[39] **viXra:1609.0393 [pdf]**
*submitted on 2016-09-28 01:15:43*

**Authors:** Sattawat Suntisurat

**Comments:** 9 Pages.

This paper describe about the elementary proof of FLT by using the graph and irrational number (complete)

**Category:** General Mathematics

[38] **viXra:1609.0382 [pdf]**
*submitted on 2016-09-27 00:52:16*

**Authors:** Sattawat Suntisurat

**Comments:** 6 Pages.

This paper describe about the elementary of the Fermat's Last Theorem by using the graph.

**Category:** General Mathematics

[37] **viXra:1609.0361 [pdf]**
*submitted on 2016-09-25 23:30:48*

**Authors:** Sattawat Suntisurat

**Comments:** 1 Page.

The paper describe about the simplest method to proove the FLT by using irrational number.

**Category:** General Mathematics

[36] **viXra:1609.0340 [pdf]**
*submitted on 2016-09-23 20:31:46*

**Authors:** Shi-YuanDong

**Comments:** 9 Pages.

On the Qin-JiuShao and Chinese remainder theorem(CRT)
秦九韶的大衍求一术及其矩阵表示

**Category:** General Mathematics

[35] **viXra:1609.0329 [pdf]**
*submitted on 2016-09-22 16:05:32*

**Authors:** Austin J. Fearnley

**Comments:** 5 pages.

Pseudo-random data are used to illustrate the relationship between errors in raw data being comparatively judged and the resulting Rasch pairs location parameters, first for data which are relatively homogeneous and second for data which have various amounts of heterogeneity. For each data type, various error sizes are used. Rasch location parameters are designed to be on an interval scale and are here demonstrated to be plotted on a more contracted scale when the objects are more homogeneous.

**Category:** General Mathematics

[34] **viXra:1609.0318 [pdf]**
*submitted on 2016-09-21 22:02:01*

**Authors:** Mithun K V, Kennath Joseph D'Couthoy, Ligin Cyrilz, Mohammed Mufeed abdullax, Kripa Suvarna

**Comments:** 12 Pages.

World is as of now concentrating on alternate material sources that are environment agreeable and biodegradable in nature. Because of the expanding natural concerns, bio composite produced out of regular fiber and polymeric resin, is one of the late advancements in the business and constitutes the present extent of experimental work. The use of composite materials field is increasing gradually in engineering. The composite consists of mainly two phases i.e. matrix and fiber. The accessibility of characteristic fiber and simplicity of assembling have enticed scientists worldwide to attempt by regional standards accessible inexpensive fiber and to learning their achievability of fortification determinations and to what degree they fulfill the obliged particulars of great strengthened polymer composite aimed at structural requisition. Fiber reinforced polymer composites has numerous preferences, for example, generally minimal effort of creation, simple to create and better quality contrast than perfect polymer tars due with this reason fiber strengthened polymer composite utilized within an assortment of provision as class of structure material. This work describe the fabrication and the mechanical behavior of banana fiber reinforced polymer composite at varying composition (25\%, 30\%, 35\%) with that of silicon carbide at 4\%, 8\%, 12\% respectively. Also the test such as the tensile test, hardness test and the bending test are carried out and the mechanical properties of the composite material are studied.

**Category:** General Mathematics

[33] **viXra:1609.0269 [pdf]**
*submitted on 2016-09-18 07:08:06*

**Authors:** W. B. Vasantha Kandasamy, K. Ilanthenral, Florentin Smarandach

**Comments:** 350 Pages.

In this book authors for the first time introduce study and develop the notion of MOD graphs, MOD directed graphs, MOD finite complex number graphs, MOD neutrosophic graphs, MOD dual number graphs and so on using edge weights from Z_n, C(Z_n),

**Category:** General Mathematics

[32] **viXra:1609.0227 [pdf]**
*submitted on 2016-09-15 07:53:48*

**Authors:** Edgar Valdebenito

**Comments:** 8 Pages.

In this note we give some formulas for pi constant: pi=3.1415926535...

**Category:** General Mathematics

[31] **viXra:1609.0225 [pdf]**
*submitted on 2016-09-15 07:58:15*

**Authors:** Edgar Valdebenito

**Comments:** 19 Pages.

This note presents a collection of Mathematical formulas involving the mathematical constant pi: pi=3.14159265...

