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New negative definiteness conditions for quadratic functions with illustration in LPV sampled-data control

AUTOMATICA 2025年 173卷

作者： Oliveira, Lucas A. L. Guelton, Kevin Motchon, Koffi M. D. Leite, Valter J. S. Univ Reims CReST UR 3804 F-51100 Reims France CEFET MG Grad Program Math & Computat Modeling PPGMMC Belo Horizonte Brazil CEFET MG Dept Mechatron Engn Campus Divinopolis Belo Horizonte Brazil

This paper addresses negative definiteness conditions of quadratic functions, common in control problems with time-varying delay systems. Existing geometric conditions, relaxed by partitioning techniques, may lack monotonic convergence, making their optimality questionable. Alternative conditions based on the generalized S-procedure are known to be necessary and sufficient when coefficients are not dependent on uncertain parameters;otherwise, they are sufficient only. We propose new approaches to mitigate these issues, unexplored in previous studies, demonstrated in stability analysis of linear parameter varying sampled-data control systems. First, we rewrite quadratic polynomial inequalities as homogeneous ones, deriving relaxed conditions using Young's inequality or Polya's theorem, offering recursive monotonic convergent relaxations. Then, based on B & eacute;zier curve equivalence and the de Casteljau algorithm, we provide further relaxed recursive monotonic convergent conditions. Two numerical examples illustrate the effectiveness and improvements over previous related studies. (c) 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://***/licenses/by-nc-nd/4.0/).

关键词： quadratic functions Negative definiteness Monotonic convergent conditions LPV systems Time-varying delay systems

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Estimation of the neighborhood of metric regularity for quadratic functions

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OPTIMIZATION 2024年第8期73卷 2527-2553页

作者： Xu, Wending Sichuan Tourism Univ Sch Big Data & Stat Chengdu Peoples R China

Metric regularity is widely concerned since its important applications in optimization and control theory. For promoting the application of metric regularity, it is valuable to study the estimation of the neighborhood which makes the regularity hold. However, it seems that no result has been established about this issue. This paper investigates the estimation of the neighborhood of metric regularity for quadratic functions. The main result gives the expression of the neighborhood of metric regularity for a kind of convex quadratic functions.

关键词： Metric regularity neighborhood estimation quadratic functions

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Re-examination of centre of buoyancy curve and its evolute for rectangular cross section, Part 2: Using quadratic functions

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BRODOGRADNJA 2023年第3期74卷 17-45页

作者： Ban, Dario Univ Split Fac Elect Engn Mech Engn & Naval Architecture Split 21000 Croatia

In this paper, exact hydrostatic particulars equations for the centre of buoyancy curve and metacentric locus curve are given for rectangular cross section using quadratic functions. Those equations have not been given for the hyperbola range of the heel angles so far, and here it is done by using basic quadratic functions and their horizontally symmetric immersion shapes, with two new methods defined: 1. Rotation of basic cross section shapes, and 2. Hydrostatic cross section area complement method that uses homothety or scaling properties of emerged and immersed areas of the rectangular cross section. Observed metacentric curve for rectangle consists of semi-cubic parabolas and Lame curve with 2/3 exponent and negative sign, resulting in the cusp discontinuities in the symmetry of those functions definition. In order to achieve above, two theorems are given: the theorem about scaling using hydrostatic cross section area complement and the theorem about parallelism of centre of buoyancy tangents with waterlines. After non-dimensional bounds are given for the existence of the swallowtail discontinuity of metacentric curve for rectangular cross section in the Part 1 of this paper, the proof of its position in the symmetry of rectangle vertex angle is given in this Part 2 of the paper, thus confirming its position from theory.

关键词： centre of buoyancy curve metacentric curve rectangular cross section quadratic functions basic geometric shapes hydrostatic area complement

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Overt and Covert Participation in an Argumentative whole-class Discussion: Spread of Ideas about quadratic functions

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INTERNATIONAL JOURNAL OF SCIENCE AND MATHEMATICS EDUCATION 2025年第3期23卷 639-661页

作者： Ofri, Ofra Tabach, Michal Tel Aviv Univ Tel Aviv Israel

Mathematical ideas are developed and spread during argumentative whole class discussions between a teacher and her students. The goal of the current study is to characterize how ideas about quadratic functions emerge and are spread during a whole-class discussion among ninth graders. To this end, we recorded both discussions between pairs of students and whole-class discussions led by the teacher. We used the approach of documenting collective activities as our methodological lens. The findings show that mathematical ideas about quadratic functions, like positive and negative range, increase and decrease range, minimum or maximum point, intersection with the axes, and more were spread in two parallel layers. Students participated in an overt layer in which they uttered their ideas in a public discussion. At the same time, they also uttered mathematical ideas privately with their peers in a covert layer. That is, whole-class discussions are not identical for all participants in that the covert layer turns these discussions into a unique experience.

