Algebraic Technique Research Articles

Motion compensated cone-beam CT reconstruction using an a priori motion model from CT simulation: a pilot study

Objective. To combat the motion artifacts present in traditional 4D-CBCT reconstruction, an iterative technique known as the motion-compensated simultaneous algebraic reconstruction technique (MC-SART) was previously developed. MC-SART employs a 4D-CBCT reconstruction to obtain an initial model, which suffers from a lack of sufficient projections in each bin. The purpose of this study is to demonstrate the feasibility of introducing a motion model acquired during CT simulation to MC-SART, coined model-based CBCT (MB-CBCT). Approach. For each of 5 patients, we acquired 5DCTs during simulation and pre-treatment CBCTs with a simultaneous breathing surrogate. We cross-calibrated the 5DCT and CBCT breathing waveforms by matching the diaphragms and employed the 5DCT motion model parameters for MC-SART. We introduced the Amplitude Reassignment Motion Modeling technique, which measures the ability of the model to control diaphragm sharpness by reassigning projection amplitudes with varying resolution. We evaluated the sharpness of tumors and compared them between MB-CBCT and 4D-CBCT. We quantified sharpness by fitting an error function across anatomical boundaries. Furthermore, we compared our MB-CBCT approach to the traditional MC-SART approach. We evaluated MB-CBCT’s robustness over time by reconstructing multiple fractions for each patient and measuring consistency in tumor centroid locations between 4D-CBCT and MB-CBCT. Main results. We found that the diaphragm sharpness rose consistently with increasing amplitude resolution for 4/5 patients. We observed consistently high image quality across multiple fractions, and observed stable tumor centroids with an average 0.74 ± 0.31 mm difference between the 4D-CBCT and MB-CBCT. Overall, vast improvements over 3D-CBCT and 4D-CBCT were demonstrated by our MB-CBCT technique in terms of both diaphragm sharpness and overall image quality. Significance. This work is an important extension of the MC-SART technique. We demonstrated the ability of a priori 5DCT models to provide motion compensation for CBCT reconstruction. We showed improvements in image quality over both 4D-CBCT and the traditional MC-SART approach.

Applying Laplace Transformation on Epidemiological Models as Caputo Derivatives

This paper delves into the application of fractional calculus, with a focus on Caputo derivatives, in epidemiological models using ordinary differential equations. It highlights the critical role Caputo derivatives play in modeling intricate systems with memory effects and assesses various epidemiological models, including SIR variants, demonstrating how Caputo derivatives capture fractional-order dynamics and memory phenomena found in real epidemics. The study showcases the utility of Laplace transformations for analyzing systems described by ordinary differential equations with Caputo derivatives. This approach facilitates both analytical and numerical methods for system analysis and parameter estimation. Additionally, the paper introduces a tabular representation for epidemiological models, enabling a visual and analytical exploration of variable relationships and dynamics. This matrix-based framework permits the application of linear algebra techniques to assess stability and equilibrium points, yielding valuable insights into long-term behavior and control strategies. In summary, this research underscores the significance of Caputo derivatives, Laplace transformations, and matrix representation in epidemiological modeling. We assume that by using this type of methodology we can get analytic solutions by hand when considering a function as constant in certain cases and it will not be necessary to search for numerical methods.

Computing the determinant of a signed graph

Abstract A signed graph is a simple graph in which every edge has a positive or negative sign. In this article, we employ several algebraic techniques to compute the determinant of a signed graph in terms of the spectrum of a vertex-deleted subgraph. Particular cases, including vertex-deleted subgraphs without repeated eigenvalues or singular vertex-deleted subgraphs are considered. As applications, an algorithm for the determinant of a signed graph with pendant edges is established, the determinant of a bicyclic graph and the determinant of a chain graph are computed. In the end, the uniqueness of the polynomial reconstruction for chain graphs is proved.

