JOURNAL OF ADVANCES IN MATHEMATICS 2021-11-04T09:56:26+00:00 Editorial Office Open Journal Systems Possibilities of Preparing Pupils for Proof Problems of Synthetic Plane Geometry Solvable by Deductive Methods 2021-11-04T09:56:26+00:00 Vojtěch Zlámal <p>Proof problems, especially the ones of the synthetic plane geometry solvable by deductive methods, play a significant role in mathematical education and due to their demanding principle also in the above-standard education including mathematical competitions. Therefore, the issue of preparing pupils for solving the proof problems is very important. This study aimed to find out if the contemporary state of the system of pupils’ preparation for synthetic plane geometry proof problems is sufficient enough for the mentioned purpose. From the full set of schools of the Czech Republic, there were 14 schools identified as the successful ones based on the results of the national round of the Mathematical Olympiad. These schools were asked questions about literature used for pupils’ preparation and the publications named in the answers were then deeply inspected. The results showed a narrow range of the literature used by the schools and the didactic-methodical inspection of stated literature detected considerable space for improvements which led the author to the main theme of his dissertation.</p> 2021-12-20T00:00:00+00:00 Copyright (c) 2021 Vojtěch Zlámal Visualization In Mathematics Teaching 2021-10-30T07:07:42+00:00 Shivkant Tiwari Dragan Obradovic Laxmi Rathour Lakshmi Narayan Mishra Vishnu Narayan Mishra <p>In recent years, there has been an increased use of information and communication technologies and mathematical software in mathematics teaching. Numerous studies of the effectiveness of mathematical learning have shown the justification and usefulness of the implementation of new teaching aids. They also showed that learning with educational software has a great impact on students' achievement in the overall acquisition of mathematical knowledge during the school year as well as in the final exam at the end of primary education. Teaching realized by using computers and software packages is interesting for students, increases their interest and active participation. It is indisputable that the use of computers and mathematical software has great benefits that have been proven and presented in their works by many researchers of effective learning. It is also indisputable that one of the main tasks of teaching mathematics is to develop constructive thinking of students. Visualization and representation of mathematical laws are of great importance in the realization of mathematics teaching. They should be applied everywhere and whenever possible.</p> 2021-11-15T00:00:00+00:00 Copyright (c) 2021 Shivkant Tiwari, Dragan Obradovic, Laxmi Rathour, Lakshmi Narayan Mishra, Vishnu Narayan Mishra New Iterative Method with Application 2019-07-19T09:47:19+00:00 O. Ababneh N. Zomot <p>In this paper, we consider iterative methods to find a simple root of a nonlinear equation</p> <p><em>f(x) = 0</em>, where <em>f : D∈R</em><em>→</em><em>R</em> for an open interval <em>D</em> is a scalar function.</p> 2021-10-15T00:00:00+00:00 Copyright (c) 2021 O. Ababneh , N. Zomot Second Order Partial Derivatives 2021-07-23T07:50:43+00:00 Shikha Pandey Dragan Obradovic Lakshmi Narayan Mishra Vishnu Narayan Mishra <p>The rules for calculating partial derivatives and differentials are the same as for calculating the derivative of a function of one variable, except that when finding partial derivatives per one variable, the other variables are considered as constants</p> 2021-09-08T00:00:00+00:00 Copyright (c) 2021 Shikha Pandey, Dragan Obradovic, Lakshmi Narayan Mishra, Vishnu Narayan Mishra A survey of topics related to Functional Analysis and Applied Sciences 2021-06-14T11:42:59+00:00 Denise Huet <p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">This survey is the result of investigations suggested by recent publications on functional analysis and applied sciences. It contains short accounts of the above theories not usually combined in a single document and completes the work of D. Huet 2017. The main topics which are dealt with involve spectrum and pseudospectra of partial differential equations, Steklov eigenproblems, harmonic Bergman spaces, rotation number and homeomorphisms of the circle, spectral flow, homogenization. Applications to different types of natural sciences such as echosystems, biology, elasticity, electromagnetisme, quantum mechanics, are also presented. It aims to be a useful tool for advanced students in mathematics and applied sciences.