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
Thousands of landslide data being taken as the nation wide statistics of sampling and the two state variables of landslide being processed with two methods described in the references, the main types of lithologica...Thousands of landslide data being taken as the nation wide statistics of sampling and the two state variables of landslide being processed with two methods described in the references, the main types of lithological groups of landslides in China have been sieved and selected.On the other hand, through the displacement table of Pascal Yanghui triangle used in the information encoding theory, the mark weight of sampling can be calculated and the main lithological groups which have close relationship with landslide occurrence can be gained.In comparison with the both results, the characteristics of main sliding lithological groups are determinated, and the main distribution regions of landslides can be prognosticated.展开更多
We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, φ(n) is an even number. But it is not true for all even number, we could find some numbers which ...We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, φ(n) is an even number. But it is not true for all even number, we could find some numbers which would not be the value of any φ(n). The Sum of two odd numbers is one even number. Gold Bach stated “Every even integer greater than 2 can be written as the sum of two primes”. Other than two, all prime numbers are odd numbers. So we can write, every even integer greater than 2 as the sum of two primes. German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio. We could find the series which is generated by one and inverse the golden ratio. Also we can note consecutive golden ratio numbers converge to the golden ratio. Lothar Collatz stated integers converge to one. It is also known as 3k + 1 problem. Tao redefined Collatz conjecture as 3k <span style="white-space:nowrap;">− 1 problem. We could not prove it directly but one parallel proof will prove this conjecture.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
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
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Thousands of landslide data being taken as the nation wide statistics of sampling and the two state variables of landslide being processed with two methods described in the references, the main types of lithological groups of landslides in China have been sieved and selected.On the other hand, through the displacement table of Pascal Yanghui triangle used in the information encoding theory, the mark weight of sampling can be calculated and the main lithological groups which have close relationship with landslide occurrence can be gained.In comparison with the both results, the characteristics of main sliding lithological groups are determinated, and the main distribution regions of landslides can be prognosticated.
文摘We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, φ(n) is an even number. But it is not true for all even number, we could find some numbers which would not be the value of any φ(n). The Sum of two odd numbers is one even number. Gold Bach stated “Every even integer greater than 2 can be written as the sum of two primes”. Other than two, all prime numbers are odd numbers. So we can write, every even integer greater than 2 as the sum of two primes. German mathematician Simon Jacob (d. 1564) noted that consecutive Fibonacci numbers converge to the golden ratio. We could find the series which is generated by one and inverse the golden ratio. Also we can note consecutive golden ratio numbers converge to the golden ratio. Lothar Collatz stated integers converge to one. It is also known as 3k + 1 problem. Tao redefined Collatz conjecture as 3k <span style="white-space:nowrap;">− 1 problem. We could not prove it directly but one parallel proof will prove this conjecture.
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.
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.
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.