Stability Analysis of Switched Linear Systems with Neural Lyapunov Functions

Neural-based, data-driven analysis and control of dynamical systems have been recently investigated and have shown great promise, e.g. for safety verification or stability analysis. Indeed, not only do neural networks allow for an entirely model-free, data-driven approach, but also for handling arbitrary complex functions via their power of representation (as opposed to, e.g. algebraic optimization techniques that are restricted to polynomial functions). Whilst classical Lyapunov techniques allow to provide a formal and robust guarantee of stability of a switched dynamical system, very little is yet known about correctness guarantees for Neural Lyapunov functions, nor about their performance (amount of data needed for a certain accuracy). We formally introduce Neural Lyapunov functions for the stability analysis of switched linear systems: we benchmark them on this paradigmatic problem, which is notoriously difficult (and in general Turing-undecidable), but which admits existing recently-developed technologies and theoretical results. Inspired by switched systems theory, we provide theoretical guarantees on the representative power of neural networks, leveraging recent results from the ML community. We additionally experimentally display how Neural Lyapunov functions compete with state-of-the-art results and techniques, while admitting a wide range of improvement, both in theory and in practice. This study intends to improve our understanding of the opportunities and current limitations of neural-based data-driven analysis and control of complex dynamical systems.

Improving depth uncertainty in plenoptic camera-based velocimetry

This work describes the development of a particle tracking velocimetry (PTV) algorithm designed to improve three-dimensional (3D), three-component velocity field measurements using a single plenoptic camera. Particular focus is on mitigating the longstanding depth uncertainty issues that have traditionally plagued plenoptic particle image velocimetry (PIV) experiments by leveraging the camera’s ability to generate multiple perspective views of a scene in order to assist both particle triangulation and tracking. 3D positions are first estimated via light field ray bundling (LFRB) whereby particle rays are projected into the measurement volume using image-to-object space mapping. Tracking is subsequently performed independently within each perspective view, providing a statistical amalgamation of each particle’s predicted motion through time in order to help guide 3D trajectory estimation while simultaneously protecting the tracking algorithm from physically unreasonable fluctuations in particle depth positions. A synthetic performance assessment revealed a reduction in the average depth errors obtained by LFRB as compared to the conventional multiplicative algebraic reconstruction technique when estimating particle locations. Further analysis using a synthetic vortex ring at a magnification of − 0.6 demonstrated plenoptic-PIV capable of maintaining the equivalent of 0.1–0.15 voxel accuracy in the depth domain at a spacing to displacement ratio of 5.3–10.5, an improvement of 84–89% compared to plenoptic-PIV. Experiments were conducted at a spacing to displacement ratio of approximately 5.8 to capture the 3D flow field around a rotor within the rotating reference frame. The resulting plenoptic-PIV/PTV vector fields were evaluated with reference to a fixed frame stereoscopic-PIV (stereo-PIV) validation experiment. A systematic depth-wise (radial) component of velocity directed toward the wingtip, consistent with observations from prior literature and stereo-PIV experiments, was captured by plenoptic-PTV at magnitudes similar to the validation data. In contrast, the plenoptic-PIV did not discern any coherent indication of radial motion. Our algorithm constitutes a significant advancement in enhancing the functionality and versatility of single-plenoptic camera flow diagnostics by directly addressing the primary limitation associated with plenoptic imaging.Graphical abstract

Differential Fault and Algebraic Equation Combined Analysis on PICO

In modern information technology, research on block cipher security is imperative. Concerning the ultra lightweight block cipher PICO, there has been only one study focused on recovering its complete master key, with a large search space of 264, and no fault analysis yet. This paper proposes a new fault analysis approach, combining differential fault and algebraic equation techniques. It achieved the recovery of PICO’s entire master key with 40 faults in an average time of 0.57 h. S-box decomposition was utilized to optimize our approach, reducing the time by a remarkable 75.83% under the identical 40-fault condition. Furthermore, PICO’s complete master key could be recovered with 28 faults in an average time of 0.78 h, indicating a significant 237 reduction in its search space compared to the previous study. This marks the first fault analysis on PICO. Compared to conventional fault analysis methods DFA (differential fault analysis) and AFA (algebraic fault analysis), our approach outperforms in recovering PICO’s entire master key, highlighting the cruciality of key expansion complexity in block cipher security. Therefore, our approach could serve to recover master keys of block ciphers with comparably complicated key expansions, and production of more secure block ciphers could result.

