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Given a rational number α ≠ 2, we establish a criterion for the existence of the general solution of an alternative quadratic functional equation of the form f (x y) + f (x y-1) = 2f (x) + 2f (y) or f (x y) + f (x y-1) = αf (x) + 2f (y), where f is a mapping from an abelian group (...) to a uniquely divisible abelian group (...). We also find the general solution in the cases when G is a 6-divisible abelian group and G is a cyclic group. [ABSTRACT FROM AUTHOR]

In this paper, some concepts of convex analysis on hyperbolic spaces are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first- and second-order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented. [ABSTRACT FROM AUTHOR]

In 1940, the stability problem of functional equations was arose due to a question of Stanisaw Ulam concerning the stability of group homomorphisms. Significant work was done by Donald H. Hyers about HYERS-ULAM STABILITY and obtained a partial affirmative answer to the question of Ulam in the context of banach spaces in the case of additive mappings. In 1978, T. M. Rassias expanded Hyers's theorem for mappings between Banach spaces by considering an unbounded Cauchy difference subject to a continuity condition upon the mapping. After that, Many Researchers had studied about Hyers-Ulam stability of an additive quadratic type functional equations. In this research article, the Hyers-Ulam stability of an additive quadratic type functional equation was discussed and obtained the generalization of Hyers-Ulam stability of an additive quadratic type functional equation f(x + ay) + af(x - y) = f(x - ay) + af(x + y) for any integer a with a ≠ 1; 0; 1 in 2-Banach space. [ABSTRACT FROM AUTHOR]

In this paper, the notion of the ravine of real-valued functions is extended from the finite-dimensional setting to an infinite-dimensional setting. Ravines of quadratic functions are studied in detail. The obtained results solve a problem raised by Professor Joachim Gwinner. In addition, it is proved that a weakly continuous real-valued convex function defined on a reflexive Banach space cannot have any ravine along the null subspace. [ABSTRACT FROM AUTHOR]

Carcass characteristics, organ weights, and intestinal biometry of broiler chickens fed diets supplemented with black velvet tamarind (Dialium guineense) stem bark (BSB) were assessed. Two hundred, day-old Ross 308 broilers were divided into 4 groups of 50 chickens, and each group replicated five times. Each group were assigned to one experimental diet in a completely randomised design designated T0 (0), T1 (0.5), T2 (1.0) and T3 (1.5 g BSB/kg feed). Data obtained on carcass characteristics, organ weights, and intestinal biometry were analysed statistically. Results showed that BSB was low in crude protein (6.42%) and high in crude fibre (30.65%) and ash (9.35%). Broiler chickens fed diet T1 had significantly higher (P < 0.05) breast and drumstick weights than those offered the other 3 diets. There were significant differences (P < 0.05) in the abdominal fat pad, liver, proventriculus, gizzard weight, and intestinal biometry of broiler chickens in all the groups. Results also showed that dietary BSB supplementation level had a quadratic effect (P < 0.05) on breast, drumstick, liver, proventriculus weight, abdominal fat pad, and intestinal biometry of broiler chickens. In contrast, dietary BSB supplementation levels had linear effect (P < 0.05) on gizzard weight. It can be concluded that BSB is rich in ash and fibre, and is suitable as a feed additive in broiler chicken diets at a level not beyond 0.5 g/kg feed for best organ weight, carcass yield, intestinal biometry. Thus, there is potential to utilize BSB for improved productivity of broiler chickens. [ABSTRACT FROM AUTHOR]

This article presents the general solution f:G→Vf:{\mathcal{G}}\to {\mathcal{V}} of the following functional equation: f(x)−4f(x+y)+6f(x+2y)−4f(x+3y)+f(x+4y)=0,x,y∈G,f\left(x)-4f\left(x+y)+6f\left(x+2y)-4f\left(x+3y)+f\left(x+4y)=0,\hspace{1.0em}x,y\in {\mathcal{G}}, where (G,+)\left({\mathcal{G}},+) is an abelian group and V{\mathcal{V}} is a linear space. We also investigate its Hyers-Ulam stability on some restricted domains. We apply the obtained results to present some asymptotic behaviors of this functional equation in the framework of normed spaces. Finally, we provide some characterizations of inner product spaces associated with the mentioned functional equation.

