This document summarizes a rule from the ancient Indian mathematics text Lilavati for computing the sides of regular polygons inscribed in a circle. The rule provides coefficients that when multiplied by the circle's diameter and divided by 12,000 yield the side lengths of triangles, squares, pentagons, etc. up to nonagons. While the rationales for some coefficients are clear, others are less accurate, possibly due to limitations in available sine tables or knowledge of exact solutions. The document discusses methods that may have been used to derive the coefficients and compares the original values to more accurate modern calculations.
This document discusses ancient Indian values of Pi that were used in mathematics. It provides 5 ancient approximations of Pi that were used in India and other parts of the world:
1. The simplest approximation of Pi = 3, which was used by the ancient Babylonians, in the Bible, and in ancient Chinese works. Similar approximations are found in ancient Indian texts.
2. The "Jaina value" of Pi = 22/7, which was frequently used in Jain texts and first appeared in India around the 1st century AD.
3. The Archimedean value of Pi = 22/7, which was recognized as a satisfactory approximation after Archimedes showed Pi is between 3 1/
1. Madhava of Sangamagrama formulated a rule equivalent to the Gregory series for calculating the inverse tangent function.
2. The rule expresses the inverse tangent as a sum of terms involving sines and cosines, equivalent to the Gregory series.
3. Ancient Indian mathematicians provided proofs of the rule, equivalent to modern proofs through term-by-term integration, showing they understood the concept of infinite series well before Western mathematicians formulated them explicitly.
1) The ancient Indian mathematician Bhaskara II first described the addition and subtraction theorems for sines in his work Jyotpatti from 1150 AD. These theorems state that the sine of the sum of two arcs is equal to the sum of the sines of individual arcs multiplied by their cosines, and the sine of the difference of two arcs is equal to the difference of the sines.
2) Bhaskara II provided two formulas based on these theorems to tabulate sines at intervals of 225 minutes and 60 minutes. These formulas allow constructing tables of 24 sines and 90 sines respectively using recurrence relations.
3) The article discusses Bhaskara II's contribution to trigon
1) Brahmagupta, an ancient Indian mathematician from the 7th century AD, developed formulas for calculating the area and diagonals of a cyclic quadrilateral using only the lengths of its four sides.
2) His formula for accurate area involved taking the square root of the product of terms involving the differences between half sums of opposite sides.
3) His expressions for the diagonals involved dividing and multiplying terms involving products of the sides, and took the square root of the results. These formulas are considered some of the most remarkable in Hindu geometry.
The document summarizes rules for calculating the perimeter and area of an ellipse from an ancient Indian mathematics text from the 9th century AD. It provides the ancient text's approximate formulas for perimeter and area, and explains how they were likely empirically derived by comparing an ellipse to simpler shapes like a semicircle or circular segment. It also gives the correct elliptic integral formulas, and shows how the ancient approximations compare to the precise solutions for a numerical example.
The document discusses addition and subtraction theorems for the sine and cosine functions in medieval Indian mathematics. It provides statements of the theorems as found in several important Indian works from the 12th to 17th centuries. These theorems are equivalent to the modern mathematical formulas for trigonometric addition and subtraction. The document also outlines various proofs and derivations of the theorems found in Indian works, which indicate how Indians understood the rationales behind the theorems.
1) Ancient Indian mathematicians developed rational approximations to trigonometric functions like sine, cosine, and versine that were found in Sanskrit works from the 7th to 17th centuries AD.
2) These approximations took the form of algebraic formulas relating the functions to angles in degrees. For example, one approximation for sine was sin(A) ≈ 16A(180-A)/(32400-A^2).
3) These formulas, which were popular in India, can be found in many original Indian mathematical works and show that such approximations were well-established by the 7th century.
Narayana Pandita, an important ancient Indian mathematician who lived in the 14th century, developed a method for approximating quadratic surds (square roots of non-perfect squares). His method involved finding integer solutions to the indeterminate equation x2 - Ny2 = 1 and taking the ratio of the solutions as the approximation. The article describes Narayana's method using the example of approximating √10 and relates his method to other ancient approaches like the binomial approximation and Newton's method. It highlights the significance of Narayana's works in the development of ancient Indian mathematics.
This document discusses ancient Indian values of Pi that were used in mathematics. It provides 5 ancient approximations of Pi that were used in India and other parts of the world:
1. The simplest approximation of Pi = 3, which was used by the ancient Babylonians, in the Bible, and in ancient Chinese works. Similar approximations are found in ancient Indian texts.
2. The "Jaina value" of Pi = 22/7, which was frequently used in Jain texts and first appeared in India around the 1st century AD.
3. The Archimedean value of Pi = 22/7, which was recognized as a satisfactory approximation after Archimedes showed Pi is between 3 1/
1. Madhava of Sangamagrama formulated a rule equivalent to the Gregory series for calculating the inverse tangent function.
2. The rule expresses the inverse tangent as a sum of terms involving sines and cosines, equivalent to the Gregory series.
3. Ancient Indian mathematicians provided proofs of the rule, equivalent to modern proofs through term-by-term integration, showing they understood the concept of infinite series well before Western mathematicians formulated them explicitly.
1) The ancient Indian mathematician Bhaskara II first described the addition and subtraction theorems for sines in his work Jyotpatti from 1150 AD. These theorems state that the sine of the sum of two arcs is equal to the sum of the sines of individual arcs multiplied by their cosines, and the sine of the difference of two arcs is equal to the difference of the sines.
2) Bhaskara II provided two formulas based on these theorems to tabulate sines at intervals of 225 minutes and 60 minutes. These formulas allow constructing tables of 24 sines and 90 sines respectively using recurrence relations.
