Algebraic Error Research Articles

Calibrating acceleration transmitters to quantify the seasonal energetic costs of activity in lake trout.

Bioenergetics models are powerful tools used to address a range of questions in fish biology. However, these models are rarely informed by free-swimming activity data, introducing error. To quantify the costs of activity in free-swimming fish, calibrations produced from standardized laboratory trials can be applied to estimate energy expenditure from sensor data for specific tags and species. Using swim tunnel respirometry, we calibrated acceleration sensor-equipped transmitting tags to estimate the aerobic metabolic rates (ṀO2) of lake trout (Salvelinus namaycush) at three environmentally relevant temperatures. Aerobic and swim performance were also assessed. Like other calibrations, we found strong relationships between ṀO2 and acceleration or swimming speed, and jackknife validations and data simulations suggest that our models accurately predict metabolic costs of activity in adult lake trout (~5% algebraic error and ~20% absolute error). Aerobic and swim performance metrics were similar to those reported in other studies, but their critical swimming speed was lower than expected. Additionally, lake trout exhibited a wide aerobic scope, suggesting that the avoidance of waters ≥15°C may be related to selection for optimal growing temperatures. The ability to quantify the free-swimming energetic costs of activity will advance our understanding of lake trout ecology and may yield improvements to bioenergetics model.

Deterministic Replacement Path Covering

In this article, we provide a unified and simplified approach to derandomize central results in the area of fault-tolerant graph algorithms. Given a graph \(G\) , a vertex pair \((s,t)\in V(G)\times V(G)\) , and a set of edge faults \(F\subseteq E(G)\) , a replacement path \(P(s,t,F)\) is an \(s\) - \(t\) shortest path in \(G\setminus F\) . For integer parameters \(L,f\) , a replacement path covering ( \(\mathsf{RPC}\) ) is a collection of subgraphs of \(G\) , denoted by \(\mathcal{G}_{L,f}=\{G_{1},\ldots,G_{r}\}\) , such that for every set \(F\) of at most \(f\) faults (i.e., \(|F|\leq f\) ) and every replacement path \(P(s,t,F)\) of at most \(L\) edges, there exists a subgraph \(G_{i}\in\mathcal{G}_{L,f}\) that contains all the edges of \(P\) and does not contain any of the edges of \(F\) . The covering value of the \(\mathsf{RPC}\) \(\mathcal{G}_{L,f}\) is then defined to be the number of subgraphs in \(\mathcal{G}_{L,f}\) . In the randomized setting, it is easy to build an \((L,f)\) - \(\mathsf{RPC}\) with covering value of \(O(\max\{L,f\}^{\min\{L,f\}}\cdot\min\{L,f\}\cdot \log n)\) , but to this date, there is no efficient deterministic algorithm with matching bounds. As noted recently by Alon et al. (ICALP 2019), this poses the key barrier for derandomizing known constructions of distance sensitivity oracles and fault-tolerant spanners. We show the following: — There exist efficient deterministic constructions of \((L,f)\) - \(\mathsf{RPC}\) s whose covering values almost match the randomized ones, for a wide range of parameters. Our time and value bounds improve considerably over the previous construction of Parter (DISC 2019). Our algorithms are based on the introduction of a novel notion of hash families that we call HM hash families. We then show how to construct these hash families from (algebraic) error correcting codes such as Reed–Solomon codes and Algebraic-Geometric codes. — For every \(L,f\) , and \(n\) , there exists an \(n\) -vertex graph \(G\) whose \((L,f)\) - \(\mathsf{RPC}\) covering value is \(\Omega(L^{f})\) . This lower bound is obtained by exploiting connections to the problem of designing sparse fault-tolerant breadth first search (BFS) structures. An application of our above deterministic constructions is the derandomization of the algebraic construction of the distance sensitivity oracle by Weimann and Yuster (FOCS 2010). The preprocessing and query time of our deterministic algorithm nearly match the randomized bounds. This resolves the open problem of Alon et al. (ICALP 2019). Additionally, we show a derandomization of the randomized construction of vertex fault-tolerant spanners by Dinitz and Krauthgamer (PODC 2011) and Braunschvig et al. (Theor. Comput. Sci., 2015). The time complexity and the size bounds of the output spanners nearly match the randomized counterparts.

