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Resumen (Objetivo) Esta investigación plantea como objetivo identificar las conexiones matemáticas que establecen futuros profesores sobre el concepto de ecuación cuadrática. (Metodología) Se empleó la técnica de entrevista de grupo focal para recolectar información. Esta consistió en la aplicación de cinco tareas a ocho futuros profesores que se encontraban cursando la Licenciatura en Matemática, área Matemática Educativa. Los participantes son de la ciudad de Chilpancingo del estado de Guerrero en México y sus edades oscilaban entre 21 y 23 años. Debido a la pandemia causada por el COVID-19, las cuatro sesiones grupales se realizaron de manera virtual con una duración de 80 minutos cada una. Se utilizó el análisis temático para examinar los datos recabados. (Resultados) Las producciones escritas y verbales de los futuros profesores indicaron que cada uno usó de forma variada las conexiones matemáticas. De manera general, las de más frecuencia fueron procedimental, características y significado; con menor frecuencia, parte-todo, modelado e implicación. Dichas conexiones corresponden a las contempladas en el marco teórico, por lo que se puede argumentar que este resulta válido y pertinente para explorar las conexiones matemáticas en futuros profesores, al resolver tareas matemáticas. (Conclusiones) Los participantes dieron significado al concepto de ecuación cuadrática en cuanto a lo que significa en contextos reales y lograron representar la función cuadrática de diversas formas, en el registro algebraico y gráfico. Sin embargo, la mayoría no estableció todas las conexiones matemáticas previstas. Abstract (Objective) This investigation seeks to identify the mathematical connections established by prospective mathematics teachers related to the concept of the quadratic equation. (Methodology) A focal group interview was used to collect information. It consisted of assigning five tasks to eight prospective mathematics teachers who are pursuing a Licentiate's degree in Mathematics in the area of Educational Mathematics. The participants are from the city of Chilpancingo in the state of Guerrero in Mexico, and were from 21 to 23 years old. Due to the COVID-19 pandemic, the four group sessions were held virtually, each lasting 80 minutes. Data were analyzed using a thematic approach. (Results) The written and verbal responses of prospective mathematics teachers indicated that each of them used mathematical connections skills in a different way. In general, the most frequent skills were related to procedures, characteristics, and meaning, and less frequently to part-whole, modeling, and implication. These connections correspond to those specified in the theoretical framework; therefore, it can be argued that this framework is valid and relevant for exploring the mathematical connections among prospective mathematics teachers when solving mathematical tasks. (Conclusions) The participants assigned meaning to the concept of quadratic equation in terms of what it represents in real contexts, and were able to present the quadratic function in different forms in the algebraic and graphical terms. However, most of them did not make the intended mathematical connections. Resumo (Objetivo) Esta pesquisa visa identificar as conexões matemáticas que os futuros professores estabelecem sobre o conceito de equação quadrática. (Metodologia) A técnica de entrevista em grupo focal foi utilizada para coletar informações e consistiu na aplicação de cinco tarefas a oito futuros professores que frequentavam o curso de Licenciatura em Matemática, área da Matemática Educativa. Os participantes são da cidade de Chilpancingo no estado de Guerrero no México e suas idades variaram de 21 a 23 anos. Devido à pandemia provocada pela COVID-19, as quatro sessões de grupo foram realizadas virtualmente com uma duração de 80 minutos cada. Foi utilizada a análise temática para examinar os dados coletados. (Resultados) As produções escritas e verbais dos futuros professores indicaram que cada um utilizou as conexões matemáticas de forma variada. De modo geral, as mais frequentes foram procedimental, características e significado; com menos frequência, fração, modelagem e implicação. Tais conexões correspondem às contempladas no quadro teórico, pelo que se pode afirmar que é válido e relevante explorar as conexões matemáticas nos futuros professores, na resolução de tarefas matemáticas. (Conclusões) Os participantes deram significado ao conceito de equação quadrática em termos do que significa em contextos reais e conseguiram representar a função quadrática de várias formas, no registo algébrico e gráfico. No entanto, a maioria não fez todas as conexões matemáticas esperadas.

