Equivalent Form Research Articles

Solving the QLY Least Squares Problem of Dual Quaternion Matrix Equation Based on STP of Dual Quaternion Matrices

Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation AXB+CYD≈E, where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the GH-representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation AXB+CYD≈E. The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method.

Algebraic knowledge for teaching test: An adaptation study

In this study, the Mathematical Knowledge for Teaching-Elementary Patterns Functions and Algebra-Content Knowledge (MKT-PFA) test, originally developed in English as part of the "Learning Mathematics for Teaching Project" at Michigan University, was adapted into Turkish. The test comprises two equivalent forms, A and B, each translated into Turkish and culturally adapted through consultations with two mathematics education academics and five secondary school math teachers pursuing doctoral studies. A total of 328 pre-service teachers at a Turkish public university's elementary school mathematics teaching department were administered form A (14 questions, 29 items) and form B (12 questions, 27 items) at a one-week interval. Psychometric analyses revealed high reliability (KR-20: A=0.712, B=0.735; Lord reliability: A=0.733, B=0.756), and strong correlations (rpbi) with the original English forms, indicating suitable adaptation. Item difficulties analyzed using a one-parameter Item Response Theory model showed a normal distribution, affirming the tests' validity for assessing pre-service teachers' algebra teaching knowledge in Türkiye.

Beyond Einstein’s General Relativity: Hybrid Metric-Palatini Gravity

It has been established that both metric and Palatini versions of f (R) gravity have interesting features, but also manifest several downsides. A hybrid combination of theories, containing elements from both formalisms, turns out to be very successful in accounting for the observed phenomenology and it is able to avoid some drawbacks of the original approaches. Here, we explore the formulation in a dynamically equivalent scalar-tensor form of this hybrid metricPalatini approach. We present several of its main achievements, such as, passing the Solar System observational tests even if the scalar field is very light or massless, and outline several applications to astrophysical and cosmological scenarios. Furthermore, we also explore the viability of generalized hybrid metric-Palatini gravitational theories.

Edge-wave phase shifts versus normal-mode phase tilts in an Eady problem with a sloping boundary

One mechanistic interpretation of baroclinic instability is that of mutual constructive interference of Rossby edge waves. The suppression of baroclinic instability over slopes has been widely established, where previous research argues that a sloping boundary modifies the properties of these Rossby edge waves, but does not provide a mechanistic explanation for the suppression that is valid over all parameter space. In the context of an Eady problem modified by the presence of a sloping boundary, we provide a mechanistic rationalization for baroclinic instability in the presence of slopes that is valid over all parameter space, via an equivalent formulation explicitly in terms of Rossby edge waves. We also highlight the differences between edge-wave phase shifts and normal-mode phase tilts, showing that the edge-wave phase shifts should be the ones that are mechanistically relevant, and normal-mode phase tilt is a potentially misleading quantity to use. Further, we present evidence that the edge-wave phase shifts but not normal-mode phase tilts are well correlated with geometric quantities diagnosed from an analysis framework based on eddy variance ellipses. The result is noteworthy in that the geometric framework makes no explicit reference to the edge-wave structures in its construction, and the correlation suggests the geometric framework can be used in problems where edge-wave structures are not so well defined or readily available. Some implications for parametrization of baroclinic instability and relevant eddy-mean feedbacks are discussed. For completeness, we also provide an explicit demonstration that the linear instability problem of the present modified Eady problem is parity-time symmetric, and speculate about some suggestive links between parity-time symmetry, shear instability, and the edge-wave interaction mechanism. Published by the American Physical Society 2024

