The document presents 49 Pythagorean triangles with one leg represented by the 9-digit Dhuruva number 554999445. Each triangle satisfies the Pythagorean equation with m and n integer coefficients. 28 of the triangles are primitive and 21 are non-primitive. Additionally, 49 more Pythagorean triangles are presented with expressions for the legs and hypotenuse equaling 554999445, with 18 being primitive and 31 non-primitive. The document examines properties and patterns of numbers in Pythagorean triangles connected to 9-digit Dhuruva numbers.
1. Mr. Mizan sold rice from three pieces of land for Tk. 25087, Tk. 16920 and Tk. 30725 respectively and gram from another piece of land for Tk. 9872, for a total of Tk. 82604.
2. Abdul Latif had Tk. 621345. He gave his elder daughter Tk. 85924, younger daughter Tk. 84790, and his son Tk. 95745, which is Tk. 9830 more than the elder daughter. The remaining money was given to his wife.
3. The sum of three numbers is 845076. Two numbers are 321674 and 286539. By subtracting the sum
The document provides solutions to calculating various statistical measures - arithmetic mean, median, mode, harmonic mean, and geometric mean - for 5 sets of data. For each data set, the document calculates the measures using the relevant formulas. The statistical measures included arithmetic mean, median, mode, harmonic mean, and geometric mean. Formulas are provided for calculating each measure.
This document contains topics related to algebra, shape, data, numbers, decimals, fractions, proportions, BIDMAS rules, factors and HCF, directed numbers, ratios, percentages, fractions and decimals, limits, multiples, and fractions. It includes examples and multi-step problems involving these mathematical concepts.
More companies in the process of recruitment, play more emphasis in the topic of numbers in numerical aptitude. Especially for AMCAT aspirants this is very much useful.
The document describes a linear programming problem to maximize the objective function Z=400A+300B subject to the constraints 2A+B≤60, A+3B≤40, and A+B≤30. The simplex method is applied to solve the primal problem, obtaining the optimal solution Z=12000 with A=30, B=0, H1=0, H2=10, H3=0. The dual problem is then formulated and also solved using the simplex method, verifying the optimal solution. A second example problem is presented following the same process.
The document contains 15 examples explaining how to solve various math problems involving number systems, equations, remainders, and arithmetic progressions. Each example shows the steps to find the number of possible solutions, values that satisfy the equations, or other numerical relationships implied by the given information.
This document contains the analysis of the forces in the reinforcing bars of several concrete beam structures. It determines the forces in each bar and whether they are in tension or compression. The analysis involves calculating the forces at each node by applying equations of equilibrium for the sum of forces in the x- and y- directions. The forces are then determined for each bar connected to that node. This is done for several nodes throughout the beams.
1. Mr. Mizan sold rice from three pieces of land for Tk. 25087, Tk. 16920 and Tk. 30725 respectively and gram from another piece of land for Tk. 9872, for a total of Tk. 82604.
2. Abdul Latif had Tk. 621345. He gave his elder daughter Tk. 85924, younger daughter Tk. 84790, and his son Tk. 95745, which is Tk. 9830 more than the elder daughter. The remaining money was given to his wife.
3. The sum of three numbers is 845076. Two numbers are 321674 and 286539. By subtracting the sum
The document provides solutions to calculating various statistical measures - arithmetic mean, median, mode, harmonic mean, and geometric mean - for 5 sets of data. For each data set, the document calculates the measures using the relevant formulas. The statistical measures included arithmetic mean, median, mode, harmonic mean, and geometric mean. Formulas are provided for calculating each measure.
This document contains topics related to algebra, shape, data, numbers, decimals, fractions, proportions, BIDMAS rules, factors and HCF, directed numbers, ratios, percentages, fractions and decimals, limits, multiples, and fractions. It includes examples and multi-step problems involving these mathematical concepts.
More companies in the process of recruitment, play more emphasis in the topic of numbers in numerical aptitude. Especially for AMCAT aspirants this is very much useful.
The document describes a linear programming problem to maximize the objective function Z=400A+300B subject to the constraints 2A+B≤60, A+3B≤40, and A+B≤30. The simplex method is applied to solve the primal problem, obtaining the optimal solution Z=12000 with A=30, B=0, H1=0, H2=10, H3=0. The dual problem is then formulated and also solved using the simplex method, verifying the optimal solution. A second example problem is presented following the same process.
The document contains 15 examples explaining how to solve various math problems involving number systems, equations, remainders, and arithmetic progressions. Each example shows the steps to find the number of possible solutions, values that satisfy the equations, or other numerical relationships implied by the given information.
This document contains the analysis of the forces in the reinforcing bars of several concrete beam structures. It determines the forces in each bar and whether they are in tension or compression. The analysis involves calculating the forces at each node by applying equations of equilibrium for the sum of forces in the x- and y- directions. The forces are then determined for each bar connected to that node. This is done for several nodes throughout the beams.
This document contains a math lesson on Roman numerals, addition, subtraction, multiplication and division of whole numbers and decimals. It includes word problems, examples worked out step-by-step and answers for students to check their work. The lesson recaps working with negative numbers and compares ordering numbers in ascending and descending order.
Statistical analysis of Crude oil prices of historical data. Tried to analyze the data using statistical models and graphs and interpreted the meaning of results found. A thorough research on the industry is done. The report uses the following mathematical calculations: Mean & its types, Median, Mode, Variance, Standard Deviation, Range, Correlation & Regression, and Kurtosis Coefficient of Skewness.
Tried to calculate it both manually and using MS Excel in built macros.
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
This document contains information about the College Algebra Real Mathematics Real People 7th Edition Larson textbook including:
- A link to download the solutions manual and test bank for the textbook
- An overview of the content covered in Chapter 2 on solving equations and inequalities, including linear equations, identities, conditionals, and more.
- 51 example problems from Chapter 2 with step-by-step solutions.
The document contains a math quiz with multiple choice and short answer questions on various math topics. Some questions involve percentages, probabilities, averages, ratios and proportions. The highest common factor, prime factors, scales, angles, inequalities and coordinate planes are also addressed.
Proton Training Solution (PTS) is a trusted Training Institute in Pune for MBA Entrance Exams (CAT | IIFT | XAT | SNAP | NMAT | CMAT | TISS | MH CET | MAT | ATMA and many others), BBA Entrance Exams (IPM-AT | DUJAT | SET | NPAT | BMCC | MIT and many others). Proton is also associated with many institutes as knowledge partner & provides Aptitude Training for Placement Preparation.
https://protons.in/
This document contains a mathematics test for Year 5 students with 40 multiple choice questions covering topics like number sense, operations, fractions, word problems, and data interpretation. The test examines students' understanding of place value, operations, fractions, and their ability to solve multi-step word problems involving addition, subtraction, multiplication and division of whole numbers, decimals and fractions.
The document describes the box or grid multiplication method for multiplying multi-digit numbers. It involves expanding the numbers, drawing a grid with the place values, placing the numbers in the grid, multiplying the numbers in each column and adding the results. Two examples are shown of multiplying a 2-digit by 2-digit number and a 3-digit by 2-digit number using this method.