</p> 2021-07-09T00:00:00+00:00 Copyright (c) 2021 Denise Huet Results on a faster iterative scheme for a generalized monotone asymptotically 2021-07-05T13:14:58+00:00 Athraa Najeb Abed Salwa Salman Abed II <p>This article devoted to present results on convergence of Fibonacci-Halpern scheme (shortly, FH) for monotone asymptotically α<sub>n-</sub>nonexpansive mapping (shortly, <em>ma α<sub>n</sub></em><em>-n</em> mapping) in partial ordered Banach space (shortly, POB space). Which are auxiliary theorem for demi-close's proof of this type of mappings, weakly convergence of increasing FFH-scheme to a fixed point with aid monotony of a norm and Σ<sub>n</sub><sup>+</sup><sub>=</sub><sup>∞</sup><sub>1 </sub>λ<sub>n</sub>= +∞, λ<sub>n </sub>=min{h<sub>n , </sub>(1-h<sub>n</sub>)} where h<sub>n </sub>⸦ (0,1) <span style="font-size: 0.875rem;"> where is associated with FH-scheme for an integer n&gt;0 more than that, convergence amounts to be strong by using Kadec-Klee property and finally, prove that this scheme is weak-<em>w</em><sup>2</sup> stable up on suitable status.</span></p> 2021-08-07T00:00:00+00:00 Copyright (c) 2021 Athraa Najeb Abed, Salwa Salman Abed II New types of almost contact metric submersions 2021-06-08T11:54:42+00:00 T.Tshikuna Matamba <p>We introduce the concept of conjugaison in contact geometry. This concept allows to define new structures which are used as base space of a Riemannian submersion. With these new structures, we study new three types of almost contact metric submersions.</p> 2021-08-07T00:00:00+00:00 Copyright (c) 2021 T.Tshikuna Matamba Metallic Ratios in Primitive Pythagorean Triples : Metallic Means embedded in Pythagorean Triangles and other Right Triangles 2021-07-12T04:04:19+00:00 Chetansing Rajput <p>The Primitive Pythagorean Triples are found to be the purest expressions of various Metallic Ratios. Each Metallic Mean is epitomized by one particular Pythagorean Triangle. Also, the Right Angled Triangles are found to be more “Metallic” than the Pentagons, Octagons or any other (n<sup>2</sup>+4)gons. The Primitive Pythagorean Triples, not the regular polygons, are the prototypical forms of all Metallic Means.</p> 2021-07-15T00:00:00+00:00 Copyright (c) 2021 Chetansing Rajput Metallic Ratios and Pascal’s Triangle : Triads of Metallic Means in the Pascal’s Triangle 2021-06-26T02:27:55+00:00 Chetansing Rajput <p>This paper introduces the close correspondence between Pascal’s Triangle and the recently published mathematical formulae those provide the precise relations between different Metallic Ratios. The precise correlations between various Metallic Means can be substantiated with Pascal’s Triangle, as described herein.</p> 2021-07-02T00:00:00+00:00 Copyright (c) 2021 Chetansing Rajput Metallic Ratio Triads, The Mathematical Relations between different Metallic Means, And Geometric Substantiation of Metallic Numbers with the Right Angled Triangles 2021-06-26T02:17:03+00:00 Chetansing Rajput <p>This paper synergize the newly discovered generalised geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper also illustrates the concept of the “TRIADS of Metallic Means”. &nbsp;The Metallic Means and their TRIADS can be geometrically substantiated, in an intriguing manner, as described in this paper.</p> 2021-07-02T00:00:00+00:00 Copyright (c) 2021 Chetansing Rajput Metallic Ratios and 3, 6, 9, The Special Significance of Numbers 3, 6, 9 in the Realm of Metallic Meanseans 2021-06-26T02:06:29+00:00 Chetansing Rajput <p>This work illustrates the intriguing relation between Metallic Means and the Numbers 3, 6 and 9. These numbers occupy special positions in the realm of Metallic Ratios, as elaborated herein.</p> 2021-07-02T00:00:00+00:00 Copyright (c) 2021 Chetansing Rajput Metallic Ratios, Pythagorean Triples & p≡1(mod 4) Primes : Metallic Means, Right Triangles and the Pythagoras Theorem 2021-06-26T07:26:25+00:00 Chetansing Rajput <p>This paper synergizes the newly discovered geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper illustrates the concept of the “Triads of Metallic Means”, and aslo the close correspondence between Metallic Ratios and the Pythagorean Triples as well as Pythagorean Primes.