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The maximum vibration transmissibility of paper honeycomb sandwich structures with different sizes of honeycomb core under various static stresses was investigated using the sine frequency sweep test. The effects of the cell length of the honeycomb, the thickness of the sandwich structure, and the static stress on the maximum vibration transmissibility were evaluated and a linear polynomial equation for evaluating the maximum vibration transmissibility was obtained. The results show that the maximum vibration transmissibility increases steadily with the increase in the cell length of the honeycomb, the thickness of the sandwich structure, and the static stress. The proposed equation for the maximum vibration transmissibility is suitable for predicting the maximum vibration transmissibility of paper honeycomb sandwich structures. In addition, the fitted three-dimensional diagrams of the effects of the factors on the maximum vibration transmissibility derived from the evaluation equation were shown to be in good agreement with the experimental results. [ABSTRACT FROM AUTHOR]

In this work, a system of non‐linear difference equations is employed to model the opinion dynamics between a small group of agents (the target group) and a very persuasive agent (the indoctrinator). Two scenarios are investigated: the indoctrination of a homogeneous target group, in which each agent grants the same weight to his (or her) partner's opinion and the indoctrination of a heterogenous target group, in which each agent may grant a different weight to his or her partner's opinion. Simulations are performed to study the required times by the indoctrinator to convince a group. Initially, these groups are in a consensus about a doctrine different to that of the ideologist. The interactions between the agents are pairwise. [ABSTRACT FROM AUTHOR]

This work is concerned with the consistency conditions for the equations in max plus algebra. The three classes of the max plus linear equations presented in the paper Shinzawa [Ultra discrete permanent and the consistency of max plus linear equations, Linear Algebra Appl. 506 (2016) 445–477] can be equivalently converted into the corresponding system of inequalities. The equivalence relation between ultra discrete permanent and maximum cycle mean is suggested. Thus, the necessary and sufficient conditions for solvability of the equations are obtained using the maximum cycle mean in this note. [ABSTRACT FROM AUTHOR]

This research addresses the issue of how to support students' representational fluency—the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper‐and‐pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth‐grade algebra students, and link to results of semi‐structured interviews with students before and after the experiment. Results of analyzing the five‐week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool‐based representations. [ABSTRACT FROM AUTHOR]

In the paper, there is analysed one particular approach to the modelling uncertainty in the project management through an original version of the fuzzy CPM (Critical Path Method). First there is shown the relevance of using the fuzzy CPM in agriculture and the related branches and present the basics of the methods used. Then, there are described the imperfections of the work which is discussed and the impacts of the previously-published approach when applied in project management practice are emphasised. In the original paper, the author uses only the discrete fuzzy numbers for activity time durations which could be considered inappropriate for the time scheduling in project management. Consecutively, the direct application of the extension principle on the comparison of continuous durations could lead to the situation when both numbers can be greater than the second one with possibility equal to one. Moreover, the simple transformation of durations to the costs by linear equations with a positive slope does not respect the current project management theory and practice. Finally, the missing comparison of project fuzzy costs among individual variants of the project is calculated. [ABSTRACT FROM AUTHOR]

This paper presents an alternative approach of teaching, learning and solving systems of two linear equations for two unknowns that yield no solutions. We conducted a desk-based research on methods that have been used by in-service mathematics teachers for teaching systems of two linear equations in ordinary secondary schools. It was found that five common methods (substitution, elimination, graphical, inverse matrix, Cramer's rule) have been used for teaching a system of two linear equations for two unknowns that yield no solution and all methods yield the same answer regardless of having different ways of approaching the system. We realized that a crossing method (alternative approach) is not found in the literature and yet not used by teachers for teaching students a system of two linear equations that yield no solution. But this crossing method yields similar answers with that resulted when using the five common methods. We present this alternative method in this paper by comparing with the answers obtained using five methods while focusing on two systems of two linear equations that yield no solution. This new approach has implications in teaching, learning and solving systems of two linear equations that yield no solutions in ordinary secondary schools, including mathematics teachers and educators can use this method for teaching students in solving systems of two linear equations for two unknowns that yield no solutions. [ABSTRACT FROM AUTHOR]

