In this paper,using the Jordan canonical form of the Pascal matrix Pn,we present a new approach for inverting the Pascal matrix plus a scalar Pn+aIn for arbitrary real number a≠1.
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
EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn i...EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn is given via the Stirling matrix of the first kind, and a recurrence relation of the elements of the matrix T, is obtained, so an open urnblem urouosed bv EI-Mikkawv[2] is solved. Some combinatorial identities are also given.展开更多
Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I ex...Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.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
基金Supported by the Natural Science Foundation of Gansu Proveince（1010RJZA049）
文摘In this paper,using the Jordan canonical form of the Pascal matrix Pn,we present a new approach for inverting the Pascal matrix plus a scalar Pn+aIn for arbitrary real number a≠1.
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the NSF of Gansu Province of China (3ZS041-A25-007)
文摘EI-Mikkawy M obtained that the symmetric Pascal matrix Qn and the Vandermonde matrix Vn are connected by the equation Qn= TnVn, where Tn is a stochastic matrix in [1]. In this paper, a decomposition of the matrix Tn is given via the Stirling matrix of the first kind, and a recurrence relation of the elements of the matrix T, is obtained, so an open urnblem urouosed bv EI-Mikkawv[2] is solved. Some combinatorial identities are also given.
文摘Generally Fibonacci series and Lucas series are the same, they converge to golden ratio. After I read Fibonacci series, I thought, is there or are there any series which converges to golden ratio. Because of that I explored the inter relations of Fibonacci series when I was intent on Fibonacci series in my difference parallelogram. In which, I found there is no degeneration on Fibonacci series. In my thought, Pascal triangle seemed like a lower triangular matrix, so I tried to find the inverse for that. In inverse form, there is no change against original form of Pascal elements matrix. One day I played with ring magnets, which forms hexagonal shapes. Number of rings which forms Hexagonal shape gives Hex series. In this paper, I give the general formula for generating various types of Fibonacci series and its non-degeneration, how Pascal elements maintain its identities and which shapes formed by hex numbers by difference and matrices.
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.