Over the past few decades, a lot of new neural network architectures and deep learning (DL)-based models have been developed to tackle problems more efficiently, rapidly, and accurately. For classification problems, it is typical to utilize fully connected layers as the network head. These dense layers used in such architectures have always remained the same – they use a linear transformation function that is a sum of the product of output vectors with weight vectors, and a trainable linear bias. In this study, we explore a different mechanism for the computation of a neuron's output. By adding a new feature, involving a product of higher order output vectors with their respective weight vectors, we transform the conventional linear function to higher order functions, involving powers over two. We compare and analyze the results obtained from six different transformation functions in terms of training and validation accuracies, on a custom neural network architecture, and with two benchmark datasets for image classification (CIFAR-10 and CIFAR-100). While the dense layers perform better in all epochs with the new functions, the best performance is observed with a quadratic transformation function. Although the final accuracy achieved by the existing and new models remain the same, initial convergence to higher accuracies is always much faster in the proposed approach, thus significantly reducing the computational time and the computational resources required. This model can improve the performance of every DL architecture that uses a dense layer, with remarkably higher improvement in larger architectures that incorporate a very high number of parameters and output classes. [ABSTRACT FROM AUTHOR]

In this paper we consider generalized monomial functions f , g : F → C (of possibly different degree) that also fulfill f (P (x)) = Q (g (x)) x ∈ F , where P ∈ F [ x ] and Q ∈ C [ x ] are given (classical) polynomials. [ABSTRACT FROM AUTHOR]

In this paper we give necessary conditions for quadratic functions f : R → R that satisfy the additional equation y 2 f (x) = x 2 f (y) under the condition x y = 1 . [ABSTRACT FROM AUTHOR]

In order to manage application of saline water in irrigated lands and coping with crisis of water resources scarcity in arid and semi-arid regions, it is logical to obtain the optimum yieldsalinity-water production functions for crops. In this way, a greenhouse experiment with three irrigation treatments including 100, 75 and 50 percent of water requirement and four salinity levels of irrigation water including 0.7, 4, 8, and 12 dS m-1, as a factorial in the form of a completely randomized block design with three replications was conducted in Kashmar region. The objective of this study was to determine the optimum yield-water-salinity production function of safflower. The crop production functions used in this study were linear, CobbDouglas, quadratic and transcendental functions. For assessing the accuracy and efficiency of these functions in predicting yield and yield components of safflower, the statistics of mean absolute error, normal root mean square error, determination coefficient, efficiency function and agreement index were used. Additionally, the plant salinity resistance threshold was determined by Mass-Hoffman linear model. The results showed among the applied functions, the quadratic function the lowest NRMSE and the highest R2, had the most efficiency in predicting yield and yield components of safflower in simultaneous salinity and drought stress conditions. According to Mass-Hoffman linear model, grain yield relative reduction by 10 percent, in salinity of 3.75 dS m-1 and grain yield relative reduction by 25 and 50 percent were occurred in salinity of 6.75 and 11.75 dS m-1, respectively. According to the findings of this research it can be concluded that the quadratic equation is recommendable as the best yieldwater-salinity production function of safflower in pot conditions. Since, crop production functions are dependent to soil type, weather and climate conditions and water management in the field, more comprehensive studies should be done in the field conditions for determining the best yield-water-salinity production function of safflower. [ABSTRACT FROM AUTHOR]

The optical reflection characteristics of asphalt pavement have an important influence on road-lighting design, and the macrotexture and microtexture of asphalt pavement significantly affect its reflection characteristics. To investigate the impact of texture parameters on the retroreflection coefficient of asphalt pavement, the texture indices of rutted plate specimens and field asphalt pavement were obtained by a pavement texture tester, including the macrotexture surface area (S1), microtexture surface area (S2), macrotexture distribution density (D1), microtexture distribution density (D2), root mean square slope (Δq), skewness (Rsk), and steepness (Rku). The corresponding retroreflective coefficient RL was measured by using a retroreflectometer. In the laboratory experiments, rutted specimens of AC-13, SMA-13, and OGFC-13 asphalt mixtures were formed. The changes in texture parameters and the retroreflection coefficient of rutting specimens before and after rolling were studied, and a factor-influence model between macro- and microtexture parameters and RL was established, along with correlation models of the texture index and RL. The results show that after the rutting test, S1, S2, D1, D2, Δq, and Rku decreased, Rsk increased, and RL increased. In the single-factor model, the parameters could be used to characterize RL with high prediction accuracy, whereas for the onsite measurements, the parameters Δq, Rsk, and Rku could well characterize RL. The nonlinear model established, based on the BP neural network algorithm, improved the prediction accuracy. This research provides ideas for optimizing the reflection characteristics of asphalt pavement and a decision-making basis for road-lighting design. [ABSTRACT FROM AUTHOR]