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The work mentioned in the title of this comments-typed paper contains several calculations that disagree with some basic chemistry concepts. These misleading calculations mainly include (i) both kinetic and isotherm modeling through linear equations and (ii) calculating the thermodynamic parameters for the adsorption processes. Thus, we remind how to correct ways allowing to make these calculations. In our opinion, it is a source of confusion for the scientific community to continue using inappropriate methods as applied in the original paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of fractional equations with nonlinear boundary condition. We establish a new comparison principle related to linear fractional equation and show the existence of extreme solution by using monotone iterative method and lower and upper solutions method. [ABSTRACT FROM AUTHOR]

This paper introduces three novel approaches to size geothermal energy piles in a MILP, offering fresh perspectives and potential solutions. The research overlooks MILP models that incorporate the sizing of a geothermal borefield. Therefore, this paper presents a new model utilizing a g-function model to regulate the power limits. Geothermal energy is an essential renewable source, particularly for heating and cooling. Complex energy systems, with their diverse sources of heating and cooling and intricate interactions, are crucial for a climate-neutral energy system. This work significantly contributes to the integration of geothermal energy as a vital energy source into the modelling of such complex systems. Borehole heat exchangers help generate heat in low-temperature energy systems. However, optimizing these exchangers using mixed-integer-linear programming (MILP), which only allows for linear equations, is complex. The current research only uses R-C, reservoir, or g-function models for pre-sized borefields. As a result, borehole heat exchangers are often represented by linear factors such as 50 W/m for extraction or injection limits. A breakthrough in the accuracy of borehole heat exchanger sizing has been achieved with the development of a new model, which has been rigorously compared to two simpler models. The geothermal system was configured for three energy systems with varying ground and bore field parameters. The results were then compared with existing geothermal system tools. The new model provides more accurate depth sizing with an error of less than 5 % compared to simpler models with an error higher than 50 %, although it requires more calculation time. The new model can lead to more accurate borefield sizing in MILP applications to optimize energy systems. This new model is especially beneficial for large-scale projects that are highly dependent on borefield size. [ABSTRACT FROM AUTHOR]

This paper is devoted to finding a necessary and sufficient condition for the solvability of the overdetermined problem for Poisson's equation with both the Dirichlet and Neumann conditions on the entire boundary. The proof is based on the boundary condition formula for the Newton potential. The obtained results are also extended to general second-order linear elliptic equations. As a byproduct, we present a characterization of the Schiffer property. It gives a definitive answer to the Schiffer problem. [ABSTRACT FROM AUTHOR]

The paper proposes a method to solve the time fractional Fokker–Planck equation using the generalized Hermite spectral method and Laplace transform. By the Laplace transform, the ordinary differential equation about the orthogonal expansion coefficient is transformed into a linear algebraic equation system, with the coefficient matrix being a tridiagonal lower triangular matrix. The orthogonal expansion coefficients of frequency domain components are obtained through this transformation. The approximate solution at any time is then obtained by using the numerical inverse Laplace transform with the Talbot algorithm. Numerical experiments have been carried out to demonstrate the high accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard's iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two. [ABSTRACT FROM AUTHOR]

Rayleigh-Ritz discriminant analysis (RRDA) is an effective algorithm for linear discriminant analysis (LDA), but there are some drawbacks in its implementation. In this paper, we first improved Rayleigh-Ritz discriminant analysis (IRRDA) to make its framework more concise, and established the equivalence theory of the solution space between our discriminant analysis and RRDA. Second, we proposed a new model based on positive definite systems of linear equations for linear discriminant analysis, and certificated the rationality of the new model. Compared with the traditional linear regression model for linear discriminant analysis, the coefficient matrix of our model avoided forming a centralized matrix or appending the original data matrix, but the original matrix itself, which greatly reduced the computational complexity. According to the size of data matrix, we designed two solution schemes for the new model based on the block conjugate gradient method. Experiments in real-world datasets demonstrated the effectiveness and efficiency of our algorithm and it showed that our method was more efficient and faster than RRDA. [ABSTRACT FROM AUTHOR]

