Vector Subtraction Research Articles

Digital Models and 3D Biomechanics Analysis in Orthodontics. Part 1: Vector Calculations

Biomechanics is essential for optimizing orthodontic appliances and controlling dental movement. Charles J. Burstone pioneered a three-dimensional (3D) approach in orthodontics, advocating for a shift beyond appliance-focused methods. Initially, biomechanics studies were constrained to two-dimensional (2D) analysis due to the complexities of 3D evaluation. Despite progress in computational tools and digital modeling, orthodontic biomechanics has largely maintained a 2D orientation. This paper advances orthodontic biomechanics into 3D, re-evaluating concepts previously limited to 2D frameworks. A dedicated software, DDP-Ortho (Ortolab, Poland), is introduced to enable orthodontists to analyze and resolve biomechanical challenges in 3D, facilitating appliance designs with precise 3D force systems. The representation and calculation of force vectors and moments in 3D are detailed, emphasizing the inherent complexity absent computational support. Key processes such as vector subtraction and addition, fundamental for assessing and refining orthodontic force systems, are explained. Additionally, the vector split (couple replacement) method, previously described in 2D, is extended to 3D, addressing the unique constraints and challenges of this approach. These tools promise to refine the accuracy and effectiveness of orthodontic treatments, setting the stage to examine the interactions between 3D force systems and dental movement, which will be addressed in a subsequent paper, to broaden the potential of contemporary orthodontic therapy.

Методика обработки сигналов, позволяющая уменьшить погрешности измерений бистатических характеристик рассеяния образцов материалов

In this paper a signal processing technique for bistatic measurements of the reflection coefficient of flat material samples is presented. This technique is based on comparing the reflected signal from the object under study and the background signal in the time domain. The use of this technique makes it possible to increase the accuracy of measurements of the characteristics of bistatic scattering of objects with a small amplitude of the reflected wave. This technique allows us to offset for the measurement error caused by the inaccuracy of antenna positioning and significantly reduce the measurement error caused by the interaction of antennas with each other at large bistatic angles. The reflection coefficient of the material sample was measured. The obtained results were processed by vector subtraction, time domain and using the developed technique. It was shown that the use of this technique allows one to increase the accuracy of measuring the reflectance of the material. This technique reduces the error in determining the minimum value of the reflection coefficient and the angle at which this minimum is achieved. The most effective is to use background synthesis and time domain together.

Subtraction games in more than one dimension

This paper concerns two-player alternating play combinatorial games (Conway 1976) in the normal-play convention, i.e. last move wins. Specifically, we study impartial vector subtraction games on tuples of nonnegative integers (Golomb 1966), with finite subtraction sets. In case of two move rulesets we find a complete solution, via a certain P-to-P principle (where P means that the previous player wins). Namely x∈P if and only if x+a+b∈P, where a and b are the two move options. Flammenkamp (1997) observed that, already in one dimension, rulesets with three moves can be hard to analyze, and still today his related conjecture remains open. Here, we solve instances of rulesets with three moves in two dimensions, and conjecture that they all have regular outcomes. Through several computer visualizations of outcomes of multi-move two-dimensional rulesets, we observe that they tend to partition the game board into periodic mosaics on very few regions/segments, which can depend on the number of moves in a ruleset. For example, we have found a five-move ruleset with an outcome segmentation into six semi-infinite slices. In this spirit, we develop a coloring automaton that generalizes the P-to-P principle. Given an initial set of colored positions, it quickly paints the P-positions in segments of the game board. Moreover, we prove that two-dimensional rulesets have row/column eventually periodic outcomes. We pose open problems on the generic hardness of two-dimensional rulesets; several regularity conjectures are provided, but we also conjecture that not all rulesets have regular outcomes.