</p> 2021-07-02T00:00:00+00:00 Copyright (c) 2021 Chetansing Rajput Metallic Means : Beyond the Golden Ratio, New Mathematics and Geometry of all Metallic Ratios based upon Right Triangles, The Formation of the Triples of Metallic Means, And their Classical Correspondence with Pythagorean Triples and p≡1(mod 4) Primes, A 2021-06-03T03:45:40+00:00 Dr. Chetansing Rajput <p>This paper brings together the newly discovered generalised geometry of all Metallic Means and the recently published mathematical formulae those provide the precise correlations between different Metallic Ratios. The paper also puts forward the concept of the “Triples of Metallic Means”. This work also introduces the close correspondence between Metallic Ratios and the Pythagorean Triples as well as Pythagorean Primes. Moreover, this work illustrates the intriguing relationship between Metallic Numbers and the Digits 3 6 9.</p> 2021-06-05T00:00:00+00:00 Copyright (c) 2021 Dr. Chetansing Rajput Heat Dissipation Performance of Micro-channel Heat Sink with Various Protrusion Designs 2021-05-15T22:07:44+00:00 Siyuan Bai Khalil Guy Yuxiang Jia Weiyi Li Qingxia Li Xinyao Yang <p>This research will focus on studying the effect of aperture size and shape of the micro-channel heat sink on heat dissipation performance for chip cooling. The micro-channel heat sink is considered to be a porous medium with fluid subject inter-facial convection. Derivation based on energy equation gives a set of governing partial differential equations describing the heat transfer through the micro-channels. Numerical simulation, including steady-state thermal analysis based on CFD software, is used to create a finite element solver to tackle the derived partial differential equations with properly defined boundary conditions related to temperature. After simulating three types of heat sinks with various protrusion designs including micro-channels fins, curly micro-channels fins, and Micro-pin fins, the result shows that the heat sink with the maximum contact area per unit volume will have the best heat dissipation performance, we will interpret the result by using the volume averaging theorem on the porous medium model of the heat sink.</p> 2021-06-08T00:00:00+00:00 Copyright (c) 2021 Siyuan Bai, Khalil Guy, Yuxiang Jia, Weiyi Li, Qingxia Li, Xinyao Yang PDE boundary conditions that eliminate quantum weirdness: a mathematical game inspired by Kurt Gödel and Alan Turing 2021-05-21T05:12:22+00:00 Jeffrey Boyd <p>Although boundary condition problems in quantum mathematics (QM) are well known, no one ever used boundary conditions technology to abolish quantum weirdness. We employ boundary conditions to build a mathematical game that is fun to learn, and by using it you will discover that quantum weirdness evaporates and vanishes. Our clever game is so designed that you can solve the boundary condition problems for a single point if-and-only-if you also solve the “weirdness” problem for all of quantum mathematics. Our approach differs radically from Dirichlet, Neumann, Robin, or Wolfram Alpha. We define domain Ω in one-dimension, on which a partial differential equation (PDE) is defined. Point <strong>α</strong> on ∂Ω is the location of a boundary condition game that involves an off-center bi-directional wave solution called<strong> Æ</strong>, an “elementary wave.” Study of this unusual, complex wave is called the Theory of Elementary Waves (TEW). We are inspired by Kurt Gödel and Alan Turing who built mathematical games that demonstrated that axiomatization of all mathematics was impossible. In our machine quantum weirdness vanishes if understood from the perspective of a single point <strong>α</strong>, because that pinpoint teaches us that nature is organized differently than we expect.