The numerical solution of initial-value problems (IVP) for ordinary differential equations (ODE) is at this time a mature subject, with many high-quality codes freely available. Second-order linear equations without singularities are an especially simple class of problems to solve, even more so if only a single scalar equation such as the Mathieu equation y ′ ′ + (a - 2 q cos 2 x) y = 0 is being considered. Nonetheless, the topic is not yet exhausted, and this paper considers the case of writing an efficient arbitrary-precision code for the solution of such equations. For this purpose, an implicit Hermite–Obreshkov method attains nearly spectral accuracy at a cost only polynomial in the number of bits of accuracy requested. This is interesting for the Mathieu equation in particular because the solutions can be highly oscillatory of variable frequency and be highly ill-conditioned. This paper reports on the details of the prototype Maple implementation of the method and summarizes the approximation theoretic results justifying the choice of a balanced Hermite–Obreshkov method including its backward stability and decent Lebesgue constants. This method may be of especial interest for the solution of so-called D-finite equations, for which Taylor series coefficients up to degree m are available at cost only O(m), instead of the more usual O (m 2) . This paper celebrates the happy occasion of the 90th birthday of John C. Butcher. [ABSTRACT FROM AUTHOR]

This is the second paper in a series of papers addressing the characteristic gluing problem for the Einstein vacuum equations. We solve the codimension-10 characteristic gluing problem for characteristic data which are close to the Minkowski data. We derive an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges that are conserved by the linearized null constraint equations and act as obstructions to the gluing problem. The gauge-dependent charges can be matched by applying angular and transversal gauge transformations of the characteristic data. By making use of a special hierarchy of radial weights of the null constraint equations, we construct the null lapse function and the conformal geometry of the characteristic hypersurface, and we show that the aforementioned charges are in fact the only obstructions to the gluing problem. Modulo the gauge-invariant charges, the resulting solution of the null constraint equations is C m + 2 for any specified integer m ≥ 0 in the tangential directions and C 2 in the transversal directions to the characteristic hypersurface. We also show that higher-order (in all directions) gluing is possible along bifurcated characteristic hypersurfaces (modulo the gauge-invariant charges). [ABSTRACT FROM AUTHOR]

The paper presents a method for determining the dynamic response of systems containing viscoelastic damping elements with uncertain design parameters. A viscoelastic material is characterized using classical and fractional rheological models. The assumption is made that the lower and upper bounds of the uncertain parameters are known and represented as interval values, which are then subjected to interval arithmetic operations. The equations of motion are transformed into the frequency domain using Laplace transformation. To evaluate the uncertain dynamic response, the frequency response function is determined by transforming the equations of motion into a system of linear interval equations. Nevertheless, direct interval arithmetic often leads to significant overestimation. To address this issue, this paper employs the element-by-element technique along with a specific transformation to minimize redundancy. The system of interval equations obtained is solved iteratively using the fixed-point iteration method. As demonstrated in the examples, this method, which combines the iterative solving of interval equations with the proposed technique of equation formulation, enables a solution to be found rapidly and significantly reduces overestimation. Notably, this approach has been applied to systems containing viscoelastic elements for the first time. Additionally, the proposed notation accommodates both parallel and series configurations of damping elements and springs within rheological models. [ABSTRACT FROM AUTHOR]

When a system of one-sided max-plus linear equations is inconsistent, the approximate solutions within an admissible error bound may be desired instead, particularly with some sparsity property. It is demonstrated in this paper that obtaining the sparsest approximate solution within a given L8 error bound may be transformed in polynomial time into the set covering problem, which is known to be NP-hard. Besides, the problem of obtaining the sparsest approximate solution within a given L1 error bound may be reformulated as a polynomial-sized mixed integer linear programming problem, which may be regarded as a special scenario of the facility location-allocation problem. By this reformulation approach, this paper reveals some interesting connections between the sparsest approximate solution problems in max-plus algebra and some well known problems in discrete and combinatorial optimization. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mixed solution of reduced biquaternion matrix equation Σni=1 AiXiBi=Ewith sub-matrix constraints. With the help of LC-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion matrices, the reduced biquaternion matrix equation (1.1) can be transformed into linear equations. A systematic method, GH-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the least squares problem of linear equations. Meanwhile, we give the necessary and suficient conditions for the compatibility of reduced biquaternion matrix equation (1.1) under sub-matrix constraints. Numerical examples are given to demonstrate the results. The method proposed in this paper is applied to color image restoration. [ABSTRACT FROM AUTHOR]

