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Paper Type |
: | Research Paper |

Title |
: | The Method of Proof of Fermat's Last Theorem |

Country |
: | Russia |

Authors |
: | Vadim N. Romanov |

: | 10.9790/5728-1403010105 |

**Abstract: **The paper considers Fermat's theorem as a special case of the problem of least (smallest) deviation.
It is proved that the theorem is true, since error can not vanish on the set of rational numbers.

**Keywords:** Number theory, natural numbers, Fermat's last theorem

[1] Romanov V.N. Elementary way of proving Fermat's theorem. // Int. Journal of Engineering Research and Application, 2018, Vol. 5,

Issue 1, pp.57 – 68.

[2] Romanov V.N. Investigation of the fundamental problems of number theory. Publishing house "Asterion", 2015 (in Russian).

[3] Wiles A. Modular Elliptic Curves and Fermat's Last Theorem. // Annals of Mathematics, 1995, Vol. 142, pp. 443-551.

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**Abstract: **Critical thinking is one of the important indicators for students to be competing in the world of work and personal life, students must have the ability to solve problems and must be able to think critically. One effort to improve critical thinking skills, especially in math lessons is to implement problem-solving learning. This article describes the implementation of problem-solving learning in improving students' critical thinking skills of grade VIII SMP Negeri 5 Bulukumba obtained through classroom action research that is conducted for 2 cycles with the result that with the application of learning problem solving can improve the ability of critical thinking as seen from the indicators the success of 35 people or 87.5% of students have completed learning mathematics and as many as 5 people or 12.5% of students have not finished learning math.........

**Keywords**: Critical Thinking, Problem-Solving

[1] Djamarah & Zain. Teaching and Learning Strategies. Jakarta: Rineka Cipta. 2006.

[2] Fachrurazi, Application of Problem Based Learning to Improve the Ability of Critical Thinking and Student Matter Communications, Jurnal UPI. Special edition 1(1), 2011.

[3] Fisher, Alec. Critical Thinking An Introduction. Jakarta : Erlangga. 2009.

[4] Fitria, Hanifah. T. "Improving the Ability of Critical Thinking of Junior High School Students through Elicitina Activities Model (MEAS)" "Approach in Mathematics Learning". Bandung : Repositori UPI.Edu.

[5] Kamdi, W dkk. (2007). Innovative Learning Models. Malang : Universitas Negeri Malang.

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Paper Type |
: | Research Paper |

Title |
: | The Geometric Least Squares Fitting Of Ellipses |

Country |
: | Saudi Arabia |

Authors |
: | Abdellatif Bettayeb |

: | 10.9790/5728-1403011218 |

**Abstract: **The problem of Fitting conic sections to given data in the plane is one which is of great interest and
arises in many applications, e.g. computer graphics, statistics, coordinate metrology, aircraft industry,
metrology, astronomy, refractometry, and petroleum engineering [7, 2, 3]. In this paper, we present several
methods which have been suggested for Fitting ellipses to data in the plane. We will look particularly at one
method, by giving examples and using Matlab to solve these problem.

[1] Dixon, L. C. W., Spedicato, E. and Szego, G. P., Nonlinear Optimization: theory and algorithms, Birkhauser Boston, 1980. ISBN 3-7643-3020-1. pp. 91-102.

[2] Fletcher, R. G., Practical Methods of Optimization, John Wiley and Sons, 1995. ISBN 0 471 91547 5 pp. 111-136.

[3] Gander, W., Golub, G. H. and Strebel, R., Fitting of circles and ellipses: least square solution,BIT, 34(1994), pp. 556-577.

[4] Huffel Sabine Van, Recent Advances in total least squares techniques and errors in variables modeling, " Orthogonal Least Squares Fitting by Conic Sections " by Helmuth Spath pp. 259-264, Library of Congress, USA, 1997. ISBN 0-89871-393-5.

[5] Kolman, B., Elementary linear algebra, 1991. ISBN 0-02-366045-7.

[6] Schwarz, H. R., Numerical Analysis: A comprehensive introduction, Great Britain, 1989. ISBN 0471 92064 9 pp. 294-329.