As a category of the recurrent neural network (RNN), zeroing neural network (ZNN) can effectively handle time-variant optimization issues. Compared with the fixed-parameter ZNN that needs to be adjusted frequently to achieve good performance, the conventional variable-parameter ZNN (VPZNN) does not require frequent adjustment, but its variable parameter will tend to infinity as time grows. Besides, the existing noise-tolerant ZNN model is not good enough to deal with time-varying noise. Therefore, a new-type segmented VPZNN (SVPZNN) for handling the dynamic quadratic minimization issue (DQMI) is presented in this work. Unlike the previous ZNNs, the SVPZNN includes an integral term and a nonlinear activation function, in addition to two specially constructed time-varying piecewise parameters. This structure keeps the time-varying parameters stable and makes the model have strong noise tolerance capability. Besides, theoretical analysis on SVPZNN is proposed to determine the upper bound of convergence time in the absence or presence of noise interference. Numerical simulations verify that SVPZNN has shorter convergence time and better robustness than existing ZNN models when handling DQMI.

Individual differences in the timing of developmental processes are often of interest in longitudinal studies, yet common statistical approaches to modeling change cannot directly estimate the timing of when change occurs. The time-to-criterion framework was recently developed to incorporate the timing of a prespecified criterion value; however, this framework has difficulty accommodating contexts where the criterion value differs across people or when the criterion value is not known a priori, such as when the interest is in individual differences in when change starts or stops. This article combines aspects of reparameterized quadratic models and multiphase models to provide information on the timing of change. We first consider the more common situation of modeling decelerating change to an offset point, defined as the point in time at which change ceases. For increasing trajectories, the offset occurs when the criterion attains its maximum ("inverted J-shaped" trajectories). For decreasing trajectories, offset instead occurs at the minimum. Our model allows for individual differences in both the timing of offset and ultimate level of the outcome. The same model, reparameterized slightly, captures accelerating change from a point of onset ("J-shaped" trajectories). We then extend the framework to accommodate "S-shaped" curves where both the onset and offset of change are within the observation window. We provide demonstrations that span neuroscience, educational psychology, developmental psychology, and cognitive science, illustrating the applicability of the modeling framework to a variety of research questions about individual differences in the timing of change. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

As a hot topic in unsupervised learning, clustering methods have been greatly developed. However, the model becomes more and more complex, and the number of parameters becomes more and more with the continuous development of clustering methods. And parameter-tuning in most methods is a laborious work due to its complexity and unpredictability. How to propose a concise and beautiful model in which the parameters can be learned adaptively becomes a very meaningful problem. Aim at tackling this problem, we develop a novel self-weighted unsupervised linear discriminative analysis method, namely SWULDA. The proposed method not only avoids adjusting parameters but also explains the link between k-means and linear discriminant analysis (LDA). To obtain superior structural performance, the idea of minimizing the within-class scatter matrix and maximizing the between-class scatter matrix is embedded in the unsupervised model. Moreover, equipped with the proposed quadratic weighted optimization framework, the parameter can be adaptively learned. The extensive experiments on several datasets are conducted to validate the effectiveness of our method.

Solving quadratic equations with radicals on a computer with floating-point arithmetic requires great care to handle correctly all possible parameters. Literature on the subject glosses over the details, often considered as important but tedious to present. As a consequence, most implementations are flawed in one way or another. After having reviewed both the literature and the actual implementations in several programming languages and applications, we present an algorithm inspired from an exposition by Pat Sterbenz from 1974, adapted to take advantage of more recent researches in the field, which leads to a robust quadratic equation solver.