1. The document describes the steps to conduct a one-way ANOVA test to compare group means. It involves stating the ANOVA model and hypotheses, determining the rejection criteria, performing computations such as calculating sums of squares, and drawing conclusions based on comparing the computed F-ratio to the critical F-value.
2. As an example, it analyzes data on hours of pain relief from 4 headache drug types. Following the steps, it is determined that the different drug types provide significantly different hours of pain relief since the computed F-ratio exceeds the critical value.
3. The key computations are calculating the sum of squares between and within groups, determining the F-ratio as the between-groups mean square over the
Econometrics is a tough subject so is its homework and assignments in general. In case you are looking for econometrics assignment help, you can rely on Economicshelpdesk. Our tutors are expert and we honour our client’s need as well as privacy.
The document provides an overview of different math topics including whole numbers, fractions, decimals, money, and percentage. It lists the key concepts and formulas for each topic. It also includes review questions to test memorization of important math facts related to time, length, mass, volume, and measurement units.
The document provides information on numerical reasoning concepts including arithmetic progression, geometric progression, formulas, ratio and proportion problems, alligation, and mixture problems. It includes 15 multi-step word problems covering these topics and their step-by-step solutions. The problems demonstrate how to set up and solve ratios, proportions, alligation and mixture scenarios to find unknown values.
This document discusses the Concrete-Picture-Abstract (CPA) instructional approach for teaching mathematical concepts. It explains that CPA uses a three-part strategy where concepts are first introduced concretely using physical objects, then pictorially using diagrams and pictures, and finally abstractly using symbolic notation. The document provides examples of using the CPA approach to teach place value, addition and subtraction of integers, and multiplication of integers. It emphasizes starting with concrete manipulatives to help students understand before moving to more abstract representations.
o level mathematics complete revision guide for quick and deep revision to solve questions with meticulous techniques and tricks with question to solve for clearance of concepts
This document is a practice test for a Grade 6 mathematics assessment. It contains 24 multiple choice questions about math concepts like geometry, measurement, numbers and operations, statistics and probability. The test is introduced with directions that explain students should choose the best answer for each question and may use scratch paper to solve problems. A reference sheet with formulas for shapes, conversions, and other information is provided for use during the test.
The document provides information about a management aptitude test and social entrepreneurship program. It discusses developing change makers and offers a free, comprehensive program in social and spiritual entrepreneurship open to all. It then provides examples of math and reasoning questions along with solutions.
The document provides examples of solving quadratic equations by taking the square root of both sides of the equation. It shows isolating the squared term, taking the square root of both sides, and solving for x. Three example equations are given and worked through as examples: x^2=324, x^2+5=23, and 2(x+5)^2-3=45. The reader is then prompted to solve the equations 3x^2=36, (x-7)^2-5=75, and (x+2)^2=28.
This document contains the questions from a mathematics examination for Class 12. It has three sections with increasing point values for the questions. Section A contains 10 single-mark questions on topics like vectors, integrals, and geometry. Section B has 8 four-mark questions involving more complex integrals. Section C contains 6 six-mark questions, including integrals, differential equations, and word problems on topics like profit maximization and probability. The examination tests a range of mathematical concepts and problem-solving abilities.
Some standard questions asked in cognizant aptitude tests recently has been sorted with answers. it will be beneficial to other company preparation aptitude also.
This document contains a mock CAT exam with 26 multiple choice questions covering topics in mathematics, data interpretation, and logical reasoning. The questions are from sections including numbers, algebra, geometry, data sufficiency, and data interpretation from tables and graphs. The level of difficulty of the questions ranges from easy to moderate.
Isotherm Studies for Performance Evaluation of Functionalized Carbon Nanotube...IJERA Editor
Composite Nanofiltration membranes were fabricated by incorporating Carbon Nanotubes (CNT) and Polyvinylidene Fluoride (PVDF) membranes. A suspension was prepared by dispersing CNT (30 minutes of sonication) in Dimethyl Sulfoxide (DMSO) solvent for 0.5 mg/ml. This suspension after vacuum filtration dispersed over PVDF membrane of pore size 0.22μm. Excess DMSO was removed by washing using Ethanol followed by Deionized water. Normal and various functionalized (Hydroxyl (-OH), Carboxylic (-COOH), Amine (-NH2)) SWCNTs and MWCNTs were purchased and the CNT based polymer composite nanofiltration membranes were fabricated by vacuum filtration method at a uniform loading of 10mg/cm2. The experimental results were used to evaluate the TDS removal efficiency of the fabricated membrane modules, which are normal and functionalized (4 composites in both SW and MW combinations) CNT. The TDS removal efficiency of all the 8 composite membranes were observed under varied operating conditions viz., varying flow rate (100,150,200 and 250 ml/hr) and influent TDS (2500, 3000, 4000 and 5000 mg/l). The flux of the membrane is varied 200 to 500 lit/m2.hr. The Isotherm studies using the experimental data on Langmuir and Freundlich isotherm models validated the results of the experiment and are indicative of the role of adsorption of the functionalized CNT based polymer composite nanofiltration membranes which removing the TDS. The results showed that the amino functionalized SWCNTs based polymer composite (PVDF/SWCNTs) nanofiltration membranes can be used as an effective adsorbent for desalination due to their high adsorption capacity.
Reliability Analysis of Slope Stability by Central Point MethodIJERA Editor
Given uncertainty and variability of the slope stability analysis parameter, the paper proceed from the perspective of probability theory and statistics based on the reliability theory. Through the central point method of reliability analysis, performance function about the reliability of slope stability analysis is established. What’s more, the central point method and conventional limit equilibrium methods do comparative analysis by calculation example. The approach’s numerical results are consistent with the traditional limit equilibrium method and meet the objective reality. The accuracy and practicality of reliability analysis is confirmed in order to provide reliability theory a scientific basis for the feasibility of slope stability analysis.
Definitief programma Raedelijn 2015-2016 Raedelijn
Raedelijn werkt met een programmatische meerjarenaanpak: formulering van programmalijnen met doelstellingen op de langere termijn (2 jaar).
De overalldoelstelling van het programma is:
- De toegevoegde waarde van de eerstelijnszorg versterken in partnership vanuit de eerste lijn met ziekenhuizen, gemeenten en welzijn
- Organiseren van goede, samenhangende en doelmatige zorg dichtbij de mensen thuis
- Zorg betaalbaar houden
This document contains a math lesson on Roman numerals, addition, subtraction, multiplication and division of whole numbers and decimals. It includes word problems, examples worked out step-by-step and answers for students to check their work. The lesson recaps working with negative numbers and compares ordering numbers in ascending and descending order.
Statistical analysis of Crude oil prices of historical data. Tried to analyze the data using statistical models and graphs and interpreted the meaning of results found. A thorough research on the industry is done. The report uses the following mathematical calculations: Mean & its types, Median, Mode, Variance, Standard Deviation, Range, Correlation & Regression, and Kurtosis Coefficient of Skewness.
Tried to calculate it both manually and using MS Excel in built macros.
