### Content: Volume 2, Issue 2

### Article

**Availability and Profit Analysis of a linear Consecutive 2-out-of-4 Repairable System with Units Exchange**

^{1}Department of Mathematics, Usmanu Danfodio University, Sokoto, Nigeria

^{2}Department of Mathematical Sciences, Faculty of Science, Bayero University, Kano, Nigeria

*American Journal of Applied Mathematics and Statistics*.

**2014**, 2(2), 83-87

**DOI:**10.12691/ajams-2-2-6

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

U.A. Ali, Ibrahim Yusuf. Availability and Profit Analysis of a linear Consecutive 2-out-of-4 Repairable System with Units Exchange.

*American Journal of Applied Mathematics and Statistics*. 2014; 2(2):83-87. doi: 10.12691/ajams-2-2-6.

Correspondence to: Ibrahim Yusuf, Department of Mathematical Sciences, Faculty of Science, Bayero University, Kano, Nigeria. Email: Ibrahimyusuffagge@gmail.com

### Abstract

In this paper, we study some reliability characteristics of a repairable linear consecutive 2-out-of-4 system. The system work when to two units in a row (consecutive) works. The system is attended by three repairmen. When an operating unit failed, a standby is switched on or an idle operating unit is exchange with the failed unit. The explicit expressions of the reliability characteristics such availability, busy period of the repairmen and profit function are derived. Some cases are analyzed graphically to investigate the impact of system parameters on availability and profit.

### Keywords

### References

[1] | A. Khatab, N. Nahas, and M. Nourelfath, Availbilty of k-out-of-n: G systems with non identical components subject to repair priorities. Reliab. Eng. Syst. Safety. 94, 2009, 142-151. | ||

[2] | I. Yusuf, Availability and profit analysis of 3-out-of-4 repairable system with preventive maintenance, International Journal of Applied Mathematical Research, 1 (4), 2012, pp 510-519. | ||

[3] | I. Yusuf, and B. Yusuf, Evaluation of reliability characteristics of two dissimilar network flow systems. Applied Mathematical Sciences, Vol. 7, No. 40, 2013, pp 1983-1999. | ||

[4] | I.Yusuf, and N. Hussaini, Evaluation of Reliability and Availability Characteristics of 2-out of -3 Standby System under a Perfect Repair Condition American Journal of Mathematics and Statistics, Vol. 2No. 5, 2012, pp 114-119. | ||

[5] | J. Shao, and L.R. Lamberson, Modeling shared-load k-out-of-n : G systems IEEE Trans. Reliab., 40: 1991. | ||

[6] | K.M. El-Said, Cost analysis of a system with preventive maintenance by using Kolmogorov’s forward equations method, American Journal of Applied Sciences, Vol. 5,No. 4, 2008, pp 405-410. | ||

[7] | K.M.El-Said, and M.S. El-Sherbeny, Evaluation of reliability and availability characteristics of two different systems by using linear first order differential equations, Journal of Mathematics, and Statistics, Vol. 1 No. 2, 2005, pp 119-123. | ||

[8] | M. Y. Haggag, “Cost analysis of a system involving common cause failure and preventive maintenance, ”Journal of Mathematics and Statistics, Vol. 5, No. 4, 2009, pp 305-310. | ||

[9] | Y. Barron, E. Frostig and B.Levikson, Analysis of r out of n systems with several repairmen, exponential life times and phase type repair times: an algorithmic approach. Eur. J. Oper. Res. Algorithm Approach 169, 2006, pp 202-225. | ||

### Article

**Recurrence Relations for Single and Product Moments of Generalized Order Statistics from Extreme Value Distribution**

^{1}Hindu College, University of Delhi, Delhi, India

*American Journal of Applied Mathematics and Statistics*.

**2014**, 2(2), 77-82

**DOI:**10.12691/ajams-2-2-5

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

Kamal Nain Kapoor. Recurrence Relations for Single and Product Moments of Generalized Order Statistics from Extreme Value Distribution.

*American Journal of Applied Mathematics and Statistics*. 2014; 2(2):77-82. doi: 10.12691/ajams-2-2-5.

Correspondence to: Kamal Nain Kapoor, Hindu College, University of Delhi, Delhi, India. Email: kamal.180968@gmail.com

### Abstract

In this paper, we establish some recurrence relations satisfied by single and product moments of Generalized Order Statistics from Extreme Value Distribution. These recurrence relations are independent of left truncated point and therefore are also applicable for Logistic as well as for half Logistic distributions studied in Balakrishnan (1985) and Saran and Pandey (2012). For a particular case these results verify the corresponding results of Saran and Pandey (2004) and Kumar (2010).

