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2016 - 1601(65) - 1602(28) - 1603(35) - 1604(39) - 1605(18) - 1606(25) - 1607(45) - 1608(36) - 1609(38)

Any replacements are listed further down

[1785] **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

[1784] **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

[1783] **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

[1782] **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

[1781] **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

[1780] **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

[1779] **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

[1778] **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

[1777] **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

[1776] **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

[1775] **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

[1774] **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

[1773] **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

[1772] **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

[1771] **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

[1770] **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

[1769] **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

[1768] **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

[1767] **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

[1766] **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

[1765] **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

[1764] **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

[1763] **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

[1762] **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

[1761] **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

[1760] **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

[1759] **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

[1758] **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

[1757] **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

[1756] **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

[1755] **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

[1754] **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

[1753] **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

[1752] **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

[1751] **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

[1750] **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

[1749] **viXra:1609.0066 [pdf]**
*submitted on 2016-09-05 18:30:15*

**Authors:** Marvin Ray Burns

**Comments:** 10 Pages. This pdf from a structured and clear Mathematica notebook is complemented with the traditional form of the mathematics language. For further research or copying formulae, download marvinrayburns.comClosedformsGelfond.nb .

First we will consider
integral(cos[pi*I*x]x^(1/x),{x,1,a})and
integral(sin[pi*I*x]x^(1/x),{x,1,a}),
where it appears that the limit of the ratio of a to a-1,as a goes to infinity,in
integral(Sin[Pi*I*x]x^(1/x),{x,1,a})/integral(Sin[Pi*I*x]x^(1/x),{x,1,a-1}) and integral(Cos[Pi*I*x]x^(1/x),{x,1,a})/integral(Cos[Pi*I*x]x^(1/x),{x,1,a-1})
is Gelfond's Constant,(e^pi). We will consider that the hypothesis and provide hints for a proof using L'Hospital's Rule since, with the hyperbolic integrals, we have indeterminate forms as a goes to infinity.
We shall compare the ratios of
integral(sin[pi*I*x]x^(1/x),{x,1,a}) and integral(sin[pi*I*x]x^(1/x),{x,1,a-1}) with
integral(sin[pi*x]x^(1/x),{x,1,a}) and integral(sin[pi*x]x^(1/x),{x,1,a-1}).
In later parts of this paper we will look at the ratio of three deceptively similar looking integrals which have the terms of MKB for integrands. (For reference we call these the "later integrals."
Finally, after finding no closed form for the ratios of the later integrals, we will settle for closed forms of their differences.

**Category:** General Mathematics

[1748] **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

[1747] **viXra:1608.0364 [pdf]**
*submitted on 2016-08-27 03:19:35*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 1 Page.

TRL Lustre Is The Light’s Cumulative Consecutive Higher Orders Orthogonal Primality Overgrowth (All Inclusive) Of It’s Own Primality.

**Category:** General Mathematics

[1746] **viXra:1608.0348 [pdf]**
*submitted on 2016-08-25 08:09:08*

**Authors:** Edgar Valdebenito

**Comments:** 14 Pages.

In this note we give some formulas for Catalan's constant G=0.915965594177...

**Category:** General Mathematics

[1745] **viXra:1608.0347 [pdf]**
*submitted on 2016-08-25 08:14:20*

**Authors:** Edgar Valdebenito

**Comments:** 13 Pages.

In this note we give some formulas related with: pi constant pi=3.1415...,Catalan's constant G=0.9159...,Euler's constant gamma=0.5772...