Literature shows a tendency to relegate the role of coefficient b to second place in the teaching of the quadratic function. We report an experience with Chilean university teachers, who designed a teaching and learning sequence with this function for construction engineering students. Our focus was on the didactic-mathematical knowledge about the effects of varying coefficient b on the graphical representation of this function that the participating teachers made evident. We constituted a focus group with 10 teachers and then qualitatively analyzed their dialogues using the mathematics teacher's didactic-mathematical knowledge and competencies model. We highlight the following results: (a) the importance of mathematical knowledge and that of the epistemic facet to interpret the effect of coefficient b on the graphical representation of the quadratic function and (b) the proposal of an interpretation for the graphical behavior of coefficient b that contributes to the teaching of the quadratic function. [ABSTRACT FROM AUTHOR]

This paper considers a class of systems called density systems. For such systems, the derivative of a quadratic function depends on some function termed the density function. The latter function is used to define the properties of the space affecting the behavior of the systems under consideration. The role of density systems in control law design is shown. Control systems are constructed for plants with known and unknown parameters. The theoretical results are illustrated by numerical simulation. [ABSTRACT FROM AUTHOR]

Visual reasoning plays an important role in problem-solving and the success of the student’s learning mathematics. This study aims to reveal the students’ visual reasoning ability in solving quadratic function problems in terms of activist, reflector, and pragmatic learning styles. This research used a qualitative approach with a case study were 30 9th-Grade students at one of the public junior high schools in Karanganyar District, Central Java, Indonesia. The Learning Style Questionnaire (LSQ) from Honey and Mumford was used to categorize the students’ learning style. In each category, the researcher selected two students purposively to be interviewed. The researchers used two quadratic equation problems to explore the students’ visual reasoning profile. Before being used, the questions were validated by two experts in mathematics education. The result showed that students with activist learning styles unable to demonstrate all indicators of visual reasoning. The students with pragmatics learning style can exhibit only one indicators of reasoning, namely interpretation. Then, students with reflector and theoretical learning style able to perform all indicators of visual reasoning. Thus, it can be concluded that the student’s learning style affect the visual reasoning ability in solving quadratic function.

This Collaborative Classroom Action Research (CAR) aims to improve learning outcomes in Quadratic Function material using the cooperative learning type TGT (Teams, Games, and Tournament) which assisted by GeoGebra and Quizizz for class X students of SMA N 3 Salatiga. This CAR is carried out in 2 cycles based on the Kemmis and Taggart Theories which consist of 4 stages namely the planning, implementation, observation, and reflection. The subjects of this study were 34 students of class X SMA N 3 Salatiga. The data collection technique of this research uses observation and tests. The data were analyzed using descriptive and quantitative analysis. The results showed that there was an increase in the average value, in the pre-cycle was 77.7; cycle I was 81.6; and cycle II was 83.4. The percentage of classical completeness also shows an increase, students who have scored above KKM (75) in the pre-cycle was 58.06% (18 students), cycle I was 81.82% (27 students), and cycle II was 87.88%. (29 students).

This article investigates the stability analysis of discrete‐time neural networks with time‐varying delays by the utilization of quadratic delay information. First, three extended negative‐definiteness lemmas for matrix‐valued quadratic function with different matrices injection are established. Second, a novel delay‐product‐type Lyapunov functional with the asymmetric summation is developed to relax the positive‐definiteness of functional. Then, the proposed negative definite approaches are utilized in combination with some typical summation inequalities to realize the construction of linear matrix inequalities. Based on these improved technologies, two delay‐dependent stability criteria with less conservatism and fewer computational burdens are derived. Finally, several numerical examples are presented to show the validity and superiority of the proposed methods. [ABSTRACT FROM AUTHOR]