[Objetivo]Esta investigación plantea como objetivo identificar las conexiones matemáticas que establecen futuros profesores sobre el concepto de ecuación cuadrática.[Metodología]Se empleó la técnica de entrevista de grupo focal para recolectar información. Esta consistió en la aplicación de cinco tareas a ocho futuros profesores que se encontraban cursando la Licenciatura en Matemática, área Matemática Educativa. Los participantes son de la ciudad de Chilpancingo del estado de Guerrero en México y sus edades oscilaban entre 21 y 23 años. Debido a la pandemia causada por el COVID-19, las cuatro sesiones grupales se realizaron de manera virtual con una duración de 80 minutos cada una. Se utilizó el análisis temático para examinar los datos recabados.[Resultados]Las producciones escritas y verbales de los futuros profesores indicaron que cada uno usó de forma variada las conexiones matemáticas. De manera general, las de más frecuencia fueron procedimental, características y significado; con menor frecuencia, parte-todo, modelado e implicación. Dichas conexiones corresponden a las contempladas en el marco teórico, por lo que se puede argumentar que este resulta válido y pertinente para explorar las conexiones matemáticas en futuros profesores, al resolver tareas matemáticas.[Conclusiones]Los participantes dieron significado al concepto de ecuación cuadrática en cuanto a lo que significa en contextos reales y lograron representar la función cuadrática de diversas formas, en el registro algebraico y gráfico. Sin embargo, la mayoría no estableció todas las conexiones matemáticas previstas.

We describe a graph parametrization of rational quadratic differentials with presence of a simple pole, whose critical trajectories form a network depending on parameters focusing on the network topological jumps. Obtained bifurcation diagrams are associated with the Stasheff polytopes.

The linearity assumption is widely used when acquiring the hydrodynamic coefficients of a floating structure. However, the linear damping is frequently underestimated, especially for the natural frequency. To investigate the sloping seafloor effects on the damping terms of a single module of a semi-submersible Very Large Floating Structure (VLFS), this paper revisits the conventional formulation and further proposes the direct integration method for obtaining the linear and quadratic damping coefficients from free-decay tests. Numerical free-decay simulations of the single module over variable bathymetry are carried out by the CFD numerical tank. Corresponding model tests are also implemented to verify and validate against the numerical solutions. The effects of the sloping seafloor, as well as the water depth, on the hydrodynamic coefficients are investigated based on the validated CFD modeling. Both numerical and experimental results indicate that the acquisition of the linear and quadratic damping coefficients is sensitive to the data-processing and identification approaches. For the case studied in present paper, the identification errors introduced by the conventional method are 1.5 % while they are 0.5 % using the direct integration method. The quadratic damping coefficient for heave mode decreases about 10.4 % when the sloping angle increases from 0 to 6 deg.

Given an undirected connected graph G = ( V , E ) , the minimum linear arrangement problem (MinLA) consists in determining a permutation π ≔ ( π 1 , … , π n ) of the node-set V = { 1 , … , n } , which minimizes the sum of edge costs | π i − π j | over all edges { i , j } ∈ E . We introduce a compact quadratic formulation, prove its correctness, and generate new mixed-integer linear formulations that require a very small number of variables and constraints. The idea behind the way we model permutations allows the development of strong valid constraints to strengthen the new formulations. We also adapt some cutting plane procedures from the literature to one of the new models. Computational experiments with benchmark and new instances are very encouraging and improve state-of-the-art solution approaches from the literature.

Knee points, characterized as a small improvement on one objective can lead to a significant degradation on at least one of the other objectives, are attractive to decision makers in multi-criterion decision-making. This paper presents a simple and effective knee point identification method to help decision makers identify solution(s) of interest from a given set of trade-off solutions thus facilitating posterior decision-making. Our basic idea is to sequentially validate whether a solution is a knee point or not by comparing its localized trade-off utility with others within its neighborhood characterized from a decomposition perspective. In particular, a solution is a knee point if and only if it has the best localized trade-off utility among its neighbors. We implement a GPU version that carries out the knee point identification in a parallel manner. This GPU version reduces the worst-case complexity from quadratic to linear. The performance of our proposed method is compared with five state-of-the-art knee point identification methods on 134 test problem instances and two real-world engineering design problems. Empirical results demonstrate its outstanding performance especially on problems with many local knee points. We further validate the usefulness of our proposed method for guiding evolutionary multi-objective optimization algorithms to search for knee points on the fly during the evolutionary process.

This article investigates the stability problem for discrete-time neural networks with a time-varying delay by focusing on developing new Lyapunov-Krasovskii (L-K) functionals. A novel L-K functional is deliberately tailored from two aspects: 1) the quadratic term and 2) the single-summation term. When the variation of the discrete-time delay is further considered, the constant matrix involved in the quadratic term is extended to be a delay-dependent one. All these innovations make a contribution to a quadratic function with respect to the delay from the forward differences of L-K functionals. Consequently, tractable stability criteria are derived that are shown to be more relaxed than existing results via numerical examples.