Motivated by the various applications of the trapping diffusion-influenced reaction theory in physics, chemistry, and biology, this paper deals with irreducible Cartesian tensor (ICT) technique within the scope of the generalized method of separation of variables (GMSV). We provide a survey from the basic concepts of the theory and highlight the distinctive features of our approach in contrast to similar techniques documented in the literature. The solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in terms of ICT. By means of proved translational addition theorem, we straightforwardly reduce the general boundary value diffusion problem for N spherical sinks to the corresponding resolving infinite set of linear algebraic equations with respect to the unknown tensor coefficients. These coefficients exhibit an explicit dependence on the arbitrary three-dimensional configurations of N sinks with different radii and surface reactivities. Our research contains all relevant mathematical details such as terminology, definitions, and geometrical structure, along with a step by step description of the GMSV algorithm with the ICT technique to solve the general diffusion boundary value problem within the scope of Smoluchowski's trapping model. [ABSTRACT FROM AUTHOR]

We introduce a novel deep‐learning architecture for image upscaling by large factors (e.g. 4x, 8x) based on examples of pristine high‐resolution images. Our target is to reconstruct high‐resolution images from their downscale versions. The proposed system performs a multi‐level progressive upscaling, starting from small factors (2x) and updating for higher factors (4x and 8x). The system is recursive as it repeats the same procedure at each level. It is also residual since we use the network to update the outputs of a classic upscaler. The network residuals are improved by Iterative Back‐Projections (IBP) computed in the features of a convolutional network. To work in multiple levels we extend the standard back‐projection algorithm using a recursion analogous to Multi‐Grid algorithms commonly used as solvers of large systems of linear equations. We finally show how the network can be interpreted as a standard upsampling‐and‐filter upscaler with a space‐variant filter that adapts to the geometry. This approach allows us to visualize how the network learns to upscale. Finally, our system reaches state of the art quality for models with relatively few number of parameters. [ABSTRACT FROM AUTHOR]

In this article, we present as a comparative result of Adomian Decomposition Method (ADM), Modified Adomian Decomposition Method (MADM) and Variational Iteration Method (VIM). These methods used for developed to find the analytical approximate solution of linear Fredholm integro-differential equations. The main purpose of this paper was to show a better method for Numerical equations which does not give easily analytical solution. So, in this paper, we find approximate solutions of linear Fredholm integro-differential equations. We explain the convergence of ADM, MADM and VIM by using examples of a deterministic model by graphs and tables. All the calculations performed by the help of MATLAB (2018) Version 9.4. [ABSTRACT FROM AUTHOR]

Combining multiple devices for localization has important applications in the military field. This paper exploits the land-based short-wave platforms and satellites for fusion localization. The ionospheric reflection height error and satellite position errors have a great impact on the short-wave localization and satellite localization accuracy, respectively. In this paper, an iterative constrained weighted least squares (ICWLS) algorithm is proposed for these two kinds of errors. The algorithm converts the nonconvex equation constraints to linear constraints using the results of the previous iteration, thus ensuring convergence to the globally optimal solution. Simulation results show that the localization accuracy of the algorithm can reach the corresponding Constrained Cramér–Rao Lower Bound (CCRLB). Finally, the localization results of the two methods are fused using Kalman filtering. Simulations show that the fused localization accuracy is improved compared to the single-means localization. [ABSTRACT FROM AUTHOR]

In this paper, new Sturmian comparison results were obtained for linear and nonlinear hyperbolic equations on a rectangular prism. The results obtained for linear equations extended those given by Kreith [Sturmian theorems on hyperbolic equations, Proc. Amer. Math. Soc., 22 (1969), 277-281] in which the Sturmian comparison theorem for linear equations was obtained on a rectangular region in the plane. For the purpose of verification, an application was described using an eigenvalue problem. [ABSTRACT FROM AUTHOR]

This paper proposes the D2Q5 Lattice Boltzmann method (LBM) method, in two dimensions with five discrete lattice velocities, for simulating linear sound wave propagation in closed rooms. A second-order linear acoustic equation obtained from the LBM method was used as the model equation. Boundary conditions at the domain boundary use the bounce-back scheme. The LBM numerical calculation algorithm in this paper is relatively simpler and easy to implement. Parallelization with the GPU CUDA was implemented to speed up the execution time. The calculation results show that the use of parallel GPU CUDA programming can accelerate the proposed simulation 27.47 times faster than serial CPU programming. The simulation results are validated with analytical solutions for acoustic pulse reflected by the flat and oblique walls, the comparisons show very good concordance, and the D2Q5 LBM has second-order accuracy. In addition, the simulation results in the form of wavefront propagation images in complicated shaped rooms are also compared with experimental photographs, and the comparison also shows excellent concordance. The numerical results of the D2Q5 LBM are promising and also demonstrate the great capability of the D2Q5 LBM for investigating room acoustics in various complexities. [ABSTRACT FROM AUTHOR]