Almost perfect nonlinear functions possess optimal resistance to differential cryptanalysis and are widely studied. Most known APN functions are defined using their representation as a polynomial over a finite field and very little is known about combinatorial constructions of them on F 2 n . In this work we propose two approaches for obtaining quadratic APN functions on F 2 n . The first approach exploits a secondary construction idea, it considers how to obtain a quadratic APN function in n + 1 variables from a given quadratic APN function in n variables using special restrictions on the new terms. The second approach is searching for quadratic APN functions that have a matrix representation partially filled with the standard basis vectors in a cyclic manner. This approach allows us to find a new APN function in 7 variables. We prove that the updated list of quadratic APN functions in dimension 7 is complete up to CCZ-equivalence. Also, we observe that the quadratic parts of some APN functions have a low differential uniformity. This observation allows us to introduce a new subclass of APN functions, the so-called stacked APN functions. These are APN functions of algebraic degree d such that eliminating monomials of degrees k + 1,..., d for any k < d results in APN functions of algebraic degree k. We provide cubic examples of stacked APN functions for dimensions up to 6. [ABSTRACT FROM AUTHOR]

The educational product presented here is the result of a research project entitled "Teaching mathematics to blind people using Monet software: Creating tactile graphics for the teaching of quadratic functions," conducted within the scope of the Professional Master's Program in Teaching Sciences, Mathematics, and Technologies at UDESC. This product consists of an activity booklet that provides representations of the knowledge object "quadratic function" in algebraic, graphic, tabular, and natural language formats, accessible through touch and vision. This accessibility is made possible by the simultaneous and synchronized presence of Braille and ink writings, as well as embossed graphics and tables. The goal of this product is to offer blind high school students’ tactile access to representations of the mentioned function while also serving as a support material for teachers, even if they are not proficient in the Braille System. Constructed with the Braille Fácil and Monet software and printed in Braille, these representations underwent evaluation and approval by two blind experts who serve as Braille text reviewers. The experimentation of the activities involved the participation of two blind students. The research was grounded in Duval's Theory of Semiotic Representation Registers and adhered to the criteria established by Cerqueira and Ferreira for the production of tactile materials. In this qualitative work, with a case study as the methodological approach, it was concluded that an appropriate material, combined with the "Global Interpretation Procedure," represents an integral mechanism for blind individuals to analyze representations of the quadratic function through their remaining senses. [ABSTRACT FROM AUTHOR]

The optical reflection characteristics of asphalt pavement have an important influence on road-lighting design, and the macrotexture and microtexture of asphalt pavement significantly affect its reflection characteristics. To investigate the impact of texture parameters on the retroreflection coefficient of asphalt pavement, the texture indices of rutted plate specimens and field asphalt pavement were obtained by a pavement texture tester, including the macrotexture surface area (S1), microtexture surface area (S2), macrotexture distribution density (D1), microtexture distribution density (D2), root mean square slope (Δq), skewness (Rsk), and steepness (Rku). The corresponding retroreflective coefficient RL was measured by using a retroreflectometer. In the laboratory experiments, rutted specimens of AC-13, SMA-13, and OGFC-13 asphalt mixtures were formed. The changes in texture parameters and the retroreflection coefficient of rutting specimens before and after rolling were studied, and a factor-influence model between macro- and microtexture parameters and RL was established, along with correlation models of the texture index and RL. The results show that after the rutting test, S1, S2, D1, D2, Δq, and Rku decreased, Rsk increased, and RL increased. In the single-factor model, the parameters could be used to characterize RL with high prediction accuracy, whereas for the onsite measurements, the parameters Δq, Rsk, and Rku could well characterize RL. The nonlinear model established, based on the BP neural network algorithm, improved the prediction accuracy. This research provides ideas for optimizing the reflection characteristics of asphalt pavement and a decision-making basis for road-lighting design.

Let G be an abelian group, A a C ∗ -algebra and M a pre-Hilbert A -module with an A -valued inner product ⟨. ,. ⟩ . We show if a function f : G → M satisfies the inequality ⟨ f (x) + f (y) , f (x) + f (y) ⟩ ≤ ⟨ f (x + y) , f (x + y) ⟩ , x , y ∈ G , then f is additive. We also prove that for functions f : G → M , the inequality ⟨ 2 f (x) + 2 f (y) - f (x - y) , 2 f (x) + 2 f (y) - f (x - y) ⟩ ≤ ⟨ f (x + y) , f (x + y) ⟩ , x , y ∈ G , implies f is quadratic. These results enable us to prove the equivalence of a functional inequality and the Drygas functional equation. In addition, we investigate the stability problem associated with these functional inequalities. Finally, we give some examples of quadratic and Drygas functions. [ABSTRACT FROM AUTHOR]