Students’ understanding of vector operations: With and without physics education technology simulation

This study aims to investigate the influence of technology integration (physics education technology simulation) in physics learning. A quantitative method with a quasi-experimental post-test-only design was applied to reveal the differences in students’ abilities to operate vectors between the group that learned and tested using simulation and the group that learned using graphics only. The test consisted of 16 questions on the addition and subtraction of two vectors divided into one and two dimensions. Student response data were analyzed using descriptive statistics and Kruskal-Wallis test. The analysis results show that the vector operation score for experimental group 1, which learned vector operations through graphic learning using simulation and tested with the same simulation (86.15), is higher than experimental group 2, which learned vector operations through graphic learning using simulation (76.59), and the control group, which learned vector operations through graphic learning without simulation assistance (71.81). All three groups have higher average scores for one-dimensional than two-dimensional vector operations. They also have higher average scores for vector addition compared to vector subtraction operations. Kruskal-Wallis test results indicate a significant difference in the average vector operation scores for the three groups (ρ=0.021). This suggests that integrating technology in the learning and testing processes leads to a significant difference. Moreover, the difference is significant for one-dimensional vector operations (ρ=0.004) and not significant for two-dimensional vector operations (ρ=0.107). These findings could support the implementation of similar approaches at the university and high school levels, especially for vector-related topics, in both learning and testing processes.

Enhancing students’ understanding of vectors through personalized adaptive learning

In introductory college physics courses, it is important that students have a thorough understanding of vectors since several topics depend on vectors. When the test of understanding of vectors (TUV) was administered to the students completing the algebra-based College Physics I course, it was found that students had considerable difficulties in certain vector concepts. To overcome these difficulties, the question-bank for personalized adaptive learning (PAL) was modified. These modifications were implemented in a later semester for the students of the same course. In the current work, we present students’ performance on 12 questions on TUV for two semesters with a focus on improvement in students’ performance on graphic representation and determination of direction and magnitude of a vector presented in unit vector notation, understanding of graphical methods for vector addition, subtraction and multiplication of a vector by a negative scalar. This study presents the development of questions in PAL assignments which may be beneficial for the teaching community.

Science and engineering students’ difficulties in understanding vector concepts

The study assesses science and engineering students’ difficulties in understanding vector concepts. Test of understanding of vectors was administered to 101 university students who were completing their second introductory physics course. Rasch measurement software (Winsteps) was used to analyze the raw data. The concepts that the students found most difficult were identified; these included the graphical representation of a unit vector, the graphical addition and subtraction of vectors, and the interpretation of the dot and cross products, among others. On the other hand, some concepts were easier for students, such as the magnitude of a vector presented in unit-vector notation, and determination of the components of vectors. No significant differences in understanding of vector concepts could be attributed to gender; however, engineering students outperformed science students in understanding vector concepts. This study offers recommendations that physics instructors and researchers can use to improve the teaching of vector concepts.

Application of coordinate systems for vessel trajectory prediction improvement using a recurrent neural networks

According to the Global Maritime Insurance annual report, among human and non-human risk factors, the number of accidents in maritime transport remains a significant issue. One of the factors is vessel collisions and anomalies at sea. Massive historical data from automatic identification systems are analyzed, and intelligent transportation systems are being developed to solve the problem of vessel trajectory prediction. The most ordinary attempt to improve accuracy is by evaluating the historical vessel behavior and learning the patterns and similarities of the predicted vessel movements. However, this paper shows that a better forecast also may be reached by choosing a different trajectory calculation strategy. The geographical or polar coordinate system values are used in a classical way, but several modifications, such as Universal Transverse Mercator (UTM), have been proposed in this study as an alternative to Mercator’s projection coordinates. Two main positioning of the vessels motion transformations were tested: by changing the coordinates to measurements of the angular distance (haversine) and displacement angle (azimuth) functions between different time steps; degree coordinates transformation into a Cartesian system using UTM with vector subtraction. The last case improves the accuracy of almost 30% in the available data sample by using the autoencoder architecture, compared to the longitude and latitude predictions even with computed delta features. The research generally compared three recurrent network architectures (with their hyperparameter – cell sizes): Autoencoder Long Short-Term Memory, Bi-directional Long Short-Term Memory, and Gated Recurrent Unit networks. The model calculations are performed in a real historical dataset, exclusively on cargo vessel type trajectories in the Netherlands (North Sea) coastal region. Also, the methods were validated in another dataset of the Baltic Sea Region.

