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QR decomposition is widely used for solving the least squares problem. However, existing materials about it may be too abstract for non-mathematicians, especially STEM students, and/or require serious background in linear algebra. The paper describes theoretical background and examples of GNU Octave compatible MATLAB scripts that give relatively simple but complete explanations about how to use QR decomposition for the least squares problem solution. Only basic knowledge of linear algebra and calculus are required. Both Givens rotations and Householder reflections usage for the linear least squares problem were considered. It was shown that the algorithm based on Givens rotations is even easier to program than explicit formation of the normal equations with subsequent usage of Gaussian elimination.

Recent rank-based attacks have reduced the security of Rainbow, which is one of the multi-layer UOV signatures, below the NIST security requirements by speeding up iterative kernel-finding operations using classical mathematics techniques. If quantum algorithms are applied to perform these iterative operations, the rank-based attacks may be more threatening to multi-layer UOV, including Rainbow. In this paper, we propose a quantum rectangular MinRank attack called the Q-rMinRank attack, the first quantum approach to key recovery attacks on multi-layer UOV signatures. Our attack is a general model applicable to multi-layer UOV signature schemes, and in this paper, we provide examples of its application to Rainbow and the Korean TTA standard, HiMQ. We design two quantum oracle circuits to find the kernel in consideration of the depth-width trade-off of quantum circuits. One is to reduce the width of the quantum circuits using qubits as a minimum, and the other is to reduce the depth using parallelization instead of using a lot of qubits. By designing quantum circuits to find kernels with fewer quantum resources and complexity by adding mathematical techniques, we achieve quadratic speedup for the MinRank attack to recover the private keys of multi-layer UOV signatures. We also estimate quantum resources for the designed quantum circuits and analyze quantum complexity based on them. The width-optimized circuit recovers the private keys of Rainbow parameter set V with only 1089 logical qubits. The depth-optimized circuit recovers the private keys of Rainbow parameter set V with a quantum complexity of 2 174 , which is lower than the complexity of 2 221 recovering the secret key of AES-192, which provides the same security level as parameter set III. [ABSTRACT FROM AUTHOR]

Purpose. As a scientific field, bioinformatics has drawn remarkable attention from various fields, such as information technology, mathematics, and modern biological sciences, in recent years. The topic models originating from the field of natural language processing have become the focus of attention with the rapid accumulation of biological datasets. Thus, this research is aimed at modeling the topic content of the bioinformatics literature presented by Iranian researchers in the Scopus Citation Database. Methodology. This research was a descriptive-exploratory study, and the studied population included 3899 papers indexed in the Scopus database, which had been indexed in this database until March 9, 2022. The topic modeling was then performed on the abstracts and titles of the papers. A combination of LDA and TF-IDF was utilized for topic modeling. Findings. The data analysis with topic modeling resulted in identifying seven main topics "Molecular Modeling," "Gene Expression," "Biomarker," "Coronavirus," "Immunoinformatics," "Cancer Bioinformatics," and "Systems Biology." Moreover, "Systems Biology" and "Coronavirus" had the largest and smallest clusters, respectively. Conclusion. The present investigation demonstrated an acceptable performance for the LDA algorithm in classifying the topics included in this field. The extracted topic clusters indicated excellent consistency and topic connection with each other. [ABSTRACT FROM AUTHOR]

This research aims to improve the rationality and intelligence of AUTOMATICALLY HIGHER MATHEMATICALLY EXAM SYSTEM (AHMES) through some AI algorithms. AHMES is an intelligent and high-quality higher math examination solution for the Department of Computer Engineering at Pai Chai University. This research redesigned the difficulty system of AHMES and used some AI algorithms for initialization and continuous adjustment. This paper describes the multiple linear regression algorithm involved in this research and the AHMES learning (AL) algorithm improved by the Q-learning algorithm. The simulation test results of the upgraded AHMES show the effectiveness of these algorithms. [ABSTRACT FROM AUTHOR]

This paper presents a modular approach to motion planning with provable stability guarantees for robots that move through changing environments via periodic locomotion behaviors. We focus on dynamic walkers as a paradigm for such systems, although the tools developed in this paper can be used to support general compositional approaches to robot motion planning with dynamic movement primitives (DMPs). By formulating the planning process as a switching system with multiple equilibria (SSME), we prove that the system's evolution remains within explicitly characterized trapping regions in the state space under suitable constraints on the frequency of switching among the DMPs. These conditions encapsulate the low-level stability limitations in a form that can be easily communicated to the planner. Furthermore, we show how the available primitives can be safely composed online in a receding horizon manner to enable the robot to react to moving obstacles. The proposed framework can be applied in a wide class of 3D bipedal walking models, and offers a modular approach for integrating readily available low-level locomotion control and high-level planning methods. [ABSTRACT FROM AUTHOR]

This paper investigates numerical solutions of pseudomontone variational inequality problems in real Hilbert spaces. Two new algorithms based on the celebrated extragradient method are introduced. Under suitable conditions, it is proved that the two algorithms achieve weak convergence and linear convergence rates. The results extend and improve upon related works in the literature on the associated extragradient algorithms. [ABSTRACT FROM AUTHOR]

