The inclusion-exclusion formula (IEF) is a fundamental tool for evaluating network reliability with known minimal paths or minimal cuts. However, the formula contains many pairs of terms which cancel. Using the notion...The inclusion-exclusion formula (IEF) is a fundamental tool for evaluating network reliability with known minimal paths or minimal cuts. However, the formula contains many pairs of terms which cancel. Using the notion of comparable node partitions some properties of canceling terms in IEF are given. With these properties and the thought of “dynamic programming” method, a simple and efficient inclusion-exclusion algorithm for evaluating the source-to-terminal reliability of a network starting with cutsets is presented. The algorithm generates all the non-canceling terms in the unreliability expression. The computational complexity of the algorithm is O(n+m3+M), where n and m are the numbers of nodes and minimal cuts of the given network respectively, M is the number of terms in the final symbolic unreliability expression that generated using the presented algorithm. Examples are shown to illustrate the effectiveness of the algorithm.展开更多
In this note, the author find an upper bound formula for the number of the p × p normalized Latin Square,the first row and column of which are both standard order 1, 2,…p.
In this article, we study the generalized Bernoulli learning model based on the probability of success pi = ai /n where i = 1,2,...n 0a1a2ann and n is positive integer. This gives the previous results given by Abdulna...In this article, we study the generalized Bernoulli learning model based on the probability of success pi = ai /n where i = 1,2,...n 0a1a2ann and n is positive integer. This gives the previous results given by Abdulnasser and Khidr [1], Rashad [2] and EL-Desouky and Mahfouz [3] as special cases, where pi = i/n pi = i2/n2 and pi = ip/np respectively. The probability function P(Wn = k) of this model is derived, some properties of the model are obtained and the limiting distribution of the model is given.展开更多
Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N（a1x12 ＋ ···＋anxn2 = bx1 ···xs） denote the number of solutions（x1,...,xn） of the equation a1x12 ＋·&#...Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N（a1x12 ＋ ···＋anxn2 = bx1 ···xs） denote the number of solutions（x1,...,xn） of the equation a1x12 ＋···＋ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F＊q,b ∈ F＊q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.展开更多
文摘The inclusion-exclusion formula (IEF) is a fundamental tool for evaluating network reliability with known minimal paths or minimal cuts. However, the formula contains many pairs of terms which cancel. Using the notion of comparable node partitions some properties of canceling terms in IEF are given. With these properties and the thought of “dynamic programming” method, a simple and efficient inclusion-exclusion algorithm for evaluating the source-to-terminal reliability of a network starting with cutsets is presented. The algorithm generates all the non-canceling terms in the unreliability expression. The computational complexity of the algorithm is O(n+m3+M), where n and m are the numbers of nodes and minimal cuts of the given network respectively, M is the number of terms in the final symbolic unreliability expression that generated using the presented algorithm. Examples are shown to illustrate the effectiveness of the algorithm.
文摘In this note, the author find an upper bound formula for the number of the p × p normalized Latin Square,the first row and column of which are both standard order 1, 2,…p.
文摘In this article, we study the generalized Bernoulli learning model based on the probability of success pi = ai /n where i = 1,2,...n 0a1a2ann and n is positive integer. This gives the previous results given by Abdulnasser and Khidr [1], Rashad [2] and EL-Desouky and Mahfouz [3] as special cases, where pi = i/n pi = i2/n2 and pi = ip/np respectively. The probability function P(Wn = k) of this model is derived, some properties of the model are obtained and the limiting distribution of the model is given.
基金Supported by the National Natural Science Foundation of China (Grant Nos.1097120510771100)
文摘Let Fq be a finite field with q = pf elements,where p is an odd prime.Let N（a1x12 ＋ ···＋anxn2 = bx1 ···xs） denote the number of solutions（x1,...,xn） of the equation a1x12 ＋···＋ anxn2 = bx1 ···xs in Fnq,where n 5,s n,and ai ∈ F＊q,b ∈ F＊q.In this paper,we solve the problem which the present authors mentioned in an earlier paper,and obtain a reduction formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xs,where n 5,3 ≤ s n,under a certain restriction on coefficients.We also obtain an explicit formula for the number of solutions of equation a1x21 ＋ ··· ＋ anxn2 = bx1 ···xn-1 in Fqn under a restriction on n and q.