Evaluation of Ocular Residual Astigmatism in Eyes with Myopia and Myopic Astigmatism and Its Interaction with Other Forms of Astigmatism.

To evaluate the prevalence, magnitude, and direction of ocular residual astigmatism (ORA) in eyes with myopia and myopic astigmatism, and its interaction with refractive, anterior corneal, posterior corneal, and true net power astigmatism. Refractive surgery candidates with myopia and myopic astigmatism were studied. Refractive astigmatism (RA) was measured using the Nidek® AR-310A autorefractometer. Anterior corneal astigmatism (ACA), posterior corneal astigmatism (PCA), and true net power astigmatism (TNP) were measured using the Wavelight® Oculyzer II. Astigmatism was converted from polar to vector notation. ORA was calculated by vector subtraction of ACA from RA vertexed to corneal plane. Compensation factor (CF) was calculated as the ratio of ORA that compensates ACA for both J0 and J45. 154 eyes of 88 patients (mean age 31.7±7.1 years) were included. With-the-rule (WTR) astigmatism was the most common for both RA (55.6%) and ACA (74%), while against-the-rule (ATR) was the most common for PCA (87.7%) and ORA (74.0%). The axes of RA and ACA were within 10° of each other in 46.8% of the eyes, and within 30° of each other in 76.0%. The mean difference in value between the axis of RA and ACA was 25.6°. 71.4% of eyes in the study had an ORA ≥ 0.5D, 44.1% had ORA ≥ 0.75D and 26% had ORA ≥ 1D. There was a statistically significant difference between ACA and each of RA and TNP. Using TNP to calculate ORA instead of ACA reduced its magnitude. RA is positively correlated to ACA and more strongly to TNP. The most common pattern of compensation between ORA and ACA was under-compensation for J0 (49%) and same-axis-augmentation for J45 (35%). ORA, PCA, and the interaction between ORA and ACA can affect results during refractive planning.

Research on high-speed and high-precision laser processing algorithm based on macro-micro platform

Aiming at the problem that XY motion platform cannot realize high-speed and high-precision machining simultaneously, a high-speed and high-precision laser processing algorithm is proposed based on the macro-micro structure of laser cutter with a laser galvanometer scanner. The velocity planning of macro platform and composite motion is carried out by applying finite impulse response (FIR) filters three times, and velocity planning of micro platform is obtained by vector subtraction method. The motion decomposition of the macro-micro platform is realized accurately. The macro platform is responsible for large-format movement, and the micro platform is responsible for fast-speed turning. Velocity overlap of adjacent segments is employed to improve the machining efficiency. The constraints are established to determine the overlapping time, so that the algorithm can realize high-speed machining without introducing contour error, and ensure that the velocity, acceleration, and jerk of the macro platform can be processed within the limit range. Experiment and simulation analysis show that compared with using only the XY platform, the total machining time of using the macro-micro platform with the algorithm proposed in this paper is reduced by more than 35%.

Leftover Astigmatism: The Missing Link Between Measured and Calculated Posterior Corneal Astigmatism.