**Category:** General Mathematics

[1744] **viXra:1608.0296 [pdf]**
*submitted on 2016-08-24 01:36:28*

**Authors:** Florentin Smarandache

**Comments:** 33 Pages.

This chapter may look like a glossary of the fusion rules and we also introduce new ones presenting their formulas and examples: Conjunc
tive, Disjunctive, Exclusive Disjunctive, Mixed Conjunctive-Disjunctive rules, Conditional rule, Dempster’s, Yager’s, Smets’ TBM rule, Dubois-Prade’s, Dezert-Smarandache classical and hybrid rules, Murphy’s average rule, Inagaki-Lefevre-Colot-Vannoorenberghe Unified Combination rules [and, as particular cases:
Iganaki’s parameterized rule, Weighted Average Operator, minC (M. Daniel), and newly Proportional Conflict Redistribution rules (Smarandache-Dezert) among which PCR5 is the most exact way of redistribution of the conflicting mass to non-empty sets following the path of the conjunctive rule], Zhang’s Center Combination rule, Consensus Operator (Jøsang), Cautious Rule (Smets), -junctions rules (Smets), etc. and three new T -norm & T -conorm rules adjusted from fuzzy and neutrosophic sets to information fusion (Tchamova-Smarandache). Introducing the degree of union and degree of inclusion with respect
to the cardinal of sets not with the fuzzy set point of view, besides that of intersection,
many fusion rules can be improved.

**Category:** General Mathematics

[1743] **viXra:1608.0294 [pdf]**
*submitted on 2016-08-24 01:38:48*

**Authors:** Florentin Smarandache, Jean Dezert

**Comments:** 18 Pages.

This chapter introduces the notion of qualitative belief assignment to model beliefs of human experts expressed in natural language (with linguistic labels).We show how qualitative beliefs can be efficiently combined using an extension of Dezert-Smarandache Theory (DSmT) of plausible and paradoxical quantitative reasoning to qualitative reasoning. We propose a new arithmetic on linguistic labels which allows a direct extension of classical or hybrid DSm fusion rules. An approximate qualitative PCR5 rule is also proposed jointly with a Qualitative Average Operator. We also show how crisp or interval mappings can be used to deal indirectly with linguistic labels. A very simple example is provided to illustrate our qualitative fusion rules.

**Category:** General Mathematics

[1742] **viXra:1608.0292 [pdf]**
*submitted on 2016-08-24 01:45:02*

**Authors:** J. Dezert, A. Tchamova, P. Konstantinova, Florentin Smarandache

**Comments:** 21 Pages.

We analyze the behavior of several combinational rules for temporal/sequential attribute data fusion for target type estimation. Our comparative analysis is based on: Dempster’s fusion rule, Proportional Conflict Redistribution rule no.
5 (PCR5), Symmetric Adaptive Combination (SAC) rule and a new fusion rule, based on fuzzy T-conorm and T-norm operators (TCN).

**Category:** General Mathematics

[1741] **viXra:1608.0290 [pdf]**
*submitted on 2016-08-24 01:49:56*

**Authors:** Xinhan Huang, Xinde Li, Min Wang

**Comments:** 22 Pages.

Characteristics of uncertainty and imprecision, even imperfection is presented from knowledge acquisition in map reconstruction of autonomous mobile robots. Especially in the course of building grid map using sonar, this characteristic
of uncertainty is especially servere.

**Category:** General Mathematics

[1740] **viXra:1608.0289 [pdf]**
*submitted on 2016-08-24 01:51:11*

**Authors:** Arnaud Martin, Christophe Osswald

**Comments:** 21 Pages.

In this chapter, we present and discuss a new generalized proportional conflict redistribution rule. The Dezert-Smarandache extension of the Dempster- Shafer theory has relaunched the studies on the combination rules especially for the management of the conflict. Many combination rules have been proposed in the last few years. We study here different combination rules and compare them in terms
of decision on didactic example and on generated data. Indeed, in real applications,
we need a reliable decision and it is the final results that matter. This chapter shows that a fine proportional conflict redistribution rule must be preferred for the combination in the belief function theory.

**Category:** General Mathematics

[1739] **viXra:1608.0288 [pdf]**
*submitted on 2016-08-24 01:55:36*

**Authors:** A. A. K. Majumdar

**Comments:** 3 Pages.

This paper gives an alternative approach to ¯nd the determinant of the right circular matrix with geometric sequence, using the known results of the circulant matrix.Keywords circulant matrix, right circulant matrix with geometric sequence.