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
This document contains information about the College Algebra Real Mathematics Real People 7th Edition Larson textbook including:
- A link to download the solutions manual and test bank for the textbook
- An overview of the content covered in Chapter 2 on solving equations and inequalities, including linear equations, identities, conditionals, and more.
- 51 example problems from Chapter 2 with step-by-step solutions.
The document contains a math quiz with multiple choice and short answer questions on various math topics. Some questions involve percentages, probabilities, averages, ratios and proportions. The highest common factor, prime factors, scales, angles, inequalities and coordinate planes are also addressed.
Proton Training Solution (PTS) is a trusted Training Institute in Pune for MBA Entrance Exams (CAT | IIFT | XAT | SNAP | NMAT | CMAT | TISS | MH CET | MAT | ATMA and many others), BBA Entrance Exams (IPM-AT | DUJAT | SET | NPAT | BMCC | MIT and many others). Proton is also associated with many institutes as knowledge partner & provides Aptitude Training for Placement Preparation.
https://protons.in/
This document contains a mathematics test for Year 5 students with 40 multiple choice questions covering topics like number sense, operations, fractions, word problems, and data interpretation. The test examines students' understanding of place value, operations, fractions, and their ability to solve multi-step word problems involving addition, subtraction, multiplication and division of whole numbers, decimals and fractions.
The document describes the box or grid multiplication method for multiplying multi-digit numbers. It involves expanding the numbers, drawing a grid with the place values, placing the numbers in the grid, multiplying the numbers in each column and adding the results. Two examples are shown of multiplying a 2-digit by 2-digit number and a 3-digit by 2-digit number using this method.
1. The document describes the steps to conduct a one-way ANOVA test to compare group means. It involves stating the ANOVA model and hypotheses, determining the rejection criteria, performing computations such as calculating sums of squares, and drawing conclusions based on comparing the computed F-ratio to the critical F-value.
2. As an example, it analyzes data on hours of pain relief from 4 headache drug types. Following the steps, it is determined that the different drug types provide significantly different hours of pain relief since the computed F-ratio exceeds the critical value.
3. The key computations are calculating the sum of squares between and within groups, determining the F-ratio as the between-groups mean square over the
Econometrics is a tough subject so is its homework and assignments in general. In case you are looking for econometrics assignment help, you can rely on Economicshelpdesk. Our tutors are expert and we honour our client’s need as well as privacy.
The document provides an overview of different math topics including whole numbers, fractions, decimals, money, and percentage. It lists the key concepts and formulas for each topic. It also includes review questions to test memorization of important math facts related to time, length, mass, volume, and measurement units.
The document provides information on numerical reasoning concepts including arithmetic progression, geometric progression, formulas, ratio and proportion problems, alligation, and mixture problems. It includes 15 multi-step word problems covering these topics and their step-by-step solutions. The problems demonstrate how to set up and solve ratios, proportions, alligation and mixture scenarios to find unknown values.
This document discusses the Concrete-Picture-Abstract (CPA) instructional approach for teaching mathematical concepts. It explains that CPA uses a three-part strategy where concepts are first introduced concretely using physical objects, then pictorially using diagrams and pictures, and finally abstractly using symbolic notation. The document provides examples of using the CPA approach to teach place value, addition and subtraction of integers, and multiplication of integers. It emphasizes starting with concrete manipulatives to help students understand before moving to more abstract representations.
o level mathematics complete revision guide for quick and deep revision to solve questions with meticulous techniques and tricks with question to solve for clearance of concepts
This document is a practice test for a Grade 6 mathematics assessment. It contains 24 multiple choice questions about math concepts like geometry, measurement, numbers and operations, statistics and probability. The test is introduced with directions that explain students should choose the best answer for each question and may use scratch paper to solve problems. A reference sheet with formulas for shapes, conversions, and other information is provided for use during the test.
The document provides information about a management aptitude test and social entrepreneurship program. It discusses developing change makers and offers a free, comprehensive program in social and spiritual entrepreneurship open to all. It then provides examples of math and reasoning questions along with solutions.
The document provides examples of solving quadratic equations by taking the square root of both sides of the equation. It shows isolating the squared term, taking the square root of both sides, and solving for x. Three example equations are given and worked through as examples: x^2=324, x^2+5=23, and 2(x+5)^2-3=45. The reader is then prompted to solve the equations 3x^2=36, (x-7)^2-5=75, and (x+2)^2=28.
This document contains the questions from a mathematics examination for Class 12. It has three sections with increasing point values for the questions. Section A contains 10 single-mark questions on topics like vectors, integrals, and geometry. Section B has 8 four-mark questions involving more complex integrals. Section C contains 6 six-mark questions, including integrals, differential equations, and word problems on topics like profit maximization and probability. The examination tests a range of mathematical concepts and problem-solving abilities.
Some standard questions asked in cognizant aptitude tests recently has been sorted with answers. it will be beneficial to other company preparation aptitude also.
This document contains a mock CAT exam with 26 multiple choice questions covering topics in mathematics, data interpretation, and logical reasoning. The questions are from sections including numbers, algebra, geometry, data sufficiency, and data interpretation from tables and graphs. The level of difficulty of the questions ranges from easy to moderate.
Isotherm Studies for Performance Evaluation of Functionalized Carbon Nanotube...IJERA Editor
Composite Nanofiltration membranes were fabricated by incorporating Carbon Nanotubes (CNT) and Polyvinylidene Fluoride (PVDF) membranes. A suspension was prepared by dispersing CNT (30 minutes of sonication) in Dimethyl Sulfoxide (DMSO) solvent for 0.5 mg/ml. This suspension after vacuum filtration dispersed over PVDF membrane of pore size 0.22μm. Excess DMSO was removed by washing using Ethanol followed by Deionized water. Normal and various functionalized (Hydroxyl (-OH), Carboxylic (-COOH), Amine (-NH2)) SWCNTs and MWCNTs were purchased and the CNT based polymer composite nanofiltration membranes were fabricated by vacuum filtration method at a uniform loading of 10mg/cm2. The experimental results were used to evaluate the TDS removal efficiency of the fabricated membrane modules, which are normal and functionalized (4 composites in both SW and MW combinations) CNT. The TDS removal efficiency of all the 8 composite membranes were observed under varied operating conditions viz., varying flow rate (100,150,200 and 250 ml/hr) and influent TDS (2500, 3000, 4000 and 5000 mg/l). The flux of the membrane is varied 200 to 500 lit/m2.hr. The Isotherm studies using the experimental data on Langmuir and Freundlich isotherm models validated the results of the experiment and are indicative of the role of adsorption of the functionalized CNT based polymer composite nanofiltration membranes which removing the TDS. The results showed that the amino functionalized SWCNTs based polymer composite (PVDF/SWCNTs) nanofiltration membranes can be used as an effective adsorbent for desalination due to their high adsorption capacity.
Reliability Analysis of Slope Stability by Central Point MethodIJERA Editor
Given uncertainty and variability of the slope stability analysis parameter, the paper proceed from the perspective of probability theory and statistics based on the reliability theory. Through the central point method of reliability analysis, performance function about the reliability of slope stability analysis is established. What’s more, the central point method and conventional limit equilibrium methods do comparative analysis by calculation example. The approach’s numerical results are consistent with the traditional limit equilibrium method and meet the objective reality. The accuracy and practicality of reliability analysis is confirmed in order to provide reliability theory a scientific basis for the feasibility of slope stability analysis.