[7] The MathWorks, Inc. MATLAB REFERENCE GUIDE, 1992. PP. 205.

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Paper Type |
: | Research Paper |

Title |
: | Universal Portfolios Generated by the f and Bregman Divergences |

Country |
: | Malaysia |

Authors |
: | Choon Peng Tan || Kee Seng Kuang |

: | 10.9790/5728-1403011925 |

**Abstract: **The 𝑓 and Bregman divergences are used to generate two universal portfolios in a model-free stock market. The logarithm of the estimated next-day wealth return is approximated by 𝑘 terms of its Taylor series. The condition for the two universal portfolios to be identical is derived. A sufficient condition for the two universal portfolios to be identical is demonstrated. The Helmbold family of universal portfolios are both 𝑓-divergence and Bregman-divergence universal portfolios. An empirical study of the Type 2 Helmbold universal portfolio is presented. There is empirical evidence that the investor's wealth can be increased by using this type of universal portfolio.

**Keywords -** Bregman divergence, 𝑓-divergence, Investment, Universal portfolio

[1] T. M. Cover, Universal Portfolios, Mathematical Finance, vol. 1, pp. 1-29, Jan 1991.

[2] T. M. Cover and E. Ordentlich, Universal portfolios with side information, IEEE Transactions on Information Theory, vol.42, no.2, pp.348-363, Mar. 1996.

[3] C. P. Tan, Performance bounds for the distribution-generated universal portfolios, Proc. 59th ISI World Statistics Congress, Hong Kong, 5327-5332, 2013.

[4] D. P. Helmbold, R. E. Shapire, Y. Singer and M. K. Warmuth, On-line portfolio selection using multiplicative updates, Mathematical Finance, vol.8, no.4, pp.325-347, Oct. 1998.

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Paper Type |
: | Research Paper |

Title |
: | Les Nombres Graphiques Et Le Problème P=NP |

Country |
: | France |

Authors |
: | M. SGHIAR |

: | 10.9790/5728-1403012629 |

**Abstract: **I will prove the existence of a function f n with values in ℕ on any Graph G/ {X1,…, Xn} with
cardinal n , such that either f n(X1) is equal to 0 or it is the sum of all hamiltonian cycles. The number of the
operations to be performed to calculate f n(X1) is of the order O(n3) and that it follows that we
have P= NP .
Résumé : Je démontre l'existence d'une fonction f n à valeurs dans ℕ sur tout Graphe G/ {X1 ,…,Xn} de
cardinal n , telle que soit f n(X1) est égal à 0 soit elle est la somme de tous les cycles hamiltoniens. Le
nombre d'opérations à effectuer pour calculer f n(X1) est de l'ordre de O(n3) et il en résulte qu'on a bien
P= NP .

**Keywords: **Graph, Hamilton cycles, P=NP, the knapsack problem, the travelling salesman problem, TSP, KP.

[1]. Michael R. Garey et David S. Johnson,A Guide to the Theory of NP -Completeness,Computers and Intractability , W. H.

Freeman, 1979,199−200, ISBN 0-7167-1045-5.

[2]. Chuzo Iwamoto et Godfried Toussaint, Finding Hamiltonian circuits in arrangements of Jordan curves is NP-complete, Information

Processing Letters, 1994, volume 52, number 15, pages183–189.

[3]. Gabriel Andrew Dirac, Some theorems on abstract graphs, Proceedings of the London Mathematical Society, 1952, 3e série,vol. 2,

pages 69–81.

[4]. Oystein Ore, A Note on Hamiltonian Circuits, American Mathematical Monthly, 1960, number 67, pages 55.

[5]. Lajos Posa, A theorem concerning hamilton lines, journal Magyar Tud. Akad. Mat. Kutato Int. Kozl., 1962, volume 7, pages 225–226.

