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### 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

### 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

### Keywords

### 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

_{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

### 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

_{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

### 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

_{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. | ||