To quantify the total eye astigmatism that is not accounted for by measurement of anterior corneal astigmatism and posterior corneal astigmatism and knowledge of intraocular lens (IOL) astigmatism and assess whether it is correlated with candidate sources of or correlates with leftover astigmatism. Vector subtraction of anterior corneal, posterior corneal, and IOL astigmatism from total eye astigmatism as represented by spectacle astigmatism to yield a value of "leftover" astigmatism that is neither corneal nor lenticular. This value was derived in a series of eyes following cataract surgery. This novel entity was examined for correlation with candidate sources of or correlates with leftover astigmatism. In 103 pseudophakic eyes with known IOL toricity, mean leftover astigmatism was 0.71 ± 0.43 diopters. This was significantly correlated with against-the-rule anterior corneal astigmatism (P < .001). Leftover astigmatism is clinically substantial. Because it is included in IOL cylinder power calculations based on refractive outcome, it may explain why methods of IOL cylinder power calculation using refractive outcome-based adjustments to anterior corneal astigmatism (previously described as adjustments for "posterior corneal astigmatism") are more successful than adjustment on the basis of measured posterior corneal astigmatism. [J Refract Surg. 2022;38(9):559-564.].

In-Motion Coarse Alignment Method for SINS/GPS Integration in Polar Region

In this article, an in-motion coarse alignment method in the polar region is proposed. Different from the traditional coarse alignment method, the vector observation is constructed with the grid frame. Thus, the calculated singularity in the polar region, which is easy to see in the navigational frame, is suppressed. Considering the GPS measurements, the velocity errors are assumed in the GPS measurements. Since the characteristics of the observation vectors, the initial velocity error always exists during the whole alignment process. To address the initial velocity error, a simple method with the vector subtraction operation is used to construct the new observation vector. In the new observation vector, the initial velocity is eliminated. Thus, the initial velocity error is suppressed. To verify the performance of the proposed method, a virtual polar region method is proposed to transform the experimental data of the field tests in the low and middle latitude region. The simulation and field tests are shown that the proposed method can get more accurate alignment results than the traditional methods.

Battle Royale Optimizer for solving binary optimization problems

<h2>Abstract</h2> Battle Royale Optimizer (BRO) is a recently proposed metaheuristic optimization algorithm used only in continuous problem spaces. The BinBRO is a binary version of BRO. The BinBRO algorithm employs a differential expression, which utilizes a dissimilarity measure between binary vectors instead of a vector subtraction operator, used in the original BRO algorithm to find the nearest neighbor. To evaluate BinBRO, we applied it to two popular benchmark datasets: the uncapacitated facility location problem (UFLP) and the maximum-cut (Max-Cut) graph problems from OR-Library. An open-source MATLAB implementation of BinBRO is available on CodeOcean and GitHub websites.

An Improved Initial Alignment Method for SINS/GPS Integration With Vectors Subtraction

SINS/GPS integrated navigation system has been used widely in many application situations. In the SINS/GPS integrated navigation system, the initial alignment process is carried out to determine the initial attitude of the vehicle. Based on the previous works, the position loci method is a robust initial alignment method. The outliers, which are contained in the GPS measurements, can be suppressed by the position loci. However, the initial velocity is used to construct the observation vectors in the previous works. When the errors are contained in the initial velocity measurements, the initial alignment results are bad. To address this problem, the vector subtraction operation is adopted in this paper. Then, the initial velocity is offset by this operation. Thus, the interferences, which are caused by the initial velocity errors, are suppressed. The simulation and field tests are designed to verify the performance of the proposed method. It can be found that the alignment results are similar with the methods, which contains small or no initial velocity error.

Quantum Pythagorean Fuzzy Evidence Theory: A Negation of Quantum Mass Function View