**Category:** General Mathematics

[30] **viXra:1609.0209 [pdf]**
*submitted on 2016-09-13 04:10:21*

**Authors:** Rajesh Singh, Prayas Sharma, Florentin Smarandache

**Comments:** 11 Pages.

Singh et al. (20009) introduced a family of exponential ratio and product type estimators in stratified random sampling. Under stratified random sampling without replacement scheme, the expressions of bias and mean square error (MSE) of Singh et al. (2009) and some other estimators, up to the first- and second-order approximations are
derived. Also, the theoretical findings are supported by a numerical example.

**Category:** General Mathematics

[29] **viXra:1609.0208 [pdf]**
*submitted on 2016-09-13 04:12:01*

**Authors:** M. Khoshnevisan, Sukanto Bhattacharya, Florentin Smarandache

**Comments:** 20 Pages.

Each individual investor is different, with different financial goals, different levels of
risk tolerance and different personal preferences. From the point of view of investment management, these characteristics are often defined as objectives and constraints. Objectives can be the type of return being sought, while constraints include factors such as time horizon, how liquid the investor is, any personal tax situation and how risk is handled.

**Category:** General Mathematics

[28] **viXra:1609.0207 [pdf]**
*submitted on 2016-09-13 04:14:28*

**Authors:** S. Bhattacharya, S. Malakar, Florentin Smarandache

**Comments:** 8 Pages.

Our paper computationally explores Ricker’s predator satiation model with the objective
of studying how the extinction dynamics of an animal species having a two-stage lifecycle
is affected by a sudden spike in mortality due to an extraneous extreme event. Our
simulation model has been designed and implemented using sockeye salmon population
data based on a stochastic version of Ricker’s model; with the shock size being reflected
by a sudden reduction in the carrying capacity of the environment for this species. Our
results show that even for a relatively marginal increase in the negative impact of an
extreme event on the carrying capacity of the environment, a species with an otherwise
stable population may be driven close to extinction.

**Category:** General Mathematics

[27] **viXra:1609.0205 [pdf]**
*submitted on 2016-09-13 04:18:03*

**Authors:** Olufadi Yunusa, Rajesh Singh, Florentin Smarandache

**Comments:** 13 Pages.

In this paper, we propose a new estimator for estimating the finite population mean using two
auxiliary variables. The expressions for the bias and mean square error of the suggested
estimator have been obtained to the first degree of approximation and some estimators are
shown to be a particular member of this estimator. Furthermore, comparison of the suggested estimator with the usual unbiased estimator and other estimators considered in this paper is carried out. In addition, an empirical study with two natural data from literature is used to expound the performance of the proposed estimator with respect to others.

**Category:** General Mathematics

[26] **viXra:1609.0201 [pdf]**
*submitted on 2016-09-13 04:23:28*

**Authors:** M. Khoshnevisan, Florentin Smarandache, Sukanto Bhattacharya

**Comments:** 28 Pages.

In this paper we take a look at a simple portfolio insurance strategy using a protective put
and computationally derive the investor’s governing utility structures underlying such a
strategy under alternative market scenarios. Investor utility is deemed to increase with an
increase in the excess equity generated by the portfolio insurance strategy over a simple
investment strategy without any insurance.

**Category:** General Mathematics

[25] **viXra:1609.0200 [pdf]**
*submitted on 2016-09-13 04:24:33*

**Authors:** Manoj Kr. Chaudhary, V. K. Singh, Rajesh Singh, Florentin Smarandache

**Comments:** 16 Pages.

This chapter presents the detailed discussion on the effect of non-response on the
estimator of population mean in a frequently used design, namely, stratified random
sampling. In this chapter, our aim is to discuss the existing allocation schemes in
presence of non-response and to suggest some new allocation schemes utilizing the
knowledge of response and non-response rates of different strata. The effects of proposed
schemes on the sampling variance of the estimator have been discussed and compared
with the usual allocation schemes, namely, proportional allocation and Neyman
allocation in presence of non-response. The empirical study has also been carried out in
support of the results.

**Category:** General Mathematics

[24] **viXra:1609.0195 [pdf]**
*submitted on 2016-09-13 04:32:47*

**Authors:** Viplav Kumar Singh, Rajesh Singh, Florentin Smarandache

**Comments:** 12 Pages.

In this article we have proposed an efficient generalised class of estimator using two
auxiliary variables for estimating unknown population variance.We have also extended our problem to the case of two phase sampling. In support of theoretical results we have included an empirical study.

**Category:** General Mathematics

[23] **viXra:1609.0194 [pdf]**
*submitted on 2016-09-13 04:33:58*

**Authors:** Hemant Verma, Rajesh Singh, Florentin Smarandache

**Comments:** 9 Pages.

In this paper, we have studied the problem of estimating the finite population mean when information on two auxiliary attributes are available. Some improved estimators in simple random sampling without replacement have been suggested and their properties are studied.