The concept of contraction mappings plays a significant role in mathematics, particularly in the study of fixed points and the existence of solutions for various equations. In this study, we described two types of enriched contractions: enriched F-contraction and enriched F'-contraction associated with u-fold averaged mapping, which are involved with Kirk's iterative technique with order u. The contractions extracted from this paper generalized and unified many previously common super contractions. Furthermore, u-fold averaged mappings can be seen as a more general form of both averaged mappings and double averaged mappings. Moreover, we demonstrated that the ufold averaged mapping with enriched contractions has a unique fixed point. Our work examined the necessary conditions for the u-fold averaged mapping and weak enriched contractions to have equal sets of fixed points. Additionally, we illustrated that an appropriate Kirk's iterative algorithm can effectively approximate a fixed point of a u-fold averaged mapping as well as the two enriched contractions. Also, we delved into the well-posedness, limit shadowing property, and Ulam-Hyers stability of the u-fold averaged mapping. Furthermore, we established necessary conditions that guaranteed the periodic point property for each of the illustrated strengthened contractions. To underscore the generality of our findings, we presented several examples that aligned with comparable results found in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, an operational matrix method based on the Bell polynomials has been presented to find approximate solutions of highorder Volterra integro-difierential equations. This method uses a simple computational manner to obtain a quite acceptable approximate solution. The main characteristic behind this method lies in the fact that on the one hand, the problem will be reduced to a system of algebraic equations and on the other hand, the effciency and accuracy of the Bell polynomials for solving these equations are acceptable. The convergence analysis of this method will be shown by preparing some theorems. Moreover, we will obtain an estimation of the error bound for this algorithm. Finally, some examples are presented to illustrate the applicability, eficiency and accuracy of this scheme in comparison with some other well-known methods such as Legendre, Bernoulli, Taylor and Bessel polynomial algorithms. [ABSTRACT FROM AUTHOR]

This article investigates subject-related research metrics from the most popular scientometrics platforms with the capability of citation count. Quantitative analyses have been carried out to determine the most- to the least-cited subject areas to indicate the most active and dynamic fields in research. Subject areas are classified based on statistical analyses and the number of citations received in the period of search. Numerous articles were analyzed from top-ranked journals in all research fields covered by SCImago journal classification. A comprehensive data set was generated in 27 research subject areas, including 313 subject categories and 81 high-ranked journals. Results showed that the highest number of citations were related to the sciences and science-related subject areas. The subject areas of computer science and mathematics had the lowest research metrics and the smallest number of highly cited publications. Web of Science performed better than Scopus for subject-related evaluations. Analysis of frequency data showed that Google Scholar cited fewer papers for low-cited publications than did Scopus or Web of Science. For highly cited publications, however, Google Scholar was found to have better performance. Web of Science showed the best consistency in citation coverage in most of the investigated subject areas. [ABSTRACT FROM AUTHOR]

The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter C st , 1 that is found not robust as the polynomial degree p increases. This is related to the H 1 stability bound of the L 2 projection onto polynomials of degree at most p and its growth C st, 1 ∝ (p + 1) 1 / 2 as p → ∞ . A similar estimate for the Galerkin projection holds with a p-robust constant C st , 2 and C st , 2 ≤ 2 for right-isosceles triangles. This paper utilizes the new inequality with the constant C st , 2 to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a p-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved L 2 error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of finding a solution of a pseudomonotone variational inequality problem with the constraints of fixed points of a finite family of demicontractive multivalued mappings. We introduce a new generalized viscosity inertial Tseng's extragradient method which uses self-adaptive step sizes. Unlike some existing results in this direction, we prove our strong convergence theorems without the sequentially weakly continuity condition of the pseudomonotone operator and without the knowledge of Lipschitz constants. Moreover, our strong convergence results do not follow the conventional "two cases" approach, which was often employed in proving strong convergence. Finally, we apply our result to convex minimization problems and present several numerical experiments to illustrate the performance of the proposed algorithms in comparison with other existing methods in the literature. [ABSTRACT FROM AUTHOR]

A given special case of NP-complete scheduling problem can be approximated by solving a special case of similar problem with the same precedence graph. We construct a metric space over a set of special cases of this problem and consider the statistical relationship between distance between a pair of special cases of the under consideration and the average error of the approximated solution. Sethi, Gabow, Coffman's and Fujii's algorithms for this problem are used. It is shown that the absolute and the relative error of the objective function decreases over the density of a graph with a fixed number of jobs. In general case, relative non-zero error value increases with the number of jobs. [ABSTRACT FROM AUTHOR]

In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality constrained optimization when a nearly feasible point is known. In particular, we first propose a new Newton-conjugate gradient (Newton-CG) method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a substantially better complexity than the Newton-CG method in [C. W. Royer, M. O'Neill, and S. J. Wright, Math. Program., 180 (2020), pp. 451-488]. We then propose a Newton-CG based augmented Lagrangian (AL) method for finding an approximate SOSP of nonconvex equality constrained optimization, in which the proposed Newton-CG method is used as a subproblem solver. We show that under a generalized linear independence constraint qualification (GLICQ), our AL method enjoys a total inner iteration complexity of O(ε7/peration complexity of O(ε7/2 min\{ n, ε3/4\}) for finding an (ε, ε)-SOSP of nonconvex equality constrained optimization with high probability, which are significantly better than the ones achieved by the proximal AL method in [Y. Xie and S. J. Wright, J. Sci. Comput., 86 (2021), pp. 1-30]. In addition, we show that it has a total inner iteration complexity of O(ε11/2) and an operation complexity of O(ε11/2 min\{ n, ε5/4\}) when the GLICQ does not hold. To the best of our knowledge, all the complexity results obtained in this paper are new for finding an approximate SOSP of nonconvex equality constrained optimization with high probability. Preliminary numerical results also demonstrate the superiority of our proposed methods over the other competing algorithms. [ABSTRACT FROM AUTHOR]