Globally Optimal Relative Pose and Scale Estimation from Only Image Correspondences with Known Vertical Direction

Installing multi-camera systems and inertial measurement units (IMUs) in self-driving cars, micro aerial vehicles, and robots is becoming increasingly common. An IMU provides the vertical direction, allowing coordinate frames to be aligned in a common direction. The degrees of freedom (DOFs) of the rotation matrix are reduced from 3 to 1. In this paper, we propose a globally optimal solver to calculate the relative poses and scale of generalized cameras with a known vertical direction. First, the cost function is established to minimize algebraic error in the least-squares sense. Then, the cost function is transformed into two polynomials with only two unknowns. Finally, the eigenvalue method is used to solve the relative rotation angle. The performance of the proposed method is verified on both simulated and KITTI datasets. Experiments show that our method is more accurate than the existing state-of-the-art solver in estimating the relative pose and scale. Compared to the best method among the comparison methods, the method proposed in this paper reduces the rotation matrix error, translation vector error, and scale error by 53%, 67%, and 90%, respectively.

Enhancing the acquisition of basic algebraic principles using algebra tiles

Introducing initial algebraic principles poses a significant challenge, often compounded by the inherent abstract nature of algebra. This article introduces an innovative pedagogical approach that promotes the use of algebra tiles, a didactic manipulative material formed by a collection of geometric pieces symbolizing distinct algebraic monomials. Additionally, this article includes the findings of a quasi-experimental study that applied this inventive teaching method. This research was carried out across two separate classes of 15 students (10 boys and five girls) of first year of compulsory secondary education. One group of students adhered to the conventional teaching approach (the control group), while the other class embraced the proposed methodology using algebra tiles (the experimental group). The disparities in algebraic proficiency observed between these two student groups, as assessed through various examinations conducted during the intervention were statistically significant, with the experimental group consistently achieving superior results. Moreover, mathematical and algebraic errors of students were assessed using eight distinct indicators. In all cases, the experimental group demonstrated lower error percentages, and these errors showed a marked decrease as the intervention progressed. In summary, this innovative methodology markedly enhanced students’ comprehension of algebra, their knowledge, and their motivation while significantly reducing mathematical errors.

On a model of online analog computation in the cell with absolute functional robustness: Algebraic characterization, function compiler and error control

The Turing completeness of continuous Chemical Reaction Networks (CRNs) states that any computable real function can be computed by a continuous CRN on a finite set of molecular species, possibly restricted to elementary reactions, i.e. with at most two reactants and mass action law kinetics. In this paper, we introduce a more stringent notion of robust online analog computation, called Absolute Functional Robustness (AFR), for the CRNs that stabilize the concentration values of some output species to the result of one function of the input species concentrations, while allowing arbitrary perturbations for intermediate and output species throughout the attraction basin. We prove that the set of real functions stabilized by a CRN with mass action law kinetics is precisely the set of real algebraic functions. Based on this result, we present a compiler which takes as input any algebraic function (defined by one polynomial and one point for selecting one branch of the algebraic curve defined by the polynomial) and generates an abstract CRN to stabilize it. Furthermore, we provide error bounds to estimate and control the error of an unperturbed system, under the assumption that the environment inputs are driven by k-Lipschitz functions.

Model reduction for stochastic systems with nonlinear drift

In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such nonlinearities in high dimensional settings occur, e.g., when stochastic reaction diffusion equations are discretized in space. We provide a brief discussion around existence, uniqueness and stability of solutions. (Almost) stability then is the basis for new concepts of Gramians that we introduce and study in this work. With the help of these Gramians, dominant subspace is identified leading to a balancing related highly accurate reduced order SDE. We provide an algebraic error criterion and an error analysis of the propose model reduction schemes. The paper is concluded by applying our method to spatially discretized reaction diffusion equations.

Exploring Learner Errors and Misconceptions in Algebraic Expressions with Grade 9 Learners Through the use of Algebra Tiles

South African learners’ performance in mathematics, both locally and internationally, has raised significant concerns, particularly in the realm of algebra. To address this issue, a mixed-methods study was conducted to explore the effectiveness of using algebra tiles to teach algebraic expressions. The study aimed to investigate algebraic expression errors and misconceptions among Grade 9 learners and evaluate the impact of an intervention involving algebra tiles on learners’ post-test results. Data were collected through tests administered to a class of 22 Grade 9 learners. The findings of the study confirmed the error types identified in the literature and demonstrated a notable improvement in performance on the post-test following the intervention using algebra tiles. The results indicated that the intervention successfully rectified pre-existing errors and misconceptions, resulting in an 18% enhancement in overall performance among participants. The study aligned with a Vygotskian sociocultural perspective, emphasizing the pivotal role of manipulatives in facilitating learning within the zone of proximal development. The use of manipulatives aids learners in constructing conceptual understanding by reinforcing abstract ideas. Therefore, the study contributes to existing research highlighting the utility of manipulatives in mathematics classrooms, underscoring their effectiveness in enhancing learning outcomes.