3) The article discusses Bhaskara II's contribution to trigon
1) Brahmagupta, an ancient Indian mathematician from the 7th century AD, developed formulas for calculating the area and diagonals of a cyclic quadrilateral using only the lengths of its four sides.
2) His formula for accurate area involved taking the square root of the product of terms involving the differences between half sums of opposite sides.
3) His expressions for the diagonals involved dividing and multiplying terms involving products of the sides, and took the square root of the results. These formulas are considered some of the most remarkable in Hindu geometry.
The document summarizes rules for calculating the perimeter and area of an ellipse from an ancient Indian mathematics text from the 9th century AD. It provides the ancient text's approximate formulas for perimeter and area, and explains how they were likely empirically derived by comparing an ellipse to simpler shapes like a semicircle or circular segment. It also gives the correct elliptic integral formulas, and shows how the ancient approximations compare to the precise solutions for a numerical example.
The document discusses addition and subtraction theorems for the sine and cosine functions in medieval Indian mathematics. It provides statements of the theorems as found in several important Indian works from the 12th to 17th centuries. These theorems are equivalent to the modern mathematical formulas for trigonometric addition and subtraction. The document also outlines various proofs and derivations of the theorems found in Indian works, which indicate how Indians understood the rationales behind the theorems.
1) Ancient Indian mathematicians developed rational approximations to trigonometric functions like sine, cosine, and versine that were found in Sanskrit works from the 7th to 17th centuries AD.
2) These approximations took the form of algebraic formulas relating the functions to angles in degrees. For example, one approximation for sine was sin(A) ≈ 16A(180-A)/(32400-A^2).
3) These formulas, which were popular in India, can be found in many original Indian mathematical works and show that such approximations were well-established by the 7th century.
Narayana Pandita, an important ancient Indian mathematician who lived in the 14th century, developed a method for approximating quadratic surds (square roots of non-perfect squares). His method involved finding integer solutions to the indeterminate equation x2 - Ny2 = 1 and taking the ratio of the solutions as the approximation. The article describes Narayana's method using the example of approximating √10 and relates his method to other ancient approaches like the binomial approximation and Newton's method. It highlights the significance of Narayana's works in the development of ancient Indian mathematics.
News international conference at almoraSohil Gupta
The document provides information about an upcoming colloquium on the history of mathematical sciences and a symposium on nonlinear analysis to be held from May 16-19, 2011 in Almora, India. The colloquium will feature talks by national and international speakers on various topics related to the history of mathematical sciences. A registration fee of $150 or 1500 rupees is required to attend. Accommodation and local transportation will be provided for registered participants. Papers on relevant topics are invited for both events.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
1. Aryabhata, an Indian astronomer and mathematician from the 5th century AD, approximated pi (π) to 3.1416 in his famous work the Aryabhatiya. This approximation correct to four decimal places was one of the most accurate approximations of pi used anywhere in the ancient world.
2. Aryabhata expressed pi as a fraction 62832/20000, which can be expressed in continued fractions as 3 + (4/16). This value of pi was later used and referenced by many Indian and foreign mathematicians and astronomers over subsequent centuries.
3. Scholars debate whether Aryabhata's value of pi was influenced by the Greeks
Manindra Agrawal and two of his students, Neeraj Kayal and Nitin Saxena, proved that primes is in P by developing a new algorithm. Their proof was a breakthrough because it provided a simple, elegant solution to a long-standing open problem. It was also accessible to undergraduate students, unlike many other advanced mathematical proofs. Within a few days of publishing their preprint, experts verified the proof and it received widespread attention, with over two million views of their website in the first ten days.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
A three day national seminar on advances in mathematics was organized by MBICT in January 2012. It received support from various organizations and over 100 mathematicians and engineers participated and presented on topics related to the history of mathematics and engineering applications. Key topics included the origins of right angles in ancient Indian mathematics, theories of equations from Newton to modern times, and using mathematics to understand nature.
1) The document discusses an ancient Indian mathematics problem involving using gnomons (vertical rods) to measure the height of a lamp post and the distance between shadow tips.
2) It presents the original rule from the Aryabhatiya, and provides examples and explanations of how to use the rule to calculate heights and distances.
3) The rule involves using ratios between shadow lengths, distance between shadow tips, and gnomon length to determine the "upright" distance and lamp post height.
The First International Conference on the History of Mathematical Sciences was held in New Delhi from December 20-23, 2001. It was organized by the Indian Society for History of Mathematics and Ramjas College in collaboration with other institutions. The conference covered all aspects of the history of mathematical sciences, especially ancient Indian history, and had participants from 11 countries as well as India. There were over 20 sessions over the 4 days covering topics from ancient to modern mathematics through invited talks and paper presentations.
Brahmagupta was an ancient Indian mathematician from the 7th century AD who made important contributions to mathematics. This document discusses one of Brahmagupta's formulas for calculating the volume of frustum-like solids (solids with parallel ends of different shapes and sizes). The formula provides a method to calculate both the practical volume and gross volume, and uses the difference between the two to find the accurate volume. The formula is shown to apply to specific cases like truncated wedges and cones. The formula demonstrates Brahmagupta's sophisticated mathematical abilities and had influence in both India and other parts of the ancient world.
The document summarizes a two-day international seminar on the history of mathematics held in New Delhi in 2012. Over 150 mathematicians from 16 countries participated in the event, which celebrated the 125th birth anniversary of Srinivasa Ramanujan and covered various aspects of mathematics history, especially ancient Indian history. There were 11 sessions over the two days featuring talks and papers from scholars. The seminar concluded with positive feedback and was considered a great success in arranging such a large international event.
1) The ancient Indian text Baudhayana's Sulba Sutra from around 800-400 BC contains a rule for approximating the square root of two. The rule increases the side of a square by one-third and one-fourth parts to get an approximation of 1.41421.