Complexity of Some Types of Cyclic Snake Graphs

The number of spanning trees in graphs (networks) is a crucial invariant, and it is also an important measure of the reliability of a network. Spanning trees are special subgraphs of a graph that have several important properties. First, <I>T</I> must span <I>G</I>, which means it must contain every vertex in graph <I>G</I>, if <I>T</I> is a spanning tree of graph <I>G</I>. <I>T</I> needs to be a subgraph of <I>G</I>, second. Stated differently, any edge present in <I>T</I> needs to be present in <I>G </I>as well. Third, <I>G</I> is the same as <I>T</I> if each edge in <I>T</I> is likewise present in <I>G</I>. In path-finding algorithms like Dijkstra's shortest path algorithm and A* search algorithm, spanning trees play an essential part. In those approaches, spanning trees are computed as component components. Protocols for network routing also take advantage of it. In numerous techniques and applications, minimum spanning trees are highly beneficial. Computer networks, electrical grids, and water networks all frequently use them. They are also utilized in significant algorithms like the min-cut max-flow algorithm and in graph issues like the travelling salesperson problem. In this paper, we use matrix analysis and linear algebra techniques to obtain simple formulas for the number of spanning trees of certain kinds of cyclic snake graphs.

Multivariate Stochastic Volatility Modeling via Integrated Nested Laplace Approximations: A Multifactor Extension

This study introduces a multivariate extension to the class of stochastic volatility models, employing integrated nested Laplace approximations (INLA) for estimation. Bayesian methods for estimating stochastic volatility models through Markov Chain Monte Carlo (MCMC) can become computationally burdensome or inefficient as the dataset size and problem complexity increase. Furthermore, issues related to chain convergence can also arise. In light of these challenges, this research aims to establish a computationally efficient approach for estimating multivariate stochastic volatility models. We propose a multifactor formulation estimated using the INLA methodology, enabling an approach that leverages sparse linear algebra and parallelization techniques. To evaluate the effectiveness of our proposed model, we conduct in-sample and out-of-sample empirical analyses of stock market index return series. Furthermore, we provide a comparative analysis with models estimated using MCMC, demonstrating the computational efficiency and goodness of fit improvements achieved with our approach.

Vertical Distribution Mapping for Methane Fugitive Emissions Using Laser Path-Integral Sensing in Non-Cooperative Open Paths.

Observing the vertical diffusion distribution of methane fugitive emissions from oil/gas facilities is significant for predicting the pollutant's spatiotemporal transport and quantifying the random emission sources. A method is proposed for methane's vertical distribution mapping by combining the laser path-integral sensing in non-non-cooperative open paths and the computer-assisted tomography (CAT) techniques. It uses a vertical-plume-mapping optical path configuration and adapts the developed dynamic relaxation and simultaneous algebraic reconstruction technique (DR-SART) into methane-emission-distribution reconstruction. A self-made miniaturized TDLAS telemetry sensor provides a reliable path to integral concentration information in non-non-cooperative open paths, with Allan variance analysis yielding a 3.59 ppm·m sensitivity. We employed a six-indexes system for the reconstruction performance analysis of four potential optical path-projection configurations and conducted the corresponding validation experiment. The results have shown that that of multiple fan-beams combined with parallel-beam modes (MFPM) is better than the other optical path-projection configurations, and its reconstruction similarity coefficient (ε) is at least 22.4% higher. For the different methane gas bag-layout schemes, the reconstruction errors of maximum concentration (γm) are consistently around 0.05, with the positional errors of maximum concentration (δ) falling within the range of 0.01 to 0.025. Moreover, considering the trade-off between scanning duration and reconstruction accuracy, it is recommended to appropriately extend the sensor measurement time on a single optical path to mitigate the impact of mechanical vibrations induced by scanning motion.

A high-order target phase approach for the station-keeping of periodic orbits

A novel high-order target phase approach (TPhA) for the station-keeping of periodic orbits is proposed in this work. The key elements of the TPhA method, the phase-angle Poincare map and high-order maneuver map, are constructed using differential algebra (DA) techniques to determine station-keeping epochs and calculate correction maneuvers. A stochastic optimization framework tailored for the TPhA-based station-keeping process is leveraged to search for fuel-optimal and error-robust TPhA parameters. Quasi-satellite orbits (QSOs) around Phobos are investigated to demonstrate the efficacy of TPhA in mutli-fidelity dynamical models. Monte Carlo simulations demonstrated that the baseline QSO of JAXA’s Martian Moons eXploration (MMX) mission could be maintained with a monthly maneuver budget of approximately 1 m/s.

