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Considering the basic role of numbers in Mathematics, Science, and Technology the author developed a new structure of numbers named as ‘Theory of Dynamics of Numbers.’ According to the Theory of Dynamics of Numbers, the author defined 0 (zero) is the starting point of any number and also defined 0 (zero) as a neutral number. The numbers can move in infinite directions from the starting point 0 (zero) and back to 0 (zero). The author has defined the three types of numbers: 1) Neutral Numbers, 2) Count Up Numbers, and 3) Count Down Numbers. These three types of numbers cover the entire numbers in the number system where there is no necessity for the concept of imaginary numbers. Introducing this new concept the author solved the quadratic equation in one unknown (say x) in the form ax2 + bx + c = 0, even if the numerical value of the discriminant b2 – 4ac < 0 in real numbers without using the concept of imaginary numbers. Already the author solved the quadratic equation x2 + 1 = 0 and proved that √ -1 = -1 by using the Theory of Dynamics of Numbers. The Theory of Dynamics of Numbers is a more powerful tool than that of the real and imaginary number system to explain the truth of nature.

Abstract Estimation of the traditional transverse electric (TE) and transverse magnetic (TM) impedances of the magnetotelluric tensor for two-dimensional structures can be decoupled from the estimation of the strike direction with significant implications when dealing with galvanic distortions. Distortion-free data are obtainable by combining a quadratic equation with the phase tensor. In the terminology of Groom–Bailey, the quadratic equation provides amplitudes and phases that are immune to twist, and the phase tensor provides phases immune to both, twist and shear. On the other hand, distortion-free strike directions can be obtained using Bahr's approach or the phase tensor. In principle, this is all that is needed to proceed to a two-dimensional (2D) interpretation. However, the resulting impedances are strike ignorant because they are invariant under coordinate system rotation, and if they are to be related to a geological strike, they must be linked to a particular direction. This is an additional ambiguity to the one of 90° arising in classic strike-determination methods, which must be resolved independently. In this work, we use the distortion model of Groom–Bailey to resolve the ambiguity by bringing back the coupling between impedances and strike in the presence of galvanic distortions. Our approach is a hybrid between existing numerical and analytical methods that reduces the problem to a binary decision, which involves associating the invariant impedances with the correct TE and TM modes. To determine the appropriate association, we present three algorithms. Two of them require optimizing the fit to the data, and the third one requires a comparison of phases. All three keep track of possible crossings of the phase curves providing a clear-cut solution. Synthetic and field data illustrate the performance of the three schemes. Graphical Abstract

A unified method for solving general polynomial equation of degree less than five that incorporate a computational formula that relate the coefficients of the depressed equation and the coefficient of the standard polynomial equation is proposed in this study. This is to ensure that this method is valid for all degree less than five. It shall apply the undetermined parameter method of auxiliary function to obtain solutions to these polynomial equations of degree less than five in one variable. In particular, the result of our work is a unification and improvement on the work of several authors in the sense that only applicable for the case of polynomial equation of degree one. Finally, our results improve and generalize the result by applying standard formula methods for solving higher degree polynomials. It is recommended that the effort should be made toward providing other variant methods that are simpler and friendly.

Students' low creative thinking skills and difficulty in drawing graphs from the given algebraic equations are the background of this study. The purpose of this study was to determine the mathematical creative thinking abilities between Android-based GeoGebra and ordinary learning. This study also investigated students’ responses to the application of Android-based GeoGebra. The research method used was quasi-experimental with a nonequivalent group pretest-posttest design. The instruments were a creative thinking test and a questionnaire. The subjects of this study were students of one of the junior high school in Cianjur selected by purposive sampling technique. The studied material was the quadratic equations as a part of algebra. The gain index data analysis technique used was the normality test and the Mann Whitney test while the questionnaire data responses were analyzed through the average percentage of students’ answers. As a result, the students' mathematical creative thinking ability with Android-based GeoGebra was better than ordinary learning. Also, the students provided positive responses to the application of Android-based GeoGebra.

The paper first reviews the pedagogic usefulness of the matrix graphic organizer. This usefulness is well established in the literature. The paper then shows how the matrix graphic organizer meets the pedagogic excellence criteria of executive function, goal-setting, and several educational hierarchies. This paper also points to the need to use a unified approach to pedagogy and assessment. Illustrative examples are provided using the topics quadratic equation and solving simultaneous linear systems.