Analysis on transient wave propagation in the soft matter of hydrogels

The precise control and structural design of the soft matter of the hydrogel subjected to dynamic loading require an in-depth understanding of its transient wave characteristics. However, for the complex solid–liquid coupling effects of the soft matter of hydrogels, the traditional single-phase wave model is no longer applicable. In this study, a transient wave propagation model that considers the solid–liquid coupling effects and its computational method is proposed. The wave-governing equations of hydrogels are derived based on the mass conservation and dynamic equilibrium equations, where the motions of solid and liquid phases are independently described. After transforming the governing equations into the equivalent weak form, numerical solutions for transient wave propagation in one- and two-dimensional hydrogels are calculated. The accuracy of the proposed model is verified by a comparison between semi-analytical and commercial finite element method solutions. We observe that the travelling and reflection processes of the two types of compression waves, P1 and P2, with different wave speeds are captured when the dynamic coefficient of permeability kf increases. Furthermore, the influence of solid–liquid coupling effects on the transient responses of hydrogels is discussed. The results show that the hydrogels exhibit the dynamic characteristics of a single-phase medium to a certain extent when kf is sufficiently small.

Boundary problems of sequential fractional differential equations having a monomial coefficient

The article examines sequential fractional-order differential equations composed of monomial coefficients defined as nonlinear boundary problems. The original problem is converted into an integral equation in an equivalent form. The contraction mapping principle and Krasnoselskii's fixed-point theorem are implemented to obtain two existence conditions. The problem of Ulam-Hyers stability is also investigated. An illustration is presented to showcase practical application of the obtained results.

An information-theoretic framework for conditional causality analysis of brain networks.

Identifying directed network models for multivariate time series is a ubiquitous problem in data science. Granger causality measure (GCM) and conditional GCM (cGCM) are widely used methods for identifying directed connections between time series. Both GCM and cGCM have frequency-domain formulations to characterize the dependence of time series in the spectral domain. However, the original methods were developed using a heuristic approach without rigorous theoretical explanations. To overcome the limitation, the minimum-entropy (ME) estimation approach was introduced in our previous work (Ning & Rathi, 2018) to generalize GCM and cGCM with more rigorous frequency-domain formulations. In this work, this information-theoretic framework is further generalized with three formulations for conditional causality analysis using techniques in control theory, such as state-space representations and spectral factorizations. The three conditional causal measures are developed based on different ME estimation procedures that are motivated by equivalent formulations of the classical minimum mean squared error estimation method. The relationship between the three formulations of conditional causality measures is analyzed theoretically. Their performance is evaluated using simulations and real neuroimaging data to analyze brain networks. The results show that the proposed methods provide more accurate network structures than the original approach.

Oscillatory behavior of second‐order nonlinear delay dynamic equations with multiple sublinear neutral terms utilizing canonical transformation

This paper aims to present innovative criteria for characterizing the oscillatory behavior of second‐order nonlinear delay dynamics expressed in noncanonical form with sublinear neutral terms. We achieve this by converting the noncanonical equation into an equivalent canonical form. The practical examples presented throughout this paper illustrate the significance of our findings and contribute to enhancing, generalizing, and augmenting the existing literature on this subject.

Development and Evaluation of the Maintenance Office Duty Test: An Alternate Form of the Charge of Quarters Duty Test for Assessing Executive Function After Mild Traumatic Brain Injury to Inform Duty-Readiness After Concussion.

After mild traumatic brain injury, service members may experience difficulty with executive functions, which could interfere with return to duty and life roles. Because performance-based multitasking assessments are sensitive to executive dysfunction, a team of military and civilian rehabilitation researchers developed the Charge of Quarters Duty Test (CQDT) to help inform duty readiness after concussion; it is a multitasking test based on a military task scenario that challenges executive functions, such as foresight and planning, set shifting, and prospective memory. Although previous study indicates that CQDT has reliability and known-groups validity, like other multitasking tests, it should not be readministered after rehabilitative care because of learning effects. The purpose of this study was to develop an alternate form of the CQDT and evaluate its equivalence to the CQDT. A measurement development study was conducted in which subject matter experts on the study team used an iterative approach to create an alternate form. To evaluate equivalence, a repeated measures design was employed in which each participant performed one or both test versions twice and acted as their own control. The study team created the Maintenance Office Duty Test (MODT), the alternate form of the CQDT. A convenience sample of 40 adults performed both the MODT and CQDT approximately 2 weeks apart; 4 had a history of military service and 16 had a history of acquired brain injury. We compared mean scores and standard deviations of each test's 4 subscores using matched-pair t-tests and found that scores were not statistically different, suggesting that the 2 tests are equivalent. In addition, matched-pair t-tests were used to compare Time 1 and Time 2 subscores when (1) participants performed the 2 test versions (CQDT and MODT) and (2) participants performed the same test twice. None of the differences in Time 1 and Time 2 subscores were statistically significant when participants performed the 2 test versions. However, when participants performed the same test twice, three of the four Time 1 and Time 2 scores were significantly different, with Time 2 reflecting improved performance and possible learning effects (performance accuracy P = .013; total number of rule breaks P = .015; performance time P = .002). The MODT appears to be an equivalent form of the CQDT that mitigates learning effects that often accompany performance of multitasking assessments. Having developed and validated the equivalence of the CDQT's alternate form, military rehabilitation clinicians have an expanded set of clinical tools by which to identify possible executive dysfunction and evaluate service members' response to rehabilitative care via pre- and post-rehabilitation testing after mild traumatic brain injury.