[M. Ma, The connectivity of exchanged hypercubes, Discrete Math., Algorithms Appl. 2(2) (2010) 213–220.] proved that the vertex connectivity and the edge connectivity of the exchanged hypercube are each equal to the minimum degree of the graph. This paper presents a proof of that result, which is substantially simpler and more intuitive. [ABSTRACT FROM AUTHOR]

Rational functions have deep system-theoretic significance. They represent the natural way of modeling linear dynamical systems in the frequency (Laplace) domain. Using rational functions, the goal of this paper to compute models that match (interpolate) given data sets of measurements. In this paper, the authors show that using special rational orthonormal systems, the Malmquist-Takenaka systems, it is possible to write the rational interpolant r(n,m), for n=N-1,m=N using only N sampling nodes (instead of 2N nodes) if the interpolating nodes are in the complex unit circle or on the upper half-plane. Moreover, the authors prove convergence results related to the rational interpolant. They give an efficient algorithm for the determination of the rational interpolant. [ABSTRACT FROM AUTHOR]

In this paper, a reliable new scheme is presented based on combining Reproducing Kernel Method (RKM) with a practical technique for the nonlinear problem to solve the System of Singularly Perturbed Boundary Value Problems (SSPBVP). The Gram-Schmidt orthogonalization process is removed in the present RKM. However, we provide error estimation for the approximate solution and its derivative. Based on the present algorithm in this paper, can also solve linear problem. Several numerical examples demonstrate that the present algorithm does have higher precision. [ABSTRACT FROM AUTHOR]

In the literature, besides the assumption of strict complementarity, superlinear convergence of implementable polynomial-time interior point algorithms using known search directions, namely, the HKM direction, its dual or the NT direction, to solve semi-definite programs (SDPs) is shown by (i) assuming that the given SDP is nondegenerate and making modifications to these algorithms [Kojima et al. Local convergence of predictor–corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Math. Program. Ser. A 80 (1998), pp. 129–160], or (ii) considering special classes of SDPs, such as the class of linear semi-definite feasibility problems (LSDFPs) and requiring the initial iterate to the algorithm to satisfy certain conditions [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621]. Otherwise, these algorithms are not easy to implement even though they are shown to have polynomial iteration complexities and superlinear convergence [Luo et al. Superlinear convergence of a symmetric primal–dual path following algorithm for semidefinite programming, SIAM J. Optim. 8 (1998), pp. 59–81]. The conditions in [Sim, Superlinear convergence of an infeasible predictor–corrector path-following interior point algorithm for a semidefinite linear complementarity problem using the Helmberg–Kojima–Monteiro direction, SIAM J. Optim. 21 (2011), pp. 102–126], [Sim, Interior point method on semi-definite linear complementarity problems using the Nesterov-Todd (NT) search direction: Polynomial complexity and local convergence, Comput. Optim. Appl. 74 (2019), pp. 583–621] that the initial iterate to the algorithm is required to satisfy to have superlinear convergence when solving LSDFPs however are not practical. In this paper, we propose a practical initial iterate to an implementable infeasible interior point algorithm that guarantees superlinear convergence when the algorithm is used to solve the homogeneous feasibility model of an LSDFP. [ABSTRACT FROM AUTHOR]

Being inspired by the biological eye, event camera is a novel asynchronous technology that poses a paradigm shift in acquisition of visual information. This paradigm enables event cameras to capture pixel-size fast motions much more naturally compared to classical cameras.In this paper, we present a new asynchronous event-driven algorithm for detection of high-frequency pixel-size periodic signals using an event camera. Development of such new algorithms to efficiently process the asynchronous information of event cameras is essential to utilize its special properties and potential, and being a main challenge in the research community.It turns out that this algorithm, which was developed in order to satisfy the new paradigm, is related to an untreated theoretical problem in probability: Let 0 ≤ τ1 ≤ τ2 ≤⋯ ≤ τm ≤ 1 originated from an unknown distribution. Let also 휖,δ ∈ ℝ, and d ∈ ℕ. What can be said about the probability Φ(m,d) of having more than d adjacent τi-s pairs that the distance between them is δ, up to an error 휖? This problem, which reminds the area of order statistic, shows how the new visualization paradigm is also an opportunity to develop new areas and problems in mathematics. [ABSTRACT FROM AUTHOR]

To improve the convergence of the gradient iterative (GI) algorithm and the Jacobi‐gradient iterative (JGI) algorithm [Bayoumi, Appl Math Inf Sci, 2021], a shift‐splitting Jacobi‐gradient iterative (SSJGI) algorithm for solving the matrix equation A풱−풱‾B=C is presented in this paper, which is based on the splitting of the coefficient matrices. The proposed algorithm converges to the exact solution for any initial value with some conditions. To demonstrate the effectiveness of the SSJGI algorithm and to compare it to the GI algorithm and the JGI algorithm [Bayoumi, Appl Math Inf Sci, 2021], numerical examples are provided. [ABSTRACT FROM AUTHOR]