This paper describes and examines students' shared construction of meanings while learning about quadratic functions via digital artifacts that simulate real-world phenomena, like the motion of a ball on an inclined plane, focusing specifically on the role of the teacher in that construction process. The study follows the interactions of twenty 15-year-old students and their teacher during the completion of three sequential digital tasks, analyzing how these interactions promote the students' ability to construct the mathematical meanings of the quadratic function. The study was guided by the theory of semiotic mediation, which treats artifacts as fundamental to cognition and views learning as the evolution from meanings connected to the use of a certain artifact to those recognizable as mathematical. The data analysis showed students progressing from the description of the real-world phenomenon toward a construction of the meaning of the quadratic function that models the phenomenon. While marking the "critical moments" in this progress, we analyze the communication strategies used by the teacher to facilitate it. Our research findings showed evidence that certain types of questions and the strategy "re-voicing" can be particularly effective in prompting students' construction of mathematical meanings. [ABSTRACT FROM AUTHOR]

Being able to understand a written text is an essential life skill, as is solving a problem, which configures a bidirectional relationship between both skills. With regard to reading comprehension, at least three specific levels are required to achieve a full understanding of a text: literal, inferential, and critical. For its part, mathematics has changed substantially in recent decades, prioritizing problem-solving as the central axis of the teaching process as it constitutes a resource for learning. This study connects both skills and aims to design and validate a mathematical problem-solving test contextualized to the quadratic function, to assess the level of reading comprehension of secondary school students. The study is nonexperimental, cross-sectional, and focused on psychometric aspects since it aims to validate a measurement instrument. It was applied on a pilot basis to three municipal secondary education courses in Chile. The content, construct, discrimination, and reliability validation tests carried out allowed us to demonstrate that the problem-solving test contextualized to the quadratic function is a reliable instrument to produce knowledge in educational research. [ABSTRACT FROM AUTHOR]

This paper focuses on portfolio selection problem solving through the application of some evolutionary algorithms such as genetic algorithms (GA), particle swarm (PS) and differential evolution (DE). Therefore we focus mainly on the performance of genetic algorithms compared to the other two techniques on data collected from the Tunis Stock Exchange over a period of 5 years. We find in this context that genetic algorithms are similarly more efficient in choice of the optimal of portfolio compared to PS and DE. [ABSTRACT FROM AUTHOR]

The aim of this article is to introduce a new conjugate direction method (CDM) which satisfies a conjugate and descent property to solve an unconstrained optimization problem. The objective function is differentiable and the gradient (first derivative of the objective function) is Lipchitz continuous. The new direction is computed based on the last direction with a new parameter as a coefficient of the next gradient of the given objective function. We consider Numerical examples, with the help of the MATLAB program, using the new method compared with the Fletcher-Reeves conjugate gradient (F-R-CGM) method to verify the effectiveness of the new CDM.Our results are reported as a table containing total iterations and function evaluation. [ABSTRACT FROM AUTHOR]

This paper takes a typical crescent-shaped sand dune in the Taklamakan Desert, the largest desert in China, as the object of study. The mathematical model of the crescent-shaped dune is established by the integral method through two calculation methods of quadratic and catenary functions. We have deconstructed a two-dimensional top view of the crescent-shaped dune volume and proposed a formula for calculating the volume of the crescent-shaped dune. Methodological guidance is provided for the field measurement and estimation of the volumetric shape of crescent-shaped sand dunes. The purpose of this study is to estimate the volume of crescent dunes by using the basic survey tool and simple function integration method. This paper proposes two methods for estimating the volume of the same dune using elementary functions. After calculation, the dune volume to be studied is about 14,000m3, and the difference between the above two calculation methods is about 630m3. This method is different from the previous use of computer scanning to obtain the volume parameter information. Instead, it focuses on the reality that the field survey cannot use the large-calculate computer, and reduces the calculation steps and difficulties. It is an effective new method for field survey and estimation of crescent dune volume. It also provides a feasible algorithm for monitoring sediment transport in crescent dunes by real-time video. [ABSTRACT FROM AUTHOR]

Abstract The main aim of this paper is to establish the Hyers–Ulam stability and hyperstability of a Jensen-type quadratic mapping in 2-Banach spaces. That is, we prove the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form g ( x + y 2 + z ) + g ( x + y 2 − z ) + g ( x − y 2 + z ) + g ( x − y 2 − z ) = g ( x ) + g ( y ) + 4 g ( z ) , $$ g \biggl( \frac{x+y}{2} + z \biggr) + g \biggl( \frac{x+y}{2} - z \biggr) + g \biggl( \frac{x-y}{2} + z \biggr) + g \biggl( \frac{x-y}{2} - z \biggr) = g(x) + g(y) + 4 g(z), $$ in 2-Banach spaces by using the Hyers direct method.