**Category:** General Mathematics

[1738] **viXra:1608.0287 [pdf]**
*submitted on 2016-08-24 01:57:02*

**Authors:** Samuel Corgne, Laurence Hubert-Moy, Gregoire Mercier, Jean Dezert

**Comments:** 13 Pages.

This chapter presents an environmental application of DSmT for the land cover prediction. The spatial prediction of land cover at the field scale in winter is useful to reduce the bare soils in agricultural intensive regions.

**Category:** General Mathematics

[1737] **viXra:1608.0286 [pdf]**
*submitted on 2016-08-24 01:58:54*

**Authors:** Milan Daniel

**Comments:** 25 Pages.

Dempster’s rule, non-normalized conjunctive rule, Yager’s rule and Dubois-Prade’s rule for belief functions combination are generalized to be applicable to hyper-power sets according to the DSm theory. A comparison of the rules with
DSm rule of combination is presented. A series of examples is included.

**Category:** General Mathematics

[1736] **viXra:1608.0283 [pdf]**
*submitted on 2016-08-24 02:02:21*

**Authors:** Andrew Schumann

**Comments:** 23 Pages.

The Dezert-Smarandache theory of plausible and paradoxical reasoning is based on the premise that some elements i of a frame have a non-empty
intersection.

**Category:** General Mathematics

[1735] **viXra:1608.0282 [pdf]**
*submitted on 2016-08-24 02:03:23*

**Authors:** Albena Tchamova, Tzvetan Semerdjiev, Jean Dezert

**Comments:** 12 Pages.

This chapter presents an approach for target behavior tendency estimation (Receding, Approaching). It is developed on the principles of Dezert-Smarandache theory (DSmT) of plausible and paradoxical reasoning applied to conventional
sonar amplitude measurements, which serve as an evidence for corresponding decision-making procedures.

**Category:** General Mathematics

[1734] **viXra:1608.0280 [pdf]**
*submitted on 2016-08-24 02:06:16*

**Authors:** Denis de Brucq

**Comments:** 20 Pages.

This chapter presents new important links between the most important theories developed in literature for managing uncertainties (i.e. probability, fuzzy sets and evidence theories).

**Category:** General Mathematics

[1733] **viXra:1608.0277 [pdf]**
*submitted on 2016-08-24 02:10:34*

**Authors:** J. Dezert, A. Tchamova, T. Semerdjiev, P. Konstantinova

**Comments:** 18 Pages.

The objective of this chapter is to present and compare different fusion rules which can be used for Generalized Data Association (GDA) for multitarget tracking (MTT) in clutter.

**Category:** General Mathematics

[1732] **viXra:1608.0274 [pdf]**
*submitted on 2016-08-24 02:14:26*

**Authors:** A. A. K. Majumdar

**Comments:** 8 Pages.

Two types of the Smarandache LCM ratio functions have been introduced by Murthy. Recently, the second type of the Smarandache LCM ratio function has been considered by Khainar, Vyawahare and Salunke. This paper establishes the relationships
between these two forms of the Smarandache LCM ratio functions, and derives some reduction
formulas and interesting properties in connection with these functions.

**Category:** General Mathematics

[1731] **viXra:1608.0265 [pdf]**
*submitted on 2016-08-23 04:07:10*

**Authors:** Jean Dezert, Florentin Smarandache, Mohammad Khoshnevisan

**Comments:** 18 Pages.

This chapter presents several classes of fusion problems which cannot be directly approached by the classical mathematical theory of evidence, also known as Dempster-Shafer Theory (DST), either because Shafer’s model for the frame of
discernment is impossible to obtain, or just because Dempster’s rule of combination fails to provide coherent results (or no result at all). We present and discuss the potentiality of the DSmT combined with its classical (or hybrid) rule of combination to attack these infinite classes of fusion problems.

**Category:** General Mathematics

[1730] **viXra:1608.0264 [pdf]**
*submitted on 2016-08-23 04:08:39*

**Authors:** Jean Dezert, Florentin Smarandache

**Comments:** 20 Pages.

In this chapter one studies, within the DSmT framework, the case when the sources of information provide imprecise belief functions/masses, and we generalize
the DSm rules of combination (classic or hybrid rules) from scalar fusion to sub-unitary interval fusion and, more generally, to any set of sub-unitary interval fusion. This work generalizes previous works available in literature which appear limited to IBS (Interval-valued Belief Structures) in the Transferable Belief Model framework. Numerical didactic examples of these new DSm fusion rules for dealing with imprecise information are also presented.