2) This approximation method of linear interpolation was popular in ancient India. The rule can be explained through a two-term interpolation formula.
3) Repeated use of this interpolation process yields the four-term approximation found in the Sulba Sutra, equivalent to methods used by later mathematicians like Neugebauer. While an improvement, the Indian approximation was still less accurate than the sexagesimal
Bhaskara II, an Indian mathematician from the 12th century, derived the formula for the surface area of a sphere. He used a crude form of integration by dividing the surface into thin circular sections and approximating their areas. This led to the formula of circumference x diameter. Earlier, Aryabhata II had provided the same formula, though Bhaskara criticized another mathematician, Lalla, for providing an incorrect formula. Bhaskara's derivation showed originality compared to Archimedes' method, though it lacked the same level of ingenuity.
Bhaskara I, an Indian mathematician from around 600 AD, developed an elegant algebraic formula for approximating trigonometric sine values. This formula, described in his text Mahabhaskariya, expresses sine as a function of the angular arc in degrees. Several subsequent Indian texts, including works by Brahmagupta, Varahamihira, and Bhaskara II, reference or derive equivalent forms of Bhaskara I's approximation formula. The document discusses the accuracy of the formula and compares it to actual sine values, providing historical context on the development and transmission of trigonometric concepts in early Indian and Western mathematics.
This document discusses fractional parts of Aryabhata's sine differences that are found in Govindasvami's commentary on Mahabhashya. Specifically:
1. Govindasvami's commentary provides the sexagesimal fractional parts (seconds and thirds) of Aryabhata's 24 tabular sine differences, allowing for a more accurate sine table with values given to the second order of sexagesimal fractions.
2. These fractional parts found in Govindasvami's commentary are subtracted from or added to Aryabhata's sine differences to improve the accuracy of the table.
3. In addition to improving Aryabhata's sine table, the
The document presents a new interpretation of a rule in the Ganita-siira-sa4graha (GSS), an ancient Indian mathematics text, for calculating the surface area of a spherical segment.
Previous interpretations by Rangacharya and Jain assumed the rule treated the spherical segment as a flat circular base. The document suggests an alternate interpretation where the Sanskrit term "viskambha" refers to the curvilinear breadth of the segment, not its diameter. This new interpretation matches the true mathematical formula and has less error than the previous views.
1. The property that the second order differences of sines are proportional to the sines themselves was known in early Indian mathematics, dating back to Aryabhata I in the 5th century AD.
2. The paper describes various formulations of this proportionality found in important Indian astronomical works from the 5th to 15th centuries. It also outlines Nilakantha Somayaji's ingenious geometric proof of the property from his commentary on Aryabhatiya in the early 16th century.
3. The Indian mathematical method of computing sine tables using this difference property, which involves an implied differential process, was considered novel by Western scholars despite being used in India since ancient times
1. According to Jaina cosmography, the Jambu Island is circular with a diameter of 100,000 yojanas.
2. Ancient Jaina texts provided approximations for the circumference of the Jambu Island using simple formulas like C=3D. More accurate values were also calculated using the formula C=√(10D2) which is better than C=3D.
3. The article examines various circumference values found in ancient Jaina texts like the Tiloya-Pannatti and expresses them using the detailed unit system from the text to compare to the precise modern calculation of the circumference.
1) Neelakantha Somaydji was an important mathematician from medieval India who lived from 1443-1543 AD and wrote several astronomical works.
2) In one of his works called Golasara, Neelakantha provides a formula for computing the length of a small circular arc given the Indian Sine and Versed Sine.
3) The formula, which is equivalent to the modern formula, enables the approximate computation of an angle when its sine or cosine is given.
News international conference at almoraSohil Gupta
The document provides information about an upcoming colloquium on the history of mathematical sciences and a symposium on nonlinear analysis to be held from May 16-19, 2011 in Almora, India. The colloquium will feature talks by national and international speakers on various topics related to the history of mathematical sciences. A registration fee of $150 or 1500 rupees is required to attend. Accommodation and local transportation will be provided for registered participants. Papers on relevant topics are invited for both events.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive function. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
1. Aryabhata, an Indian astronomer and mathematician from the 5th century AD, approximated pi (π) to 3.1416 in his famous work the Aryabhatiya. This approximation correct to four decimal places was one of the most accurate approximations of pi used anywhere in the ancient world.
2. Aryabhata expressed pi as a fraction 62832/20000, which can be expressed in continued fractions as 3 + (4/16). This value of pi was later used and referenced by many Indian and foreign mathematicians and astronomers over subsequent centuries.
3. Scholars debate whether Aryabhata's value of pi was influenced by the Greeks
Manindra Agrawal and two of his students, Neeraj Kayal and Nitin Saxena, proved that primes is in P by developing a new algorithm. Their proof was a breakthrough because it provided a simple, elegant solution to a long-standing open problem. It was also accessible to undergraduate students, unlike many other advanced mathematical proofs. Within a few days of publishing their preprint, experts verified the proof and it received widespread attention, with over two million views of their website in the first ten days.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
A three day national seminar on advances in mathematics was organized by MBICT in January 2012. It received support from various organizations and over 100 mathematicians and engineers participated and presented on topics related to the history of mathematics and engineering applications. Key topics included the origins of right angles in ancient Indian mathematics, theories of equations from Newton to modern times, and using mathematics to understand nature.
1) The document discusses an ancient Indian mathematics problem involving using gnomons (vertical rods) to measure the height of a lamp post and the distance between shadow tips.
2) It presents the original rule from the Aryabhatiya, and provides examples and explanations of how to use the rule to calculate heights and distances.
3) The rule involves using ratios between shadow lengths, distance between shadow tips, and gnomon length to determine the "upright" distance and lamp post height.