MUYSC: an end-to-end muography simulation toolbox

SUMMARY Muography is an imaging technique that relies on the attenuation of the muon flux traversing geological or anthropogenic structures. Several simulation frameworks help to perform muography studies by combining specialized codes: for muon generation through muon transport to muon detector performance. This methodology is precise but requires significant computational resources and time. We present an end-to-end python-based MUographY Simulation Code, which implements a muography simulation framework capable of rapidly estimating muograms of any geological structure worldwide. This framework considers the generated muon flux as the observation point; the energy loss of muons passing through the geological target; the integrated muon flux detected by the telescope and estimates the 3-D density distribution of the target using algebraic reconstruction techniques. The simulations ignore the relatively small muon flux variance caused by geomagnetic effects, solar modulation and atmospheric conditions. We validate the code performance by comparing our simulation results with data from other frameworks.

Surface science insight note: A linear algebraic approach to elucidate native films on Fe3O4 surface

Standard materials are often used to obtain spectra that can be compared to those from unknown samples. Spectra measured from these known substances are also used as a means of computing sensitivity factors to allow quantification by X‐ray photoelectron spectroscopy (XPS) of less well‐defined materials. Spectra from known materials also provide line shapes suitable for inclusion in spectral models which, when fitted to spectra, permit the chemical state for a sample to be assessed. Both types of information depend on isolating photoemission signals from the inelastically scattered signal. In this Insight note, technical issues associated with the use of XPS of as received Fe3O4 powder sample surface are discussed. The Insight note is designed to show how linear algebraic techniques applied to data collected from a sample marketed as pure Fe3O4 powder are used to verify that XPS has been performed on chemistry representative of the sample. The methods described in this Insight note can further be utilized in elucidating complex XPS data obtained from thin films formed or evolved during cyclic/non‐steady use of complex (electro)catalyst surfaces, especially in the presence of contaminants.

Estimation of pollutant source using source-receptor response matrix from maximum-length sequence pulse technique

Accurate estimation of pollutant sources in built environments is crucial for monitoring air quality and preventing harm to health and property. This study presented an improved method for estimating the dynamical pollutant source strength using the concentration signal monitored at the receptor point. We employed a built environment system subjected to a pollutant pulse signal in accordance with the maximum-length sequence (m-sequence) to acquire response concentration. Subsequently, the source-receptor response matrix was inversely identified based on the known pulse signal and the response concentration. Then, the unknown source release rate was inversely estimated by the monitored concentration signal and previously identified response matrix. In addition, the performances of different inverse algorithms and m-sequence orders were also analyzed. Here both simulation and application to a real experiment of a wind duct were implemented, and the results demonstrated the higher-order m-sequence method exhibited better noise resistance compared to the traditional unit pulse method. In both simulation and experiment, the m-sequence pulse method reduced the root mean square error (RMSE) by 39.9 % and 70.7 %, respectively, compared to the unit pulse method. In experiment, the simultaneous multiplicative algebraic reconstruction technique (SMART) combined with the Tikhonov regularization (TR) method showed relatively stable performance.

Characteristics of restricted selection indices and geometrical interpretation of restricted breeding values.

Restricted selection is used to control genetic changes in one or more characters. Three main selection indices are adopted for this purpose. First, Kempthorne's index is used to maximize aggregate breeding value (BV) with changes in some traits restricted to zero; second, Harville's index is used to maximize aggregate BV with proportional changes for some traits; and third, Yamada's index is mathematically used to achieve the relative desired changes for all traits. Kempthorne's index is equivalent to Harville's index. However, the relationship between Kempthorne's and Yamada's indices has not been clarified. In addition, the characteristics of restricted selection indices and the relationship between BV and restricted BV (RBV) are also unknown. The aim of this study was to clarify the characteristics of restricted selection indices and describe the relationship between BV and RBV by using linear algebra and geometric techniques, respectively. First, I proved that Yamada's index is part of Kempthorne's index. Second, I investigated the relationship between BVs that were estimated using an ordinary selection index (EBVs) and RBVs estimated using a restricted selection index (ERBVs) and proved that the ERBVs of the restricted traits are proportional to the relative desired changes. Third, I proved that RBV is represented by a linear function of BV and geometrically represented the relationship between BV and RBV. In this study, new findings on restricted selection indices and RBV were obtained. This useful clarification of the relationship between BV and RBV will make it possible to evaluate the response to selection using not only a restricted selection index, but also a restricted BLUP in computer simulation studies.