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This study investigates the way students perceive and express change in an exponential situation with different reasoning methods for continuous change - for example, students, who use chunky reasoning, construct a new function representing the change of exponential distance-time function. Using case study approach, the study presents the results of teaching experiments conducted on three high school freshmen students. It may be difficult to generalize the results since the sample size was limited; however, the findings have the potential to inform researchers in mathematics education about students’ conceptual knowledge of continuous change. The information about the process of constructing the speed-time function from the distance-time function based on the students' understanding of continuous change may also be helpful in interpreting student thinking in differential concept.

Abstract In this paper, we introduce the functional equations f ( 2 x − y ) + f ( x + 2 y ) = 5 [ f ( x ) + f ( y ) ] , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + 4 f ( y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 5 f ( x ) + f ( 2 y ) + f ( − y ) , f ( 2 x − y ) + f ( x + 2 y ) = 4 [ f ( x ) + f ( y ) ] + [ f ( − x ) + f ( − y ) ] . $$\begin{aligned} f(2x-y)+f(x+2y)&=5\bigl[f(x)+f(y)\bigr], \\ f(2x-y)+f(x+2y)&=5f(x)+4f(y)+f(-y), \\ f(2x-y)+f(x+2y)&=5f(x)+f(2y)+f(-y), \\ f(2x-y)+f(x+2y)&=4\bigl[f(x)+f(y)\bigr]+\bigl[f(-x)+f(-y)\bigr]. \end{aligned}$$ We show that these functional equations are quadratic and apply them to characterization of inner product spaces. We also investigate the stability problem on restricted domains. These results are applied to study the asymptotic behaviors of these quadratic functions in complete β-normed spaces.

In realistic power system state estimation, the distribution of measurement noise is usually assumed to be Gaussian while many researcher have verified that it can be non-Gaussian. In this paper, a new robust state estimator based on exponential absolute value function is proposed to address the non-Gaussian measurement noise and outliers. The influence function, a robust statistics tool, is used to obtain the state estimates to reduce its computational burden. A state estimation mean squared error formula of the proposed robust estimator is derived which can be used as a reference in the wide area monitoring system design or upgrade. Simulation results obtained from the IEEE 30-bus, 118-bus and 300-bus systems verify the effectiveness and robustness of the proposed robust estimator. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to establish the Hyers–Ulam stability and hyperstability of a Jensen-type quadratic mapping in 2-Banach spaces. That is, we prove the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form g (x + y 2 + z) + g (x + y 2 − z) + g (x − y 2 + z) + g (x − y 2 − z) = g (x) + g (y) + 4 g (z) , in 2-Banach spaces by using the Hyers direct method. [ABSTRACT FROM AUTHOR]

Let f be a function from an Abelian normed group G to an inner product space E such that f(0) = 0. It is proved that if f satisfies the functional inequality ‖ 2 f (x) + f (y) + f (- y) - f (x - y) ‖ ⩽ ‖ f (x + y) ‖ , ‖ x ‖ + ‖ y ‖ ⩾ d for some d > 0 , then f is a solution of the Drygas functional equation. The equivalence of the functional inequality ‖ f (x) + f (y) ‖ ⩽ ‖ f (x + y) ‖ and additive functional equation f (x + y) = f (x) + f (y) has also been investigated under some restricted conditions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of j-hom-derivation, j ∈ {1, 2} and solve the new generalized additive-quadratic functional equations in the sense of ternary Banach algebras. Moreover, using the fixed point method, we prove its Hyers-Ulam stability. [ABSTRACT FROM AUTHOR]

Copyright of Educación y Humanismo is the property of Universidad Simon Bolivar and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The purpose of this study is to investigate how two 7th grade students with different concepts of ratio reason about change in covarying quantities in quadratic function problems and how these differences affect the process of expressing algebraic equations. To this end, we selected two 7th grade students who had no formal instruction in quadratic functions and four clinical interviews (each of which lasted about 90 minutes) were conducted every week from December 2019 to January 2020 with each of them. Our analysis revealed that one student whose conception of ratio was at the level of additive comparison attended to differences in amount of change in quadratic problem situations and interpreted the problems additively by repetitively adding the differences, whereas another student who operated at the interiorized ratio level could perceive the problem situations as the differences in amount of change per unit time, which resulted in their distinctive ways of expressing quadratic functions as equations. We have finalized our study by suggesting an explanatory hypothesis about students' ratio concepts that seem highly pertinent to their ability to interpret quantitatively and to express equations of quadratic functions.