Dempster–Shafer (D-S) evidence theory is an effective methodology to handle unknown and imprecise information because it can assign probability into the power set. However, the process of obtaining information is a complex task, which can consider the rational, conscious, objective evaluation of utility with behavioral effects. Besides, in most cases, information can be obtained from different angles at the same time. The quantum model of mass function (QM) uses amplitude and phase angle to easily express those properties of information that can extend D-S evidence theory to the unit circle in a complex plane. Moreover, everything in nature will have its opposite, which is a kind of universality. The Bayes theorem is essentially the process of negation. However, in most cases, decisions can be made by only fully considering the known information without considering the other side of the information. Hence, considering the negation of information is a question to be investigated deeply, which can analyze information from the other point. This article proposes negation of QM by using the subtraction of vectors in the unit circle, which can degenerate into negation proposed by Yager in standard probability theory and negation proposed by Yin <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in D-S evidence theory. Negation can provide us more information to consider the problem from both positive and negative aspects. In this article, negation can be understood information, which does not belong to event <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$A$</tex-math></inline-formula> , that is to say, negation can be regarded as nonmembership by using the fuzzy terms. Based on the above discussion, this article proposes the quantum pythagorean fuzzy evidence theory (QPFET), which is the novel work to consider QPFET from the point of negation. Besides, there are some numerical examples to explain the proposed method. In order to explore the applications of QPFET, this article discusses the possibility of the VI <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\check{s}$</tex-math></inline-formula> ekriterijumsko Kompromisno Rangiranje method under QPFET to handle multicriteria decision-making that enables us to capture 2-D data, considering not only amplitude but also phase angle.

A novel fast SVPWM method based on two-stage transformation for three-level inverter

Compared with the two-level inverter, the three-level inverter has excellent performance in pressure voltage switching loss and harmonic applied to the fields of the speed regulation and new energy generation. However, the computational complexity of the space vector pulse width modulation (SVPWM) method for multilevel inverter restricts the implementation on low cost microprocessor. This paper proposes a novel fast SVPWM method for three-level inverter to convert the space voltage vector in the three-level space to the two-level by vector subtraction with the central vector. The conduction time of each switch can be computed quickly so that the runtime is reduced significantly. Finally, the experimental results verify the effectiveness and practicability of the new space vector modulation algorithm based on two-stage transformation.

A Cross-Media Deep Relationship Classification Method Using Discrimination Information

Relation classification is one of the most fundamental tasks in the area of cross-media, which is essential for many practical applications such as information extraction, question&answer system, and knowledge base construction. In the cross-media semantic retrieval task, in order to meet the needs of cross-media uniform representation and semantic analysis, it is necessary to analyze the semantic potential relationship and construct semantic-related cross-media knowledge graph. The relationship classification technology is an important part of solving semantic correlation classification. Most of existing methods regard relation classification as a multi-classification task, without considering the correlation between different relationships. However, two relationships in the opposite directions are usually not independent of each other. Hence, this kind of relationships are easily confused in the traditional way. In order to solve the problem of confusing the relationships of the same semantic with different entity directions, this paper proposes a neural network fusing discrimination information for relation classification. In the proposed model, discrimination information is used to distinguish the relationship of the same semantic with different entity directions, the direction of entity in space is transformed into the direction of vector in mathematics by the method of entity vector subtraction, and the result of entity vector subtraction is used as discrimination information. The model consists of three modules: sentence representation module, relation discrimination module and discrimination fusion module. Moreover, two fusion methods are used for feature fusion. One is a Cascade-based feature fusion method, and another is a feature fusion method based on convolution neural network. In addition, this paper uses the new function added by cross-entropy function and deformed Max-Margin function as the loss function of the model. The experimental results show that the proposed discriminant feature is effective in distinguishing confusing relationships, and the proposed loss function can improve the performance of the model to a certain extent. Finally, the proposed model achieves 84.8% of the F1 value without any additional features or NLP analysis tools. Hence, the proposed method has a promising prospect of being incorporated in various cross-media systems.