Fault-prognosability, [formula omitted]-step prognosis and [formula omitted]-step predictive diagnosis in partially observed petri nets by means of algebraic techniques

In this paper, we explore some issues relevant to fault monitoring in discrete event systems modeled by partially observed LPNs with, possibly indistinguishable observable events and acyclic unobservable subnet. Firstly, we address the (offline) fault-prognosability analysis problem. Subsequently, we tackle the two online problems of K-step fault prognosis and K-step predictive diagnosis. We propose algebraic formulations and solutions to these problems. Namely, a necessary and sufficient condition for fault-prognosability is established based on solving an integer optimization problem. The proposed approach is applicable for bounded Petri nets. As for the K-step prognosis and K-step predictive diagnosis, algebraic approaches based on state estimation on a sliding horizon are elaborated to produce relevant verdicts. The established results for K-step prognosis and K-step predictive diagnosis are applicable for both bounded and unbounded Petri nets.

Generating function and root properties for chromatic polynomials

The chromatic polynomial P(G,t) related to graphs is a major area of algebraic graph theory. These polynomials allow graphs to be extracted using algebraic techniques and contain important information about their structure and properties. In particular, the coefficients, roots, and derivatives of graphing polynomials at certain points and their values are frequently interpreted in important ways. A graph with vertices V and a chromatic polynomial chi_{G}(n) is called G. In the paper, we discover some properties by defining h_{G}(z) as the generating function of the chromatic polynomial of the graph along with properties of the roots. A real number t is called a chromatic root of the generating function for the chromatic polynomial of a graph G if h(G, t) = 0. Since the chromatic polynomial enumerates the appropriate colors of a graph, it is natural to see how color roots behave in light of the assumptions related to them.

Improvement of the longitudinal phase space tomography at the J-PARC synchrotrons

The longitudinal phase space tomography, which reconstructs the phase space distribution from the one-dimensional bunch profiles, is used in various accelerators to measure longitudinal beam parameters. At the J-PARC, an implementation of the phase space tomography based on the Convolution Back Projection method (CBP) has been used to measure the momentum spread of the injected beam. The method assumes that the beam distribution rotates without significant deformation during the synchrotron oscillation. Because of the nonlinearity of synchrotron motion with sinusoidal RF voltage, the method can be used only in limited situations such as small amplitude synchrotron oscillation. Algebraic Reconstruction Techniques (ART) in conjunction with particle tracking, which is implemented in the CERN’s tomography code, allows accurate reconstructions even for nonlinear large amplitude synchrotron oscillations. We present the overview of the application of the CERN’s tomography code to the J-PARC synchrotrons. The results of benchmarking are also reported.

The altered Hermite matrix: implications and ramifications

<p>Matrix theory is essential for addressing practical problems and executing computational tasks. Matrices related to Hermite polynomials are essential due to their applications in quantum mechanics, numerical analysis, probability, and signal processing. Their orthogonality, recurrence relations, and spectral properties make them a valuable tool for both theoretical research and practical applications. From a different perspective, we introduced a variant of the Hermite matrix that incorporates triple factorials and demonstrated that this matrix satisfies various properties. By utilizing effective matrix algebra techniques, various algebraic properties of this matrix have been determined, including the product formula, inverse matrix and eigenvalues. Additionally, we extended this matrix to a more generalized form and derived several identities.</p>

Higher Overtone Vibrational Frequencies in Naphthalene Using the Lie Algebraic Technique

Higher Overtone Vibrational Frequencies in Naphthalene Using the Lie Algebraic Technique

Bandwidth estimation in network probing techniques utilizing min-plus algebraic methods

Bandwidth estimation in network probing techniques plays a crucial role in modern networks, particularly in ensuring superior Quality of Service (QoS). This is largely influenced by advancements in IEEE 802.11 standards, which aim to enhance service performance. Existing methodologies are designed to assess the available resources on a given channel, providing valuable support for bandwidth-constrained applications. However, these methods encounter limitations due to potential non-linearities within networks. The current approach leverages min-plus algebra from network calculus, which, although effective, falls short in certain dynamic scenarios. In this research, we propose a novel methodology for bandwidth estimation by integrating other state-of-the-art QoS protocols while maintaining the foundational principles of the min-plus algebra technique. This approach aims to develop a more versatile and adaptable system, better suited for complex network environments.