For agronomic and agricultural economic modelling and analysis, a good representation of response functions is vital. In this article, the perspective is considered by different regression methods. This is due to the fact that if least squares regression is used, observations (outliers) might lead to misleading conclusions. Robust regression approaches such as M estimation, MM estimation, S estimation, and LTS estimation are used to solve this problem since they are unaffected by outliers. The quadratic function (QF) and Square root function (SF) are used to study the production as a response variable with labour and capital as explanatory variables of the maize data. The use of robust regression rather than least square regression results in more appropriate shifts in coefficient estimates and their degree of significance, as well as higher levels of goodness of fit. The proposed robust method after handling the outliers having the same precision of M estimation. The QF used to study the maize production found to be best as compared to the SF on the basis of cross validation techniques. [ABSTRACT FROM AUTHOR]

In the following manuscript we will show as a starting point a theoretical analysis of the gradient method, known as one of the first descent methods, and from this we will identify the strength of the conjugate gradient methods. Taking an objective function, we will determine the values that optimize it by means of different methods, indicating the differences of geometric type that these have. Different systems will be used, in order to serve as a test, obtaining their solution in each case and finding the speed at which they converge in accordance with the conjugate gradient methods proposed by Hestenes-Stiefel and Fletcher-Reeves. [ABSTRACT FROM AUTHOR]

To obtain chaotic systems with good properties to help chaos based image encryption, this paper first study the condition under which quadratic functions can be used as pseudo-random sequence generator. We give a necessary and sufficient condition for quadratic functions topologically conjugate with the most frequently used Logistic map and Tent map. Since topological conjugacy maintains chaotic characteristics, more quadratic functions can be used as pseudo-random sequence generators in the digital image encryption. Then, it analyzes a diffusion algorithm with a good effect but poor in resisting the chosen plaintext attack. It is also verified by simulation. The quadratic function chaotic system is applied in the improvement of the diffusion algorithm. Furthermore, the improved algorithm contains plaintext related and cyclic shift operations. Finally, the performance analysis shows that the improved scheme has good security and high efficiency.

In this paper, novel geometry-based negative conditions are presented for a quadratic function with time-varying delays. The system performance could be optimal by adjusting a parameter value. New stability criteria for the time-varying delayed systems are derived.

Being able to understand a written text is an essential life skill, as is solving a problem, which configures a bidirectional relationship between both skills. With regard to reading comprehension, at least three specific levels are required to achieve a full understanding of a text: literal, inferential, and critical. For its part, mathematics has changed substantially in recent decades, prioritizing problem-solving as the central axis of the teaching process as it constitutes a resource for learning. This study connects both skills and aims to design and validate a mathematical problem-solving test contextualized to the quadratic function, to assess the level of reading comprehension of secondary school students. The study is nonexperimental, cross-sectional, and focused on psychometric aspects since it aims to validate a measurement instrument. It was applied on a pilot basis to three municipal secondary education courses in Chile. The content, construct, discrimination, and reliability validation tests carried out allowed us to demonstrate that the problem-solving test contextualized to the quadratic function is a reliable instrument to produce knowledge in educational research.

This study was conducted determine the effect of groundnut shell (GNS) on growth rate, body condition score (BCS) and blood mineral status of confined unsexed mixed cattle breed during the winter season in North West province, South Africa. Thirty-five cattle were divided into five groups (T1, T2, T3, T4, and T5) and each group assigned to GNS at 0, 500, 700, 1000 and 1200 gkg-1 feed in a completely randomized design. Cattle were fed basal diet (blue buffalo grass; BBG) and results showed that BBG and GNS were moderate in nutrients. Cattle on diet T3 had improved (p<0.05) live weight, average daily gain and BCS. Animals on diets T3 and T4 had increased (p<0.05) plasma magnesium and cobalt value than those on diets T1 and T2. Cattle offered diet T3 had increased (p<0.05) plasma manganese, iron, carbon, copper and chromium concentrations when compared to those on other four diets. GNS supplementation had no effect (p>0.05) on plasma zinc and selenium. GNS supplementation had quadratic effect (p<0.05) on plasma minerals. It is concluded that GNS contains moderate amount of essential minerals and therefore suitable for use in cattle production during the winter season in North West Province of South Africa. [ABSTRACT FROM AUTHOR]