Definitief programma Raedelijn 2015-2016 Raedelijn
Raedelijn werkt met een programmatische meerjarenaanpak: formulering van programmalijnen met doelstellingen op de langere termijn (2 jaar).
De overalldoelstelling van het programma is:
- De toegevoegde waarde van de eerstelijnszorg versterken in partnership vanuit de eerste lijn met ziekenhuizen, gemeenten en welzijn
- Organiseren van goede, samenhangende en doelmatige zorg dichtbij de mensen thuis
- Zorg betaalbaar houden
This document describes an experiment investigating heat transfer using twisted aluminum tape. The experiment aimed to increase the efficiency of a heat exchanger by inducing turbulence in liquid flow using a twisted aluminum tape inserted into the inner pipe. This increased the Reynolds number and heat transfer coefficient compared to without the tape. Temperature measurements found that the outlet temperature of the hot fluid decreased with the tape, indicating increased heat transfer. The increased turbulence from the tape also provided some resistance to fouling buildup. Overall, inserting twisted aluminum tape was found to enhance heat transfer in the heat exchanger.
A review on TIG welding for optimizing process parameters on dissimilar jointsIJERA Editor
Tungsten Inert Gas Welding (TIG) is relatively high strength welding technique. This technique are mostly used in fabrication and other industries to join the either similar or dissimilar materials. In particular, it can be used to join high-quality strength of metal and alloys.In this paper we discuss abouttheTungsten Inert Gas welding of joining heat treatableof stainless steel and mild steel.These welded joints have higher tensile strength to weight ratio and finer micro structure. Tungsten Inert Gas Weldingofdissimilar material such as stainless steel and mild steel have the potential to hold good mechanical and metallurgical properties.
This document provides a summary of Hazem Said's work experience, education, skills, and languages. Some of his work experience includes preparing time schedules, quantities take-off, shop drawings, and BOQs. His education includes a bachelor's degree in civil engineering from Alazhar University. His skills include AutoCAD, SAP2000, Revit, Primavera P6, MS Project, and Adobe software. He is proficient in Arabic and English.
This document describes the design, fabrication, and performance evaluation of an oil-fired crucible furnace using locally sourced materials in Nigeria. Key components of the furnace include a furnace casing, crucible, burner housing, and refractory lining. Experimental tests found the furnace had a heating rate of 43.9°C/min, attained a maximum temperature of 1386°C, and had melting rates of 333.3g/min for aluminum and 115.8 g/min for copper. The furnace efficiency was determined to be 10.34% due to energy wastage from the open furnace design.
Investigation of various orthogonal wavelets for precise analysis of X-ray im...IJERA Editor
Now-a-days X-rays are playing very important role in medicine. One of the most important applications of Xray
is detecting fractures in bones. X-ray provides important information about the type and location of the
fracture. Sometimes it is not possible to detect the fractures in X-rays with naked eye. So it needs further
processing to detect the fractures even at minute levels. To detect minute fractures, in this paper various edge
feature extraction methods are analyzed which helps medical practitioners to study the bone structure, detects
the bone fracture, measurement of fracture treatment, and treatment planning prior to surgery. The classical
derivative edge detection operators such as Roberts, Prewitt, sobel, Laplacian of Gaussian can be used as edge
detectors, but a lot of false edge information will be extracted. Therefore a technique based on orthogonal
wavelet transforms like Haar, daubechies, coiflet, symlets are applied to detect the edges and are compared.
Among all the methods, Haar wavelet transform method performs well in detecting the edges with better
quality. The various performance metrics like Ratio of Edge pixels to size of image (REPS), peak signal to noise
ratio (PSNR) and computation time are compared for various wavelets.
Discretizing of linear systems with time-delay Using method of Euler’s and Tu...IJERA Editor
Delays deteriorate the control performance and could destabilize the overall system in the theory of discretetime
signals and dynamic systems. Whenever a computer is used in measurement, signal processing or control
applications, the data as seen from the computer and systems involved are naturally discrete-time because a
computer executes program code at discrete points of time. Theory of discrete-time dynamic signals and systems
is useful in design and analysis of control systems, signal filters, state estimators and model estimation from
time-series of process data system identification. In this paper, a new approximated discretization method and
digital design for control systems with delays is proposed. System is transformed to a discrete-time model with
time delays. To implement the digital modeling, we used the z-transfer functions matrix which is a useful model
type of discrete-time systems, being analogous to the Laplace-transform for continuous-time systems. The most
important use of the z-transform is for defining z-transfer functions matrix is employed to obtain an extended
discrete-time. The proposed method can closely approximate the step response of the original continuous timedelayed
control system by choosing various of energy loss level. Illustrative example is simulated to demonstrate
the effectiveness of the developed method.\
The Influence Of Infinite Impedance Flanges On The Electromagnetic Field Of A...IJERA Editor
The document summarizes research investigating the influence of infinite impedance flanges on the electromagnetic field of a plane waveguide. It presents the theoretical background, describing the problem setup and boundary conditions considered. It then outlines the solution method, detailing the integral equations derived to relate the unknown tangential electric field on the waveguide surface. Numerical results are presented analyzing the effects of impedance flanges and edge features on the electromagnetic field and radiation patterns.
Research on Linking between Bridges and Existing Road Network of Mountainous ...IJERA Editor
Because of the complex topography of mountainous city, there is usually large difference between bridge and existing road network in elevation. In order to realize rapid traffic of vehicles, reasonable interchange ramp can be designed to realize rapid dividing of vehicles at bridgehead. This paper mainly introduces a typical interchange at bridgehead — Caiyuanba Interchange, located in Chongqing, a mountainous city, explains difficulties encountered in engineering design and realizes rapid dividing of vehicles by designing reasonable interchange at bridgehead, so as to accumulate certain experience in the construction of interchange at bridgehead.
Recent research shows the tremendous potential for the development of optical devices viz. photo-detector, optical sources, connectors and applications etc. This is mainly because of the success of optical communication in the recent for gigabit transmission and is intended for terabits transmission in future. In this paper, mathematical model for the optical dependence of I-V, C-V characteristics of MISFET structure (to be used as photo-detector) is reported. Model is based on solution of Poisson‟s and current continuity equation. Proposed structure of MISFET includes, In0.53Ga0.47As used as substrate material and InP as insulator. Light is made to incident perpendicular to the surface. Drain current can be controlled optically by means of varying light intensity of incident radiation. There is significant effect of intensity modulation on IV and CV characteristics of MISFET. As a result of intensity modulation, drain current increases significantly in presence of illumination mainly due to change in carrier concentration of channel results from photo-generated carriers. Simulation of mathematical model is carried out in MATLAB.