</p> 2021-06-05T00:00:00+00:00 Copyright (c) 2021 Jeffrey Boyd On the existence of positive solutions for a nonlinear elliptic class of equations in R2 and R3 2021-05-20T17:21:35+00:00 Jose Quintero <p>We study the existence of positive solutions for an elliptic equation in R<sup>N</sup> for N = 2, 3 which is related with the existence of standing (localized) waves and the existence of the ground state solutions for some physical model or systems in fluid mechanics to describe the evolution of weakly nonlinear water waves. We use a variational approach and the well-known principle of concentration-compactness due to P. Lions to obtain the existence of this type of solutions, even in the case that the nonlinear term g is a non-homogeneous function or an operator defined in H<sup>1</sup>(R<sup>N</sup>) with values in R.</p> 2021-06-04T00:00:00+00:00 Copyright (c) 2021 Jose Quintero Golden Ratio, Silver Ratio and other Metallic Means ; Geometric Substantiation of all Metallic Ratios 2021-05-06T06:53:21+00:00 Chetansing Rajput <p>This paper introduces the concept of special right angled triangles those epitomize the different Metallic Ratios. These right triangles not only have the precise Metallic Means embedded in all their geometric features, but they also provide the most accurate geometric substantiation of all Metallic Means. These special right triangles manifest the corresponding Metallic Ratios more holistically than the regular pentagon, octagon or tridecagon, etc</p> 2021-05-23T00:00:00+00:00 Copyright (c) 2021 Chetansing Rajput Metallic Means; The Geometric Substantiation of all Metallic Ratios 2021-04-27T15:48:09+00:00 Chetansing Rajput <p>This paper introduces certain new geometric aspects of the Metallic Ratios. Each Metallic Ratio is observed to be closely associated with a special right triangle, which provides the precise fractional expression of that Ratio. This work explicates the geometric substantiation of Metallic Ratio, on basis of the right angled triangle which is the quintessential form of that particular ratio.</p> 2021-05-23T00:00:00+00:00 Copyright (c) 2021 Chetansing Rajput Metallic Ratios : The Mathematical Relationships between different Metallic Means 2021-04-22T18:22:59+00:00 Chetansing Rajput <p>This paper introduces the precise mathematical relationships between different Metallic Ratios. Certain distinctive mathematical correlations are found to exist between various Metallic Means. This work presents the empirical formulae those provide the exact mathematical relationships between different Metallic Ratios. &nbsp;&nbsp;</p> 2021-05-23T00:00:00+00:00 Copyright (c) 2021 Chetansing Rajput Expectation of Rice Pod Production in Iraq by Using Time Series 2021-04-27T10:38:08+00:00 Gorgees Shaheed Mohammad <p>The research aims to shed light on the reality of the production of Rice pods in Iraq during the period of time (1943-2019) and its development with time, then predict the production of Rice pods based on three Models of prediction Models, which are the time regression Model on production, in addition to studying the effect of harvested area on production quantities. Then forecasting the production of the Rice pods according to the Model of the regression of the harvested area on the production, the Autoregression Model, and the integrative moving averages (Box Jenkins Models), and in the end the comparison between the expected values ​​of production through the three Models to know the best Model to represent the time series of production of the Rice pods , through the use of the statistical program (SPSS (, Based on annual secondary data represented by the quantities of Rice pods, and the size of the harvested areas of this material in Iraq for the period from 1945 until 2019 obtained from (Central Statistical Organization, Iraq, 2020)</p> 2021-05-07T00:00:00+00:00 Copyright (c) 2021 Gorgees Shaheed Mohammad Some Properties of Chaotic Modified of Bogdanov Map 2021-04-19T05:57:44+00:00 Wafaa H Al-Hilli Rehab Amer Kamel <p>In this research to the modified dynamics of Bogdanov's map studied, and the found sensitivity to the initial conditions of the modified map found as well as the Lyapunov exponent .the general characteristics of the map by the diffeomorpism. Finally we boosted my research with matlab to find chaotic areas</p> 2021-05-07T00:00:00+00:00 Copyright (c) 2021 Wafaa H Al-Hilli, Rehab Amer Kamel Coefficient Bounds and Fekete-Szeg¨o inequality for a Certain Families of Bi-Prestarlike Functions Defined by (M,N)-Lucas Polynomials 2021-03-09T09:13:28+00:00 Najah Ali Jiben Al-Ziadi Abbas Kareem Wanas <p>In the current work, we use the (M,N)-Lucas Polynomials to introduce a new families of holomorphic and bi-Prestarlike functions defined in the unit disk O and establish upper bounds for the second and third coefficients of the Taylor-Maclaurin series expansions of functions belonging to these families. Also, we debate Fekete-Szeg¨o problem for these<br />families. Further, we point out several certain special cases for our results.