### Keywords

order statistics, record values, generalized order statistics, single moment, product moments, recurrence relations, extreme value distribution

### References

[1] | Athar, H., Kheaja S. K. and Nayabuddin (2012). Expectation identities of Pareto distribution based on generalized order statistics. American Journal of Applied Mathematics and Mathematical Sciences, 1, 23-29. | ||

[2] | Balakrishnan, N. (1985). Order statistics from the Half logistic distribution. J. Statist. Comp. Simul., 20, 287-309. | ||

[3] | Balakrishnan, N. and Joshi, P.C. (1984). Product moments of order statistics from doubly truncated exponential distribution. Naval Res. Logist. Quart, 31, 27-31. | ||

[4] | Balakrishnan, N. and Malik, H. J. (1986). Order statistics from linear exponential distribution, Part I: Increasing hazard rate case. Commun. Statist. – Theor. Meth. 15, 179-203. | ||

[5] | Joshi, P. C. (1978). Recurrence relations between moments of order statistics from exponential and truncated exponential distributions. Sankhya, Ser. B, 39, 362-371. | ||

[6] | Joshi, P. C. (1982). A note on mixed moments of order statistics from exponential and truncated exponential distributions. J. Statist. Plann. Inf., 6, 13-16. | ||

[7] | Kamps, U. (1995a). A concept of generalized order statistics. B. G. Teubner, Stuttgart. | ||

[8] | Kamps, U. (1995b). A concept of generalized order statistics. J. Statist. Plann. Inf., 48, 1-23. | ||

[9] | Kumar, D. (2010). Recurrence relations for single and product moments of generalized order statistics from p^{th }order exponential distribution and its characterization, J. Statist. Res. Iran 7, 201-212. | ||

[10] | Mohie El-Din, M. M., Mahmoud, M. A. W., Abu-Youssef, S. E. and Sultan, K. S. (1997). Order statistics from the doubly truncated linear exponential distribution and its characterizations. Commun. Statist.- Simul. Comput. 26, 281-290. | ||

[11] | Nain, K. (2010 a). Recurrence relations for single and product moments of k^{th} record values from generalized Weibull distribution and a characterization. International Mathematical Forum, 5, No. 33, 1645-1652. | ||

[12] | Nain, K. (2010 b). Recurrence relations for single and product moments of ordinary order statistics from p^{th} order exponential distribution. International Mathematical Forum, 5, No. 34, 1653 – 1662. | ||

[13] | Pawlas, P. and Szynal, D. (2001). Recurrence relations for single and product moments of generalized order statistics from Pareto, Generalized Pareto and Burr distributions. Commun. Statist. Theor. Meth., 30, 739-746. | ||

[14] | Saran, J. and Nain, K. (2012a). Recurrence relations for single and product moments of generalized order statistics from doubly truncated p^{th }order Exponential Distribution, JKSA 23. | ||

[15] | Saran, J. and Nain, K. (2012b). Relationships for moments of kth record values from doubly truncated pth order exponential and generalized Weibull distributions. ProbStat Forum., 05, 142-149. | ||

[16] | Saran, J. and Nain, K. (2012c). Relationships for moments of kth record values from doubly truncated pth order exponential and generalized Weibull distributions. ProbStat Forum., 05, 142-149. | ||

[17] | Saran, J and Nain, K (2013). Explicit Expressions for Single and Product Moments of Generalized Order Statistics from a New Class of Exponential Distributions Characterization, JKSA 24, 37-52. | ||

[18] | Saran, J. and Pandey, A. (2004). Recurrence relations for single and product moments of generalized order statistics from linear exponential distribution. Journal of Applied Statistical Science, 13, 323-333. | ||

[19] | Saran, J. and Pandey, A. (2009). Recurrence relations for single and product moments of generalized order statistics from linear exponential and Burr distributions. Journal of Statistical Theory and Applications, 8, No. 3, 383-391. | ||

[20] | Saran, J. and Pandey, A. (2011). Recurrence relations for marginal and joint moment generating functions of dual generalized order statistics from inverse Weibull distribution. Journal of Statistical Studies, 30, 65-72. | ||

[21] | Saran J. and Pande V.(2012). Recurrence relation for moments of progressively type-II right censored order statistics from half logistic distribution. J. Statistical Theory and Applications, 11, 87-96. | ||

[22] | Saran, J. and Pushkarna, N. (1999). Moments of order statistics from doubly truncated linear exponential distribution. J. Korean Statist. Soc., 28, 279-296. | ||

### Article

**Some New Generalizations of Fuzzy Average Code Word Length and their Bonds**

^{1}University of Kashmir, Hazratbal, Srinagar, India

*American Journal of Applied Mathematics and Statistics*.