Boolean functions have very nice applications in coding theory and cryptography. In coding theory, Boolean functions have been used to construct linear codes in different ways. The objective of this paper is to construct binary linear codes with few weights using the defining-set approach. The defining sets of the codes presented in this paper are defined by some special Boolean functions and some additional restrictions. First, two families of binary linear codes with at most three or four weights from Boolean functions with at most three Walsh transform values are constructed and the parameters of their duals are also determined. Then several classes of binary linear codes with explicit weight enumerators are produced. Some of the binary linear codes are optimal or almost optimal according to the tables of best codes known maintained at http://www.codetables.de, and the duals of some of them are distance-optimal with respect to the sphere packing bound. [ABSTRACT FROM AUTHOR]

E.G. Belousov and V.G. Andronov [Solvability and Stability of Problems of Polynomial Programming (in Russian), Publishing House of the Moscow University, Moscow, p. 4, 1993] have observed that the notions of structural convexity and ravine of a polynomial function, which were introduced by themselves, are useful for studying stability and solvability of convex polynomial mathematical programming problems, as well as for the investigation of the distribution of integer points in convex sets. The results given in Chapters 3-5 of the book justify their observations. This paper presents some facts about structural convexity and ravines of quadratic functions. Among other things, we obtain a verifiable criterion for structural convexity of a quadratic function and show that such a function cannot have ravines along linear subspaces. [ABSTRACT FROM AUTHOR]

This paper probes into the synchronization for memristor-based hybrid neural networks via nonlinear coupling. At first, a new condition is established to judge whether quadratic functions are negative or not on a closed interval regardless of their concavity or convexity. Then, by utilizing Legendre orthogonal polynomials, a recent extended integral inequality with free matrices is popularized to get tighter lower bound of some integral terms. Next, based on a novel Lyapunov functional, by applying our new integral inequality with free matrices, linear convex combination method and the new criterion, a new delay-dependent condition is gained to reach the global synchronization for the considered neural networks. At last, an example is presented to account for the validity of our results. [ABSTRACT FROM AUTHOR]

Mathematical representation has an essential role in solving mathematical problems. However, there are still many mathematics education students who have difficulty in representing ill-structured problems. Even though the ill-structured-problem-solving tasks designed to help mathematics education students understand the relevance and meaningfulness of what they learn, they also are connected with their prior knowledge. The focus of this research is exploring the used of mathematical representations in solving ill-structured problems involving quadratic functions. The topic of quadratic functions is considered necessary in mathematics teaching and learning in higher education. It's because many mathematics education students have difficulty in understanding these matters, and they also didn’t appreciate their advantage and application in daily life. The researchers' explored mathematical representation as used by two subjects from fifty-four mathematics education students at the University of Nusantara PGRI Kediri by using a qualitative approach. We were selected due to their completed all steps for solving the ill-structured problem, and there have different ways of solving these problems. Mathematical representation explored through an analytical framework of solving ill-structured issues such as representing problems, developing alternative solutions, creating solution justifications, monitoring, and evaluating. The data analysis used technique triangulation. The results show that verbal and symbolic representations used both subjects to calculate, detect, correct errors, and justify their answers. However, the visual representation used only by the first subject to detect and correct errors.

This study aims to describe the analysis of students' critical thinking abilities with high mathematics abilities in solving functional problems at SMAN 1 Pamekasan. This research was conducted at SMAN 1 Pamekasan with the research subjects of 3 students with high math abilities from 36 students of class X-F. This research is a case study with a qualitative approach. Data is collected through test methods, interviews, and documentation. The results of the study explain that in solving the function problems in the form of a story about the linear function of the three subjects able to fulfill all indicators of critical thinking, namely interpretation, analysis, evaluation, inference, explanation, and self-regulation. However, the three subjects have different critical thinking abilities when solving problems in the form of stories about quadratic function. The difference is the first subject meets all indicators of critical thinking, the second subject meets all critical thinking indicators except inference and self-regulation indicators, while the third subject only fulfills evaluation and explanation indicators. Information about the description of students' critical thinking abilities with high mathematics ability can be used as information to design the right learning process in the world of education.