Seismic Response of RC Framed Buildings with Open Ground StoreyIJERA Editor
RC framed buildings are generally designed without considering the structural action of masonry infill walls. These masonry infill walls are widely used as partitions and considered as non-structural elements. But they affect both the structural and non-structural performance of RC buildings during earthquake. RC framed building with open ground storey is known as soft storey, which performs poorly during earthquake. In order to study this total 144 RC framed buildings having bare frame, full infill frame and open ground storey frame were analyzed by seismic coefficient method and response spectrum method for various seismic hazards. The present study deals with the comparison of base shear for medium rise RC framed buildings having P+5, P+7, P+9 and P+11 storeys for various seismic zones (III, IV & V) and for various soil conditions (Hard & Medium) as per IS 1893(part 1): 2002. This work helps in understanding the effect of earthquake with increase in height of RC framed buildings on base shear for various seismic zones and soil conditions. The result shows that the effect of infill stiffness on structural response is significant under lateral loads. It is found that the presence infill walls increases the base shear by 60-65% more than bare frame by both seismic coefficient method and response spectrum method.
Blue cowboy boots can give your ankles a stylish embrace. Cowboy boots have a rich history dating back thousands of years. Modern cowboy boots often feature a Cuban heel, rounded or pointed toe, high shaft, and no lacing. Blue cowboy boots are a bold choice that stands out from more common shades. While blue may not be a typical color for cowboy boots, it can make a elegant statement when paired with the right outfit. For quality blue or other colored cowboy boots at affordable prices, consider online retailers like Mensusa.
This document analyzes various wavelet transforms for edge detection in X-ray bone images. It begins with an introduction to edge detection and its importance in medical imaging. Classical derivative operators can detect edges but also extract false information and are sensitive to noise. Wavelet transforms offer multi-resolution analysis to detect edges at different scales. The document then provides background on wavelet theory and discrete wavelet transforms. It analyzes applying various orthogonal wavelets like Haar, Daubechies, and Coiflet to X-ray images and compares their performance in edge detection based on metrics like edge detection accuracy and computation time. Haar wavelets performed best at detecting edges with better quality in less time.
The document summarizes a youth summit hosted in Cape Town, South Africa in June 2016 by the Mayoral Urban Regeneration Programme (MURP) in collaboration with other city departments. The summit targeted youth ages 13 to 25 from the Bishop Lavis, Bonteheuwel and Valhalla Park areas. Both members of youth organizations and individual candidates could participate based on a selection criterion. The goal was to deepen democracy through youth participation in community issues in line with city and national policies.
Synchronizing Parallel Tasks Using STMIJERA Editor
The past few years have marked the start of a historic transition from sequential to parallel computation. The necessity to write parallel programs is increasing as systems are getting more complex while processor speed increases are slowing down. Current parallel programming uses low-level programming constructs like threads and explicit synchronization using locks to coordinate thread execution. Parallel programs written with these constructs are difficult to design, program and debug. Also locks have many drawbacks which make them a suboptimal solution. One such drawback is that locks should be only used to enclose the critical section of the parallel-processing code. If locks are used to enclose the entire code then the performance of the code drastically decreases. Software Transactional Memory (STM) is a promising new approach to programming shared-memory parallel processors. It is a concurrency control mechanism that is widely considered to be easier to use by programmers than locking. It allows portions of a program to execute in isolation, without regard to other, concurrently executing tasks. A programmer can reason about the correctness of code within a transaction and need not worry about complex interactions with other, concurrently executing parts of the program. If STM is used to enclose the entire code then the performance of the code is the same as that of the code in which STM is used to enclose the critical section only and is far better than code in which locks have been used to enclose the entire code. So STM is easier to use than locks as critical section does not need to be identified in case of STM. This paper shows the concept of writing code using Software Transactional Memory (STM) and the performance comparison of codes using locks with those using STM. It also shows why the use of STM in parallel-processing code is better than the use of locks.
Shoot 3
Time Monday 23rd
March
Tuesday 24th March Wednesday 25th
March
Thursday 26th March Friday 27th March Saturday 28th March Sunday 29th March
P1:
P2: Shoot 4
Breaktime:
P3:
Lunchtime: Shoot 5
P4:
P5:
Evening:
Special pairs of pythagorean triangles and dhuruva numbers of orders 4, 5 and 6eSAT Journals
Abstract
This paper concerns with the problem of obtaining pairs of pythagorean triangles, where, in each pair, the difference between
their perimeters is represented by 6174(Dhuruva number of order 4),61974,82962,63954(Dhuruva number of order 5),
631764(Dhuruva number of order 6). Also, we present the number of pairs of primitive and non-primitive Pythagorean triangles.
Key words: Pairs of Pythagorean triangles, Dhuruva number, primitive and non-primitive Pythagorean triangles.
2010 Mathematics Subject Classification: 11D09, 11Y50, 11-04.
1 Euler Polynomial P(n) = n + n^2 + 41:
I have further elaborated on this quadratic equation devised by Leonhard Euler in 1772. This resulted in my own formulas with regard to this quadratic equation.
2 Legendre Polynomial P(n) = 2n^2 + 29:
This quadratic equation, once discovered or invented by Adrien-Marie Legendre, generates many primes. I have further elaborated on this equation, resulting in my own formulas.
3 Prime-Generating Polynomial P(n) = 6n^2 + 6n + 31: Self-made quadratic equation and related equations that produce prime numbers.
4 Various Prime-Generating Quadratic Polynomials: Self-invented figures with differently arranged numbers, creating patterns of prime numbers. I came up with a number of equations that describe the patterns of these prime numbers.
5 Franklin's Magic Squares:
Benjamin Franklin has created some world-famous magic squares. I have been thinking of some methods that he could possibly have used to make these magic squares. This also applies to the rediscovered magic squares of Franklin by Paul C. Pasles.
6 Magic Constant Triangle and RS:
Triangle with the numbers entered in zigzag, along with my self-invented formula RS = 2n^2 + 3 (for n ≥ 2).
7 Magic Squares:
I made these magic squares with, among other things, prime numbers.
8 Pi Triangle:
I made this pi triangle by entering the numbers of pi in a zigzag pattern, resulting in several interesting patterns and remarkable observations.
9 Ulam Normal and Zigzag:
In 1963, Stanislaw Ulam devised a spiral that, to his surprise, arranged many prime numbers in a line. I have entered the numbers in the Ulam spiral in a zigzag which reveals other patterns.
10 Various Figures:
Self-invented figures with different ways of arranging the numbers.
11 Generalized Collatz Problem:
This mathematical problem has been around for years and is still unsolved after 80 years.
I have provided my own take on this with some possible ways to approach this problem without pretending to have found the solution.
Generation of Pythagorean triangle with Area/ Perimeter as a Wagstaff prime n...IRJET Journal
This document discusses generating Pythagorean triangles where the ratio of the triangle's area to perimeter equals various Wagstaff prime numbers. It defines Pythagorean triangles and Wagstaff prime numbers. The method expresses the ratio of area to perimeter as m2(m1-m2)/2 equal to different Wagstaff primes, generating Pythagorean triangles. Several cases are presented where this ratio equals a 1, 2, 3, 4, 5, or 6 digit Wagstaff prime. Observations of the generated triangles are provided, such as some being primitive or non-primitive and other relationships between the sides.
