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
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
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
The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the hist...The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the historical backgrounds of these concepts, we propose an algorithm that will generate the elements of these matrices. In fact, we will show that the indicated algorithm can be used to construct the elements of any power series matrix generated by any polynomial (see Definition 1), and hence, it is a generalization of the specific algorithms that give us the Pascal and the Fibonacci matrices.展开更多
In this paper, we will see that some k?-Fibonacci sequences are related to the classical Fibonacci sequence of such way that we can express the terms of a k -Fibonacci sequence in function of some terms of the classic...In this paper, we will see that some k?-Fibonacci sequences are related to the classical Fibonacci sequence of such way that we can express the terms of a k -Fibonacci sequence in function of some terms of the classical Fibonacci sequence. And the formulas will apply to any sequence of a certain set of k'?-Fibonacci sequences. Thus we find k -Fibonacci sequences relating to other k -Fibonacci sequences when?σ'k?is linearly dependent of?.展开更多
We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, <em>φ</em>(<em>n</em>) is an even number. But it is not true for all even...We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, <em>φ</em>(<em>n</em>) 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 <em>φ</em>(<em>n</em>). 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;">−</span> 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
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.
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.
文摘The Pascal matrix and the Fibonacci matrix are among the most well-known and the most widely-used tools in elementary algebra. In this paper, after a brief introduction where we give the basic definitions and the historical backgrounds of these concepts, we propose an algorithm that will generate the elements of these matrices. In fact, we will show that the indicated algorithm can be used to construct the elements of any power series matrix generated by any polynomial (see Definition 1), and hence, it is a generalization of the specific algorithms that give us the Pascal and the Fibonacci matrices.
文摘In this paper, we will see that some k?-Fibonacci sequences are related to the classical Fibonacci sequence of such way that we can express the terms of a k -Fibonacci sequence in function of some terms of the classical Fibonacci sequence. And the formulas will apply to any sequence of a certain set of k'?-Fibonacci sequences. Thus we find k -Fibonacci sequences relating to other k -Fibonacci sequences when?σ'k?is linearly dependent of?.
文摘We know Pascal’s triangle and planer graphs. They are mutually connected with each other. For any positive integer n, <em>φ</em>(<em>n</em>) 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 <em>φ</em>(<em>n</em>). 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;">−</span> 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.