关键词： Documenting collective activity Knowledge shifts Overt and covert participation quadratic functions

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Gradient descent for quadratic functions using geometric mean and the Kai Fang method

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CONCURRENCY AND COMPUTATION-PRACTICE & EXPERIENCE 2023年第16期35卷

作者： Rim, KwangCheol Kim, Pankoo Ko, Hoon Dongsin Univ Coll Basic & Gen Educ Seoul South Korea Chosun Univ Dept Comp Engn Gwangju South Korea Chungbuk Natl Univ Res Inst Comp & Informat Commun RICIC Cheongju South Korea

The geometric mean is typically used to measure the mean of inflation rate and population fluctuation. It is also used in the description and analysis of singularities and geometric distance spaces. Gradient descent is an integral part of artificial intelligence. In this study, we transform the gradient calculation from conventional quadratic gradient descent algorithms into a root extraction calculation using geometric means. To eliminate the computational complexity of differential operations in gradient calculation and to easily calculate roots using only fundamental arithmetic operations, we introduce the Kai Fang method, the East Asian traditional root extraction method. To do this, we propose a new quadratic gradient descent method based on geometric means and we apply the Kai Fang method with geometric means to create an improved quadratic gradient descent method. The proposed method shows improved computational ease over conventional methods.

关键词： geometric mean gradient descent Kai Fang method quadratic functions

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Coordinating visual and algebraic reasoning with quadratic functions

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MATHEMATICS EDUCATION RESEARCH JOURNAL 2024年第1期36卷 33-69页

作者： Wilkie, Karina J. Monash Univ Fac Educ 47-49 Moorooduc Highway Frankston Vic 3199 Australia

quadratics provide a foundational context for making sense of many important algebraic concepts, such as variables and parameters, nonlinear rates of change, and views of function. Yet researchers have highlighted students' difficulties in connecting such concepts. This in-depth qualitative study with two pairs of Year 10 (15 or 16-year-old) students investigated the potential of figural pattern generalisation-a context not traditionally used for teaching quadratics-to stimulate students' coordination of visual and algebraic reasoning and attention to quadratic function concepts. Theorisations of embodied visualisation, algebraic thinking, and student noticing were drawn on to analyse the pairs responding to 19 quadratic figural pattern generalisation tasks interspersed throughout their class topic on quadratic equations. It was found that students became adept at connecting the generality of different types of structural aspects of figures (square, rectangular, linear, constant/invariant) to their symbolic expression in quadratic equations. Students' construction of numeric instantiations of figural aspects was found to support pairs in moving towards symbolic generalisation. Task prompts to find different (but equivalent) algebraic equations for the same pattern evidenced pairs beginning to distinguish among general, factorised and standard forms of quadratic equations. One pair's attention to first and second differences (between total quantities of figures in a sequence) highlighted both the difficulty of and potential for connecting quadratic rate-of-change concepts and parameters visually. Implications for including figural pattern generalisation when teaching quadratics and suggestions for further research are shared.

关键词： Algebraic thinking Visualisation quadratic functions Figural pattern generalisation Student noticing

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Structures and representations used by 6th graders when working with quadratic functions

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ZDM-MATHEMATICS EDUCATION 2022年第6期54卷 1393-1406页

作者： Ramirez, Rafael Canadas, Maria C. Damian, Alba Univ Granada Granada Spain

This study lies within the field of early-age algebraic thinking and focuses on describing the functional thinking exhibited by six sixth-graders (11- to 12-year-olds) enrolled in a curricular enhancement program. To accomplish the goals of this research, the structures the students established and the representations they used to express the generalization of the functional relationship were analyzed. A questionnaire was designed with three geometric tasks involving the use of continuous variables in quadratic functions. The students were asked to calculate the areas of certain figures for which some data were known, and subsequently to formulate the general rule. The results show that the participating students had difficulties expressing structures involving quadratic functions. However, they displayed the potential to use different types of representations to establish the functional relationship. The originality of this study lies in the differences observed in the process of generalization with discrete variables, since, in the case of continuous variables, students could recognize the general expression from analyzing the set of values that can be attributed to the variables in an interval.