This document discusses polygonal numbers and square numbers. It defines polygonal numbers as numbers that can be represented by points in polygonal shapes, with the first number being 1 and subsequent numbers determined by adding points. Some examples of polygonal numbers given are triangles, squares, pentagons, and hexagons. It then defines square numbers as natural numbers that are the square of another natural number, providing the formula k2. It notes some properties of square numbers like their placement in the multiplication table and that the last digits of square numbers are limited to 0, 1, 4, 5, 6, or 9.
The aim of this research paper was to derive the value of, and analyze π using purely basic
geometrical and trigonometric methods. The derivation has been proposed after studying regular polygons and
observing the similarity of an infinite sided regular polygon with a circle. Several test cases with the formula
derived have been checked to verify this derivation and apart from raw calculation of data, an alternate
verification using calculus has also been shown
1. A multiple regression model was run to analyze the relationship between a dependent variable (Y) and 3 independent variables (X1, X2, X3) based on data from 1987-2016.
2. The results showed that a one unit increase in X1 would increase Y by 0.16, a one unit increase in X2 would increase Y by 0.13, and a one unit increase in X3 would increase Y by 0.007. However, none of these relationships were statistically significant.
3. Additional regression runs between the dependent variable and each independent variable individually did not show any statistically significant relationships either. This suggests the independent variables are not good predictors of the dependent variable based on this
Using Japanese Patent Information to Identify Strategic AlliancesAlan Engel
This 1998 presentation made at the Detroit SCIP conference details university-industry cooperation by Professor Isao Karube of the Tokyo Institute of Technology between 1973 and 1998, as evidenced by assignees on patents for which he is listed as inventor.
The document simulates gym capacity at the Töölö Unisport gym in Helsinki based on arrival rates. It uses an exponential distribution to simulate customer inter-arrival times based on historical crowdedness data. It assigns customers to different exercise machines and tracks their waiting times. The simulation is run multiple times to analyze average waiting times and system capacity under different arrival rates.
The Shapiro-Wilk test is a test of normality used in statistics to determine if a sample comes from a normally distributed population. It was developed in 1965 by Samuel Shapiro and Martin Wilk. The test calculates a test statistic W that is compared to critical values, with smaller values of W indicating stronger evidence that the sample is not from a normal distribution. An example calculation is provided to demonstrate how to perform the test on a sample and interpret the results against critical values to determine if the sample appears to be normally distributed.
This document contains a table listing chi-square distribution values for different degrees of freedom and probability levels. The table includes:
- Degrees of freedom (ν) ranging from 1 to 100+ in the left column.
- Various probability levels (p) from 0.001 to 0.5 across the top row.
- The chi-square distribution values at each intersection of ν and p.
This document contains a table listing chi-square distribution values for different degrees of freedom and probability levels. The table includes:
- Degrees of freedom (ν) ranging from 1 to 100+ in the left column.
- Various probability levels (p) from 0.001 to 0.5 across the top row.
- The body of the table provides the chi-square distribution critical values corresponding to each combination of degrees of freedom and probability level.
This document contains a table listing chi-square distribution values for different degrees of freedom and probability levels. The table includes:
- Degrees of freedom (ν) ranging from 1 to 100+ in the left column.
- Probability levels (p) from 0.001 to 0.5 across the top row.
- The corresponding chi-square distribution critical values within the table.
This document contains a table listing chi-square distribution values for different degrees of freedom and probability levels. The table includes:
- Degrees of freedom (ν) ranging from 1 to 100+ in the left column.
- Various probability levels (p) from 0.001 to 0.5 across the top row.
- The corresponding chi-square distribution critical values where the probability of exceeding that value is equal to the probability level.
This document contains a table listing chi-square distribution values for different degrees of freedom and probability levels. The table includes:
- Degrees of freedom (ν) ranging from 1 to 100+ in the left column.
- Various probability levels (p) from 0.001 to 0.5 across the top row.
- The chi-square distribution values at each intersection of ν and p.
This document contains a table listing chi-square distribution values for different degrees of freedom and probability levels. The table includes:
- Degrees of freedom (ν) ranging from 1 to 100+ in the left column.
- Various probability levels (p) from 0.001 to 0.5 across the top row.
- The chi-square distribution values at each intersection of ν and p.
This document contains a table listing chi-square distribution values for different degrees of freedom and probability levels. The table includes:
- Degrees of freedom (ν) ranging from 1 to 100+ in the left column.
- Various probability levels (p) from 0.001 to 0.5 across the top row.
- The chi-square distribution values at each intersection of ν and p.
Numerical and Statistical Quantifications of Biodiversity: Two-At-A-Time Equa...IJAEMSJORNAL
The ecological concept of biodiversity is a challenging environmental problem that requires a sound mathematical reasoning. We have used the method of a numerical simulation that is indexed by a numerical scheme to predict biodiversity loss and biodiversity gain due to a decreasing and increasing variations of the intrinsic growth rates together. The novel results that we have obtained that we have not seen elsewhere, but do complement other similar numerical predictions of biodiversity are presented and discussed quantitatively.
The problem asks to find the first ten digits of the sum of 100 50-digit numbers. It provides the 100 numbers. The numbers are summed and the first ten digits of the very large resulting sum are reported as the answer.
This document contains MATLAB code that:
1) Defines functions to calculate the value of a quadratic equation and the distance of a falling object;
2) Plots the distance over time for a falling object;
3) Performs matrix operations like replacing submatrices and calculating the inverse.
The correlation coefficient between random walk and time has a characteristic shape of histogram or density function. Some findings has been revealed and it is desirable to be investigated more.
Similar to Special Pythagorean Triangles And 9-Digit Dhuruva Numbers (20)
Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
This paper describes a speed control device for generating electrical energy on an electricity network based on the doubly fed induction generator (DFIG) used for wind power conversion systems. At first, a double-fed induction generator model was constructed. A control law is formulated to govern the flow of energy between the stator of a DFIG and the energy network using three types of controllers: proportional integral (PI), sliding mode controller (SMC) and second order sliding mode controller (SOSMC). Their different results in terms of power reference tracking, reaction to unexpected speed fluctuations, sensitivity to perturbations, and resilience against machine parameter alterations are compared. MATLAB/Simulink was used to conduct the simulations for the preceding study. Multiple simulations have shown very satisfying results, and the investigations demonstrate the efficacy and power-enhancing capabilities of the suggested control system.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Special Pythagorean Triangles And 9-Digit Dhuruva Numbers
1. M.A.Gopalan et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 3, ( Part -1) March 2015, pp.159-165
www.ijera.com 159 | P a g e
Special Pythagorean Triangles And 9-Digit Dhuruva Numbers
M.A.Gopalan1
, K.Geetha2
, Manju Somanath3
1
Professor, Department of Mathematics, Shrimathi Indira Gandhi College, Trichy,
2
Assiatant Professor, Department of Mathematics, Cauvery College for Women, Trichy,
3
Assiatant Professor, Department of Mathematics, National College, Trichy.
Abstract
Pythagorean triangles, each with a leg represented by a 9-digit Dhuruva number are obtained. A few interesting
results are given.