**Category:** General Mathematics

[1729] **viXra:1608.0263 [pdf]**
*submitted on 2016-08-23 04:09:58*

**Authors:** Ion P ATRASCU, Florentin SMARANDACHE

**Comments:** 4 Pages.

In this paper, the notion of Apollonius circle of rank k is introduced and a number of results related to the classical Apollonius circles are generalized.

**Category:** General Mathematics

[1728] **viXra:1608.0262 [pdf]**
*submitted on 2016-08-23 04:12:13*

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

This short paper presents mathematical rebuses.

**Category:** General Mathematics

[1727] **viXra:1608.0261 [pdf]**
*submitted on 2016-08-23 04:14:00*

**Authors:** Jean Dezert, Florentin Smarandache, Albena Tchamova

**Comments:** 13 Pages.

Modern multitarget-multisensor tracking systems involve the development of reliable methods for the data association and the fusion of multiple sensor information, and more specifically the partitioning of observations into tracks. This
chapter discusses and compares the application of Dempster-Shafer Theory (DST) and the Dezert-Smarandache Theory (DSmT) methods to the fusion of multiple sensor attributes for target identification purpose. We focus our attention on the paradoxical Blackman’s association problem and propose several approaches to outperform Blackman’s solution. We clarify some preconceived ideas about the use of degree of
conflict between sources as potential criterion for partitioning evidences.

**Category:** General Mathematics

[1726] **viXra:1608.0259 [pdf]**
*submitted on 2016-08-23 04:17:08*

**Authors:** Jean Dezert, Florentin Smarandache

**Comments:** 13 Pages.

In this chapter, we examine several issues for ordering or partially ordering elements of hyper-power sets involved in the DSmT. We will show the benefit of some of these issues to obtain a nice and interesting structure of matrix representation
of belief functions.

**Category:** General Mathematics

[1725] **viXra:1608.0258 [pdf]**
*submitted on 2016-08-23 04:18:43*

**Authors:** Jean Dezert, Florentin Smarandache

**Comments:** 34 Pages.

This chapter presents a general overview and foundations of the DSmT, i.e. the recent theory of plausible and paradoxical reasoning developed by the authors, specially for the static or dynamic fusion of information arising from several independent but potentially highly conflicting, uncertain and imprecise sources of
evidence. We introduce and justify here the basis of the DSmT framework with respect to the Dempster-Shafer Theory (DST), a mathematical theory of evidence developed in 1976 by Glenn Shafer. We present the DSm combination rules and provide some simple illustrative examples and comparisons with other main rules of
combination available in the literature for the combination of information for simple
fusion problems. Detailed presentations on recent advances and applications of DSmT are presented in the next chapters of this book.

**Category:** General Mathematics

[1724] **viXra:1608.0257 [pdf]**
*submitted on 2016-08-23 04:19:57*

**Authors:** Jean Dezert, Florentin Smarandache

**Comments:** 13 Pages.

The development of DSmT is based on the notion of Dedekind’s lattice, called also hyper-power set in the DSmT framework, on which is defined the general basic belief assignments to be combined. In this chapter, we explain the structure of the hyper-power set, give some examples of hyper-power sets and show how they can be generated from isotone Boolean functions. We also show the interest to work with the hyper-power set rather than the power set of the refined frame of discernment in
terms of complexity.

**Category:** General Mathematics

[1723] **viXra:1608.0249 [pdf]**
*submitted on 2016-08-23 04:03:04*

**Authors:** Jean Dezert, Florentin Smarandache, Milan Daniel

**Comments:** 12 Pages.

This chapter introduces a generalized pignistic transformation (GPT)developed in the DSmT framework as a tool for decision-making at the pignistic level. The GPT allows to construct quite easily a subjective probability measure
from any generalized basic belief assignment provided by any corpus of evidence. We focus our presentation on the 3D case and we provide the full result obtained by the proposed GPT and its validation drawn from the probability theory.