Simple Summary: Cancer is basically a tough condition on a patient's body where cell grows uncontrollably. Normal cells are affected, which destroys the health of the patient. The main problem in cancer is spreading from one part to another. Therefore, the mathematical modeling of cancerous tumors integrates to check overall stability. A novel approach is introduced such as Bernstein polynomial with combination of genetic algorithm, sliding mode controller, and synergetic control. The proposed solution has easily eliminated cancerous cells within five days using synergetic control. In addition, five cases are incorporated to evaluate error function. In addition, a brief comparative study is added to contrast the simulation results with theoretical modeling. Cancerous tumor cells divide uncontrollably, which results in either tumor or harm to the immune system of the body. Due to the destructive effects of chemotherapy, optimal medications are needed. Therefore, possible treatment methods should be controlled to maintain the constant/continuous dose for affecting the spreading of cancerous tumor cells. Rapid growth of cells is classified into primary and secondary types. In giving a proper response, the immune system plays an important role. This is considered a natural process while fighting against tumors. In recent days, achieving a better method to treat tumors is the prime focus of researchers. Mathematical modeling of tumors uses combined immune, vaccine, and chemotherapies to check performance stability. In this research paper, mathematical modeling is utilized with reference to cancerous tumor growth, the immune system, and normal cells, which are directly affected by the process of chemotherapy. This paper presents novel techniques, which include Bernstein polynomial (BSP) with genetic algorithm (GA), sliding mode controller (SMC), and synergetic control (SC), for giving a possible solution to the cancerous tumor cells (CCs) model. Through GA, random population is generated to evaluate fitness. SMC is used for the continuous exponential dose of chemotherapy to reduce CCs in about forty-five days. In addition, error function consists of five cases that include normal cells (NCs), immune cells (ICs), CCs, and chemotherapy. Furthermore, the drug control process is explained in all the cases. In simulation results, utilizing SC has completely eliminated CCs in nearly five days. The proposed approach reduces CCs as early as possible. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the rank of a block-structured symbolic matrix (a generic partitioned matrix) A = (A α β x α β) , where A α β is a 2 × 2 matrix over a field F and x α β is an indeterminate for α = 1 , 2 , ⋯ , μ and β = 1 , 2 , ⋯ , ν . This problem can be viewed as an algebraic generalization of the bipartite matching problem and was considered by Iwata and Murota (SIAM J Matrix Anal Appl 16(3):719–734, 1995). Recent interests in this problem lie in the connection with non-commutative Edmonds' problem by Ivanyos et al. (Comput Complex 27:561–593, 2018) and Garg et al. (Found. Comput. Math. 20:223–290, 2020), where a result by Iwata and Murota implicitly states that the rank and non-commutative rank (nc-rank) are the same for this class of symbolic matrices. The main result of this paper is a simple and combinatorial O ((μ ν) 2 min { μ , ν }) -time algorithm for computing the symbolic rank of a (2 × 2) -type generic partitioned matrix of size 2 μ × 2 ν . Our algorithm is inspired by the Wong sequence algorithm by Ivanyos et al. for the nc-rank of a general symbolic matrix, and requires no blow-up operation, no field extension, and no additional care for bounding the bit-size. Moreover it naturally provides a maximum rank completion of A for an arbitrary field F . [ABSTRACT FROM AUTHOR]

With the rapid development of artificial intelligence (AI) concept technology, it promotes the innovation of educational concept. Mostly for the education information analysis in the class of mathematics in the university, it should be based on a big data-driven system to promote the quality of teaching in the classroom. In the method of teaching math in university, teachers should take full advantage of the benefits of a big data-driven system powered by AI, grow a good teaching model for students, promote education through big data, progressively teach students according to their aptitude, develop in a tailored direction, increase teaching quality and effectiveness, and finally create more great talents for our country. For the sake of improving the resource sharing and the management level of the curriculum which teaches advanced knowledge about mathematics teaching, based on a particle swarm optimization algorithm, an advanced math teaching system is proposed in this paper. The fusion model that can be used in the teaching process of math in university is constructed, the adaptive scheduling of the curriculum which teaches advanced knowledge about mathematics auxiliary teaching resources is realized by an optimization algorithm used for fusion particle swarm, the autocorrelation feature of the curriculum which teaches advanced knowledge about mathematics auxiliary teaching resources is extracted, and the adaptive optimization of the curriculum which teaches advanced knowledge about mathematics auxiliary teaching resource fusion is realized by fuzzy correlation feature matching and statistical analysis. In the process of particle swarm optimization, the combination of statistical features is studied and managed, the resource scheduling and information fusion are realized, and the management capability of the curriculum which teaches advanced knowledge about mathematics auxiliary teaching is promoted, and the experimental results demonstrate that the designed system has a high integration of teaching information resources and strong information scheduling ability and improved the management level of the curriculum which teaches advanced knowledge about mathematics complementary teaching. [ABSTRACT FROM AUTHOR]

In the early years of school, pupils have a very positive approach towards any new topic. Very often, the first contact with mathematics in the Italian school risks giving a limited and somehow negative impression, generating long-lasting effects. In this paper, we report on a series of experiences carried out in a second-year class of a primary school that allowed us to introduce new concepts and approaches through games and puzzles. The aim was to stimulate the creativity and empower pupils. We have used simple materials and applied the Puzzle-Based Learning method. Despite the limited time devoted to this experience, the outcomes have been extremely positive. [ABSTRACT FROM AUTHOR]

There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we study a discrete momentum consensus-based optimization (Momentum-CBO) algorithm which corresponds to a second-order generalization of the discrete first-order CBO [S.-Y. Ha, S. Jin and D. Kim, Convergence of a first-order consensus-based global optimization algorithm, Math. Models Methods Appl. Sci. 30 (2020) 2417–2444]. The proposed algorithm can be understood as the modification of ADAM-CBO, replacing the normalization term by unity. For the proposed Momentum-CBO, we provide a sufficient framework which guarantees the convergence of algorithm toward a global minimum of the objective function. Moreover, we present several experimental results showing that Momentum-CBO has an improved success rate of finding the global minimum compared to vanilla-CBO and show the stability of Momentum-CBO under different initialization schemes. We also show that Momentum-CBO can be used as the alternative of ADAM-CBO which does not have a proper convergence analysis. Finally, we give an application of Momentum-CBO for Lyapunov function approximation using symbolic regression techniques. [ABSTRACT FROM AUTHOR]