An uncertain programming model for fixed charge transportation problem with item sampling rates

This paper analyzes the transportation issue involving multiple objectives and items with fixed costs amid uncertainty, which aims to increase net profit while minimizing carbon emissions, to determine an optimal product shipping strategy. This paper introduces the use of uncertain theory to address the transportation dilemma, considering various challenges such as potential uncertainties during the actual transport process. It involves defining variables such as supply, demand and the rate of product sampling qualification as uncertain factors, constructing mathematical models, and deriving the corresponding model as well as the respective equivalent form by means of uncertainty theory. A linear weighted method is adopted to reflect the significance of each objective as identified by policymakers and suggest a sparrow optimization algorithm combined with butterfly search for numerical experiments to discover the optimal solution. This demonstrates the practicality and effectiveness of the proposed models.

Open-source synthetic photoplethysmographic signal generator with analog output.

The photoplethysmographic (PPG) signal of the finger is being used to create embedded devices that estimate physiological variables. This project outlines an innovative method for developing a synthetic PPG generator that produces both actual reference digital signals and their equivalent analog signals using open-source technology. A series of PPG profiles is synthesized using three variant Gaussian functions. A low-frequency trend induced by respiratory frequency and background noise are then added. To generate a diverse range of continuously variable PPG profiles within specified boundaries and customizable levels of interference, all parameters undergo random fluctuations on a cycle-by-cycle basis, as per user-defined constraints. The generated signal is then converted into its equivalent analog form through the use of an RC filter that low-frequency filters a Pulse-Width Modulation square wave that is modulated directly by the generated signal. The software returns different PPG profiles and allows the signal comparison before vs after the addition of different-intensity modulated respiratory trends and background noise. The digital signal is faithfully converted into an equivalent analog voltage signal capable of reproducing not only the waveform profile but also the respiratory trend and various levels of noise.

Gauss-Seidel Type Iterative Algorithm for a Generalized System of Extended Non-linear Variational Inequalities

This study is focused on a new generalized system of extended non-linear variational inequalities (GSENVI, for short) involving 3k-distinct non-linear relaxed cocoercive operators. We give the equivalent formulation of GSENVI in a more convenient form by using auxiliary principal technique. Through the projection technique, we demonstrate that the non-linear projection equations are analogous to equivalent form of GSENVI. By the use of alternative fixed point formulations, we proposed the k-steps Gauss-Seidel type iterative algorithms to obtain an approximate solution of the GSENVI. Further, we discuss the convergence of proposed k-step Gauss Seidel type iterative algorithms. Several special cases of GSENVI are discussed for the reliability of our findings.

Revisiting Possibilistic Fuzzy C-Means Clustering Using the Majorization-Minimization Method.