**Category:** General Mathematics

[1722] **viXra:1608.0248 [pdf]**
*submitted on 2016-08-23 04:05:22*

**Authors:** Jean Dezert, Florentin Smarandache

**Comments:** 44 Pages.

This chapter presents a general method for combining uncertain and paradoxical (i.e. highly conflicting) sources of evidence for a wide class of fusion problems. From the foundations of the DSmT we show how the DSm rule of combination can be extended to take into account all possible integrity constraints (if any) of the problem under consideration due to the true nature of elements/concepts
involved into it. We show how Shafer’s model can be considered as a specific hybrid DSm model and can be easily handled by the DSmT and one presents here a new efficient alternative to Dempster’s rule of combination, following steps of previous researchers towards this quest. Several simple didactic examples are also provided to show the efficiency and the generality of the approach proposed in this work.

**Category:** General Mathematics

[1721] **viXra:1608.0196 [pdf]**
*submitted on 2016-08-19 04:32:18*

**Authors:** Victor VLADAREANU, Ovidiu-Ilie SANDRU, Mihnea Alexandru MOISESCU, Florentin SMARANDACHE, Hongnian YU

**Comments:** 6 Pages.

The paper presents an application of Extenics Engineering principles to a two-dimensional robotic workspace. This provides the mathematical basis and considerations for obtaining the trajectory tracking reference in robotic applications. A brief history and overview of the relevant theoretical concepts is provided.

**Category:** General Mathematics

[1720] **viXra:1608.0194 [pdf]**
*submitted on 2016-08-19 04:42:48*

**Authors:** Mumtaz Ali, Florentin Smarandache, Jian-qiang Wang

**Comments:** 8 Pages.

Fuzzy sets and intuitionistic fuzzy sets can’t
handle imprecise, indeterminate, inconsistent, and incomplete information. Neutrosophic sets play an important role to overcome this difficulty. A neutrosophic set has a truth membership function, indeterminate membership function, and a falsehood membership functions that can handle all types of ambiguous information. New type of union and intersection has been proposed in this paper.

**Category:** General Mathematics

[1719] **viXra:1608.0193 [pdf]**
*submitted on 2016-08-19 04:45:46*

**Authors:** Young Bae Jun, Florentin Smarandache, Chang Su Kim

**Comments:** 3 Pages.

R-unions and R-intersections of T-external (Iexternal,F-external) neutrosophic cubic sets are considered.Examples to show that the R-intersection and R-union of Texternal
(I-external, F-external) neutrosophic cubic sets may not be a T-external (I-external, F-external) neutrosophic cubic setare provided. Conditions for the R-union and R-intersection of T-external (I-external, F-external) neutrosophic cubic sets to be a T-external (I-external, F-external) neutrosophic cubic set are discussed.

**Category:** General Mathematics

[1718] **viXra:1608.0192 [pdf]**
*submitted on 2016-08-19 04:47:23*

**Authors:** Said Broumi, Mohamed Talea, Florentin Smarandache, Assia Bakali

**Comments:** 8 Pages.

The single valued neutrosophic graph is a new
version of graph theory presented recently as a generalization of fuzzy graph and intuitionistic fuzzy graph. The single valued
neutrosophic graph (SVN-graph) is used when the relation between nodes (or vertices) in problems are indeterminate. In this paper, we examine the properties of various types of degrees, order and size of single valued neutrosophic graphs and a new definition for regular single valued neutrosophic graph is given.

**Category:** General Mathematics

[1717] **viXra:1608.0191 [pdf]**
*submitted on 2016-08-19 04:48:59*

**Authors:** Irfan Deli, Yusuf Subas, Florentin Smarandache, Mumtaz Ali

**Comments:** 8 Pages.

Interval valued bipolar fuzzy weighted
neutrosophic set(IVBFWN-set) is a new generalization of fuzzy set, bipolar fuzzy set, neutrosophic set and bipolar neutrosophic set so that it can handle uncertain information more flexibly in the process of decision making.