Background: Readability metrics provide us with an objective and efficient way to assess the quality of educational texts. We can use the readability measures for finding assessment items that are difficult to read for a given grade level. Hard‐to‐read math word problems can put some students at a disadvantage if they are behind in their literacy learning. Despite their math abilities, these students can perform poorly on difficult‐to‐read word problems because of their poor reading skills. Less readable math tests can create equity issues for students who are relatively new to the language of assessment. Less readable test items can also affect the assessment's construct validity by partially measuring reading comprehension. Objectives: This study shows how large language models help us improve the readability of math assessment items. Methods: We analysed 250 test items from grades 3 to 5 of EngageNY, an open‐source curriculum. We used the GPT‐3 AI system to simplify the text of these math word problems. We used text prompts and the few‐shot learning method for the simplification task. Results and Conclusions: On average, GPT‐3 AI produced output passages that showed improvements in readability metrics, but the outputs had a large amount of noise and were often unrelated to the input. We used thresholds over text similarity metrics and changes in readability measures to filter out the noise. We found meaningful simplifications that can be given to item authors as suggestions for improvement. Takeaways: GPT‐3 AI is capable of simplifying hard‐to‐read math word problems. The model generates noisy simplifications using text prompts or few‐shot learning methods. The noise can be filtered using text similarity and readability measures. The meaningful simplifications AI produces are sound but not ready to be used as a direct replacement for the original items. To improve test quality, simplifications can be suggested to item authors at the time of digital question authoring. Lay Description: What is Known About the Subject: Difficult to read math assessment items cause measurement and equity issuesReadability of math word problems negatively correlated with outcomes What Our Paper Adds: GPT‐3 AI system is capable to simplify math word problemsPrompt‐based and few‐shot learning‐based approaches are able to create meaningful simplifications, but with a very low accuracy rateText similarity and readability measures can be used to filter out noisy outputs and discover interesting simplifications What are the Implications for Practitioners: GPT‐3 AI is capable of generating relevant math world problem simplificationsSimplified text can be used as suggestions to question authors during the digital item authoring process [ABSTRACT FROM AUTHOR]

Let a,b ∈ Rn be column vectors, and (u, v) be the inner product of vectors u and v on Rn. Let G ⊂ GL(n,R) be a compact matrix group. For A ∈ G and a continue function f on G, the integral ∫G f (A)dA is the invariant integral of the compact group G. In this paper, we study the inequality ∀x ∈ Rn ∫G e(Aa, x)dA ≥ ∫G e(Ab, x)dA. We prove that the above inequality holds if and only if b ∈Conv(Ga) . This work follows a series of results, that is, Muirhead (1903), Hardy, Littlewood and P`olya (1932), Rado (1952), Daykin (1971), Kimelfeld (1995) and Schulman (2009). Furthermore, We construct an determining algorithm when G is finite. Compared with other effective algorithms, this one is symbolic and easy to implement on computer. [ABSTRACT FROM AUTHOR]

In this paper, fast numerical methods for solving the real quasi-symmetric Toeplitz linear system are studied in two stages. First, based on an order-reduction algorithm and the factorization of Toeplitz matrix inversion, a sequence of linear systems with a constant symmetric Toeplitz matrix are solved. Second, two new fast algorithms are employed to solve the real quasi-symmetric Toeplitz linear system. Furthermore, we show a fast algorithm for quasi-symmetric Toeplitz matrix-vector multiplication. In addition, the stability analysis of the splitting symmetric Toeplitz inversion is discussed. In mathematical or engineering problems, the proposed algorithms are extraordinarily effective for solving a sequence of linear systems with a constant symmetric Toeplitz matrix. Fast matrix-vector multiplication and a quasi-symmetric Toeplitz linear solver are proven to be suitable for image encryption and decryption. [ABSTRACT FROM AUTHOR]

One of the most useful mathematical tools for examining the relationships among objects is the concept of relation. Besides, it can also be necessary to throw light on uncertainties in these relationships. Soft set theory, in which different approaches used in defining the notions bring about different applications in many areas, enables to overcome uncertainties. The purpose of this paper is to define soft relation in a different way and to give a decision making method using the concept of soft relation. For this purpose, firstly, the soft relations are defined on the collection of soft elements, unlike the previous ones. After their basic properties are provided, the correspondence between the soft and classical relations is investigated and some examples are given. Finally, an algorithm is proposed using the soft relation for solving decision making problems, where the decision is related to other circumstances, and given an illustrative example. [ABSTRACT FROM AUTHOR]