The First International Conference on the History of Mathematical Sciences was held in New Delhi from December 20-23, 2001. It was organized by the Indian Society for History of Mathematics and Ramjas College in collaboration with other institutions. The conference covered all aspects of the history of mathematical sciences, especially ancient Indian history, and had participants from 11 countries as well as India. There were over 20 sessions over the 4 days covering topics from ancient to modern mathematics through invited talks and paper presentations.
Brahmagupta was an ancient Indian mathematician from the 7th century AD who made important contributions to mathematics. This document discusses one of Brahmagupta's formulas for calculating the volume of frustum-like solids (solids with parallel ends of different shapes and sizes). The formula provides a method to calculate both the practical volume and gross volume, and uses the difference between the two to find the accurate volume. The formula is shown to apply to specific cases like truncated wedges and cones. The formula demonstrates Brahmagupta's sophisticated mathematical abilities and had influence in both India and other parts of the ancient world.
The document summarizes a two-day international seminar on the history of mathematics held in New Delhi in 2012. Over 150 mathematicians from 16 countries participated in the event, which celebrated the 125th birth anniversary of Srinivasa Ramanujan and covered various aspects of mathematics history, especially ancient Indian history. There were 11 sessions over the two days featuring talks and papers from scholars. The seminar concluded with positive feedback and was considered a great success in arranging such a large international event.
1) The ancient Indian text Baudhayana's Sulba Sutra from around 800-400 BC contains a rule for approximating the square root of two. The rule increases the side of a square by one-third and one-fourth parts to get an approximation of 1.41421.
2) This approximation method of linear interpolation was popular in ancient India. The rule can be explained through a two-term interpolation formula.
3) Repeated use of this interpolation process yields the four-term approximation found in the Sulba Sutra, equivalent to methods used by later mathematicians like Neugebauer. While an improvement, the Indian approximation was still less accurate than the sexagesimal
Bhaskara II, an Indian mathematician from the 12th century, derived the formula for the surface area of a sphere. He used a crude form of integration by dividing the surface into thin circular sections and approximating their areas. This led to the formula of circumference x diameter. Earlier, Aryabhata II had provided the same formula, though Bhaskara criticized another mathematician, Lalla, for providing an incorrect formula. Bhaskara's derivation showed originality compared to Archimedes' method, though it lacked the same level of ingenuity.
Bhaskara I, an Indian mathematician from around 600 AD, developed an elegant algebraic formula for approximating trigonometric sine values. This formula, described in his text Mahabhaskariya, expresses sine as a function of the angular arc in degrees. Several subsequent Indian texts, including works by Brahmagupta, Varahamihira, and Bhaskara II, reference or derive equivalent forms of Bhaskara I's approximation formula. The document discusses the accuracy of the formula and compares it to actual sine values, providing historical context on the development and transmission of trigonometric concepts in early Indian and Western mathematics.
This document discusses fractional parts of Aryabhata's sine differences that are found in Govindasvami's commentary on Mahabhashya. Specifically:
1. Govindasvami's commentary provides the sexagesimal fractional parts (seconds and thirds) of Aryabhata's 24 tabular sine differences, allowing for a more accurate sine table with values given to the second order of sexagesimal fractions.
2. These fractional parts found in Govindasvami's commentary are subtracted from or added to Aryabhata's sine differences to improve the accuracy of the table.
3. In addition to improving Aryabhata's sine table, the
The document presents a new interpretation of a rule in the Ganita-siira-sa4graha (GSS), an ancient Indian mathematics text, for calculating the surface area of a spherical segment.
Previous interpretations by Rangacharya and Jain assumed the rule treated the spherical segment as a flat circular base. The document suggests an alternate interpretation where the Sanskrit term "viskambha" refers to the curvilinear breadth of the segment, not its diameter. This new interpretation matches the true mathematical formula and has less error than the previous views.
1. The property that the second order differences of sines are proportional to the sines themselves was known in early Indian mathematics, dating back to Aryabhata I in the 5th century AD.
2. The paper describes various formulations of this proportionality found in important Indian astronomical works from the 5th to 15th centuries. It also outlines Nilakantha Somayaji's ingenious geometric proof of the property from his commentary on Aryabhatiya in the early 16th century.
3. The Indian mathematical method of computing sine tables using this difference property, which involves an implied differential process, was considered novel by Western scholars despite being used in India since ancient times
1. According to Jaina cosmography, the Jambu Island is circular with a diameter of 100,000 yojanas.
2. Ancient Jaina texts provided approximations for the circumference of the Jambu Island using simple formulas like C=3D. More accurate values were also calculated using the formula C=√(10D2) which is better than C=3D.
3. The article examines various circumference values found in ancient Jaina texts like the Tiloya-Pannatti and expresses them using the detailed unit system from the text to compare to the precise modern calculation of the circumference.
1) Neelakantha Somaydji was an important mathematician from medieval India who lived from 1443-1543 AD and wrote several astronomical works.
2) In one of his works called Golasara, Neelakantha provides a formula for computing the length of a small circular arc given the Indian Sine and Versed Sine.
3) The formula, which is equivalent to the modern formula, enables the approximate computation of an angle when its sine or cosine is given.
This document contains questions for an optical fiber communication examination. It asks students to:
1. Classify and explain different types of attenuation in optical fibers, and compare different fiber types.
2. Consider a multimode fiber example and calculate modal dispersion after 10 km.
3. Explain an LED light source, PIN and APD photodetector structures, and expanded beam fiber optic connectors.
A new six point finite difference scheme for nonlinear waves interaction modelAlexander Decker
1) The document presents a new six point finite difference scheme for modeling nonlinear wave interaction using the coupled 1D Klein-Gordon-Zakharov system of equations.