Keywords: Pythagorean triangles, 9-digit Dhuruva numbers
2010 Mathematics Subject Classification: 11D09, 11Y50, 11-04.
I. Introduction:
The fascinating branch of mathematics is the
theory of numbers where in Pythagorean triangles
have been a matter of interest to various
mathematicians and to the lovers of mathematics,
because it is a treasure house in which the search for
many hidden connection is a treasure hunt. For a rich
variety of fascinating problems one may refer [1-
17].A careful observer of patterns may note that there
is a one to one correspondence between the
polygonal numbers and the number of sides of the
polygon. Apart from the above patterns we have
some more fascinating patterns of numbers namely
Jarasandha numbers, Nasty numbers and Dhuruva
numbers. These numbers have been presented in [18-
21].
In [22-24], special Pythagorean triangles
connected with polygonal numbers and Nasty
numbers are obtained. Recently in [25], special
Pythagorean triangles in connection with Hardy
Ramanujan number 1729 and in [26], special
Pythagorean triangles connection with 5 digit duruva
number are exhibited. Thus the main objective of this
paper is to find out the special Pythagorean triangles
in connection with 9-digit Dhuruva numbers.
In this communication, we have presented
Pythagorean triangles with each leg represented by 9-
Digit Dhuruva numbers 554999445, 864197532
respectively. Also a few interesting results are
obtained.
II. BASIC DEFINITONS:
Definition 2.1:
The ternary quadratic Diophantine equation
given by
222
zyx is known as Pythagorean
equation where x, y, z are natural numbers. The
above equation is also referred to as Pythagorean
triangle and denote it by T(x,y,z).
Also, in Pythagorean triangle T(x,y,z) :
222
zyx , x and y are called its legs and z its
hypotenuse.
Definition 2.2:
Most cited solution of the Pythagorean equation
is
2222
,2, nmzmnynmx , where
0 nm . This solution is called primitive, if m,n
are of opposite parity and gcd(m,n)=1.
Definition 2.3: Dhuruva numbers
The numbers which do not change when we
perform a single operation or a sequence of
operations are known as Dhuruva numbers.
III. METHOD OF ANALYSIS:
Section A:
In this section, we exhibit Pythagorean triangles,
each with a leg represented by the nine digit Dhuruva
number 554999445 and denote this number by N.
To start with, it is noted that the leg y can not be
represented by N as y is even and N is odd. Also z
can not be written as sum of two squares. Since a
positive integer P can be written as a sum of two
integer squares iff the canonical prime factorization
rr
r
rr
pppP .........21
21 , (where ip are distinct
primes) satisfies the condition if )4(mod3ip
then ir is even. A prime )4(mod1p can be
written as
22
bap
Now, consider x=N
2 2
554999445m n
which is a binary quadratic Diophantine equation.
Solving the above equation for m,n, we get 49 integer
solutions and thus, we have 49 pythagorean triangles,
each having the leg x and the expression
RESEARCH ARTICLE OPEN ACCESS
5. M.A.Gopalan et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 3, ( Part -1) March 2015, pp.159-165
www.ijera.com 163 | P a g e
On following the procedure of section A, we get 15 integer solutions and thus, we have 15 pythagorean
triangles, each having the leg y to be represented by the nine digit Dhuruva number N=864197532 as shown in
the table below:
S.No m n x y z
1 216049383 2 46677335894680700 864197532 46677335894680700
2 144032922 3 20745482619858100 864197532 20745482619858100
3 72016461 6 5186370654964480 864197532 5186370654964560
4 48010974 9 2305053624428590 864197532 2305053624428760
5 39281709 11 1543052661960560 864197598 1543052661960800
6 24005487 18 576263406106845 864197532 576263406107493
7 19640853 22 385763106567125 864197532 385763106568093
8 16003658 27 256117069380235 864197532 256117069381693
9 13093902 33 171450269584515 864197532 171450269586693
10 8001829 54 64029267342325 864197532 64029267348157
11 6546951 66 42862567392045 864197532 42862567400757
12 4364634 99 19050029944155 864197532 19050029963757
13 2387286 181 5699134413035 864197532 5699134478557
14 2182317 198 4762507449285 864197532 4762507527693
15 727439 594 529167145885 864197532 529167851557
Note that there are 8 primitive and 7 non-primitive pythagorean triangles.
Also, it is noted that the expressions
2
z x y
,
2A
x
P
are represented respectively by the dhuruva
number 864197532 as shown in table below:
S.
No
m n x y z
2
z x y
,
2A
x
P
1 43209876
6
43209876
4 1728395072 373418685429050000
37341868542905000
0 864197532
2 28806584
4
28806584
1 1728395056 165963859230470000
16596385923047000
0 864197532
3 21604938
3
21604937
9 1728395048 93354670060966300 93354670060966300 864197532
4 14403292
2
14403291
6 1728395028 41490963511321100 41490963511321100 864197532
5 96021948 96021939 1728394984 18440427267034300 18440427267034400 864197598
6 78563412 78563401 1728394943 12344417681768400 12344417681768500 864197532
7 72016461 72016449 1728394920 10372739581534000 10372739581534100 864197532
8 48010974 48010956 1728394740 4610105520462290 4610105520462610 864197532
9 39281706 39281684 1728394580 3086103124145810 3086103124146290 864197532
10 32007316 32007289 1728394335 2048934826652650 2048934826653380 864197532
11 26187804 26187771 1728393975 1371600428289770 1371600428290860 864197532
12 24005487 24005451 1728393768 1152525083819270 1152525083820570 864197532
13 19640853 19640809 1728393128 771524484740154 771524484742090 864197532
14 16003658 16003604 1728392148 512232410366864 512232410369780 864197532
15 13093902 13093836 1728390708 342898810776144 342898810780500 864197532
16 8729268 8729169 1728385263 152398511236584 152398511246385 864197532
17 8001829 8001721 1728383400 128056806295418 128056806307082 864197532
18 6546951 6546819 1728377640 85723406397738 85723406415162 864197532
19 4774572 4774391 1728362303 45591347171304 45591347204065 864197532
20 4364634 4364436 1728355860 38098331512848 38098331552052 864197532
21 727439 726251 1726983720 1056606602378 1056608013722 864197532
22 2182317 2181921 1728238248 9523286581914 9523286738730 864197532
Note that there are 6 primitive and 16 non-primitive pythagorean triangles.
Also it is noted that leg x to be represented by the nine digit Dhuruva number N=864197532 as shown in
the table below:
S.No. m n x y z
6. M.A.Gopalan et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 3, ( Part -1) March 2015, pp.159-165
www.ijera.com 164 | P a g e
1 72016464 72016458 864197532 10372741309929000 10372741309929100
2 24005496 24005478 864197532 1152526812214180 1152526812214500
3 8001856 8001802 864197532 128058534689024 128058534691940
The above 3 are non-primitive pythagorean triangles.
Also, it is noted that the expressions
2
y x z
,
2A
P
are represented respectively by the dhuruva number
864197532 as shown in table below:
S.