Possibilistic fuzzy c-means (PFCM) clustering is a kind of hybrid clustering method based on fuzzy c-means (FCM) and possibilistic c-means (PCM), which not only has the stability of FCM but also partly inherits the robustness of PCM. However, as an extension of FCM on the objective function, PFCM tends to find a suboptimal local minimum, which affects its performance. In this paper, we rederive PFCM using the majorization-minimization (MM) method, which is a new derivation approach not seen in other studies. In addition, we propose an effective optimization method to solve the above problem, called MMPFCM. Firstly, by eliminating the variable V∈Rp×c, the original optimization problem is transformed into a simplified model with fewer variables but a proportional term. Therefore, we introduce a new intermediate variable s∈Rc to convert the model with the proportional term into an easily solvable equivalent form. Subsequently, we design an iterative sub-problem using the MM method. The complexity analysis indicates that MMPFCM and PFCM share the same computational complexity. However, MMPFCM requires less memory per iteration. Extensive experiments, including objective function value comparison and clustering performance comparison, demonstrate that MMPFCM converges to a better local minimum compared to PFCM.

A Case Study on Formal Sequential Equivalence Checking based Hierarchical Flow Setup towards Faster Convergence of Complex SOC Designs

A Case Study on Formal Sequential Equivalence Checking based Hierarchical Flow Setup towards Faster Convergence of Complex SOC Designs

Optimal synchronization with [formula omitted]-gain performance: An adaptive dynamic programming approach

This paper studies an optimal synchronous control protocol design for nonlinear multi-agent systems under partially known dynamics and uncertain external disturbance. Under some mild assumptions, Hamilton–Jacobi–Isaacs equation is derived by the performance index function and system dynamics, which serves as an equivalent formulation. Distributed policy iteration adaptive dynamic programming is developed to obtain the numerical solution to the Hamilton–Jacobi–Isaacs equation. Three theoretical results are given about the proposed algorithm. First, the iterative variables is proved to converge to the solution to Hamilton–Jacobi–Isaacs equation. Second, the L2-gain performance of the closed loop system is achieved. As a special case, the origin of the nominal system is asymptotically stable. Third, the obtained control protocol constitutes an Nash equilibrium solution. Neural network-based implementation is designed following the main results. Finally, two numerical examples are provided to verify the effectiveness of the proposed method.

Spectral approximation and error analysis for the transmission eigenvalue problem with an isotropic inhomogeneous medium

In this paper, we propose an effective Legendre-Fourier spectral method for the transmission eigenvalue problem in polar geometry with an isotropic inhomogeneous medium. The basic idea of this methodology is to rewrite the initial problem into its equivalent form by using polar coordinates and some specially designed polar conditions. A variational method and its discrete version (i.e., Legendre-Fourier spectral method) are then presented within a class of weighted Sobolev spaces. With the help of the spectral theory of compact operators and the approximation properties of some specially designed projections in non-uniformly weighted Sobolev spaces, error estimates with spectral accuracy of the Legendre-Fourier spectral method for both the eigenvalue and eigenfunction approximations are established. Numerical experiments are presented to confirm the theoretical findings and the efficiency of our algorithm.

A novel multi-fidelity optimization framework for high-altitude propellers

A novel optimization methodology is presented for optimizing high-altitude propellers operating in the lower stratosphere. The proposed methodology combines a Bayesian optimization approach for most design variables and Vortex Theory for the optimization of twist distribution and rotational velocity. For the Bayesian optimization part of the problem, a multi-fidelity approach is employed with three levels of modeling fidelity. The considered fidelity levels, from lower to higher, are: Vortex Theory, and 3D Reynolds-Averaged Navier-Stokes (RANS) with the use of γ−Reθ transition model, converged with first-order upwind, and second-order upwind for the momentum equations. Additionally, the optimization problem involves two objectives: maximizing the cruise efficiency and minimizing the total volume of the propeller. To enhance the performance of multi-fidelity, multi-objective (MFMO) acquisition functions, a novel algorithm is proposed. The proposed optimization methodology demonstrates its effectiveness in generating several high-performing propeller designs and proves to be more efficient than an equivalent problem formulation relying solely on Bayesian optimization. Post-processing of optimization data reveals new design variable bounds that correspond to more efficient blade shapes. These findings provide insights into some important geometric features of high-altitude propellers and establish guidelines for future optimization efforts within the specified thrust and power consumption ranges. Finally, simulations of the best-performing designs across a range of advance ratios and altitudes confirm their high performance in various operating conditions.