The single-valued neutrosophic hesitant fuzzy set (SV-NHFS) is a hybrid structure of the single-valued neutrosophic set and the hesitant fuzzy set that is designed for some incomplete, uncertain, and inconsistent situations in which each element has a few different values designed by the truth membership hesitant function, indeterminacy membership hesitant function, and falsity membership hesitant function. A strategic decision-making technique can help the decision-maker accomplish and analyze the information in an effcient manner. However, in our real lives, uncertainty will play a dominant role during the information collection phase. To handle such uncertainties in the data, we present a decision-making algorithm in the SV-NHFS environment. In this paper, we first presented the basic operational laws for SV-NHF information under Einstein's t-norm and tconorm. Furthermore, important properties of Einstein operators, including the Einstein sum, product, and scalar multiplication, are done under SV-NHFSs. Then, we proposed a list of novel aggregation operators' names: Single-valued neutrosophic hesitant fuzzy Einstein weighted averaging, weighted geometric, order weighted averaging, and order weighted geometric aggregation operators. Finally, we discuss a multi-attribute decision-making (MADM) algorithm based on the proposed operators to address the problems in the SV-NHF environment. A numerical example is given to illustrate the work and compare the results with the results of the existing studies. Also, the sensitivity analysis and advantages of the stated algorithm are given in the work to verify and strengthen the study. [ABSTRACT FROM AUTHOR]

In this paper, a new accelerated fixed point algorithm for finding a common fixed point of two countable families of nonexpansive mappings in a Hilbert space is introduced. The weak convergence of the proposed algorithm is analyzed under some suitable conditions. We apply our proposed algorithm to solve the convex optimization problems in the form of the sum of two lower semi-continuous and convex functions. Some numerical experiments are conducted to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In the paper, we introduce two accelerated cyclic iterative algorithms for solving the multipleset split common fixed-point problem of quasi-nonexpansive operators in real Hilbert spaces. Inspired by the primal-dual algorithm, our proposed algorithms combine inertial technique with the self-adaptive stepsizes such that the implementation of the algorithms does not need any prior information about bounded linear operator norms. The weak and strong convergence of the proposed algorithms are established under suitable assumptions. As applications, we obtain several iterative algorithms to solve the multiple-set split feasibility problem. Finally, numerical results are included to demonstrate the efficiency of the proposed iterative algorithms. [ABSTRACT FROM AUTHOR]

Let k 1 be an integer, and let G be a finite and simple graph with vertex set V (G). A signed total Italian k-dominating function on a graph G is a function f: V (G) ! f1; 1; 2g such that P u2N(v) f(u) ≥ k for every v 2 V (G), where N(v) is the neighborhood of v, and each vertex u with f(u) = 1 is adjacent to a vertex v with f(v) = 2 or to two vertices w and z with f(w) = f(z) = 1. A set ff1; f2;: ::; fdg of distinct signed total Italian k-dominating functions on G with the property that Pd i=1 fi(v) ≤ k for each v 2 V (G), is called a signed total Italian k-dominating family (of functions) on G. The maximum number of functions in a signed total Italian k-dominating family on G is the signed total Italian k-domatic number of G, denoted by dk stI (G). In this paper we initiate the study of signed total Italian k-domatic numbers in graphs, and we present sharp bounds for dk stI (G). In addition, we determine the signed total Italian k-domatic number of some graphs. [ABSTRACT FROM AUTHOR]

In the 1610s and 1620s, a new computational technology took hold in England: printed mathematical tables for compound interest and discounting ("present value") problems. Historians of finance and accounting have long recognized these paper tools as predecessors of essential modern techniques like "discounted cash flow." Yet the history of these tables remains hazy. What did early seventeenth-century users do with them? Who used them? Why did they appear when they did? This article turns to one obscure but influential text—Ambrose Acroyd's Tables of Leasses and Interest (1628)—as a guide to these questions. Two key facts emerge. First, despite the prominence of similar calculations in financial applications today, these early tables were not confined to England's nascent financial sector. Rather, their foremost use related to agricultural property, specifically in assessing certain payments ("fines") landlords charged tenants for farm leases. Second, among the leading "early adopters" were institutions of the Church of England. Amidst the inflation of the early modern "price revolution," bishops, cathedrals, and colleges confronted a complex of economic, political, and social pressures. Mathematical tables like Acroyd's emerged out of long-running conflicts between church landlords and tenants over how to determine just and reasonable fines on church lands. Discounting tables were thus not tools of instrumental rationality evincing a new capitalist mentality, but tools of social accommodation and products of the era's "economy of obligation." This early modern tale offers a vivid example of how and why one community turned to a mathematical algorithm to resolve conflicts about fairness. [ABSTRACT FROM AUTHOR]

Traditional blood flow mapping algorithms are not effective enough when the region of interest contains the sources of artifacts. This paper examines the signatures of blood flow signals, as well as noise and artifact signals. Mathematical rules are proposed on the basis of these features to discriminate the blood flow signal from other signals, including the Doppler twinkling artifact. The diagnostic efficiency of the proposed decision rules is assessed using the previously collected database of raw ultrasound Doppler signals. The study results showed that taking account of the spatial stability of blood flow velocity can increase diagnostic efficiency to 91.6%. [ABSTRACT FROM AUTHOR]

Blood pressure (BP) determines whether a person has hypertension and offers implications as to whether he or she could be affected by cardiovascular disease. Cuff-based sphygmomanometers have traditionally provided both accuracy and reliability, but they require bulky equipment and relevant skills to obtain precise measurements. BP measurement from photoplethysmography (PPG) signals has become a promising alternative for convenient and unobtrusive BP monitoring. Moreover, the recent developments in remote photoplethysmography (rPPG) algorithms have enabled new innovations for contactless BP measurement. This paper illustrates the evolution of BP measurement techniques from the biophysical theory, through the development of contact-based BP measurement from PPG signals, and to the modern innovations of contactless BP measurement from rPPG signals. We consolidate knowledge from a diverse background of academic research to highlight the importance of multi-feature analysis for improving measurement accuracy. We conclude with the ongoing challenges, opportunities, and possible future directions in this emerging field of research. [ABSTRACT FROM AUTHOR]