This paper deals with the experimental procedures of lock-in thermography (LIT) for polyetheretherketone (PEEK), which is used as a lightweight material in various industrial fields. The LIT has limitations due to non-uniform heating by external optic sources and the non-uniformity correction (NUC) of the infrared (IR) camera. It is generating unintended contrast in the IR image in thermal imaging inspection, reducing detection performance. In this study, the non-uniformity effect was primarily improved by producing an equivalent array halogen lamp. Then, we presented absolute temperature compensation (ATC) and temperature ratio compensation (TRC) techniques, which can equalize the thermal contrast of the test samples by compensating for them using reference samples. By applying compensation techniques to data acquired from the test samples, defect detectability improvement was quantitatively presented. In addition, binarization was performed and detection performance was verified by evaluating the roundness of the detected defects. As a result, the contrast of the IR image was greatly improved by applying the compensation technique. In particular, raw data were enhanced by up to 54% using the ATC compensation technique. Additionally, due to improved contrast, the signal-to-noise ratio (SNR) was improved by 7.93%, and the R2 value of the linear trend equation exceeded 0.99, demonstrating improved proportionality between the defect condition and SNR. [ABSTRACT FROM AUTHOR]

We derive an exponential decay rate of energy for the cubic Schrödinger equation on a dissipative waveguide with a damping term that is effective locally on a neighborhood at the boundary and at infinity. In this paper, we give a new example for which the geometric control condition is sufficient to obtain an exponential decay of energy Schrödinger equation. The proof is based on Strichartz's estimates and the smoothing effects argument associated with the linear Schrödinger equation(LS). [ABSTRACT FROM AUTHOR]

This paper revisits the set-invariance for linear delay-difference equations and proposes novel delay-dependent notions in contrast with the existing delay-independent constructions available in the literature. The main tool in this endeavour is the model transformation by means of a matrix parametrization, which opens the way to the elaboration of delay-dependent conditions for set invariance. For the class of polyhedral sets, it will be shown how these transformed models enable particularly structured invariance conditions. Aside from the mathematical developments, a discussion of the delay-dependent conditions with respect to the confinement of the state trajectories of the original system will be presented. The relationship will be established by means of constraints involving the initial states for the time-delay systems. The characterization of this set of admissible initial conditions can be analysed in terms of complexity and conservativeness with respect to the classical delay-independent set-invariance. An illustrative example is provided to underline that confinement of state trajectories in a set can be achieved even though this is not invariant according to the classical definition. [ABSTRACT FROM AUTHOR]

Iteration is one of the most important topics in computer science and attentions are paid to functional equations involving iterates, one of which is the linear combination of alternative iterates. Recently existence, uniqueness and dependence for increasing Lipschitzian solutions were obtained for linear combination of alternative iterates on $ [0, 1] $ [ 0 , 1 ] and $ \mathbb {R} $ R in the case that all given functions are strictly increasing. In this paper we work in various cases of monotonicity (increasing and decreasing) among given functions and the unknown function. [ABSTRACT FROM AUTHOR]

This study purposes to extract some fractional analytical solutions of the Peyrard-Bishop-Deoxyribo-Nucleic-Acid (β -PBDNA) dynamic model with the beta-derivative by the unified Riccati equation expansion method (UREEM). Furthermore, we examine the role of various parameters of the fractional model on the soliton dynamic. The research focuses on computational biophysics and materials science, examining the impact of various parameters on the fractional model. This paper contributes to understanding soliton solutions and the β -PBDNA dynamic model, demonstrating the applicability of the UREEM method to various fractional models. Some soliton solutions of the model are successfully generated by applying the UREEM. Implementing the UREEM, we take a fractional wave transformation to convert the model into a nonlinear ordinary differential equation. So, a linear equation system is generated. After the system is solved, the soliton solutions are gained for the appropriate solution sets. Finally, 3D, 2D and contour graphs of diverse solutions are depicted at suitable values of parameters. In addition, this paper presents 3D, 2D and contour graphs of various solutions with suitable parameter values. The results are beneficial for interpreting the model in future work and confirm that UREEM is effectively applicable to diverse fractional models, coupled with a comprehensive graphical analysis of how different parameters influence these solutions. [ABSTRACT FROM AUTHOR]