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Paper Type |
: | Research Paper |

Title |
: | On Scalar Quasi - Weak Commutative Algebras |

Country |
: | India |

Authors |
: | G.Gopalakrishnamoorthy || S.Anitha || M.Kamaraj |

: | 10.9790/5728-1403013037 |

**Abstract: **The concepts of scalar commutativity defined in an Algebra A over a commutative ring R and Quasi - weak commutativity defined in a near- ring are mixed together to coin the concept of scalar quasi weak commutativity in an algebra A over a commutative ring R and its various properties are studied

[1]. R.Coughlin and M.Rich, On scalar dependent algebras, Canad.J.Math,24(1972), 696-702.

[2]. R.Coughlin, K.Kleinfield and M.Rich, Scalar dependent algebras, Proc, Amer.Math.Soc, 39(1973), 69-73.

[3]. G.Gopalakrishnamoorthy,M.Kamaraj and S.Geetha,On Quasi weak commutative

[4]. Near-rings,International Jour. of Math Research,Volume 5(5),(2013),431-440.

[5]. G.Gopalakrishnamoorthy,S.Geetha and S.Anitha ,On Quasi - weak m power commutative Near-rings and Quasi-weak (m,n) power commutative Near-rings, IOSR Jour. of Mathematics,Vol12(4),(2016),87-90.

[6]. K.Koh, J.Luh and M.S.Putcha, On the associativity and commutativity of algebras over commutative rings, Pacific.J.Math,63,No2,(1976),423-430.

[7]. Pliz Glinter, Near-Rngs, North Holland, Aneterdam;(1983)

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**Abstract: **In this paper, for a single grade marketing organization with non- instantaneous shortage in
manpower (wastage) due to policy decisions, classified according to their intensity of attrition, the moments of
time taken for recruitment are obtained using a suitable policy for recruitment when (i) the system has a control
limit for alertness (referred as optional threshold), in addition to the breakdown threshold (referred as
mandatory threshold) for cumulative shortage in manpower (ii) the shortage process of manpower due to exits
form a geometric process (iii) both the thresholds have an additional component for the cumulative shortage in
manpower due to frequent breaks (iv) the exits form an ordinary renewal process with exponential inter-exits,
according as the inter-policy decision times form a geometric process or an order statistics.

**Keywords:** Single grade marketing organization; Classified policy decisions; Non-instantaneous exits;
Geometric process; optional and mandatory thresholds with two components and Policy of recruitment

[1]. D. J. Bartholomew, Stochastic model for social processes, (John Wiley and Sons, New York, 1973).

[2]. D. J. Bartholomew and F. Andrew Forbes, Statistical techniques for manpower planning, (John Wiley and Sons, New York, 1979).

[3]. R. C. Grinold and K.T. Marshall, Manpower planning models, (North-Holland, New York, 1977).

[4]. J. B. Esther Clara, Contributions to the study on some stochastic models in manpower planning, doctoral diss., Bharathidasan

University, Tiruchirappalli, 2012.

[5]. A. Devi and A. Srinivasan, Variance of time to recruitment for single grade manpower system with different epochs for decisions

and exits, International Journal of Research in Mathematics and Computations, 2(1), Jan. – Jun. 2014, 23 – 27

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Paper Type |
: | Research Paper |

Title |
: | A Toolbox of Commutative Bi-Complex Algebra |

Country |
: | USA |

Authors |
: | Alexei V. Smirnov |

: | 10.9790/5728-1403014449 |

**Abstract: **Hyper-complex numbers forming four-dimensional quaternion-scalar space are considered.
Corresponding complementary algebra is introduced as an additional non-vector extension over the field of
complex numbers. Similarly to conventional complex numbers this commutative 4th-rank algebra possesses
division, conjugation, rooting and factorization along with direct analog of Euler's formula. Rotations can be
represented consistently within this algebra as well. Some of direct applications include electromagnetic wave
theory, beam and accelerator physics.