**Category:** General Mathematics

[22] **viXra:1609.0193 [pdf]**
*submitted on 2016-09-13 04:36:02*

**Authors:** Rajesh Singh, Sachin Malik, Florentin Smarandache

**Comments:** 12 Pages.

In the present article, we proposed a family of estimators for estimating population means using known value of some population parameters. Khoshnevisan et al.proposed a
general family of estimators for estimating population means using known value of some
population parameter(s) which after some substitutions led to some ratio and product estimators initially proposed by Sisodia and Dwivedi, Singh and Tailor, Pandey and Dubey.

**Category:** General Mathematics

[21] **viXra:1609.0192 [pdf]**
*submitted on 2016-09-13 04:45:37*

**Authors:** Jayant Singh, Hansraj Yadav, Rajesh Singh, Florentin Smarandache

**Comments:** 19 Pages.

Jaipur urban area has grown tremendously in last three decades. Composition of People
migrating due to various reasons has display a meticulous trend. Dominance of people moving due to marriages is getting sturdy whereas Jaipur city is losing its lustre in attractingpersons for education and business.

**Category:** General Mathematics

[20] **viXra:1609.0191 [pdf]**
*submitted on 2016-09-13 04:47:19*

**Authors:** Prayas Sharma, Rajesh Singh, Florentin Smarandache

**Comments:** 18 Pages.

Chakrabarty (1979), Khoshnevisan et al. (2007), Sahai and Ray (1980), Ismail et al. (2011) and Solanki et al. (2012) proposed estimators for estimating population mean Y. Up to the first order of approximation and under optimum conditions, the minimum mean squared error (MSE) of all the above estimators is equal to the MSE of the regression estimator. In this paper, we have tried to found out the second order biases and mean square errors of these estimators using information on auxiliary variable based on simple random sampling. Finally, we have compared the performance of these estimators with some numerical illustration.

**Category:** General Mathematics

[19] **viXra:1609.0190 [pdf]**
*submitted on 2016-09-13 04:48:14*

**Authors:** Sukanto Bhattacharya, Florentin Smarandache, M. Khoshnevisan

**Comments:** 12 Pages.

In our present paper, we have explored the possibilities and developed arguments for an
application of principles of neutrosophic game theory as a generalization of the fuzzy
game theory model to a better understanding of the Israel-Palestine problem in terms of
the goals and governing strategies of either side. We build on an earlier attempted
justification of a game theoretic explanation of this problem by Yakir Plessner (2001) and
go on to argue in favour of a neutrosophic adaptation of the standard 2x2 zero-sum game
theoretic model in order to identify an optimal outcome.

**Category:** General Mathematics

[18] **viXra:1609.0189 [pdf]**
*submitted on 2016-09-13 04:49:49*

**Authors:** Rajesh Singh, Shweta Maurya, Ashish K. Singh

**Comments:** 20 Pages.

In this chapter an attempt has been made to study the effect of certain pollutants on water of Ramgarh Lake of Rajasthan. Time series analysis of the observed data has been done using trend, single exponential smoothing and double exponential smoothing methods.

**Category:** General Mathematics

[17] **viXra:1609.0188 [pdf]**
*submitted on 2016-09-13 04:51:09*

**Authors:** Manoj Kr. Chaudhary, Anil Prajapati, Rajesh Singh, Florentin Smarandache

**Comments:** 15 Pages.

The present paper presents the detail discussion on estimation of population mean in
simple random sampling in the presence of non-response. Motivated by Gupta and Shabbir
(2008), we have suggested the class of estimators of population mean using an auxiliary variable under non-response. A theoretical study is carried out using two-phase sampling scheme when the population mean of auxiliary variable is not known. An empirical study has also been done in the support of theoretical results.

**Category:** General Mathematics

[16] **viXra:1609.0187 [pdf]**
*submitted on 2016-09-13 04:52:30*

**Authors:** Neeraj Kumar, Sanjey Kumar, Florentin Smarandache

**Comments:** 17 Pages.

Fuzzy set theory is primarily concerned with how to quantitatively deal with imprecision and uncertainty, and offers the decision maker another tool in addition to the classical deterministic and probabilistic mathematical tools that are used in modeling real-world problems. The present study investigates a fuzzy economic order quantity model for two storage facility.