**2014**, 2(2), 73-76

**DOI:**10.12691/ajams-2-2-4

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

M.A.K. Baig, Mohd Afzal Bhat, Mohd Javid Dar. Some New Generalizations of Fuzzy Average Code Word Length and their Bonds.

*American Journal of Applied Mathematics and Statistics*. 2014; 2(2):73-76. doi: 10.12691/ajams-2-2-4.

Correspondence to: M.A.K. Baig, University of Kashmir, Hazratbal, Srinagar, India. Email: baigmak62@gmail.com

### Abstract

In this communication, we propose a new generalizations of fuzzy average codeword length L

_{a}and study its particular cases. The results obtained not only generalize the existing fuzzy average code word length but all the known results are the particular cases of the proposed length. Some new fuzzy coding theorems have also been proved.### Keywords

fuzzy set, generalized fuzzy entropy, generalized fuzzy average, godeword length, information Bounds

### References

[1] | Bhandari, N. R. Pal, Some new information measures for fuzzy sets, Information Sciences 1993; Vol. 67, No. 3: pp. 209-228. | ||

[2] | Campbell, L.L., A coding theorem and Renyi’s entropy, Information and Control 1965; Vol. 8: pp. 423-429. | ||

[3] | De Luca, S. Termini, A Definition of Non-probabilistic Entropy in the Setting of fuzzy sets theory, Information and Control 1972; Vol.20: pp.301-312. | ||

[4] | Havrada, J. H., Charvat, F., Quantification methods of classificatory processes, the concepts of structural α entropy, Kybernetika 1967; Vol.3: pp. 30-35. | ||

[5] | J.N.Kapur, Measures of Fuzzy Information, Mathematical Science Trust Society, New Delhi; 1997. | ||

[6] | Kapur, J. N., A generalization of Campbell’s noiseless coding theorem, Jour. Bihar Math, Society 1986; Vol.10: pp.1-10. | ||

[7] | Kapur, J. N., Entropy and Coding, Mathematical Science Trust Society, New Delhi; 1998. | ||

[8] | Lowen, R., Fuzzy Set Theory–Basic Concepts, Techniques and Bibliography, Kluwer Academic Publication. Applied Intelligence 1996; Vol. 31, No. 3: pp.283-291. | ||

[9] | Mathai, A.M., Rathie, P.N., Basic Concept in Information Theory and Statistics. Wiley Eastern Limited, New Delhi; 1975. | ||

[10] | Pal, Bezdek, Measuring Fuzzy Uncertainty, IEEE Trans. of fuzzy systems 1994; Vol. 2, No. 2: pp.107-118. | ||

[11] | Renyi, A., On measures of entropy and information. Proceedings 4^{th} Berkeley Symposium on Mathematical Statistics and Probability 1961; Vol.1: pp.541-561. | ||

[12] | Shannon, C. E., A mathematical theory of communication. Bell System Technical Journal 1948; Vol.27: pp.379-423, 623-659. | ||

[13] | Sharma, B.D., Taneja, I. J., Entropies of typeα, β and other generalized measures of information theory, Matrika 1975; Vol.22: pp. 205-215. | ||

[14] | Zadeh, L. A., Fuzzy Sets, Inform, and Control 1966; Vol.8: pp.94-102. | ||

### Article

**Increment Primes**

^{1}Doctor of Engineering Science, Academician of RANS, member of EANS, Volga Region State Technological University, Russia

*American Journal of Applied Mathematics and Statistics*.

**2014**, 2(2), 66-72

**DOI:**10.12691/ajams-2-2-3

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

P.M. Mazurkin. Increment Primes.

*American Journal of Applied Mathematics and Statistics*. 2014; 2(2):66-72. doi: 10.12691/ajams-2-2-3.