No.
m n x y z
2
y x z
,
2A
P
1 432098768 432098766 1728395072 373418688885841000 373418688885841000 864197532
2 288065844 288065841 1728395056 165963859230470000 165963859230470000 864197532
3 216049383 216049379 1728395048 93354670060966300 93354670060966300 864197532
4 144032928 144032922 1728395100 41490966968111200 41490966968111300 864197532
5 96021957 96021948 1728395144 18440430723824500 18440430723824600 864197598
6 78563423 78563412 1728395185 12344421138558600 12344421138558700 864197532
7 72016461 72016449 1728394920 10372739581534000 10372739581534100 864197532
8 48010992 48010974 1728395388 4610108977252420 4610108977252740 864197532
9 39281728 39281706 1728395548 3086106580935940 3086106580936420 864197532
10 32007343 32007316 1728395793 2048938283442780 2048938283443500 864197532
11 26187837 26187804 1728396153 1371603885079900 1371603885080980 864197532
12 24005523 24005487 1728396360 1152528540609400 1152528540610700 864197532
13 19640897 19640853 1728397000 771527941530282 771527941532218 864197532
14 16003712 16003658 1728397980 512235867156992 512235867159908 864197532
15 13093968 13093902 1728399420 342902267566272 342902267570628 864197532
16 8729367 8729268 1728404865 152401968026712 152401968036513 864197532
17 8001829 8001721 1728383400 128056806295418 128056806307082 864197532
18 6546951 6546819 1728377640 85723406397738 85723406415162 864197532
19 4774572 4774391 1728362303 45591347171304 45591347204065 864197532
20 4364634 4364436 1728355860 38098331512848 38098331552052 864197532
21 727439 726251 1726983720 1056606602378 1056608013722 864197532
22 2182317 432098766 1728238248 9523286581914 9523286738730 864197532
Note that there are 7 primitive and 15 non-primitive pythagorean triangles.
IV. CONCLUSION
One may search for the connections between
Pythagorean triangles and other Dhuruva numbers.
REFERENCES:
[1]. W. Sierpinski, Pythagorean triangles, Dover
publications, INC, Newyork, 2003.
[2]. M. A. Gopalan and G.Janaki, “Pythagorean
triangle with area/perimeter as a special
polygonal number”, Bulletin of Pure and
Applied Science, Vol.27E (No.2), 393-402,
2008.
[3]. M. A. Gopalan and A. Vijayasankar,
“Observations on a Pythagorean problem”,
Acta Ciencia Indica, Vol. XXXVI M, No 4,
517-520, 2010.
[4]. M. A. Gopalan and S. Leelavathi,
“Pythagorean triangle with area/perimeter
as a square integer”, International Journal
of Mathematics, Computer sciences and
information Technology, Vol.1, No.2, 199-
204, 2008.
[5]. M.A.Gopalan and A.Gnanam, “Pairs of
Pythagorean triangles with equal
perimeters”, Impact J.Sci.Tech., Vol 1(2),
67-70, 2007.
[6]. M.A.Gopalan and S.Leelavathi,
“Pythagorean triangle with 2 area/perimeter
as a cubic integer”, Bulletin of Pure and
Applied Science, Vol.26E (No.2), 197-
200,2007.
[7]. M.A.Gopalan and A.Gnanam, “A special
Pythagorean problem”, Acta Ciencia Indica,
Vol. XXXIII M, No 4, 1435-1439,2007.
[8]. M. A. Gopalan, A. Gnanam and G.Janaki,
“A Remarkable Pythagorean problem”, Acta
Ciencia Indica, Vol. XXXIII M, No 4, 1429-
1434,2007.
[9]. M.A.Gopalan, and S. Devibala, “On a
Pythagorean problem”, Acta Ciencia Indica,
Vol. XXXII M, No 4, 1451-1452, 2006.
[10]. M.A.Gopalan and B.Sivakami, “Special
Pythagorean triangles generated through
the integral solutions of the equation
7. M.A.Gopalan et al. Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 5, Issue 3, ( Part -1) March 2015, pp.159-165
www.ijera.com 165 | P a g e
1)2( 222
xkky ”, Diophantus J.
Math., Vol 2(1), 25-30, 2013.
[11]. M. A. Gopalan and A. Gnanam,
“Pythagorean triangles and Polygonal
numbers”, International Journal of
Mathematical Sciences, Vol 9, No. 1-2, 211-
215,2010.
[12]. K. Meena, S. Vidhyalakshmi, B. Geetha, A.
Vijayasankar and M. A. Gopalan,” Relations
between special polygonal numbers
generated through the solutions of
Pythagorean equation”,IJISM, Vol.
2(2),257-258,2014.
[13]. M.A.Gopalan and G.Janaki, “Pythagorean
triangle with perimeter as Pentagonal
number”, Antarctica J.Math., Vol 5(2), 15-
18, 2008.
[14]. M. A. Gopalan and G. Sangeetha,
“Pythagorean triangle with perimeter as
triangular number”,GJ-AMMS,Vol. 3, No
1-2,93-97,2010.
[15]. M.A.Gopalan, Manjusomanath and K.
Geetha,” Pythagorean triangle with
area/perimeter as a Special polygonal
number”, IOSR-JM, Vol. 7(3),52-62,2013.
[16]. M.A. Gopalan and V. Geetha,” Pythagorean
triangle with area/perimeter as a Special
polygonal number”, IRJES, Vol.2(7),28-
34,2013.
[17]. M. A. Gopalan and B. Sivakami,
“Pythagorean triangle with hypotenuse
minus 2(area/ perimeter) as a square
integer”, Archimedes J.Math., Vol 2(2),
153-166, 2012.
[18]. J.N.Kapur, Dhuruva numbers, Fasinating
world of Mathematics and Mathematical
sciences, Trust society, Vol 17, 1997.
[19]. Bert Miller, Nasty numbers, The
mathematics teacher, No.9,Vol 73,649,1980.
[20]. Charles Bown. K, Nasties are primitives,
The mathematics teacher, No.9, Vol 74,502-
504, 1981.
[21]. P.S.N.Sastry, Jarasandha numbers, The
mathematics teacher, No.9,Vol 37,issues 3
& 4 , 2001.
[22]. M. A. Gopalan V. Sangeetha and
Manjusomanath, “Pythagorean triangle and
Polygonal number”, Cayley J.Math., Vol
2(2),151-156, 2013.
[23]. M.A.Gopalan and G.Janaki, “ Pythagorean
triangle with nasty number as a leg”,Journal
of applied Mathematical Analysis and
Applications, Vol 4,No 1-2,13-17,2008.
[24]. M.A.Gopalan and S.Devibala, “Pythagorean
triangle with Triangular number as a leg”,
Impact J.Sci.Tech., Vol 2(4), 195-199, 2008.
[25]. Dr.Mita Darbari, A connection between
Hardy-Ramanujan number and special
Pythagorean triangle,” Bulletin of society
for Mathematical services & Standards, Vol
3, No.2, 71-73, 2014.
[26]. M. A. Gopalan, S. Vidhyalakshmi, E.
Premalatha and K. Presenna, “Special
Pythagorean triangles and 5-digit dhuruva
number”, International Research Journal of
Mathematics, Engineering and IT, Vol.1,
Issue.4, 29-33, Aug 2014.