**Keywords:** Qquaternion, hyper-complex, bi-complex algebra.

[1]. G.A. Korn, T.M. Korn, Mathematical Handbook for Scientists and Engineers. McGraw Hill (1968)

[2]. http://aanda.u-strasbg.fr:2002/articles/astro/full/2000/09/h1201/node12.html

[3]. G. H. Hoffstaetter, in arXiv:physics/0006008 V 1, 5 Jun 2000

[4]. K. Heinemann and G. H. Hoffstatter, in Physical Review E, 54 (1996) 4240; Official DESY Report 96-078.

[5]. A.V. Smirnov, in Nuclear Instruments and Methods in Physics Research, NIM A, 469 (1) (2001) 21

[6]. L. Lewin, Theory of Waveguides, London, Newnes-Butterworths (1979).

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**Abstract: **is proved that there is a unique common fixed point for the mapping in the 2 b -metric space under the condition of satisfying such contraction type. We raised a weaker five-element function class and construct the sequence by shrinking conditions.The results of this paper generalize the theory of fixed points in many Fixed point theories..

**Keywords:** b -metric space five-element function class shrink condition Cauchy sequence fixed point

[1]. Zead Mustafa. b2-Metric spaces and some fixed point theory and application. Spring Open Journal. 2014:144:1-4.

[2]. Yong-jie Piao. A family of self-maps having an unique common fixed point in 2-metric space. Chinese Journal of Mathematical Physics. 2012,32(A):1-6.

[3]. Yong-jie Piao. A quasi-shrinking non-exchange self-map family with unique common fixed points on a 2-metric space.Journal of

Heilongjiang University (Natural Science Edition), 2006,23(5): 655-657.

[4]. Yong-jie Piao. The unique common fixed points of the mapping family with the same shrinkage conditions on the 2-metric space. The

semi-annual journal of Nanjing University, 2010,27(1):82-87.

[5]. Abbas M, Junk G. Common fixed point results for non-commuting mappings without continuity in core metric spaces. J.Math.Anal. App,
2008, 341(1):416-420.

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**Abstract: **We examine the probability density function of the present value ofperpetuity, subject to a stochastic Brownian motion rate of return of an investment and then show that its inverse is a gamma distribution. The derivation uses the scale function and martingale results from the theory of calculus.

**Keywords:** Stochastic calculus; scale function; perpetuity

[1]. Beekman, J. A. and C. P. Fuelling (1991). Extra randomness in certain annuity models. Insurance:Mathematics and Economics 10,

275-287

[2]. Christiansen, M. C. (2013): Gaussian and Affine Approximation of Stochastic Diffusion Models for Interest and Mortality Rates.

Risks 1, 81-100.

[3]. DeSchepper, A., M. Teunen and M. Goovaerts (1994). An analytical inversion of a Laplace transforms related annuities certain.

Insurance:Mathematics and Economics 14, 33-37

[4]. Giaccotto, C. (1998). Stochastic modeling of interest rates: Actuarial vs. equilibrium approach. Journal of risk and insurance 53 (3),

435-453

[5]. Karatzas, I. and S. E. Shreve (1991). Brownian motion and Stochastic Calculus, 2nd ed. Springer verlag New York, Inc..

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Paper Type |
: | Research Paper |

Title |
: | Optimal Control Strategy for Alcoholism Model with Two Infected Compartments |

Country |
: | Indonesia |

Authors |
: | Khozin Mu'tamar |

: | 10.9790/5728-1403015867 |

**Abstract: **This article discussed mathematical model of alcoholism with optimal control.Model is designed using system of differential equations based on Susceptible, Infectible and Resistant (SIR) model. Infected individuals is divided into two compartments, admitted and non-admitted to alcoholism. Optimal control is used to prevent interaction between susceptible individual and infected individuals. Stability analysis is done locally using Routh-Hurwitz criteria. It can be shown that optimal control determines stability of the system. In the end of article, numerical simulation is given to illustrate uncontrolled and controlled system. The results show optimal control succeed to reduce infected individuals. Controlled system has higher susceptible individuals and has lower infected individuals than uncontrolled system..

**Keywords:** Alcoholism model, local stability, optimal control, Routh-Hurwitz criteria, SIR model

[1]. Http://www.bbc.com/indonesia/indonesia-43693188; accessed on April, 16th 2018.