</p> 2021-03-31T00:00:00+00:00 Copyright (c) 2021 Najah Ali Jiben Al-Ziadi, Abbas Kareem Wanas Coefficient Estimates for Some Subclasses of m-Fold Symmetric Bi-univalent Functions Defined by Linear Operator 2021-02-20T18:42:35+00:00 Dhirgam Allawy Hussein Hussein Sahar Jaafar Mahmood <p> The articles introduces and investigates "two new subclasses of the bi-univalent functions ." These are analytical functions related to the m-fold symmetric function and . We calculate the initial coefficients for all the functions that belong to them, as well as the coefficients for the functions that belong to a field where finding these coefficients requires a complicated method. Between the remaining results, the upper bounds for "the initial coefficients "are found in our study as well as several examples. We also provide a general formula for the function and its inverse in the m-field. A function is called analytical if it does not take the same values twice . It is called a univalent function if it is analytical at all its points, and the function is called a bi-univalent if it and its inverse are univalent functions together. We also discuss other concepts and important terms. .</p> 2021-03-25T00:00:00+00:00 Copyright (c) 2021 Dhirgam Allawy Hussein Hussein , Sahar Jaafar Mahmood Coefficient Bounds for a New Subclasses of Bi-Univalent Functions Associated with Horadam Polynomials 2021-02-12T05:37:21+00:00 Najah Ali Jiben Al-Ziadi <p>\In this work we present and investigate three new subclasses of the function class of bi-univalent functions in the open unit disk defined by means of the Horadam polynomials. Furthermore, for functions in each of the subclasses introduced here, we obtain upper bounds for the initial coefficients and . Also, we debate Fekete-Szegӧ inequality for functions belongs to these subclasses. </p> 2021-03-19T00:00:00+00:00 Copyright (c) 2021 Najah Ali Jiben Al-Ziadi A Variable Structural Control for a Hybrid Hyperbolic Dynamic System 2021-02-23T06:21:17+00:00 Xuezhang Hou <p>Abstract: In this paper, we are concerned with a hybrid hyperbolic dynamic system formulated by partial differential equations with initial and boundary conditions. First, the system is transformed to an abstract evolution system in an appropriate Hilbert space, and spectral analysis and semigroup generation of the system operator is discussed. Subsequently, a variable structural control problem is proposed and investigated, and an equivalent control method is introduced and applied to the system. Finally, a significant result that the state of the system can be approximated by the ideal variable structural mode under control in any accuracy is derived and examined.</p> 2021-03-11T00:00:00+00:00 Copyright (c) 2021 Xuezhang Hou Comparison among Some Methods for Estimating the Parameters of Truncated Normal Distribution 2020-12-12T08:59:51+00:00 Hilmi kittani Mohammad Alaesa Gharib Gharib <p>The aim of this study is to investigate the effect of different truncation combinations on the estimation of the normal distribution parameters. In addition, is to study methods used to estimate these parameters, including MLE, moments, and L-moment methods. On the other hand, the study discusses methods to estimate the mean and variance of the truncated normal distribution, which includes sampling from normal distribution, sampling from truncated normal distribution and censored sampling from normal distribution. We compare these methods based on the mean square errors, and the amount of bias. It turns out that the MLE method is the best method to estimate the mean and variance in most cases and the L-moment method has a performance in some cases.