. In the interest of routing airplanes, this paper uses time series analysis on successive weather forecasts to predict the optimal path and fuel burn of wind-based, fuel-burn networks with stochastic correlated arcs. Networks are populated with either deterministic or ensemble-based weather data, and the two data sources with and without time series analysis are compared. Methods were compared by fuel burn prediction accuracy and ability to predict a future optimal path. Of the four options, the ensemble-based methods were …
Equiangular Tight Frames That Contain Regular Simplices, Matthew C. Fickus, John Jasper, Emily J. King, Dustin G. Mixon
Equiangular Tight Frames That Contain Regular Simplices, Matthew C. Fickus, John Jasper, Emily J. King, Dustin G. Mixon
Faculty Publications
An equiangular tight frame (ETF) is a type of optimal packing of lines in Euclidean space. A regular simplex is a special type of ETF in which the number of vectors is one more than the dimension of the space they span. In this paper, we consider ETFs that contain a regular simplex, that is, have the property that a subset of its vectors forms a regular simplex. As we explain, such ETFs are characterized as those that achieve equality in a certain well-known bound from the theory of compressed sensing. We then consider the so-called binder of such an …
Equiangular Tight Frames From Group Divisible Designs, Matthew C. Fickus, John Jasper
Equiangular Tight Frames From Group Divisible Designs, Matthew C. Fickus, John Jasper
Faculty Publications
An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem in many cases. In this paper, we observe that many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known …