Hp-Robust multigrid solver on locally refined meshes for FEM discretizations of symmetric elliptic PDEs

In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that the iterative solver contracts the algebraic error robustly with respect to the polynomial degree p ≥ 1 and the (local) mesh size h. We further prove that the built-in algebraic error estimator which comes with the solver is hp-robustly equivalent to the algebraic error. The application of the solver within the framework of adaptive finite element methods with quasi-optimal computational cost is outlined. Numerical experiments confirm the theoretical findings.

Scaffolding for Algebra Errors From the Perspective of Emotional Quotient

Scaffolding for Algebra Errors From the Perspective of Emotional Quotient

Preservice teachers’ errors in the creation and extension of sequences using color tiles: An exploratory study

AbstractRecognizing, building, describing, extending, and analyzing patterns are key components of algebraic reasoning. Oftentimes pattern‐based instruction is used to support the understanding of functions and variables. In this study, elementary preservice teachers (n = 23) were asked to explore, extend, explain, create equations for, and analyze linear, geometric, and quadratic integer patterns using color tiles. Then they created, extended, described, and analyzed their own patterns while working in groups (n = 8). The research question was: What algebraic errors do PSTs demonstrate in the process of constructing, explaining, and analyzing their own linear, geometric, and quadratic patterns using color tiles? The analysis, which focused on errors and misunderstandings preservice teachers demonstrated in their final unit, revealed three specific areas of difficulties: figural errors, explanation challenges, and improper use of algebraic language. Participants failed to connect their algebraic images in a coherent manner, there was confusion in articulating their algebraic thought processes, and algebraic terminology was improperly used. Implications for teaching and further research are described.

Deep Ritz method with adaptive quadrature for linear elasticity

In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the H1/2(ΓD) norm is introduced and analyzed for linear elasticity equation in order to deal with the (essential) Dirichlet boundary condition. We show that the resulting deep Ritz method provides the best approximation among the set of deep neural network (DNN) functions with respect to the “energy” norm. Furthermore, we demonstrate that the total error of the deep Ritz simulation is bounded by the sum of the network approximation error and the numerical integration error, disregarding the algebraic error. To effectively control the numerical integration error, we propose an adaptive quadrature-based numerical integration technique with a residual-based local error indicator. This approach enables efficient approximation of the modified energy functional. Through numerical experiments involving smooth and singular problems, as well as problems with stress concentration, we validate the effectiveness and efficiency of the proposed deep Ritz method with adaptive quadrature.

Analisis Kesalahan Hasil Belajar Mahasiswa pada Operasi Matriks Mata Kuliah Aljabar Linear dan Matriks

In solving math problems, students including many students experience difficulties, so there is a high probability of experiencing errors in answering questions. This study aims to analyze student errors in solving mathematical problems in matrix material, linear algebra and matrix courses and describe student learning outcomes errors in linear algebra and matrix courses, then identify factors that affect student learning outcomes errors in linear algebra and matrix courses. This type of research is qualitative descriptive research with a case study method. Sampling was done using purposive sampling technique with tests and interviews. The sample of this study consisted of 37 4th semester students who had an average that was still below the limit of completeness in mathematics courses in the previous semester. The results showed that the factors that influence student learning outcomes errors are a) less careful in understanding the problem, b) lack of time management when working on problems, c) lack of focus when working, d) lack of mastery of concepts, formulas and calculation procedures in matrix operations, e) lack of mastery of basic mathematical operations, f) lack of caution and thoroughness.

Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors

We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.

Guaranteed contraction of adaptive inexact hp-refinement strategies with realistic stopping criteria