</p> 2021-03-07T00:00:00+00:00 Copyright (c) 2021 Hilmi kittani, Mohammad Alaesa, Gharib Gharib On Regional Boundary Gradient Strategic Sensors In Diffusion Systems 2021-01-26T05:41:28+00:00 Raheam Al-Saphory Ahlam Y Al-Shaya <p>This paper is aimed at investigating and introducing the main results regarding the concept of Regional Boundary Gradient Strategic Sensors (<em>RBGS-sensors</em> the in Diffusion Distributed Parameter Systems (<em>DDP-Systems</em> . Hence, such a method is characterized by Parabolic Differential Equations (<em>PDEs</em> in which the behavior of the dynamic is created by a Semigroup ( of Strongly Continuous type (<em>SCSG</em> in a Hilbert Space (<em>HS)</em> . Additionally , the grantee conditions which ensure the description for such sensors are given respectively to together with the Regional Boundary Gradient Observability (<em>RBG-Observability</em> can be studied and achieved . Finally , the results gotten are applied to different situations with altered sensors positions are undertaken and examined.</p> 2021-02-28T00:00:00+00:00 Copyright (c) 2021 Raheam Al-Saphory, Ahlam Y Al-Shaya Coincidence points in θ - metric spaceS 2020-12-03T05:01:36+00:00 Maha Mousa Salwa Salman Abed <p>In this paper, inspired by the concept of metric space, two fixed point theorems for α−set-valued mapping <em>T</em>:₳ → CB(₳), h θ (Tp,Tq) ≤ α(dθ(p,q)) dθ(p,q), where α: (0,∞) → (0, 1] such that α(r) &lt; 1, ∀ t ∈ [0,∞) ) are given in complete θ −metric and then extended for two mappings with R-weakly commuting property to obtain a common coincidence point.</p> 2021-02-14T00:00:00+00:00 Copyright (c) 2021 maha mousa, Salwa Salman Abed Squared prime numbers 2021-01-23T15:55:14+00:00 Dr Gunnar Appelqvist <p>My investigation shows that there is a regularity even by the prime numbers. This structure is obvious when a prime square is created. The squared prime numbers.</p> <p>1.<u> Connections in a prime square</u></p> <p>A <em>prime square </em>(or origin square) is defined as a square consisting of as many boxes as the origin prim squared. This prime settle every side of the square. So, for example, the origin square 17 have got four sides with 17 boxes along every side. The prime numbers in each of the 289 boxes are filled with primes when a prime number occur in the number series (1,2,3,4,5,6,7,8,9 and so on) and then is noted in that very box.</p> <p>If a box is occupied in the origin square A this prime number could be transferred to the corresponding box in a second square B, and thereafter the counting and noting continue in the first square A. Eventually we get two filled prime squares. Analyzing these squares, you leave out the right vertical line, representing only the origin prime number,</p> <p>When a square is filled with primes you subdivide it into four corner squares, as big as possible, denoted a, b, c and d clockwise. You also get a center line between the left and right vertical sides.</p> <p>Irrespective of what kind of constellation you activate this is what you find:</p> <ol> <li>Every constellation in the corner square a and/or d added to a corresponding constellation in the corner square b and/or c is<strong> evenly divisible with the origin prime</strong>.</li> </ol> <ol start="2"> <li>Every constellation in the corner square a and/or b added to a corresponding constellation in the corner square d and/or c is <strong><em>not</em> evenly divisible with the origin prime</strong>.</li> </ol> <ol start="3"> <li>Every reflecting constellation inside two of the opposed diagonal corner squares, possibly summarized with any optional reflecting constellation inside the two other diagonal corner squares, is <strong>evenly divisible with the origin prime squared</strong>. You may even add a reflection inside the center line and get this result.</li> </ol> <p>My <strong>Conjecture 1</strong> is that this applies to every prime square without end.</p> <p>&nbsp;</p> <ol start="2"> <li><u>A formula giving all prime numbers endless</u></li> </ol> <p>&nbsp;</p> <p>In the second prime square the prime numbers are always higher than in the first square if you compare a specific box. There is a mathematic connection between the prime numbers in the first and second square. This connection appears when you square and double the origin prime and thereafter add this number to the prime you investigate. A new higher prime is found after <em>n</em> additions.