Abstract This paper is devoted to studying some systems of quadratic differential and integral equations with Hadamard-type fractional order integral operators. We concentrate on general growth conditions for functions generating right-hand side of considered systems, which leads to the study of Hadamard-type fractional operators on Orlicz spaces. Thus we need to prove some properties of such type of operators. In contrast to the case of Caputo or Riemann–Liouville type of fractional operators, it is not a convolution-type operator, so we need to study some of their new properties. Some more general problems than systems of quadratic integral equations are also studied, and the results are new even in the context of a single integral equation with the Hadamard fractional operator. The paper concludes with illustrative examples.

A functional equation is called: generalized if all functional variables are pairwise different; ternary if all its functional variables are ternary; quadratic if each individual variable has exactly two appearances; quasigroup if its solutions are studied only on invertible functions. A length of a functional equation is the number of all its functional variables. A complete classification up to parastrophically primary equivalence of generalized quadratic quasigroup functional equations of the length three is given. Solution sets of a full family of representatives of the equivalence are found.

Fixed point theory has been used very successfully to obtain properties of solutions of integral and integro-differential equations of the form\[x(t) = g(t) +\int^t_0 A(t-s) v(t,s,x(s))ds\]because under general conditions the integral term may map bounded sets of continuous functions into equi-continuous sets. But quadratic integral equations have a coefficient of the integral terms of the form\[f(t,x(t))\int^t_0 A(t-s)v(t,s,x(s))ds\]which destroys the compactness of the map. Investigators have resorted to deep solutions often involving measures of non-compactness and Darbo's fixed point theorem. In an effort to obtain some elementary approaches, in this paper we develop an apparently new technique by showing that by using progressive contractions we can show conditions under which the product of two contractions is a contraction. We focus on integro-differential equations and use direct fixed point mappings which convert Lipschitz conditions into progressive contraction conditions.

In the work is given the solution of nonaxisymmetric stationary heat conduction problem for profiled polar-orthotropic annular plates with thermally insulated bases. The dependence of the thermophysical characteristics of the plate material of the temperature is taken into account. Temperature values are set on the contours of the annular plate: temperature T0∗ is constant on the internal contour, and on the outer contour on several arcs with length li (i = 1, k) – temperature is T1∗ (T1∗ > T0∗). The temperature distribution in such a plate is nonaxisymmetric. It is assumed that the radial λr and tangential λθ heat conduction coefficients are linearly dependent on the temperature T(r, θ): λr(T) = λ(0)r(1 - γT(r, θ)), λθ(T) = λ(0)θ(1 - γT(r, θ)) here the parameter γ > 1; the constants λ(0)r, λ(0)θ are determined experimentally at the primary temperature T0. The primary nonlinear differential heat equation is reduced to a linear differential equation of the 2nd kind in partial derivatives when a new function Z(r, θ) = [T(r, θ) - γ/2T2(r, θ)] is introduced in consideration.

Background: The study of differential growth in various animals suggests that a similar growth pattern occurs throughout the bilateral animals. This growth pattern is based on the assumption that a quadratic equation describes the relationship between two body measurements, yielding a quadratic parabola in a graphic presentation. Aim: Can human growth be studied by comparing body measurements? If the growth of one body part has a definite relation to the growth of another body part and if this relation can be expressed in a quadratic formula, then important conclusions can be reached. Subjects and methods: An official database of the mean measurements of the Belgian population has been used. Results: (1) The growth in human length is, from the beginning, constantly and negatively influenced by the growth in girth, so it is evident that growth has to stop; (2) The influence of the growth in girth is stronger in females, resulting in shorter females; (3) The growth of the human head is, from the beginning, constantly and negatively influenced by the growth in body length (both sexes show a very similar pattern); (4) Not all comparisons resulted in a quadratic parabola: the nipple distance is constantly at ∼24% of the thoracic girth in males and young females. Conclusion: The study of differential growth by using a quadratic parabola gives the answers to the questions “Why do we stop growing?” and “Why are women shorter than men?”.

This is an expository paper containing remarks on solutions to some functional equations of a form, that could be called of the radical type. Simple natural examples of them are the following two functional equations fn √xn + yn= f(x) + f(y),f n √xn + yn+ fn p|xn −yn|= 2f(x) + 2f(y) considered recently in several papers, for real functions and with given positive integer n, in connection with the notion of Ulam (or Hyers-Ulam) stability. We provide a general method allowing to determine solutions to them.

In this study, our goal is to apply a new fixed point method to prove the Hyers-Ulam-Rassias stability of a quadratic functional equation in normed spaces which are not necessarily Banach spaces. The results of the present paper improve and extend some previous results.