The purpose of this contribution it to theoretically analyze the adaptive refinement strategies for conforming hp-finite element approximations of elliptic problems proposed for exact algebraic solvers in (P. Daniel, A. Ern, I. Smears and M. Vohralk, Comput. Math. Appl. 76 (2018) 967–983.) and for inexact algebraic solvers in (P. Daniel, A. Ern, and M. Vohralk, Comput. Methods Appl. Mech. Eng. 359 (2020) 112607.). Both of these strategies are driven by guaranteed equilibrated flux energy error estimators. The employed hp-refinement criterion stems from solving two separate local residual problems posed only on the patches of elements around marked vertices selected by a bulk-chasing criterion. In the above references, we have derived a fully computable guaranteed bound on the ratio of the error on two successive steps of the hp-adaptive loop. Here, our focus is to prove that this ratio is uniformly smaller than one, and thus the convergence of the adaptive and adaptive inexact hp-refinement strategies. To be able to achieve this goal, we have to introduce some additional assumptions on the h- and p-refinements, namely an extension of the marked region, as well as a sufficient h- or p-refinement of each marked patch. We investigate two such strategies, where one ensures a polynomial-degree-robust guaranteed contraction. In the inexact case, a sufficiently precise stopping criterion for the algebraic solver is requested, but this criterion remains fully computable and also realistic in the sense that in our numerical experiments, it does not request the algebraic error to be excessively small in comparison with the total error.

Globally Optimal Relative Pose Estimation Using Affine Correspondences With Known Vertical Direction

For self-driving cars, smartphones, and unmanned aerial vehicles, cameras are typically used in combination with inertial measurement units (IMUs). The vertical direction of camera views can be determined based on partial IMU measurements—that is, the relative orientation of two views is reduced to a single degree of freedom. In this paper, we propose a globally optimal solver using affine correspondences to estimate the relative pose of two views with a known vertical direction. First, the relative pose estimation problem is formalized as a cost function optimization, which minimizes algebraic error in the least-squares sense. Then, the cost function optimization is converted into two polynomials with only two unknowns, which are represented by the parameter of the rotation angle. Finally, we use the eigenvalue polynomial solver to calculate the parameter of the rotation angle. The proposed solver is evaluated using synthetic data and real-world scenes from the KITTI and EuRoC benchmarks. The experimental results show that our method performs better than existing state-of-the-art solvers.

Estimation of Discharge Capacities Using Generalized Peukert’s Equation for Saft Industrial Standby Batteries

Saft Industrial Standby (ISD) battery utilizes robust nickel-cadmium alkaline electrochemistry that delivers reliable performance and long life of at least 15 years over a wide range of temperature (-20 °C to 50 °C). The battery is designed with nickel based positive electrode and plastic bonded cadmium electrode in a vented prismatic cell. Typically, telecom applications used to require long duration backup power sources. With the advent of 4G/5G, the duration for backup power has been reduced to 1 hour or less. Therefore, new ISD battery was designed for power applications like 174 W/cell discharge for 1 hour and at the same time it can deliver high energy density up to 100 Wh/l. Additionally, low water consumption due to the high charge efficiency makes this battery ideal for the maintenance free remote installations. All these features render the battery aptly suitable for providing back-up power in telecom networks (4G and 5G) including remote cabinets at the lowest total cost of ownership.The ISD battery (Tel.X-Plus) is compliant to the Telecordia GR (Generic Requirement)-3020 CORE standard, which is recommended for the secondary alkaline batteries used in the telecom back-up application. This standard requires declaration of minimum 28 discharge currents entailing different durations and end voltages for a specific battery capacity. Hence, correct estimation of deliverable capacities for any definite duration and end voltage can reduce significant number of test trials. In addition, it helps sizing the battery for different power and energy installations. There are a few analytical techniques, such as Peukert’s and Shepherd’s equations that can be used to model the discharge characteristics of batteries. There are three different forms of generalized Peukert’s equations: algebraic, hyperbolic tangent and complementary error functions. In this study, to evaluate the discharge capacities we investigated the complementary error function of the generalized Peukert’s equations described in the equation 1: C= (Cm/erfc(-1/n)) * efrc(((i/ik)-1)/n) ... (1)where, C is the capacity of the battery during discharge, Cm is the highest discharge capacity, i is the discharge current, ik and n are statistical parameters. We know that phase transition is key to the discharge of a battery. Therefore, we selected the equation which also defines the phase transition process. The discharge capacities were calculated between 24-hour and 0.20-hour discharge durations for 1 V/cell, 1.05 V/cell, 1.10 V/cell and 1.14 V/cell end points using the equation 1. Here, the discharge capacity that was experimentally determined at the nominal C/24 rate was used as the maximum discharge capacity, Cm. The value of the statistical parameter n is within 0 and 1 to satisfy the capacity limits when discharge current tends to infinite and zero. For example, n equals to 0.6 is used for the calculations pertaining to the 180 Ah rated battery. The other statistical parameter ik was empirically selected for each end point voltage e.g., 250 A for 1.1 V/cell. Figure 1 shows the results for the calculated discharge capacities along with the experimental values at the 1.1 V/cell cut-off. The calculated numbers closely match with the experimentally obtained values with the r2 of 0.998 (Figure 1)Results for the other cut-off voltages (1.0 V/cell, 1.05 V/cell and 1.14 V/cell) were also obtained using the equation 1 and closely match the experimental values. Thus, the phase transition form of generalized Peukert’s equation (complementary error function) can be used for evaluating the capacities between 24-hour and 0.20-hour of discharge durations. Consequently, it enhances the test efficiency remarkably and sets the stage for battery sizing. Future studies include but not limited to the significance of n and ik, algebraic form of Peukert’s equation and Shepherd’s equation to understand the effectivity of each model with a focus on the discharge durations below 0.20-hour. Figure 1