2) The scheme is derived by discretizing the system equations with finite differences. It is shown to be equivalent to a multi-symplectic integrator.
3) Numerical simulations are presented applying the new six point scheme to solve the Klein-Gordon-Zakharov equations modeling the interaction of Langmuir waves and ion sound waves in plasma.
1. The document contains questions from various engineering subjects like control systems, microcontrollers, HDL and probability. It asks the reader to attempt 5 questions by selecting at least 2 questions from each part A and B.
2. The questions cover topics like drawing block diagrams and signal flow graphs to find transfer functions, writing programs in VHDL and Verilog for combinational and sequential circuits, explaining addressing modes of microcontrollers, solving problems based on probability distributions and testing hypotheses.
3. Solutions to some questions require writing code, drawing diagrams or deriving mathematical expressions while others involve explaining concepts or deriving specifications from given data. The questions test a variety of engineering skills ranging from circuit analysis and programming to probability and control
1. The document appears to be an examination paper for a surveying course, containing multiple choice and numerical problems related to surveying techniques and calculations.
2. Questions cover topics like theodolite measurements, angle and distance measurements, triangulation, trilateration, traversing, and curve setting.
3. Students are required to attempt five questions total, selecting at least two from each part. Formulas, assumptions, and tables are permitted.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document appears to be an examination paper for a third semester engineering degree. It contains multiple choice and numerical questions on topics related to engineering mathematics, manufacturing processes, basic thermodynamics, and materials science. Some key questions assess understanding of metal casting processes, welding techniques, thermodynamic equilibrium, intensive/extensive properties, and the Joule paddle wheel experiment.
This document contains questions from an Advanced Mathematics exam for a fourth semester Bachelor's degree. It includes questions on topics such as vectors, lines and planes, motion, vector calculus, and Laplace transforms. Students were instructed to answer 15 full questions choosing from the total of 18 questions provided.
This document contains questions from a Material Science and Metallurgy exam. It covers various topics:
- Crystal structures of BCC, FCC and HCP lattices and their properties. Diffusion of iron atoms in BCC lattice.
- Mechanical properties in the plastic region from stress-strain diagrams. True and conventional strain expressions. Twinning mechanism of plastic deformation.
- Fracture mechanisms based on Griffith's theory of brittle fracture. Factors affecting creep. Fatigue testing and S-N curves for materials.
- Solidification process and expression for critical nucleus radius. Cast metal structures. Solid solutions and Hume-Rothery rules. Phase diagrams and Gibbs phase rule.
This document contains questions from a third semester B.E. degree examination in engineering mathematics, logic design, analog electronic circuits, and other subjects. It includes questions ranging from expansions of functions to solving differential equations to designing combinational logic circuits. Students are instructed to answer five questions total, selecting at least two from each part. The questions cover a wide range of engineering topics and require mathematical, analytical, and design skills to solve fully.
1. A force system acting on a beam produces a clockwise moment of 20 kN-m at B and an anticlockwise moment of 15 kN-m at C. Determine the magnitude and direction of the resultant force.
2. The centroid of the shaded area shown in Fig. Q7(b) is located. The moment of inertia of the section about its centroidal axes is calculated.
3. A ladder resting against a vertical wall is supported by friction at the wall and floor. The reactions A and B are determined. The minimum coefficient of friction required to prevent slipping is computed.
This document contains the questions from an engineering mathematics exam for the third semester B.E. degree. It includes 10 multiple choice questions covering topics like Fourier series, Fourier transforms, differential equations, and linear programming. It also contains longer questions on topics like heat transfer, interpolation, eigenvectors, Poisson's equation, and Z-transforms. The exam tests knowledge of concepts and computational skills in engineering mathematics.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help boost feelings of calmness, happiness and focus.
Digital Marketing Trends in 2024 | Guide for Staying AheadWask
https://www.wask.co/ebooks/digital-marketing-trends-in-2024
Feeling lost in the digital marketing whirlwind of 2024? Technology is changing, consumer habits are evolving, and staying ahead of the curve feels like a never-ending pursuit. This e-book is your compass. Dive into actionable insights to handle the complexities of modern marketing. From hyper-personalization to the power of user-generated content, learn how to build long-term relationships with your audience and unlock the secrets to success in the ever-shifting digital landscape.
Let's Integrate MuleSoft RPA, COMPOSER, APM with AWS IDP along with Slackshyamraj55
Discover the seamless integration of RPA (Robotic Process Automation), COMPOSER, and APM with AWS IDP enhanced with Slack notifications. Explore how these technologies converge to streamline workflows, optimize performance, and ensure secure access, all while leveraging the power of AWS IDP and real-time communication via Slack notifications.
Have you ever been confused by the myriad of choices offered by AWS for hosting a website or an API?
Lambda, Elastic Beanstalk, Lightsail, Amplify, S3 (and more!) can each host websites + APIs. But which one should we choose?
Which one is cheapest? Which one is fastest? Which one will scale to meet our needs?
Join me in this session as we dive into each AWS hosting service to determine which one is best for your scenario and explain why!
zkStudyClub - LatticeFold: A Lattice-based Folding Scheme and its Application...Alex Pruden
Folding is a recent technique for building efficient recursive SNARKs. Several elegant folding protocols have been proposed, such as Nova, Supernova, Hypernova, Protostar, and others. However, all of them rely on an additively homomorphic commitment scheme based on discrete log, and are therefore not post-quantum secure. In this work we present LatticeFold, the first lattice-based folding protocol based on the Module SIS problem. This folding protocol naturally leads to an efficient recursive lattice-based SNARK and an efficient PCD scheme. LatticeFold supports folding low-degree relations, such as R1CS, as well as high-degree relations, such as CCS. The key challenge is to construct a secure folding protocol that works with the Ajtai commitment scheme. The difficulty, is ensuring that extracted witnesses are low norm through many rounds of folding. We present a novel technique using the sumcheck protocol to ensure that extracted witnesses are always low norm no matter how many rounds of folding are used. Our evaluation of the final proof system suggests that it is as performant as Hypernova, while providing post-quantum security.