In this crossover experiment, we investigated the impact of a statement bank, enabling the reuse of previously written feedback (SA condition), on 45 math teachers' feedback for 60 completed linear equation tests, compared to traditional pen‐and‐paper feedback (PP condition). In the SA condition, teachers were encouraged to use atomic feedback, a set of formulation requirements that makes feedback items significantly more reusable. A previous study found that significantly more feedback was written in the SA condition but did not investigate the content of the feedback. To address this gap, we employed a novel approach of combining text mining with qualitative methods. Results indicate similar wording and sentiments in both conditions. However, SA feedback was more elaborate yet general, focusing on major and minor strengths and deficits, while PP feedback was shorter but more concrete, emphasising main issues. Despite low feedback quality in both conditions, the statement bank led to less effective diagnostic activities, implying that teachers' careless use of statement banks, although convenient, might lead to lower‐quality feedback. Practitioner notesWhat is already known about this topic High‐quality feedback should strike a balance between the volume and focus on the main issues, as more feedback does not necessarily equate to better feedback. Feedback should analyse a student's solution whenever possible: interpreting mistakes and communicating that interpretation as feedback.Text mining identifies meaningful patterns and new insights in text using computer algorithms.When teachers can reuse already given feedback using a software tool (statement bank), they tend to write more feedback instead of saving time.What this paper adds Feedback is compared when teachers could use a tool to reuse already given feedback (referred to as 'statement banks') versus a scenario without such a tool. Both approaches observed similar word frequencies, sentiments and amounts of erroneous, descriptive and corrective feedback. However, feedback with a statement bank tended to be more elaborate yet less specific to individual student solutions. In contrast, feedback without the tool was shorter but more concrete, focusing on main issues. Overall, the tool for reusing feedback directed teachers towards less effective diagnostic activities.The paper introduces a novel methodological approach by combining text mining with qualitative techniques in educational research. While text mining provides an overall understanding of differences and similarities in feedback approaches, qualitative methods are essential for in‐depth analysis of content characteristics and feedback quality.Implications for practice and/or policy Statement banks can support teachers by giving more feedback, but in order to improve feedback quality, further measures are necessary (eg, improving pedagogical content knowledge).Teachers may not confuse handiness with quality: statement banks can help, but when used carelessly, teachers tend to describe and correct students' work instead of analysing underlying (mis‐)conceptions using it. Continued attention to feedback quality remains necessary when using such tools. [ABSTRACT FROM AUTHOR]

In this paper, we suggest joint application of computational probabilistic analysis and the Monte Carlo method for numerical stochastic modeling problems. We use all the capabilities of computational probabilistic analysis while maintaining all the advantages of the Monte Carlo method. Our approach allows us to efficiently implement a computational hybrid scheme. In this way, we reduce the computation time and present the results in the form of distributions. The crucial new points of our method are arithmetic operations on probability density functions and procedures for constructing on the probabilistic extensions. Relying on specific numerical examples of solving systems of linear algebraic equations with random coefficients, we present the advantages of our approach. [ABSTRACT FROM AUTHOR]

Conjugate gradient (CG) methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems. In this paper, a new hybrid conjugate gradient (CG) method is presented and analyzed for solving unconstrained optimization problems, where the parameter β k is a convex combination of β k WYL and β k CD . Under the strong Wolfe line search, the new method possesses the sufficient descent condition and the global convergence properties. The preliminary numerical results show the efficiency of our method in comparison with other CG methods. Furthermore, the proposed algorithm HWYLCD was extended to solve the problem of a mode function. [ABSTRACT FROM AUTHOR]

The iterative decreasing dimension method (IDDM) is an iterative method used to solve the linear algebraic system A x = f . Such systems are important in modeling many problems in applied sciences. For a number of reasons, such as estimated measurements made for modeling, errors arising from floating point calculations, and approximation methods used for solutions, it becomes necessary to study intervals in the solutions of systems of linear equations. The objective of this paper is to utilize IDDM to achieve resolution in the interval linear system (ILS). During the calculations, the Kaucher space is considered an extended classical interval space. The solutions of Barth-Nuding and Hansen interval linear systems, which are commonly used in the literature to test the solutions of ILSs, are obtained with the interval iterative decreasing dimension method for interval linear systems (I-IDDM). Since IDDM is a variation method of Gaussian elimination, a comparative analysis of the results with the interval Gaussian elimination method (I-GEM) is performed. It has been demonstrated that our approach, I-IDDM, produces better outcomes than I-GEM. I-IDDM is also used to investigate the analog circuit problem, where interval analysis is crucial. [ABSTRACT FROM AUTHOR]

