Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal ...Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal triangle elements and pascal matrix formation, etc. This paper explains about its functions and properties of N-Summet-k. The result of variation between N and k is shown in tabulation.展开更多
We give a survey on the history, the main mathematical results and applications of the Mathematics of Harmony as a new interdisciplinary direction of modern science. In its origins, this direction goes back to Euclid...We give a survey on the history, the main mathematical results and applications of the Mathematics of Harmony as a new interdisciplinary direction of modern science. In its origins, this direction goes back to Euclid’s “Elements”. According to “Proclus hypothesis”, the main goal of Euclid was to create a full geometric theory of Platonic solids, associated with the ancient conception of the “Universe Harmony”. We consider the main periods in the development of the “Mathematics of Harmony” and its main mathematical results: algorithmic measurement theory, number systems with irrational bases and their applications in computer science, the hyperbolic Fibonacci functions, following from Binet’s formulas, and the hyperbolic Fibonacci l-functions (l = 1, 2, 3, …), following from Gazale’s formulas, and their applications for hyperbolic geometry, in particular, for the solution of Hilbert’s Fourth Problem.展开更多
After posing the axiom of linear algebra, the author develops how this allows the calculation of arbitrary base powers, which provides an instantaneous calculation of powers in a particular base such as base ten;first...After posing the axiom of linear algebra, the author develops how this allows the calculation of arbitrary base powers, which provides an instantaneous calculation of powers in a particular base such as base ten;first of all by developing the any base calculation of these powers, then by calculating triangles following the example of the “arithmetical” triangle of Pascal and showing how the formula of the binomial of Newton is driving the construction. The author also develops the consequences of the axiom of linear algebra for the decimal writing of numbers and the result that this provides for the calculation of infinite sums of the inverse of integers to successive powers. Then the implications of these new forms of calculation on calculator technologies, with in particular the storage of triangles which calculate powers in any base and the use of a multiplication table in a very large canonical base are discussed.展开更多
This article proposes a new approach based on linear programming optimization to solve the problem of determining the color of a complex fractal carpet pattern.The principle is aimed at finding suitable dyes for mixin...This article proposes a new approach based on linear programming optimization to solve the problem of determining the color of a complex fractal carpet pattern.The principle is aimed at finding suitable dyes for mixing and their exact concentrations,which,when applied correctly,gives the desired color.The objective function and all constraints of the model are expressed linearly according to the solution variables.Carpet design has become an emerging technological field known for its creativity,science and technology.Many carpet design concepts have been analyzed in terms of color,contrast,brightness,as well as other mathematical concepts such as geometric changes and formulas.These concepts represent a common process in the carpet industry.This article discusses the use of complex fractal images in carpet design and simplex optimization in color selection.展开更多
How to quickly compute the number of points on an Elliptic Curve （EC） has been a longstanding challenge. The computational complexity of the algorithm usually employed makes it highly inefficient. Unlike the general...How to quickly compute the number of points on an Elliptic Curve （EC） has been a longstanding challenge. The computational complexity of the algorithm usually employed makes it highly inefficient. Unlike the general EC, a simple method called the Weil theorem can be used to compute the order of an EC characterized by a small prime number, such as the Kobltiz EC characterized by two. The fifteen secure ECs recommended by the National Institute of Standards and Technology （NIST） Digital Signature Standard contain five Koblitz ECs whose maximum base domain reaches 571 bits. Experimental results show that the computation speed decreases for base domains exceeding 600 bits. In this paper, we propose a simple method that combines the Weil theorem with Pascals triangle, which greatly reduces the computational complexity. We have validated the performance of this method for base fields ranging from 2l^100 to 2^1000. Furthermore, this new method can be generalized to any ECs characterized by any small prime number.展开更多
文摘Summetor is an operator used in the mathematics to calculate the special numbers like binomial coefficients and combinations of group elements. It has many applications in algebra, matrices like calculation of pascal triangle elements and pascal matrix formation, etc. This paper explains about its functions and properties of N-Summet-k. The result of variation between N and k is shown in tabulation.
文摘We give a survey on the history, the main mathematical results and applications of the Mathematics of Harmony as a new interdisciplinary direction of modern science. In its origins, this direction goes back to Euclid’s “Elements”. According to “Proclus hypothesis”, the main goal of Euclid was to create a full geometric theory of Platonic solids, associated with the ancient conception of the “Universe Harmony”. We consider the main periods in the development of the “Mathematics of Harmony” and its main mathematical results: algorithmic measurement theory, number systems with irrational bases and their applications in computer science, the hyperbolic Fibonacci functions, following from Binet’s formulas, and the hyperbolic Fibonacci l-functions (l = 1, 2, 3, …), following from Gazale’s formulas, and their applications for hyperbolic geometry, in particular, for the solution of Hilbert’s Fourth Problem.
文摘After posing the axiom of linear algebra, the author develops how this allows the calculation of arbitrary base powers, which provides an instantaneous calculation of powers in a particular base such as base ten;first of all by developing the any base calculation of these powers, then by calculating triangles following the example of the “arithmetical” triangle of Pascal and showing how the formula of the binomial of Newton is driving the construction. The author also develops the consequences of the axiom of linear algebra for the decimal writing of numbers and the result that this provides for the calculation of infinite sums of the inverse of integers to successive powers. Then the implications of these new forms of calculation on calculator technologies, with in particular the storage of triangles which calculate powers in any base and the use of a multiplication table in a very large canonical base are discussed.
文摘This article proposes a new approach based on linear programming optimization to solve the problem of determining the color of a complex fractal carpet pattern.The principle is aimed at finding suitable dyes for mixing and their exact concentrations,which,when applied correctly,gives the desired color.The objective function and all constraints of the model are expressed linearly according to the solution variables.Carpet design has become an emerging technological field known for its creativity,science and technology.Many carpet design concepts have been analyzed in terms of color,contrast,brightness,as well as other mathematical concepts such as geometric changes and formulas.These concepts represent a common process in the carpet industry.This article discusses the use of complex fractal images in carpet design and simplex optimization in color selection.
基金supported by the National Natura Science Foundation of China (Nos.61332019 61572304, 61272056, and 60970006)the Innovation Grant of Shanghai Municipal Education Commission (No.14ZZ089)Shanghai Key Laboratory of Specialty Fiber Optics and Optical Access Networks (No.SKLSFO2014-06)
文摘How to quickly compute the number of points on an Elliptic Curve （EC） has been a longstanding challenge. The computational complexity of the algorithm usually employed makes it highly inefficient. Unlike the general EC, a simple method called the Weil theorem can be used to compute the order of an EC characterized by a small prime number, such as the Kobltiz EC characterized by two. The fifteen secure ECs recommended by the National Institute of Standards and Technology （NIST） Digital Signature Standard contain five Koblitz ECs whose maximum base domain reaches 571 bits. Experimental results show that the computation speed decreases for base domains exceeding 600 bits. In this paper, we propose a simple method that combines the Weil theorem with Pascals triangle, which greatly reduces the computational complexity. We have validated the performance of this method for base fields ranging from 2l^100 to 2^1000. Furthermore, this new method can be generalized to any ECs characterized by any small prime number.
