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1. Surface Registration with Eigenvalues and Eigenvectors

   Hamidian, Hajar ; Zhong, Zichun ; Fotouhi, Farshad ; Hua, Jing ( January 2019 , IEEE transactions on visualization and computer graphics)

   This paper presents a novel surface registration technique using the spectrum of the shapes, which can facilitate accurate localization and visualization of non-isometric deformations of the surfaces. In order to register two surfaces, we map both eigenvalues and eigenvectors of the Laplace-Beltrami of the shapes through optimizing an energy function. The function is defined by the integration of a smoothness term to align the eigenvalues and a distance term between the eigenvectors at feature points to align the eigenvectors. The feature points are generated using the static points of certain eigenvectors of the surfaces. By using both the eigenvalues and the eigenvectors on these feature points, the computational efficiency is improved considerably without losing the accuracy in comparison to the approaches that use the eigenvectors for all vertices. In our technique, the variation of the shape is expressed using a scale function defined at each vertex. Consequently, the total energy function to align the two given surfaces can be defined using the linear interpolation of the scale function derivatives. Through the optimization of the energy function, the scale function can be solved and the alignment is achieved. After the alignment, the eigenvectors can be employed to calculate the point-to-point correspondence of the surfaces. Therefore, the proposed method can accurately define the displacement of the vertices. We evaluate our method by conducting experiments on synthetic and real data using hippocampus, heart, and hand models. We also compare our method with non-rigid Iterative Closest Point (ICP) and a similar spectrum-based methods. These experiments demonstrate the advantages and accuracy of our method 

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2. Multicomponent diffusion in a basaltic melt: Temperature dependence

   https://doi.org/10.1016/j.chemgeo.2020.119700  

   Guo, Chenghuan Zhang ( September 2020 , Chemical geology)

   Eighteen successful diffusion couple experiments in 8-component SiO2–TiO2–Al2O3–FeO–MgO–CaO–Na2O–K2O basaltic melts were conducted at 1260°C and 0.5 GPa and at 1500°C and 1.0 GPa. These experiments are combined with previous data at 1350°C and 1.0 GPa (Guo and Zhang, 2018) to study the temperature dependence of multicomponent diffusion in basaltic melts. Effective binary diffusion coefficients of components with monotonic diffusion profiles were extracted and show a strong dependence on their counter-diffusing component even though the average (or interface) compositions are the same. The diffusion matrix at 1260°C was obtained by simultaneously fitting diffusion profiles of all diffusion couple experiments as well as appropriate data from the literature. All features of concentration profiles in both diffusion couples and mineral dissolution are well reproduced by this new diffusion matrix. At 1500°C, only diffusion couple experiments are used to obtain the diffusion matrix. Eigenvectors of the diffusion matrix are used to discuss the diffusion (exchange) mechanism, and eigenvalues characterize the diffusion rate. Diffusion mechanisms at both 1260 and 1500°C are inferred from eigenvectors of diffusion matrices and compared with those at 1350°C reported in Guo and Zhang (2018). There is indication that diffusion eigenvectors in basaltic melts do not depend much on temperature, but complexity is present for some eigenvectors. The two slowest eigenvectors involve the exchange of SiO2 and/or Al2O3 with nonalkalis. The third slowest eigenvector is due to the exchange of divalent oxides with other oxides. The fastest eigenvector is due to the exchange of Na2O with other oxide components. Some eigenvalues differ from each other by less than 1/3, and their eigenvectors are less well defined. We define small difference in eigenvalues as near degeneracy. In strict mathematical degeneracy, eigenvectors are not uniquely defined because any linear combination of two eigenvectors is also an eigenvector. In the case of near degeneracy, more constraints either in terms of higher data quality or more experiments are needed to resolve the eigenvectors. We made a trial effort to generate a set of average eigenvectors, which are assumed to be constant as temperature varies. The corresponding eigenvalues are roughly Arrhenian. Thus, the temperature-dependent diffusion matrix can be roughly predicted. The method is applied to predict experimental diffusion profiles in basaltic melts during olivine and anorthite dissolution at ~1400°C with preliminary success. We further applied our diffusion matrix to investigate multicomponent diffusion during magma mixing in the Bushveld Complex and found such diffusion may result in an increased likelihood of sulfide and Fe-Ti oxide mineralization. 