Despite the success of stochastic variance-reduced gradient (SVRG) algorithms in solving large-scale problems, their stochastic gradient complexity often scales linearly with data size and is expensive for huge data. Accordingly, we propose a hybrid stochastic-deterministic minibatch proximal gradient~(HSDMPG) algorithm for strongly convex problems with linear prediction structure, e.g.~least squares and logistic/softmax regression. HSDMPG~enjoys improved computational complexity that is data-size-independent for large-scale problems. It iteratively samples an evolving~minibatch of individual losses to estimate the original problem, and efficiently minimizes the sampled smaller-sized subproblems. For strongly convex loss of n components, HSDMPG~attains an $\epsilon$ -optimization-error within $\mathcal{O} \left(\kappa\log^{\zeta+1}\left(\frac{1}{\epsilon}\right)\frac{1}{\epsilon}\bigwedge n\log^{\zeta}\left(\frac{1}{\epsilon}\right)\right)$ stochastic gradient evaluations, where $\kappa$ is condition number, $\zeta=1$ for quadratic loss and $\zeta=2$ for generic loss. For large-scale problems, our complexity outperforms those of SVRG-type algorithms with/without dependence on data size. Particularly, when $\epsilon=\mathcal{O}(1/\sqrt{n})$ which matches the intrinsic excess error of a learning model and is sufficient for generalization, our complexity for quadratic and generic losses is respectively $\mathcal{O} (n^{0.5}\log^{2}(n))$ and $\mathcal{O} (n^{0.5}\log^{3}(n))$ , which for the first time achieves optimal generalization in less than a single pass over data. Besides, we extend HSDMPG~to online strongly convex problems and prove its higher efficiency over the prior algorithms. Numerical results demonstrate the computational advantages of~HSDMPG.

Currently, DNA strand displacement is often used to build neural networks or solve logical problems. While there are few studies on the use of DNA strand displacement to solve the higher order equations. In this paper, the catalysis, degradation, annihilation and adjusted reaction modules are built through DNA strand displacement. The chemical reaction networks of the corresponding higher order equations and simultaneous equations are established through these modules, and these chemical reaction networks can be used to build analog circuits to solve binary primary simultaneous equations and binary quadratic simultaneous equations. Finally, through Visual DSD software verification, this design can realize the solution of binary primary simultaneous equations and binary quadratic simultaneous equations, which provides a reference for DNA computation in the future.

We determine those imaginary quadratic number fields whose 2-class groups have rank 3 and 4-rank ≤2 and such that the 2-class groups of their Hilbert 2-class fields have rank 2. This result, along with previous work, gives a complete determination of all complex quadratic number fields for which the 2-class groups of their 2-class fields have rank at most 2.

We consider some types of complex continued fraction expansions and their associated maps of the Euclidean fields Q ( − d ) , d = 1 , 2 , 3 , 7 and 11 introduced by Hurwitz (1887) [7] , Lakein (1973) [10] and Shiokawa et al. (1975) [20] . We show that there exists the Legendre constant for every type of continued fraction expansions introduced by them improving the proof by Ei et al. (2019) [4] in the case of the one by A. Hurwitz with Q ( − 1 ) .

For a given odd positive integer $n$ and an odd prime $p$, we construct an infinite family of quadruples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$ and $\mathbb{Q}(\sqrt{d+4p^2})$ with $d\in \mathbb{Z}$ such that the class number of each of them is divisible by $n$. Subsequently, we show that there is an infinite family of quintuples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$, $\mathbb{Q}(\sqrt{d+4})$, $\mathbb{Q}(\sqrt{d+36})$ and $\mathbb{Q}(\sqrt{d+100})$ with $d\in \mathbb{Z}$ whose class numbers are all divisible by $n$. Our results provide a complete proof of Iizuka's conjecture (in fact a generalization of it) for the case $m=1$. Our results also affirmatively answer a weaker version of (a generalization of) Iizuka's conjecture for $m\geq 4$., Comment: 10 pages; some minor typos have been fixed as per the suggestions from the referee; to appear in 'Journal of Number Theory'