关键词： Functional thinking Geometric formulas quadratic functions Representation Structure

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Functional encryption with application to machine learning: simple conversions from generic functions to quadratic functions

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PEER-TO-PEER NETWORKING AND APPLICATIONS 2020年第6期13卷 2334-2341页

作者： Wang, Huige Chen, Kefei Zhang, Yuan Zhao, Yunlei Anhui Sci & Technol Univ Dept Comp Bengbu 233030 Peoples R China Fudan Univ Sch Comp Sci Shanghai 200433 Peoples R China Hangzhou Normal Univ Dept Math Hangzhou 311121 Peoples R China Westone Cryptol Res Ctr Beijing 100070 Peoples R China Univ Elect Sci & Technol China Sch Comp Sci & Engn Chengdu 611731 Peoples R China

Functional encryption (FE) and predicate encryption (PE) can be utilized in deploying and executing machine learning (ML) algorithms to improve efficiency. However, most of existing FE and PE algorithms only consider generic functions. Actually, quadratic-functions-based FE and PE can be used to further reduce the computation costs significantly. In this paper, we present a functional encryption scheme for quadratic functions from those for generic functions. In our constructions, ciphertexts are associated with a pair of vectors (x,y)is an element of DOUBLE-STRUCK CAPITAL ZqnxDOUBLE-STRUCK CAPITAL Zqm, private keys are associated with a quadratic function, and the decryption of ciphertexts CT(x,y) with a private key sk(F), where F is a n x m-dimensional matrix, recovers (x)TFy is an element of DOUBLE-STRUCK CAPITAL Zq. Compared with Baltico et al.'s FEs for quadratic functions (at Crypto 2017), our schemes could obtain almost the same ciphertexts size of O((n+m)logq) as their schemes (in contrast to O(n) in Baltico et al.'s schemes), and the computation for quadratic functions in our scheme does not rely on bilinear maps, while their schemes must rely on this assumption. In particular, our schemes under the standard assumptions achieve adaptive security, while Baltico et al.'s scheme only obtains selective security. Moreover, beyond the MDDH and GGM assumptions, our schemes allow for instantiations under standard assumptions such as LWE, LPN, and etc.

关键词： quadratic functions Functional encryption Adaptive security Generic conversions Machine learning

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Combinatorial t-designs from quadratic functions

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DESIGNS CODES AND CRYPTOGRAPHY 2020年第3期88卷 553-565页

作者： Xiang, Can Ling, Xin Wang, Qi S China Agr Univ Coll Math Informat Guangzhou Guangdong Peoples R China China w Normal Univ Sch Math Informat Nanchong Sichuan Peoples R China So Univ Sci Dept Comp Sci Engn Technol Shenzhen Guangdong Peoples R China

Combinatorial t-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a t-design. Till now only a small amount of work on constructing t-designs from special polynomials has been done, and it is in general hard to determine their parameters. In this paper, we investigate this idea further by using quadratic functions over finite fields, thereby obtain infinite families of 2-designs, and explicitly determine their parameters. The obtained designs cover some earlier 2-designs as special cases. Furthermore, we confirm Conjecture 3 in Ding and Tang (, 2019).

关键词： Polynomial quadratic functions t-Design

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On the spherical quasi-convexity of quadratic functions

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LINEAR ALGEBRA AND ITS APPLICATIONS 2019年 562卷 205-222页

作者： Ferreira, O. P. Nemeth, S. Z. Xiao, L. IME UFG Ave Esperanca S-NCampus 2 BR-74690900 Goiania Go Brazil Univ Birmingham Sch Math Watson Bldg Birmingham B15 2TT W Midlands England

In this paper the spherical quasi-convexity of quadratic functions on spherically convex sets is studied. Several conditions characterizing the spherical quasi-convexity of quadratic functions are presented. In particular, conditions implying spherical quasi-convexity of quadratic functions on the spherical positive orthant are given. Some examples are provided as applications of the obtained results. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Sphere Spherical quasi-convexity quadratic functions Positive orthant

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