Spin and charge transports with thermodynamic electron–hole correlation in nearly compensated metals

The long spin-relaxation time (τs) toward the large transport distance of spin current (SC) is desirable for practical applications of spintronic devices because spatially and temporally large spin coherence makes spin manipulation easier. Here, we propose that nearly compensated metals (CMs) with electron–hole collisions accompanied by spin flipping have a potential mechanism that makes τs associated with the antiparallel SC extraordinarily long, where antiparallel SC is defined as the vector subtraction of hole and electron SCs. The present theoretical study on spin and charge transports in nearly CMs shows that (i) it is antiparallel SC that satisfies the Onsager reciprocal relation in combination with conventional charge current, (ii) both longitudinal and Hall resistivities are influenced by the enhancement of τs and are classified into two specific mechanisms, i.e., the conventional and the resonance Hall effect accompanied by a sustaining mode of antiparallel SC, and (iii) the resonance Hall effect is not influenced by the spin–orbit interaction unlike the non-resonance case. These findings indicate the potential of nearly CMs in application of spintronic materials.

Higher Order Compact Schemes for the Spatial Discretization of Linear Parabolic Partial Differential Equations

A system of compact schemes used, to approximate the partial derivative of Linear Parabolic Partial Differential Equations (LPPDE), on the non-boundary nodes. Euler time integration for the temporal derivative, the Crank-Nicholson (CN)scheme for the spatial derivatives and the source term are used. The higher order spatial accuracy of the developed, central difference based compact schemes for the one and two-dimensional diffusion equations are validated, by solving numerically two test problems. The compact scheme based calculations involve a tridiagonal matrix vector multiplication and a vector vector subtraction. The interest is to demonstrate the higher order spatial accuracy and the better rate of convergence of the solution produced using the developed compact 4th order scheme, when compared with the same produced, using the conventional 2nd order scheme. The assessment is made, in terms of the discrete or norm of the true error of the converged numerical solution.

Predefined-Time Polynomial-Function-Based Synchronization of Chaotic Systems via a Novel Sliding Mode Control

In the context of chaotic secure communication, this paper is concerned with the predefined-time polynomial-function-based synchronization of chaotic systems via sliding mode control. Firstly, a novel hybrid synchronization scheme among multiple chaotic systems based on polynomial function is defined. Subsequently, based on a new predefined-time stability criterion, a novel multi-power integral terminal sliding mode control algorithm is designed to realize the predefined-time polynomial-function-based synchronization. Finally, the secure communication simulation is presented to verify the feasibility and efficiency of the proposed synchronization scheme. The polynomial-function-based synchronization not only uses the addition and subtraction of vectors, but also uses the power multiplication of vectors, which makes the nonlinear structure of the composed drive system more complex, so that the communication scheme is more secure. Applying the sliding mode control algorithm designed in this work, the synchronization time can be preset off-line without estimation, moreover, the ratio between the formation time and the convergence time of sliding mode can also be distributed in advance, which is more flexible.

Examination of Thai freshmen’s understanding on vectors using a model analysis technique

This study focused on an investigation of Thai students’ comprehension of vectors using the model analysis technique, which is based on the matrix representation of quantum physics. This technique displays students’ knowledge states and the probability of applying each mental model to solve a given vector concept. A well-known vector concept test in physics education research named the Test of Understanding of Vectors (TUV) was translated into Thai language and validated by a group of physics lecturers. It was administered to 651 first-year engineering students at Prince of Songkla University via the online system at the beginning of the first semester. The three common mental states of vectors were identified as correct, popular incorrect, and other models. Model estimation of the model analysis revealed that although 54% of the students correctly answered vector addition, they still had difficulty with vector subtraction. Theoretically, both topics are considered as identical concepts. The students had inconsistent ideas on vector subtraction as reported by eigenvalues < 0.65. Moreover, they demonstrated equal distribution in using the three mental models to solve the dot product of vectors. The most common incorrect idea was that a dot product resultant was a vector quantity. The initial knowledge states of the cross product were in an incorrect model region involving confusion between calculating magnitudes of the cross product and the dot product, and incorrectly applying the right-hand rule to locate a direction of the cross product. Overall, the findings will guide instructors which common vector ideas should be revised and how. The vector concept plays an important role in physics learning since it is embedded in several physics topics as a learning mechanism.