The paper is concerned with error identities for a class of parabolic equations. One side of such an identity is a natural measure of the distance between a function in the corresponding energy class and the exact solution of the problem in question. Another side is either directly computable or serves as a source of fully computable error bounds. Particular forms of the identities can be viewed as analogs of the hypercircle identity well known for elliptic problems. It is shown that identities possess an important consistency property. Therefore, the identities and the corresponding error estimates can be used in quantitative analysis of direct and inverse problems associated with parabolic equations. The first part of the paper deals with linear parabolic equations. A class of nonlinear problems is considered in the second part. In particular, this class includes problems whose spatial parts are presented by the α-Laplacian operator. [ABSTRACT FROM AUTHOR]

A promising approach to investigating high-dimensional problems is to identify their intrinsically low-dimensional features, which can be achieved through recently developed techniques for effective low-dimensional representation of functions such as machine learning. Based on available finite-dimensional approximate solution manifolds, this paper proposes a novel model reduction framework for kinetic equations. The method employs projections onto tangent bundles of approximate manifolds, naturally resulting in first-order hyperbolic systems. Under certain conditions on the approximate manifolds, the reduced models preserve several crucial properties, including hyperbolicity, conservation laws, entropy dissipation, finite propagation speed, and linear stability. For the first time, this paper rigorously discusses the relation between the H-theorem of kinetic equations and the linear stability conditions of reduced systems, determining the choice of Riemannian metrics involved in the model reduction. The framework is widely applicable for the model reduction of many models in kinetic theory. [ABSTRACT FROM AUTHOR]

In rotating machinery with a multi-stage assembled rotor, such as is found in aero engines, any unbalance present will undergo unknown changes at each stage when rotating the assembly phases of the rotor. Repeated disassembly and adjustments are often required to meet the rotor's residual unbalance specifications. Therefore, developing a prediction model of this two-sided unbalance for a multi-stage assembled rotor is crucial for improving the first-time assembly pass rate and assembly efficiency. In this paper, we propose a prediction model of the two-sided unbalance seen in the multi-stage assembled rotor of an aero engine. Firstly, a method was proposed to unify the mass feature parameters of each stage's rotor into a geometric measurement coordinate system, achieving the synchronous transmission of geometric and mass feature parameters during the assembly process of the multi-stage rotor. Building upon this, a linear parameter equation of the actual rotation axis of the multi-stage rotor was established. Based on this axis, the mass eccentricity errors of the rotor were calculated at each stage, further enabling the accurate prediction of two-sided unbalance and its action phase in a multi-stage rotor. The experimental results indicate that the maximum prediction errors of the two-sided unbalance and its action phase for a four-stage rotor are 9.6% and 2.5%, respectively, when using this model, which is a reduction of 53.0% and 38.1% compared to the existing model. [ABSTRACT FROM AUTHOR]

F-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the F-concavities preserved by the Dirichlet heat flow in convex domains on {\mathbb R}^n, and complete the study of preservation of concavity properties by the Dirichlet heat flow, started by Brascamp and Lieb in 1976 and developed in some recent papers. More precisely, we discover hot-concavity, which is the strongest F-concavity preserved by the Dirichlet heat flow; we show that log-concavity is the weakest F-concavity preserved by the Dirichlet heat flow; quasi-concavity is also preserved only for n=1; we prove that if F-concavity is strictly weaker than log-concavity and n\ge 2, then there exists an F-concave initial datum such that the corresponding solution to the Dirichlet heat flow is not even quasi-concave, hence losing any reminiscence of concavity. Furthermore, we find a sufficient and necessary condition for F-concavity to be preserved by the Dirichlet heat flow. We also study the preservation of concavity properties by solutions of the Cauchy–Dirichlet problem for linear parabolic equations with variable coefficients and for nonlinear parabolic equations such as semilinear heat equations, the porous medium equation, and the parabolic p-Laplace equation. [ABSTRACT FROM AUTHOR]