Paper Link: https://eprint.iacr.org/2024/257
Driving Business Innovation: Latest Generative AI Advancements & Success StorySafe Software
Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
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5th LF Energy Power Grid Model Meet-up SlidesDanBrown980551
5th Power Grid Model Meet-up
It is with great pleasure that we extend to you an invitation to the 5th Power Grid Model Meet-up, scheduled for 6th June 2024. This event will adopt a hybrid format, allowing participants to join us either through an online Mircosoft Teams session or in person at TU/e located at Den Dolech 2, Eindhoven, Netherlands. The meet-up will be hosted by Eindhoven University of Technology (TU/e), a research university specializing in engineering science & technology.
Power Grid Model
The global energy transition is placing new and unprecedented demands on Distribution System Operators (DSOs). Alongside upgrades to grid capacity, processes such as digitization, capacity optimization, and congestion management are becoming vital for delivering reliable services.
Power Grid Model is an open source project from Linux Foundation Energy and provides a calculation engine that is increasingly essential for DSOs. It offers a standards-based foundation enabling real-time power systems analysis, simulations of electrical power grids, and sophisticated what-if analysis. In addition, it enables in-depth studies and analysis of the electrical power grid’s behavior and performance. This comprehensive model incorporates essential factors such as power generation capacity, electrical losses, voltage levels, power flows, and system stability.
Power Grid Model is currently being applied in a wide variety of use cases, including grid planning, expansion, reliability, and congestion studies. It can also help in analyzing the impact of renewable energy integration, assessing the effects of disturbances or faults, and developing strategies for grid control and optimization.
What to expect
For the upcoming meetup we are organizing, we have an exciting lineup of activities planned:
-Insightful presentations covering two practical applications of the Power Grid Model.
-An update on the latest advancements in Power Grid -Model technology during the first and second quarters of 2024.
-An interactive brainstorming session to discuss and propose new feature requests.
-An opportunity to connect with fellow Power Grid Model enthusiasts and users.
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HCL Notes und Domino Lizenzkostenreduzierung in der Welt von DLAUpanagenda
Webinar Recording: https://www.panagenda.com/webinars/hcl-notes-und-domino-lizenzkostenreduzierung-in-der-welt-von-dlau/
DLAU und die Lizenzen nach dem CCB- und CCX-Modell sind für viele in der HCL-Community seit letztem Jahr ein heißes Thema. Als Notes- oder Domino-Kunde haben Sie vielleicht mit unerwartet hohen Benutzerzahlen und Lizenzgebühren zu kämpfen. Sie fragen sich vielleicht, wie diese neue Art der Lizenzierung funktioniert und welchen Nutzen sie Ihnen bringt. Vor allem wollen Sie sicherlich Ihr Budget einhalten und Kosten sparen, wo immer möglich. Das verstehen wir und wir möchten Ihnen dabei helfen!
Wir erklären Ihnen, wie Sie häufige Konfigurationsprobleme lösen können, die dazu führen können, dass mehr Benutzer gezählt werden als nötig, und wie Sie überflüssige oder ungenutzte Konten identifizieren und entfernen können, um Geld zu sparen. Es gibt auch einige Ansätze, die zu unnötigen Ausgaben führen können, z. B. wenn ein Personendokument anstelle eines Mail-Ins für geteilte Mailboxen verwendet wird. Wir zeigen Ihnen solche Fälle und deren Lösungen. Und natürlich erklären wir Ihnen das neue Lizenzmodell.
Nehmen Sie an diesem Webinar teil, bei dem HCL-Ambassador Marc Thomas und Gastredner Franz Walder Ihnen diese neue Welt näherbringen. Es vermittelt Ihnen die Tools und das Know-how, um den Überblick zu bewahren. Sie werden in der Lage sein, Ihre Kosten durch eine optimierte Domino-Konfiguration zu reduzieren und auch in Zukunft gering zu halten.
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In the realm of cybersecurity, offensive security practices act as a critical shield. By simulating real-world attacks in a controlled environment, these techniques expose vulnerabilities before malicious actors can exploit them. This proactive approach allows manufacturers to identify and fix weaknesses, significantly enhancing system security.
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The presentation culminates in a live demonstration. We'll showcase the manipulation of Galileo's Open Service pilot signal, simulating an attack on various software and hardware systems. This practical demonstration serves to highlight the potential consequences of unaddressed vulnerabilities, emphasizing the importance of offensive security practices in safeguarding critical infrastructure.
Generating privacy-protected synthetic data using Secludy and MilvusZilliz
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Generating privacy-protected synthetic data using Secludy and Milvus
Gupta1975f
1. The Mathematics Educatron SEC'I'ION B
Vol. I X , N o . 2 , J u n e 1975
GL IM P S E S OF A N C IE N T INDI A N M A T HE M A T I CS .1 4
NO
Ttre Lffavati rrle for cornputing sldes of
reE:ular polyEonsl
b2 R. G. Gupta, Llcnber,International Commis.rion History of Mathematics)
on Department
of
, Birla Instituteof 7'eohnologP.O. Mesra, RANCHI (India)
Methematic.e
( Re ce ive d lB .P ri l 1975)
1. Introduction
Coming from the pen of the famous Bhlskarrcarya (efea<fWd), the Lil.ivati
(d tvf adl) is t he m o s t p o p u l a r w o rk o f a n c i e n t Indi an mathemati cs. The cel ebratedauthor
belonging to the twelfth century A D., was a great Indian astronomer and mathenratician
who wrote several other works alsor: He is now usually designated as BlrtrskaraII (son of
Mahedvara) to distinguish him from his name sake Bhlskara I who lived in the seventh
ce n t ur y of our er a . T h e a u th o r o f L i l i v a ti w as born i n daka 1036 (or A .D . l l l 4) and
wrote the work abolt the middle of the twelfth century. Written in lucid Sanskrit, it is
devoted to arithrnetic, geometry, mensuratiort, and some other topic of elementary
rn a t hem at ic s .