Development of conductive carbon-based interdigitated electrode array integrated with microfluidic channel for blood glucose sensing

PurposeThis study aims to provide a microfluidic blood glucose sensing platform based on integrated interdigitated electrode arrays (IDEAs) on a flexible quartz glass substrate, adhering closely to pertinent electrochemical characterizations.Design/methodology/approachSensors are the key elements of the modern electronics era through which all the possible physical quantities can be detected and converted into their equivalent electrical form and processed further. But to make the sensing environment better, various types of innovative architectures are being developed nowadays and among them interdigitated electrodes are quite remarkable in terms of their sensing capability. They are a well-qualified candidate in the field of gas sensing and biosensing, but even their sensitivities are getting saturated due to their physical dimensions. Most of the thin film IDEAs fabricated by conventional optical lithographic techniques do not possess a high surface-to-volume ratio to detect the target specified and that reduces their sensitivity factor. In this context, a classic conductive carbon-based highly sensitive three dimensional (3D) IDEA-enabled biosensing system has been conceived on a transparent and flexible substrate to measure the amount of glucose concentration present in human blood. 3D IDEA possesses a way better capacitive sensing behavior compared to conventional thin film microcapacitive electrodes. To transmit the target biological analyte sample property for the detection purpose to the interdigitated array-based sensing platform, the design of a microfluidic channel is initiated on the same substrate. The complex 3D Inter Digital array structure improves the overall capacitance of the entire sensing platform and the reactive surface area as well. The manufactured integrated device displays a decent value of sensitivity in the order of 5.6 µA mM−1 cm−2.FindingsDevelopment of a low-cost array-based integrated and highly flexible microfluidic biochip to extract the quantity of glucose present in human blood.Originality/valuePotential future research opportunities in the realm of integrated miniaturized, low-cost smart biosensing systems may arise from this study.

Lattice Boltzmann method for the linear complementarity problem arising from American option pricing

In this paper, a lattice Boltzmann method is proposed for solving the linear complementarity problem (LCP) arising in single and multi-asset American put option pricing. The LCP for American option is a variable coefficient parabolic model defined on an unbounded domain. Initially, using the far field estimate method and the penalty method respectively, the LCP could be reformulated into a nonlinear parabolic partial differential equation on a bounded domain. To construct a unified lattice Boltzmann model for the option pricing problems, the above transformation equations are rewritten into an equivalent divergence form. Then, through the incorporation of an amending function into the evolution equation, which assists in recovering the source term and eliminating the error term, the lattice Boltzmann model with spatial second-order accuracy is constructed. Finally, the present model is validated using numerical simulations, and the numerical results agree well with the option values obtained by existing methods, which indicates that the present lattice Boltzmann model is efficient for solving the American put option pricing problem.

Unmanned aerial vehicle variable radius set covering problem for emergency wireless network

The utilization of unmanned aerial vehicles (UAVs) has garnered increasing interest across various disciplines. The idea of smart cities led to the exploration of using UAVs in the public service industry, with a focus on integrating all aspects of the system through information and physical items. In this paper, we introduce and model the operational problem of using UAVs to construct an emergency wireless network in a disaster situation using the set covering approach. Unique to our study is the consideration that UAVs do not require predetermined positioning or altitude specifications. Consequently, the problem described in this paper can be classified as a planar variable radius covering problem, which involves a nonconvex continuous relaxation feasible set. We introduce the application of the Dantzig–Wolfe decomposition alongside a branch-and-price algorithm to tackle this problem. Additionally, the pricing subproblem’s solvable equivalent formulation is ingeniously derived from existing theories on the minimum covering circle. For large-scale instances, we propose a heuristic derived from an extended formulation. Comparative computational experiments demonstrate that our proposed algorithms significantly surpass the efficiency of the benchmark genetic algorithm previously reported in the literature.