[2]. C.P. Bhunu, A mathematical analysis of alcoholism, World Journal of Modeling and Simulation, Vol 8 No 2, pp. 124-134, 2012.

[3]. G. Mulone, B. Straughan, Modeling binge drinking, International Journal of Biomathematics, Vol 5 No 1, 2012.

[4]. X.Y. Wang, H.F. Huo, Q.K. Kong, W.X. Shi, Optimal control strategies in an alcoholism model, Abstract and Applied Analysis,

Vol 2014 Article ID 954069, Hindawi, 2016.

[5]. J. Kahuru, L. S. Luboobi, and Y.N. Gyekye, Optimal Control Techniques on a Mathematical Model for the Dynamics of Tungiasis

in a Community, International Journal of Mathematics and Mathematical Sciences, vol. 2017, Article ID 4804897, 19 pages, 2017.

https://doi.org/10.1155/2017/4804897..

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Paper Type |
: | Research Paper |

Title |
: | On The Linear Systems over Non Commutative Rhotrices |

Country |
: | Nigeria |

Authors |
: | Okon, Ubong E || Galadima Dauda J || Malachy Markus |

: | 10.9790/5728-1403016872 |

**Abstract: **Rhotrices n P , n Q and n R were considered with the binary operation of non-commutative method of rhotrix multiplication defined by Sani(2007) to study linear systems of the form Pn Qn Rn . This work identified conditions necessary for the solvability of the system and also presented procedure for computing the square root of a rhotrix..

**Keywords:** Rhotrix; Linear system; Row-column multiplication

[1]. A.O. Ajibade, The concept of rhotrix in mathematical enrichment, International Journal of MathematicsEducation ,Science and

Technology .34 (2003) , pp 175-179.

[2]. A. Aminu, On the linear systems of rhotricesNNTDM 15 (2009), 4, 7-

[3]. K.T. Atanassov, A.G. Shannon, Matrix-tertions and matrix noitrets: exercises in mathematical. enrichment, Int. J. Math.

Educ. Sci. Technol. 29 (1998), pp 98-903

[4]. B. Sani, An alternative method for multiplication of rhotrices, Journa4l of MathematicsEducation ,Science and Technology.

35.(2004), pp 777-781.

[5]. B. Sani, The row-column multiplication for high dimensional rhotrices, Journal of MathematicsEducation ,Science and Technology.

38 (2007) pp 657-662...

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Paper Type |
: | Research Paper |

Title |
: | On Interval Valued Fuzzy Det-Norm Matrix with Partitions |

Country |
: | India |

Authors |
: | C.Loganathan || V.Pushpalatha |

: | 10.9790/5728-1403017383 |

**Abstract: **In this paper, interval valued fuzzy det-norm matrices using the structure of 𝑀𝑛∗(F) and interval valued fuzzy det-norm ordering with interval valued fuzzy matrices are defined with examples. We prove that det-norm ordering is a partition ordering on the set of all idempotent matrices in 𝑀𝑛∗(F). Some theorems of det-norm ordering with interval valued fuzzy matrices are proved.

**Keywords:** det-norm matrix, det-norm ordering, fuzzy matrix, idempotent, interval valued fuzzy matrix

[1]. Bertoiuzza, On the distributivity of t–norm and t–conorms, 2nd IEEE Interval on Fuzzy Systems, IEEE press, Piscataway, NJ, 1993, 140–147.

[2]. Jian Miao Chen, Fuzzy matrix partial ordering and generalized inverse, Fuzzy Sets System 105, 1982, 453–458.

[3]. Meenakshi A.R. and Cokilavany R., On fuzzy 2 normed linear spaces, Journal of Fuzzy Mathematics, 9(2), 2001, 345–351.

[4]. Nagoorgani A. and Kalyani G., Binormed sequences in fuzzy matrices. Bulletin of Pure and Applied Science, 22E(2), 2003, 445–451.

[5]. Nagoorgani A. and Kalyani G., Fuzzy matrix M–ordering, Science Letters, 27(3) and (4), 2004..