</p> <p>You start with lowest applicable prime number 3 and its square 3². Double it and you get 18. We add 18 to the six next prime numbers 5, 7, 11, 13, 17 and 19 in any order. After a few adds you get a prime and after another few adds you get another higher one. In this way you continue as long as you want to. The primes are creating themselves.</p> <p><strong>A formula giving all prime numbers is</strong>:</p> <p>&nbsp; 5+18×n, +18×n, +18×n … without end</p> <p>&nbsp; 7+18×n, +18×n, +18×n … without end</p> <p>11+18×n, +18×n, +18×n … without end</p> <p>13+18×n, +18×n, +18×n … without end</p> <p>17+18×n, +18×n, +18×n … without end</p> <p>19+18×n, +18×n, +18×n … without end</p> <p>The letter <em>n</em> in the formula stands for how many 18-adds you must do until the next prime is found.</p> <p>My <strong>Conjecture 2</strong> is that this you find every prime number by adding 18 to the primes 5, 7, 11, 13, 17 and 19 one by one endless.</p> <ol start="3"> <li><u>A method giving all prime numbers endless</u></li> </ol> <p>&nbsp;There is still a possibility to even more precise all prime numbers. You start a 5-number series derived from the start primes 7, 17, 19, 11, 13 and 5 in that very order. Factorized these number always begin with number 5. When each of these numbers are divided with five the quotient is either a prime number or a composite number containing of two or some more prime numbers in the nearby. By sorting out all the composite quotients you get all the prime numbers endless and in order.</p> <p>Every composite quotient starts with a prime from 5 and up, squared. Thereafter the quotients starting with that prime show up periodically according to a pattern of short and long sequences. The position for each new prime beginning the composite quotient is this prime squared and multiplied with 5. Thereafter the short sequence is this prime multiplied with 10, while the long sequence is this prime multiplied with 20.</p> <p>When all the composite quotients are deleted there are left several 5-numbers which divided with 5 give all prime numbers, and you even see clearly the distance between the prime numbers which for instance explain why the prime twins occur as they do.</p> <p>My <strong>Conjecture 3</strong> is that this is an exact method giving all prime numbers endless and in order.</p> 2021-02-14T00:00:00+00:00 Copyright (c) 2021 Gunnar Appelqvist Golden Ratio 2020-12-30T04:06:07+00:00 Dr. Chetansing Rajput <p>This paper introduces the unique geometric features of 1:2: right triangle, which is observed to be the quintessential form of Golden Ratio (φ). The 1:2: triangle, with all its peculiar geometric attributes described herein, turns out to be the real ‘Golden Ratio Triangle’ in every sense of the term. This special right triangle also reveals the fundamental Pi:Phi (π:φ) correlation, in terms of precise geometric ratios, with an extreme level of precision. Further, this 1:2: triangle is found to have a classical geometric relationship with 3-4-5 Pythagorean triple. The perfect complementary relationship between1:2: <strong> </strong>triangle and 3-4-5 triangle not only unveils several new aspects of Golden Ratio, but it also imparts the most accurate π:φ correlation, which is firmly premised upon the classical geometric principles. Moreover, this paper introduces the concept of special right triangles; those provide the generalised geometric substantiation of all Metallic Means.</p> 2021-01-17T00:00:00+00:00 Copyright (c) 2021 Dr. Chetansing Rajput On Pointwise Product Vector Measure Duality 2020-11-04T03:12:58+00:00 Levi Otanga Olwamba Maurice Oduor <p>This article is devoted to the study of pointwise product vector measure duality. The properties of Hilbert function space of integrable functions and pointwise sections of measurable sets are considered through the application of integral representation of product vector measures, inner product functions and products of measurable sets.</p> 2021-01-06T00:00:00+00:00 Copyright (c) 2021 Levi Otanga Olwamba, Maurice Oduor For the Fourier transform of the convolution in and D' and Z' 2020-11-29T11:42:41+00:00 Vasko Rechkoski Bedrije Bedzeti Vesna Manova Erakovikj <p>In this paper, we give another proof of the known lemma considering the Fourier transform of the convolution of a distribution and a function. Also, we give its application in the mentioned spaces.</p> 2021-01-06T00:00:00+00:00 Copyright (c) 2021 Vasko Rechkoski, Bedrije Bedzeti, Vesna Manova Erakovikj