This paper presents an innovative mathematical model of solid fuel gasification, which is not described in the available literature. The calculation of the components of the heterogeneous phase (including both solid and gaseous phases) as well as the calculation of the homogeneous phase (only gaseous components) is based on the balance of the total amounts of carbon, oxygen, hydrogen, and nitrogen entering the reactor space. Additionally, this paper introduces a new method for calculating the composition of the gaseous phase, based on reducing the heterogeneous mixture (composed of solid and gaseous phases) to a homogeneous gaseous phase. This approach to calculating the gaseous phase composition in the solid fuel gasification process has not been found by the authors in the cited literature. This paper also presents a model for calculating the heterogeneous and gaseous phases using the number of moles that participate in the assumed chemical reactions of the solid fuel gasification process. This approach to calculating the composition of the heterogeneous and gaseous phases of the solid fuel gasification process is also not represented in the cited literature. For comparison with the literature data, municipal solid waste (MSW) and cashew nut shell (Cashew Shell Char (CNSC)) were used as fuels in the calculation of gasification composition. The results of the calculation of the gaseous phase composition using the model presented in the paper show good agreement with the data from the literature. The calculation of the composition of the heterogeneous mixture during the steam gasification of MSW (α = 0.4) shows the presence of a solid phase (carbon) up to approximately 735 °C. At that temperature, the synthetic gas contains only gaseous components: CO = 33.10%, H2 = 52.70%, CH4 = 2.54%, CO2 = 4.97, H2O = 5.93% and N2 = 0.76%. Increasing the temperature above 735 °C eliminates the solid phase from the equilibrium mixture. The literature data on solid fuel gasification generally do not consider the proportion of the solid phase (carbon) in the equilibrium mixture. To satisfy the material balance at the input and output of the gasification reactor, it is necessary to determine the proportion of the solid phase (carbon) in the equilibrium mixture. Since the proportion of the solid phase (carbon) in the heterogeneous equilibrium mixture can only be determined through measurement, the development and application of a mathematical model in engineering practice is of great importance, so this developed model can be considered a useful tool for simulating the influence of process parameters on gas characteristics.

A recent paper by Schulze et al. (Math Methods Oper Res 92(1):107–132, 2020) presented the Rectangular Knapsack Problem (Rkp) as a crucial subproblem in the study on the Cardinality-constrained Bi-objective Knapsack Problem (Cbkp). To this end, they started an investigation into its complexity and approximability. The key results are an NP -hardness proof for a more general scenario than Rkp, and a 4.5-approximation for Rkp, raising the question of improvements for either result. In this note we settle both questions conclusively: we show that (a) Rkp is indeed NP -hard in the considered setting (and even in more restricted settings), and (b) there exists both a pseudopolynomial algorithm and a fully-polynomial time approximation scheme (i.e., efficient approximability within any desired ratio α > 1 ) for Rkp. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the application of the probabilistic method (PM) for the numerical solution of the Dirichlet generalized harmonic problem in axisymmetric finite homogeneous and isotropic bodies with a right circular cylindrical hole. The term "generalized" indicates that a boundary function has a finite number of first kind discontinuity curves. The suggested algorithm for the numerical solution of boundary problems consists of the following main stages: a) application of the PM, which in turn is based on the computer modeling of the Wiener process; b) finding the intersection point of the trajectory of the simulated Wiener process and the surface of the problem domain; c) development of a code for the numerical implementation and checking the reliability of obtained results; d) finding the probabilistic solution of generalized problems at any fixed points in the considered domains. The algorithm does not require the approximation of a boundary function. The PM is tested on an explicit analytical solution from the literature. To illustrate the effectiveness and simplicity of the proposed method several examples are considered. Numerical results are presented and discussed. [ABSTRACT FROM AUTHOR]

Malignant melanoma is one of the most threatening cancers to human health. Only 14% of patients with malignant melanoma have a remaining life span of 5 years. At present, there have been some studies looking for potential prognostic indicators of esophageal cancer from the level of genes and infiltrating immune cells, but there are still some problems that need to be resolved urgently. This paper proposes IGHG3 as the immune infiltration of malignant melanoma, which takes into account the computational mathematics. It aims to deduce the characteristics of immune cell infiltration in malignant melanoma and study the relationship between different immune cell infiltration characteristics and prognosis. The method in this article is to establish a computational mathematical model for the immunotherapy of melanoma, then study the method of identification of the affinity of the IGHG3 reagent, and finally obtain the gene expression of immune infiltration. The functions of these methods are, respectively, to predict the dynamic behavior of T cells with two different specificities through mathematical models and to test the matching degree of different concentrations of IGHG3 reagent with the human body. Then use the ssGSEA algorithm to obtain immune infiltration related data and calculate the difference between the weighted empirical cumulative distribution function of all genes in the effect of IGHG3 on melanoma that was carried out. The experimental results showed the computational mathematical method genome and all the remaining genes. In this study, a computational mathematical method to detect the IGHG3 gene expression had a significant inhibitory effect on A375 cells in the experimental group, and the knockdown efficiency reached 85.6%. [ABSTRACT FROM AUTHOR]

In this paper, the theory of viscoelastic material relaxation modulus testing, testing method, and material relaxation modulus parameter solution method is studied. Based on the IoT sensor test devices and the Hertzian mathematics theory, the relationship between the compression amount, pressure, and relaxation modulus of a rigid spherical indenter pressed into a flat viscoelastic material is studied. In combination with the Boltzmann superposition principle, the relaxation modulus test method and theory for viscoelastic materials are proposed. A good fitting curve was obtained to solve the relaxation modulus parameters of various materials under the three-element Maxwell model. The experimental setup and the results of the parameter fitting were analyzed to verify the feasibility and accuracy of the results. [ABSTRACT FROM AUTHOR]