The aim of this research is developing student activity sheets (LKS) with model problem creating to improve students’ ability to think critically. This research is a developmental research with Plomp’s development model. This development model consists of three stages that are (1) preliminary research; (2) prototype phase; (3) assessment phase. The test subjects involved 27 students of class VIII of SMPN 1 Gambut. Based on the results of research from the design of LKS consists of cover, time allocation, learning objectives, content of LKS and assessment. Activities in the LKS consists of five stages of model problem creating that are learning objectives, problem context, create the problem, anticipate students’ solutions and reflect. The contents of the LKS consists questions that guide students to find mathematical concepts or principles, issues that need to be spent students and guide students to critical thinking that are basic clarification, basic for decision, inference and advanced clarification. The validations results from three validators showed that the LKS satisfied valid criteria. As a result on field testing in SMPN 1 Gambut students’ responses were positive. In addition, this learning instrument can increase students’ ability to think critically by 74%. Penelitian ini bertujuan untuk mengembangkan LKS dengan model problem creating yang dapat meningkatkan kemampuan berpikir kritis. Penelitian ini termasuk pengembangan dengan model pengembangan Plomp, yaitu (1) preliminary research (penelitian awal); (2) prototyping phase (fase pengembangan); (3) assessment phase (fase penilaian). Subjek uji coba melibatkan 27 siswa kelas VIII SMPN 1 Gambut. Desain LKS terdiri atas cover, alokasi waktu, tujuan pembelajaran, isi LKS, dan penilaian. Kegiatan dalam LKS terdiri atas lima tahapan model problem creating, yaitu menyampaikan tujuan pembelajaran, konteks masalah, menciptakan masalah, antisipasi jawaban, dan refleksi. Isi LKS terdiri dari pertanyaan-pertanyaan yang menuntun siswa menemukan konsep atau prinsip matematika, masalah yang perlu diselesaikan siswa, dan memandu siswa untuk berpikir kritis antara lain memberikan penjelasan dasar, membangun keterampilan dasar, menyimpulkan dan memberikan penjelasan lanjut. Hasil validasi LKS dari tiga validator menunjukkan bahwa LKS memenuhi kriteria valid. Berdasarkan uji coba lapangan di SMPN 1 Gambut respon siswa terhadap LKS positif. Selain itu, peningkatan kemampuan berpikir kritis siswa sebesar 74%.

We consider the stability problem for a class of functional equations related to the Popoviciu equation.

Background: Automated calculations by laboratory information system (LIS) are efficient and accurate ways of providing calculated laboratory test results. Due to the lack of established advanced mathematical functions and equation logic in LIS software, calculations beyond simple arithmetic functions require a tedious workaround. Free and bioavailable testosterone (BT) calculations require a quadratic solver currently unavailable as ready to use the function on most commercial LIS platforms. We aimed to develop a module within the Epic Beaker LIS to enable automatic quadratic equation solving capability and real-time reporting of calculated free and BT values. Materials and Methods: We developed and implemented an advanced calculation module from the ground up using existing basic calculation programming functions in the Epic Beaker LIS. A set of calculation variables were created, and mathematical logic and functions were used to link the variables and perform the actual quadratic equation based calculations. Calculations were performed in real-time during result entry events, and calculated results populated the result components in LIS automatically. Results: Free and BT were calculated using instrument measured results of total testosterone, sex hormone binding globulin, and/or serum albumin, by applying equations widely adopted in laboratory medicine for endocrine diseases and disorders. Calculated results in Epic Beaker LIS were then compared and confirmed by manual calculations using Microsoft Excel spreadsheets and scientific calculators to have no discrepancies. Conclusions: Automated calculations of free and BT were successfully implemented and validated, the first of such implementation for the Epic Beaker LIS platform, eliminating the need of offline manual calculations, potential transcription error, and with improved turnaround time. It may serve as a model to build similarly complex equations when the clinical need arises.

The computer network service provider has to observe, evaluate and plan continually its services in order to provide reliable systems. One of the efforts in planning process is forecasting. There are many methods in it, and this paper proposes to use Quadratic Equation Formula as a forecasting method. It is used because of the simplicity and it is suitable to solve problems with parabolic characteristic that can not be solved by linier computational method. It also compares characteristic of Quadratic Equation Formula with Linear Regresion forecasting.The result of this research shows that the computer network performance can be forecasted with Quadratic Equation Formula which has similar case characteristic with it. The standard deviation error of forecast value of the Quadratic Equation Formula is 370 of 3080 forecasted throughput maximum value. While the standard deviation error of forecast value of the linear regresion method is 282 of 3518 forecasted throughput maximum value. But in the forecasted throughput of linear regresion methhod, the output value will allways increase as the increasing of the input variable value. As the result, the error distribution calculation is not interesting anymore because there is no peak value for the forecasted result.