Correspondence to: P.M. Mazurkin, Doctor of Engineering Science, Academician of RANS, member of EANS, Volga Region State Technological University, Russia. Email: kaf_po@mail.ru

### Abstract

The increment of prime numbers was a clear indication. Increase - the number increases, the addition of something. If the number of prime numbers, figuratively called the "ladder of Gauss-Riemann", the increase may well be likened to the steps, separated from the ladder itself. We prove that the law is obeyed z

_{2}(i_{2}=2)=1/2-1/2cos(πP(n)/2) in the critical line i_{2}=2 of the second digit binary number system. This functional model was stable and in other quantities of prime numbers (3000 and 100 000). The critical line is the Riemann column i_{2}=2 binary matrix of a prime rate. Not all non-trivial zeros lie on it. There is also a line of frames, the initial rate (yields patterns of symmetry) and left the envelope binary number 1. Cryptographers cannot worry: even on the critical line growth of prime numbers z_{2}_{i}=1/2-1/2cos(πP_{j}/2) contain the irrational number π=3.14159….### Keywords

### References

[1] | Don Zagier. The first 50 million prime numbers. URL: http://www.ega-math.narod.ru/Liv/Zagier.htm. | ||

[2] | Mazurkin P.M. Biotechnical principle and stable distribution laws // Successes of modern natural sciences. 2009. № 9, 93-97. | ||

### Article

**Series Primes in Binary**

^{1}Doctor of Engineering Science, Academician of RANS, member of EANS, Volga Region State Technological University, Russia

*American Journal of Applied Mathematics and Statistics*.

**2014**, 2(2), 60-65

**DOI:**10.12691/ajams-2-2-2

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

P.M. Mazurkin. Series Primes in Binary.

*American Journal of Applied Mathematics and Statistics*. 2014; 2(2):60-65. doi: 10.12691/ajams-2-2-2.

Correspondence to: P.M. Mazurkin, Doctor of Engineering Science, Academician of RANS, member of EANS, Volga Region State Technological University, Russia. Email: kaf_po@mail.ru

### Abstract

To prove the famous Riemann hypothesis, that the real part of the root is always exactly equal to 1/2, a series of 500 and the other prime numbers has been converted from decimal to binary number system. At the same time was a clear non-trivial zeros. Any prime number can be represented as quantized into binary digital signal. Quantization step to not dilute a number of prime numbers is 1. Number of levels (binary digits) depends on the power of the quantized number of primes. As a result, we get two types of zeros - the trivial and nontrivial. Capacity of a finite number of primes must be taken based on the completeness of block incidence matrix. Average statistical indicator is a binary number, and influencing variable - itself a prime number. The binary representation allows to visualize and geometric patterns in the full range of prime numbers.

### Keywords

### References

[1] | Gashkov S.B. Number systems and their applications. M. MCCME, 2004. 52. | ||

[2] | Signal. URL: http://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%B3%D0%BD%D0%B0%D0%BB. | ||

### Article

**Proof the Riemann Hypothesis**

^{1}Doctor of Engineering Science, Academician of RANS, member of EANS, Volga Region State Technological University, Russia

*American Journal of Applied Mathematics and Statistics*.

**2014**, 2(2), 53-59

**DOI:**10.12691/ajams-2-2-1

**Copyright © 2014 Science and Education Publishing**

**Cite this paper:**

P.M. Mazurkin. Proof the Riemann Hypothesis.

*American Journal of Applied Mathematics and Statistics*. 2014; 2(2):53-59. doi: 10.12691/ajams-2-2-1.

Correspondence to: P.M. Mazurkin, Doctor of Engineering Science, Academician of RANS, member of EANS, Volga Region State Technological University, Russia. Email: kaf_po@mail.ru

### Abstract

In the proof of the correctness of the Riemann hypothesis held strong Godel's incompleteness theorem. In keeping with the ideas of Poja and Hadamard's mathematical inventions, we decided to go beyond the modern achievements of the Gauss law of prime numbers and Riemann transformations in the complex numbers, knowing that at equipotent prime natural numbers will be sufficient mathematical transformations in real numbers. In simple numbers on the top left corner of the incidence matrix blocks are of the frame. When they move, a jump of the prime rate. Capacity of a number of prime numbers can be controlled by a frame, and they will be more reliable digits. In the column i=1 there is only one non-trivial zero on j=(0,∞). By the implicit Gaussian "normal" distribution , where P

_{j}- a number of prime numbers with the order-rank j. On the critical line of the formula for prime numbers . By "the famous Riemann hypothesis is that the real part of the root is always exactly equal to 1/2" is obtained - the vibration frequency of a series of prime numbers is equal π/2, and the shift of the wave - π/4.### Keywords

### References

[1] | Don Zagier. The first 50 million prime numbers. URL: http://www.ega-math.narod.ru/Liv/Zagier.htm. | ||

[2] | Number. URL: http://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE. | ||

[3] | Mazurkin P.M. Biotechnical principle and sustainable laws of distribution // Successes of modern natural sciences. 2009. № 9, 93-97. | ||

[4] | Mazurkin PM The statistical model of the periodic system of chemical elements D.I. Mendeleev. Yoshkar-Ola: MarSTU, 2006. 152. | ||