**Category:** General Mathematics

[15] **viXra:1609.0186 [pdf]**
*submitted on 2016-09-13 04:53:54*

**Authors:** Manoj K. Chaudhary, Sachin Malik, Rajesh Singh, Florentin Smarandache

**Comments:** 13 Pages.

In this paper we have adapted Singh and Shukla (1987) estimator in systematic sampling using auxiliary information in the presence of non-response. The properties of the suggested family have been discussed. Expressions for the bias and mean square error (MSE) of the suggested family have been derived. The comparative study of the optimum estimator
of the family with ratio, product, dual to ratio and sample mean estimators in systematic
sampling under non-response has also been done. One numerical illustration is carried out to
verify the theoretical results.

**Category:** General Mathematics

[14] **viXra:1609.0185 [pdf]**
*submitted on 2016-09-13 04:55:11*

**Authors:** M. Khoshnevisan, Florentin Smarandache, Sukanto Bhattacharya

**Comments:** 21 Pages.

In this paper we have devised an alternative methodological approach for quantifying
utility in terms of expected information content of the decision-maker’s choice set. We
have proposed an extension to the concept of utility by incorporating extrinsic utility;
which we have defined as the utility derived from the element of choice afforded to the
decision-maker by the availability of an object within his or her object set. We have
subsequently applied this extended utility concept to the case of investor utility derived
from a structured, financial product – an custom-made investment portfolio incorporating
an endogenous capital-guarantee through inclusion of cash as a risk-free asset, based on the Black-Scholes derivative-pricing formulation.

**Category:** General Mathematics

[13] **viXra:1609.0181 [pdf]**
*submitted on 2016-09-13 03:26:08*

**Authors:** Rajesh Singh, Mukesh Kumar, Sachin Malik, Manoj K. Chaudhary, Florentin Smarandache

**Comments:** 9 Pages.

In this chapter, we have suggested an improved estimator for estimating the population mean in stratified sampling in presence of auxiliary information. The mean square error (MSE) of the proposed estimator have been derived under large sample approximation. Besides, considering the minimum case of the MSE equation, the efficient conditions between the proposed and existing estimators are obtained. These theoretical findings are supported by a numerical example.

**Category:** General Mathematics

[12] **viXra:1609.0180 [pdf]**
*submitted on 2016-09-13 03:27:30*

**Authors:** Mohammad Khoshnevisan, Housila P. Singh, Florentin Smarandache

**Comments:** 19 Pages.

This paper proposes a family of estimators of population mean using information on several auxiliary variables and analyzes its properties in the presence of measurement errors.

**Category:** General Mathematics

[11] **viXra:1609.0179 [pdf]**
*submitted on 2016-09-13 03:28:56*

**Authors:** Rajesh Singh, Mukesh Kumar, Ashish K. Singh, Florentin Smarandache

**Comments:** 9 Pages.

This chapter proposes some estimators for the population variance of the variable under study, which make use of information regarding the population proportion possessing certain attribute. Under simple random sampling without replacement (SRSWOR) scheme, the mean squared error (MSE) up to the first order of approximation is derived. The results have been illustrated numerically by taking some empirical population considered in the literature.

**Category:** General Mathematics

[10] **viXra:1609.0178 [pdf]**
*submitted on 2016-09-13 03:29:58*

**Authors:** Hemant K. Verma, Rajesh Singh, Florentin Smarandache

**Comments:** 11 Pages.

In this paper we have proposed a median based estimator using known value of some population parameter(s) in simple random sampling. Various existing estimators are shown particular members of the proposed estimator. The bias and mean squared error of the proposed estimator is
obtained up to the first order of approximation under simple random sampling without replacement.
An empirical study is carried out to judge the superiority of proposed estimator over others.

**Category:** General Mathematics

[9] **viXra:1609.0177 [pdf]**
*submitted on 2016-09-13 03:31:38*

**Authors:** Mohammad Khoshnevisan, Housila P. Singh, Sarjinder Singh, Florentin Smarandache

**Comments:** 19 Pages.

In this paper we have suggested two classes of estimators for population median MY of the study character Y using information on two auxiliary characters X and Z in double sampling. It has been shown that the suggested classes of estimators are more efficient than the one suggested by Singh et al (2001). Estimators based on estimated optimum values have been also considered with their properties. The optimum values of the first phase and second phase sample sizes are also obtained for the fixed cost of survey.

**Category:** General Mathematics

[8] **viXra:1609.0176 [pdf]**
*submitted on 2016-09-13 03:33:36*

**Authors:** Rajesh Singh, Pankaj Chauhan, Nirmala Sawan, Florentin Smarandache

**Comments:** 10 Pages.

A general family of estimators for estimating the population variance of the variable under study, which make use of known value of certain population parameter(s),is proposed. Some well known estimators have been shown as particular member of this family. It has been shown that the suggested estimator is better than the usual unbiased estimator, Isaki’s (1983) ratio estimator, Upadhyaya and Singh’s (1999) estimator and Kadilar and Cingi (2006). An empirical study is carried out to illustrate the performance of the constructed estimator over others.