Ever since its composition, the Lilrivatl has inspired a number of commentaries,
translations, arrd editions in various [ndian languages throughout the past 800 years. It
wa s r ender ed int o Pe rs i a n b 1 ' F a i z i (1 5 8 7 A .D .) under the patronape of ki nd A kbar.
Amo ng t he E nglis h tra n s l a ti o n so f th e w o rk , th e one by H .T. C oi ebrooke (London, l 8l 7)
i s well- k nowr r t . T h e re c e n t (1 9 7 5 ) e d i ti o n o f the w ork by D r. K .V . S arnra i s val uabl e be-
cause it includes an important and elaborate sixteenth century South lndian conrmentarl3,
T her e is a t h ri l l i n g s to ry 4 a c c o rd i n g to whi ch LILA V A TI (' beauti ful ' ) l as the name
of Bhdskra's only daughter and that he titled the work after her name in the hope of consol-
ing her for art accident r,vhichprevented her marriage. But whether rhe romantic story
has any historical basis or not, it is stated to be found narrated even in the Preface to
Lildvati's translation by Faizi (sixtecn century)5.
The Rule For Flnding the Sides.
In the present article we shall discussa rule from the Lilirvati about the numerical
computation of the sides of regtrlar polygons (upto nine-sided) inscribed in any circle ol'
diameter D. The original Sanskrit text as commonly found in the Lil:ivati ksetravyavah?ira,
2. 26 THE M ATI:ITM ATI C E ED U G AT IO N
206-2C8 as followsG
is :
flaaqe+rfiqcqsg;f:faarqrsetmcefq:
I
?erficerqersdqq
ts(alqFrr€: fiHr( ltRo!ll
erliEtoerqtlsqkldc;?{qrqt: t
S(Iqsqldsq qf,aqrdue'r6* slQoetl
tq€?qrrrrddqt erq;t mRnl gwr: r
1tt;ae+agatqi aatat'd UqTTq{nRoctl
Tridvyarikagni - nabha"{candraih tribi=,tt.lsta yuedstabl'ih /
-
Vedagnibauaidca khi( vaiSca khakhabhr. ibhra-rasaih kramdt I 1206
II
Banesunakhabinaiica dvidvinandesu-s:lgaraih/
KurlmadaJca'vedaiica vt'ttavyasgsanrihate 1 207 l
l
Khakhakhibhrlrka sa4rbhakte labhyante kramio bhujih /
Vlttiinatas-tryasra-pt1rrInirlr navdsantam prthak-prthak I 12081
|
This may be translated thus :
'Iltultiply the diameter of the (given) circle, in order, by (the coefficients) 103923,
84853, 70534,6 0 C 0 0 ,5 2 0 5 5 , 5 9 2 2 ,a n d 4 1031. On di vi di ng (each of the products j ust
4
o bt ained) by 1 2 0 0 0 0 ,th e re a ro o b ta i n e d th e si desrespecti vl yof thr: (cel rri l atcral )tri angl e to
the (regular), nonagon (inscribed in the cilcle separately.'
That is, tire side of the inscribed regular polygon of n sicles given by
is
r" :(D /1 2 0 0 0 0 ). & " (l )
where the seven coefficientskn, forn equal to 3 upto 7, are separately given in the above
verbal rule. It is clear from (t) that when D is taken cqual to 120000, v;e shall have sn
equal to &o itself. Thus it may be said that Bh'iskara'sccefficients represent the sides of
regular polygons inscribed in a circle of radius 60u00.
T he Lila v a t, w a s e q u a l l y p o p u l a r i n the l ate A ryabhata S chool . B ut the ori gi nal
taxt seemsto be changed at scveral places apparently to improve rrpon it. It is therefrrre
no surprise that same of the above coefficientshave different values in the taxt c f the rule
as published alon.gwith the Kriylkramatcari (|fr,+t*'e+'il) commentary (sixteenth) centrrry
belonging to the School?.
We present the two sets of cofficients in the form of a table rvhich also contain the
corresponding modcrn or actual values for the sake of cornparasion.
3. R. C. GUPT A
27
TABI,E
(Sides polygorrs
of inscribed a circleof redius
in 60(,00)
No . o f O r iginz ri Kri y z k ra makari Modern value
si de s Lr llv at i re a d i n g ( t o r r , 'a r e s t i n t t 'g e r )
v alue
.l 103923 I 0 3 92 2, k r 03923
+ 8+853 s am e same
5 70i3+ s am e same
o 6000rJ s am e same
5205s 52C67 s2066
a 45922 s ant e same
9 41031 a. lt L2 41042
J us t af t er s t a ti n g th e a b o v e ru l e , th e a u th or has gi ven and w orked out the fol i ow i ng
example :
' I n a c ir c le of d i a me te r 2 0 0 0 , te l l me s e p aratel ythe si des of the (i nscri bed) equi l a-
teral triangle and etc .