This paper contains several novel definitions including neutrosophic E β metric space, neutrosophic quasi- S β -metric space, neutrosophic pseudo- S β -metric space, neutrosophic quasi- E -metric space and neutrosophic pseudo- E β -metric space. Further, we present some generalized fixed point results with non-trivial examples and the decomposition theorem in the setting of the neutrosophic pseudo- E β -metric space. Moreover, by using the main result, we examine the existence and uniqueness of the solution to an integral equation, a system of linear equations, and nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate time-dependent Kantowski–Sachs spherically symmetric teleparallel F (T) gravity in vacuum and in a perfect isotropic fluid. We begin by finding the field equations and solve for new teleparallel F (T) solutions. With a power-law ansatz for the co-frame functions, we find new non-trivial teleparallel F (T) vacuum solutions. We then proceed to find new non-trivial teleparallel F (T) solutions in a perfect isotropic fluid with both linear and non-linear equations of state. We find a great number of new exact and approximated teleparallel F (T) solutions. These classes of new solutions are relevant for future cosmological applications. [ABSTRACT FROM AUTHOR]

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Photoacoustic tomography (PAT) is an emerging imaging technique with great potential for a wide range of biomedical imaging applications. The transducers impulse response (TIR) is a key factor affecting the performance of photoacoustic imaging (PAI). It is customary in PAI to assume that TIR is known or obtain it from experiments. In this paper, we investigate the possibility of obtaining TIR in another way. A new method is proposed to extract TIR from observed optoacoustic signal (OPAS) data, without prior knowledge, as a known condition. It is based on the relation between the OPAS data and the photoacoustic pressure signal (PAPS) at transducer positions. The relation can be expressed as a homogeneous linear equation. The TIR is solved by solving the homogeneous equation. The numerical test verifies the effectiveness of the presented method. This article also discusses the effect of calculation parameters on the extracting precision of TIR. [ABSTRACT FROM AUTHOR]

In this paper, we study the evasion game of high speed evader involving two pursuers and a single evader with geometric constraints on the control parameters of the players in the plane. The game is described by linear equations. Evasion is said to be possible if the state of the evader doesn’t coincide with the state of any pursuer for all time. We construct an evasion strategy for the evader which ensure completion the evasion game for any initial positions of players. In addition, we introduce a new concept of approach times and demonstrate that the number of approach times does not exceed 3. [ABSTRACT FROM AUTHOR]

In this paper, a system of coupled quasi-linear and linear wave equations with a finite memory term is concerned. By constructing an appropriate Lyapunov function, we prove that the total energy associated with the system is stable under suitable conditions on memory kernel. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a novel category of radial basis functions that incorporate smoothing techniques. Initially, we employ the power augmented and shape parameter schemes to create the radial basis functions. Subsequently, we apply the newly-constructed radial basis functions using the traditional collocation method and singular values decomposition algorithm to solve the corresponding linear system equations. Finally, we analyze several pairs of radial basis functions in depth to address physical problems linked to thermal science that are governed by partial differential equations. The numerical results demonstrate that the radial basis functions constructed using the power augmented and shape parameter schemes exhibit remarkable performance. [ABSTRACT FROM AUTHOR]

The excavation of a foundation pit can exert a notable impact on the underlying tunnel. This research paper aims to analyze and synthesize measured deformation outcomes caused by foundation pit excavation on the underlying tunnel. The paper employs a two-stage analysis approach to derive the calculation formula for additional stress and the deformation control equation of the adjacent tunnel under the influence of foundation pit excavation. Subsequently, the Hermite spectrum method is applied to transform the deformation control equation of the underlying tunnel into a set of linear equations, enabling the determination of the deformation curve. To verify the precision of the theoretical calculation method, a comparative study is conducted between theoretical results and actual measurements. Moreover, a sensitivity analysis of crucial project factors is performed. The research findings reveal minimal disparity between theoretical calculation outcomes and measured deformation of the underlying tunnel, thus affirming the accuracy and rationality of the theoretical calculation formula. The excavation of the foundation pit leads to an uplift deformation in the underlying tunnel, resulting in an "n"-shaped deformation profile. Notably, the stiffness of the foundation soil and the depth at which the tunnel is buried emerge as pivotal factors influencing the deformation of the underlying tunnel. As the stiffness of the foundation soil and the depth of tunnel burial increase, the uplift deformation gradually diminishes, albeit within a restricted range of reduction. [ABSTRACT FROM AUTHOR]