**Category:** General Mathematics

[7] **viXra:1609.0171 [pdf]**
*submitted on 2016-09-13 03:42:29*

**Authors:** Rajesh Singh, Sachin Malik, Florentin Smarandache

**Comments:** 11 Pages.

In this paper, we suggest an estimator using two auxiliary variables in stratified random sampling following Malik and Singh. The propose estimator has an improvement over mean per unit
estimator as well as some other considered estimators. Expressions for bias and MSE of the
estimator are derived up to first degree of approximation. Moreover, these theoretical findings are supported by a numerical example with original data.

**Category:** General Mathematics

[6] **viXra:1609.0168 [pdf]**
*submitted on 2016-09-13 03:51:22*

**Authors:** Hemant K. Verma, Rajesh Singh, Florentin Smarandache

**Comments:** 21 Pages.

In this paper we have suggested difference-type estimator for estimation of population mean of the study variable y in the presence of measurement error using auxiliary information. The optimum estimator in the suggested estimator has been identified along with
its mean square error formula. It has been shown that the suggested estimator performs more efficient then other existing estimators. An empirical study is also carried out to illustrate the merits of proposed method over other traditional methods.

**Category:** General Mathematics

[5] **viXra:1609.0165 [pdf]**
*submitted on 2016-09-13 04:05:46*

**Authors:** Empirical Study in Finite Correlation Coefficient in Two Phase Estimation

**Comments:** 16 Pages.

This paper proposes a class of estimators for population correlation coefficient when information about the population mean and population variance of one of the variables is not available but information about these parameters of another variable (auxiliary) is available, in two phase sampling and analyzes its properties. Optimum estimator in the class is identified with its variance formula.

**Category:** General Mathematics

[4] **viXra:1609.0164 [pdf]**
*submitted on 2016-09-13 04:07:05*

**Authors:** Manoj Kr. Chaudhary, V. K. Singh, Rajesh Singh, Florentin Smarandache

**Comments:** 17 Pages.

The present chapter deals with the study of general family of factor-type estimators for estimating population mean of stratified population in the presence of nonresponse
whenever information on an auxiliary variable are available. The proposed family includes separate ratio, product, dual to ratio and usual sample mean estimators as its particular cases and exhibits some nice properties as regards to locate the optimum estimator belonging to the family. Choice of appropriate estimator in the family in order to get a desired level of accuracy even if non-response is high, is also discussed. The empirical study has been carried out in support of the results.

**Category:** General Mathematics

[3] **viXra:1609.0163 [pdf]**
*submitted on 2016-09-13 04:09:06*

**Authors:** Housila P. Singh, Sharad Saxena, Mohammad Khoshnevisan, Sarjinder Singh, Florentin Smarandache

**Comments:** 22 Pages.

This paper is speculated to propose a class of shrinkage estimators for shape parameter
β in failure censored samples from two-parameter Weibull distribution when some ‘apriori’ or guessed interval containing the parameter β is available in addition to sample information and analyses their properties. Some estimators are generated from the proposed class and compared with the minimum mean squared error (MMSE) estimator. Numerical computations in terms of percent relative efficiency and absolute relative bias indicate that certain of these estimators substantially improve the MMSE estimator in some guessed interval of the parameter space of β ,
especially for censored samples with small sizes. Subsequently, a modified class of shrinkage estimators is proposed with its properties.

**Category:** General Mathematics

[2] **viXra:1609.0066 [pdf]**
*replaced on 2016-09-15 19:17:54*

**Authors:** Marvin Ray Burns

**Comments:** 10 Pages.

First we will follow the path the author took to find out that for
integral(cos[pi*I*x]x^(1/x),{x,1,a}) and
integral(sin[pi*I*x]x^(1/x),{x,1,a}), the limit of the ratio of a to a-1,as a goes to infinity is Gelfond's Constant, (e^pi).
We will consider that the hypothesis and provide hints for a proof using L'Hospital's Rule (since we have indeterminate forms as a goes to infinity).
We find we there is no limit of the ratio of the previous forms of integrals when the "I" is left out, and give a small proof for them.

**Category:** General Mathematics

[1] **viXra:1609.0057 [pdf]**
*submitted on 2016-09-05 08:08:46*

**Authors:** Edgar Valdebenito

**Comments:** 11 Pages.

In this note we give some formulas for pi constant: pi=3.14159265...

**Category:** General Mathematics