ANALISIS KESALAHAN KONSEP MAHASISWA PADA PERKULIAHAN KONSEP DASAR MATEMATIKA

This study analyzes conceptual errors in the Basic Concepts of Mathematics course by STIT Al-Hilal Sigli students in the PGMI study program. The purpose of this study was to determine the mathematical concept errors that occurred during lectures. The samples in this study were students of the third semester of PGMI study program who were taking the basic concepts of mathematics course, totaling 73 students consisting of unit I, unit II, and unit III. Data collection techniques are carried out by monitoring each answer from exercises, quizzes, assignments, and student UTS answers. The data analysis technique was carried out by descriptive analysis of students' misconceptions about mathematical concepts. The results of the study found several conceptual errors that occurred, including: (1) conceptual errors in arithmetic operations involving negative numbers, (2) conceptual errors in fractional operations, (3) algebraic concept errors, especially those involving one or two variables, and (4 ) does not complete the units for area and length.

Algebraic Zero Error Training Method for Neural Networks Achieving Least Upper Bounds on Neurons and Layers

In the domain of artificial neural networks, it is important to know what their representation, classification and generalization capabilities are. There is also a need for time and resource-efficient training algorithms. Here, a new zero-error training method is derived for digital computers and single hidden layer networks. This method is the least upper bound on the number of hidden neurons as well. The bound states that if there are N input vectors expressed as rational numbers, a network having N − 1 neurons in the hidden layer and M neurons at the output represents a bounded function F: RD→RM for all input vectors. Such a network has massively shared weights calculated by 1 + M regular systems of linear equations. Compared to similar approaches, this new method achieves a theoretical least upper bound, is fast, robust, adapted to floating-point data, and uses few free parameters. This is documented by theoretical analyses and comparative tests. In theory, this method provides a new constructional proof of the least upper bound on the number of hidden neurons, extends the classes of supported activation functions, and relaxes conditions for mapping functions. Practically, it is a non-iterative zero-error training algorithm providing a minimum number of neurons and layers.

Computing Riemann–Roch spaces via Puiseux expansions

Computing large Riemann–Roch spaces for plane projective curves still constitutes a major algorithmic and practical challenge. Seminal applications concern the construction of arbitrarily large algebraic geometry error correcting codes over alphabets with bounded cardinality. Nowadays such codes are increasingly involved in new areas of computer science such as cryptographic protocols and “interactive oracle proofs”. In this paper, we design a new probabilistic algorithm of Las Vegas type for computing Riemann–Roch spaces of smooth divisors, in characteristic zero, and with expected complexity exponent 2.373 (a feasible exponent for linear algebra) in terms of the input size.

Perceptual lateralization on the Rod-and-Frame Test in young and older adults

Introduction There is an overall left visual field/right hemisphere advantage in young adults for masked, tachistoscopically presented images on the Rod-and-Frame Test (RFT). This study explored potential age-related lateralization differences in processing of visual context on the RFT. Methods The 35 young and 33 older adults aligned a rod surrounded either by no frame, a vertical, or leftward/rightward tilted frame to their perceived vertical. Algebraic errors of rod alignment were used to derive the rod-and-frame effect (RFE) and asymmetry index. Results Young adults had frequent indirect effects, mostly to the right-tilted frame, while older adults hardly produced any. Compared with nontilted frames, young adults displayed larger alignment errors with left-tilted frames; however, older adults exhibited this same effect for both frame tilt conditions. Young adults had smaller RFE values than older adults for the right-tilted frame, with no age-related difference in RFE for the left-tilted frame or asymmetry index. The negative asymmetry index was statistically different from the true vertical only in young adults. Conclusion There is an age-related reduction in the right hemisphere processing of left-sided visual contexts on the RFT. Such findings can assist clinicians to improve interpretation of RFT findings in clinical patients.