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3. A Fast and Effective Memristor-Based Method for Finding Approximate Eigenvalues and Eigenvectors of Non-negative Matrices

   https://doi.org/10.1109/ISVLSI.2018.00108  

   Wang, Chenghong ; Jalali, Zeinab S. ; Ding, Caiwen ; Wang, Yanzhi ; Soundarajan, Sucheta ( July 2018 , 2018 IEEE Computer Society Annual Symposium on VLSI (ISVLSI))

   Throughout many scientific and engineering fields, including control theory, quantum mechanics, advanced dynamics, and network theory, a great many important applications rely on the spectral decomposition of matrices. Traditional methods such as the power iteration method, Jacobi eigenvalue method, and QR decomposition are commonly used to compute the eigenvalues and eigenvectors of a square and symmetric matrix. However, these methods suffer from certain drawbacks: in particular, the power iteration method can only find the leading eigen-pair (i.e., the largest eigenvalue and its corresponding eigenvector), while the Jacobi and QR decomposition methods face significant performance limitations when facing with large scale matrices. Typically, even producing approximate eigenpairs of a general square matrix requires at least O(N^3) time complexity, where N is the number of rows of the matrix. In this work, we exploit the newly developed memristor technology to propose a low-complexity, scalable memristor-based method for deriving a set of dominant eigenvalues and eigenvectors for real symmetric non-negative matrices. The time complexity for our proposed algorithm is O(N^2 /Δ) (where Δ governs the accuracy). We present experimental studies to simulate the memristor-supporting algorithm, with results demonstrating that the average error for our method is within 4%, while its performance is up to 1.78X better than traditional methods. 

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4. A Tutorial on Matrix Perturbation Theory (using compact matrix notation)

   https://doi.org/10.48550/arXiv.2002.05001  

   Bamieh, Bassam ( October 2020 , ArXivorg)

   Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In this note, we give an introduction to this method that is independent of any physics notions, and relies purely on concepts from linear algebra. An additional feature of this presentation is that matrix notation and methods are used throughout. In particular, we formulate the equations for each term of the analytic expansions of eigenvalues and eigenvectors as {\em matrix equations}, namely Sylvester equations in particular. Solvability conditions and explicit expressions for solutions of such matrix equations are given, and expressions for each term in the analytic expansions are given in terms of those solutions. This unified treatment simplifies somewhat the complex notation that is commonly seen in the literature, and in particular, provides relatively compact expressions for the non-Hermitian and degenerate cases, as well as for higher order terms. 

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5. 66361 TL1 Team Approach to Predicting Response to Spinal Cord Stimulation for Chronic Low Back Pain

   https://doi.org/10.1017/cts.2021.685  

   See, Kyle ; Ho, Rachel ; Coombes, Stephen ; Fang, Ruogu ( March 2021 , Journal of Clinical and Translational Science)

   ABSTRACT IMPACT: Understanding how spinal cord stimulation works and who it works best for will improve clinical trial efficacy and prevent unnecessary surgeries. OBJECTIVES/GOALS: Spinal cord stimulation (SCS) is an intervention for chronic low back pain where standard interventions fail to provide relief. However, estimates suggest only 58% of patients achieve at least 50% reduction in their pain. There is no non-invasive method for predicting relief provided by SCS. We hypothesize neural activity in the brain can fill this gap. METHODS/STUDY POPULATION: We tested SCS patients at 3 times points: baseline (pre-surgery), at day 7 during the trial period (post-trial), and 6 months after a permanent system had been implanted. At each time point participants completed 10 minutes of eyes closed, resting electroencephalography (EEG) and self-reported their pain. EEG was collected with the ActiveTwo system and a 128-electrode cap. Patients were grouped based on the percentage change of their pain from baseline to the final visit using a median split (super responders > average responders). Spectral density powerbands were extracted from resting EEG to use as input features for machine learning analyses. We used support vector machines to predict response to SCS. RESULTS/ANTICIPATED RESULTS: Baseline and post-trial EEG data predicted SCS response at 6-months with 95.56% and 100% accuracy, respectively. The gamma band had the highest performance in differentiating responders. Post-trial EEG data best differentiated the groups with feature weighted dipoles being more highly localized in sensorimotor cortex. DISCUSSION/SIGNIFICANCE OF FINDINGS: Understanding how SCS works and who it works best for is the long-term objective of our collaborative research program. These data provide an important first step towards this goal. 

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