3. Retionales of the Rule
T he r net hod o f d e ri v i n g th e s ec o e ffi c i e n t s not gi ven i n the Li l i vati . The comi nel -
ta to r Ganc s a ( 1545 ) me n ti o n s tw o m e th o d s o f obtai ni ng them (pp. 207-208). The fi rst i s
b a se dor r r r s inga t ab l e o f Si n e s to o b ta i n
k" - 120000sin (190ln) (2)
F or th i s p u rp ()s e n e s a u s e d
Ga
va l u e s f i' om t he t abl e o f S i n e s (fo r rl te ra d i u s 343t1) w hi ch i s founds i n B h:tskar:rII' s
astrr'nomical work called Siddhzint;r iiromani ifvarr;afwrlq|qr). But the cooefficients
o b ta i ned ir r t his lv ay (u s i n g l i n e a r i n te l p o l a ti o n w here neceesary) are so rought that mast
of them do not agree lvith tlrr;segiven in the original text. Horvever, a secondorcier inter-
polation does help in this resf,ect (seebelow).
Ofcoures, more :rccrlrateSine tables can be rrsedto derive the values of the coffici-
ents to the supposedor irnlied degree ol'accuracy. But it is doubtul whether Bhiskara II
had anv such table ready at hi-sdisposal although he knewe a method of constructing a table
of 90 Sines (that is, with a tabular interval of one degree) which could serve the prrrpose.
Moreover, two of the coefficientsare far from being accrrrateto the same dcgree as others.
This indicates the possibility of sonre different method.
The second method given by the commer.tator GaneSa consistsof finding the sides
T hc las t d i g i t r c a c l i n g d vi ( two .; is sta tcd to b e co lr cctcd to tri (three) i n one of the manuscri pts :
4. 28 IIIE I.f,T Ir IIIIT tOg E D go^|rIOX
of the inscribed triangle, sguare, hexagon, and octagon geometrically by the usual method
ofemploying tl,e so-called Pythagoream theorem (see Colebrooke's translation, pp.
120-12l). However, he remarks that the proof of the sides of the regular pentagon, hep-
tagon, and nonagon cannot be given in a similar (simple and elementarr) manner.
This method is cssentially equivalent to findin61of Sines of the type (2) geometrically
for n equal to3r 4,6, and B in which cases the exact values can be easily obtained by
employing elementary mathematical operations upto the extraction of square roots. The
accuracy of the text values in these casespoints out that it was possibly this very method
which was followed by Bhakara II. He also knew the exact value of the Sine of 36 degrees
which explains the accuracy of his cofficient for n equal to 5 (pentagon)r0.
The ramaining cases(septagon and nonagon) are difficult and the lack of knowledge
of the exact solutions is reflected in the much less accurate text-values in these two cases.
But how did Bh{skara got even these approximate values ? One possibility is that he used
his tabular Sines (as indicated by Ganeda) but employed Brahmagupta's (A.D. 62S)
technique of second order interpolation which he knew and which is equivalent to the
modern Newton-Stirling intcrpolation formula upto the second orderrr. By tbis method
the result for z equal to 9 (nonagon) tallies almost fully, but in the only remaing case of
heptagon (n equal to 7), the most tedious onc where even the argumental angle is not
expressiblein whole degreesor minutes, a small difference is f,rund.
Refcrenceg and Notcs
t. For a brief description of his works, see R.C. Gupta,"Bhiskara II's Derivation for the
Surface of a Sphere" (Glimpses of Ancient Indian Mathcmatics No.6), The Malhema-
tics Education, Vol. VIII, No. 2 (June, 1973), sec. B, pp. 49-52.
2. C,rlebrooke'sEnglish translation, with nots by H.C. Banerji, has been recently reprinted
by M/s Kitab Mahal, Allahabad, 1967.
3. K.V. Sarma leditor): Lll;ruatl with Krilt-,kramakariof sa,rkaraand Nitilar.ta, Vishveshara-
nand Vedic Research Institute, Hoshiarpur, 1975.
4. Edna E. Krarner ; The Main Strearn of Malhematics. Oxford univ. Press, N.Y., 1951,
P p. 3- 5.
Also see the present author's note on LILAVATI published in Tfu Hindustan Timet,
New Delh i , V o l .5 l , N o . l 2 l , p . 5 (d e ted the l 9th May 1974).
5. R.E. Moritz : On Mathematics, 164. Dover, New York, 1958.
p,
6. See the Lilivatr with the commentaries of Ga4eSa and Mahidhara edite<iby D.V. Apte,
5. R. G. GUPTT 29
Part II, pp. 207-208,Poona, 1937 (Anandasram Sanskrit Series No. 107). In Cole-
b ro ok e' st r ans lat io rr(p . 1 2 0 ), th e s es ta n z a sa re numbered as 2(9-211.
7. Sarma (editor), op. cit., pp. 204-206.
B, Bapudeva Sastri (editor) t Siddhinta Siromani, Graha Ganita, II, 3-6, pp. 39-40
(Benares,1929).
This table appeared earlier h tbe Mah,isidhanta Aryabhata II (960 A. D). 'fhe Sine
table of Aryabhata I (born 476 A.D) and ,SltrTa-siddantaslightly different.
is
9. See R.C. Gupta. "Addition and Subtraction Theorems for the Sine aud their use ir.r
computing Tabular Sines" (Glimpses of Ancient Indian Mathematics No. ll),Thc
Malhemalics
Edacatinn,
Vol. VIII, No.3 (September 1974), Sec. B, pp. +3-46.
10. See his Jlotpatli, verses 7-B in Bapudeva Sastri (editor), op. cit., p. 28l.
It. Sid. iiir. Graha Ganita, II, l6 (Bapudeva's edition cited above, p.42). For details see
.tl
R. C. Gupta, ttSeconder Order Interpolation in Indian Mathematics etc.", Indian
J. or
Htst. Sciencc,Vol. 4 (1969), pp. 86-89.
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