In this paper, an efficient numerical algorithm is proposed for the valuation of unilateral American better-of options with two underlying assets. The pricing model can be described as a backward parabolic variational inequality with variable coefficients on a two-dimensional unbounded domain. It can be transformed into a one-dimensional bounded free boundary problem by some conventional transformations and the far-field truncation technique. With appropriate boundary conditions on the free boundary, a bounded linear complementary problem corresponding to the option pricing is established. Furthermore, the full discretization scheme is obtained by applying the backward Euler method and the finite element method in temporal and spatial directions, respectively. Based on the symmetric positive definite property of the discretized matrix, the value of the option and the free boundary are obtained simultaneously by the primal-dual active-set method. The error estimation is established by the variational theory. Numerical experiments are carried out to verify the efficiency of our method at the end. [ABSTRACT FROM AUTHOR]

This paper proposes an innovative statistical method to measure the impact of the class/school on student achievements in multiple subjects. We propose a semiparametric model for a bivariate response variable with random coefficients, that are assumed to follow a discrete distribution with an unknown number of support points, together with an Expectation-Maximization algorithm—called BSPEM algorithm—to estimate its parameters. In the case study, we apply the BSPEM algorithm to data about Italian middle schools, considering students nested within classes, and we identify subpopulations of classes, standing on their effects on student achievements in reading and mathematics. The proposed model is extremely informative in exploring the correlation between multiple class effects, which are typical of the educational production function. The estimated class effects on reading and mathematics student achievements are then explained in terms of various class and school level characteristics selected by means of a LASSO regression. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new iterative algorithm and analyze it in detail inasmuch as convergence, stability, and data dependency for the class of almost contraction mappings. We also consider another iterative algorithm called F ∗ iterative algorithm proposed by Ali et al. (Comp. Appl. Math. 39, 267 (2020)) and derive some new algorithms from this with the aim of giving an affirmative answer to an open question raised by the same authors. Our results considerably improve the corresponding results in Ali et al. (Comp. Appl. Math. 39, 267 (2020)). We submit some non-trivial numerical examples to illustrate the robustness, feasibility, and effectiveness of our findings. [ABSTRACT FROM AUTHOR]

In order to improve the effect of classroom teaching and realize the improvement of mathematics teaching quality, this article combines data-driven decision-making technology to design a mathematics teaching plan system and applies data-driven decision-making technology to the design of mathematics teaching plan by improving the big data algorithm. In addition, this paper designs a mathematics teaching plan design system based on data-driven decision-making technology. The system learning module displays the knowledge points in a chapter-sequential navigation mode and stores data or information in the nodes, which are connected to form a network structure through chains. Finally, this paper verifies the designed system with experiments. From the results of the experimental research data, it can be seen that the mathematical teaching plan design system based on the data-driven decision-making technology constructed in this article has good practical effects. On this basis, the system constructed in this article can be verified through further practice. [ABSTRACT FROM AUTHOR]

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With the increasing importance of mathematics in basic education, how to evaluate and analyze the intelligent effect of mathematics teaching classroom through scientific methods has become one of the indicators to evaluate the intelligent classroom. This paper studies the design and application of mathematics teaching intelligent classroom based on the PCA-NN (principal component analysis-neural network) algorithm. Firstly, this paper briefly describes the current research status of mathematics teaching intelligent classroom design and PCA-NN algorithm. Secondly, combined with the key factors of mathematics teaching, it formulates specific standards and puts forward an adaptive strategy of intelligent and personalized intelligent mathematics teaching classroom. Finally, the algorithm is verified by experiments. The results show that, for students with different mathematics basic levels, the mathematics teaching intelligent classroom based on the PCA-NN algorithm can effectively improve the quality of mathematics classroom teaching. Through the research on the factors such as teaching quality, effect, and interaction mode involved in the process of mathematics teaching classroom design, the intelligent classroom design factors affecting teaching quality are determined. This paper analyzes and studies the system from different angles. The research results provide some help for the current quality evaluation of classroom teaching and use the PCA-NN algorithm to make quantitative analysis and multivariate verification of mathematics classroom teaching effect. [ABSTRACT FROM AUTHOR]

Research presented in this paper was initiated by the publication [A. D. Agaltsov and R. G. Novikov, Examples of solving the inverse scattering problem and the equations of the Veselov–Novikov hierarchy from the scattering data of point potentials, Russian Math. Surveys74 (2019), 3, 373–386] and is based on its results. Two sets of the complex monopole scattering coefficients are distinguished among the possible values of these coefficients for nonabsorbing inhomogeneities. These sets differ in phases of the scattering coefficients. In order to analyze the features and possibilities of reconstructing the inhomogeneities of both sets, on the one hand, the inverse problem is solved for each given value of the monopole scattering coefficient using the Novikov functional algorithm. On the other hand, the scatterer is selected in the form of a homogeneous cylinder with the monopole scattering coefficient that coincides with the given one. The results obtained for the monopole inhomogeneity and for the corresponding cylindrical scatterer are compared in the coordinate and spatial-spectral spaces. The physical reasons for the similarities and differences in these results are discussed when the amplitude of the scattering coefficient changes, as well as when passing from one set to another. [ABSTRACT FROM AUTHOR]