**Category:** General Mathematics

[1716] **viXra:1608.0190 [pdf]**
*submitted on 2016-08-19 04:50:30*

**Authors:** Mirela Teodorescu, Daniela Gîfu, Florentin Smarandache

**Comments:** 8 Pages.

This study introduces the concept of uncertainty analysis of Neutrosophic Theory in the sphere of Maintenance Operating System
(MOS). The aim of this study is to underline the importance of uncertainties solving in Maintenance Operating System. In maintenance
process appear ambiguous states that can’t be assimilated neither true, nor false, meaning that the threshold state is a neutral one, being defined as fault in most of cases. In this regard, this uncertaintymaking decision
process can be associated as functioning according to rules of Neutrosophy, and can be evaluated using elements of Neutrosophic
Structures, Pareto Charts, maintenance metrics. Identification of uncertainties and study of their impact on maintenance, making the right
decision, is the main focus of this paper.

**Category:** General Mathematics

[1715] **viXra:1608.0189 [pdf]**
*submitted on 2016-08-19 04:52:32*

**Authors:** Azeddine Elhassouny, Florentin Smarandache

**Comments:** 7 Pages.

In this paper, (1) one simplified the standard
TOPSIS to new Multi-Criteria Decision-Making (MCDM) called Simplified-TOPSIS. Simplified-TOPSIS gives the same results and simplifies the calculation of the classical TOPSIS. An
example is presented distinctions between Simplified-TOPSIS and classical TOPSIS are underlined. (2) extend the new Simplified-TOPSIS method to Neutrosophic-simplified-TOPSIS
using single valued Neutrosophic information. An example showing the interest of Neutrosophic-simplified-TOPSIS to manipulate the uncertainty linked to information presented in Multi-Criteria Decision-Making.

**Category:** General Mathematics

[1714] **viXra:1608.0042 [pdf]**
*submitted on 2016-08-04 10:59:34*

**Authors:** Mohsen Soori, Behrooz Arezoo, Mohsen Habibi

**Comments:** 11 pages, Journal of Manufacturing Science and Engineering, 138(8), 081005, 2016

Virtual manufacturing systems carry out the simulation of manufacturing processes in digital environment in order to increase accuracy as well as productivity in part production. There are different error sources in machine tools, such as tool deflection, geometrical deviations of moving axis, and thermal distortions of machine tool structures. The errors due to tool deflection are caused by cutting forces and have direct effects on dimensional accuracy, surface roughness of the parts, and efficient life of the cutting tool, holder, and spindle. This paper presents an application of virtual machining systems in order to improve the accuracy and productivity of part manufacturing by monitoring and minimizing the tool deflection error. The tool deflection error along machining paths is monitored to present a useful methodology in controlling the produced parts with regard to desired tolerances. Suitable tool and spindle can also be selected due to the ability of error monitoring. In order to minimize the error, optimization technique based on genetic algorithms is used to determine optimized machining parameters. Free-form profile of virtual and real machined parts with tool deflection error is compared in order to validate reliability as well as accuracy of the software.

**Category:** General Mathematics

[1713] **viXra:1608.0010 [pdf]**
*submitted on 2016-08-02 08:58:48*

**Authors:** Edgar Valdebenito

**Comments:** 4 Pages.

In this note we give some formulas for Catalan's constant: G=0.915965594177...

**Category:** General Mathematics

[1712] **viXra:1608.0009 [pdf]**
*submitted on 2016-08-02 09:02:04*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

In this note we show some identities with hyperbolic functions

**Category:** General Mathematics

[1711] **viXra:1607.0544 [pdf]**
*submitted on 2016-07-29 01:16:14*

**Authors:** A.Nellai Murugan, P.Iyadurai Selvaraj

**Comments:** 9 Pages.

A graph G is a finite non-empty set of objects called vertices together with a set of unordered
pairs of distinct vertices of G which is called edges.

**Category:** General Mathematics

[1710] **viXra:1607.0543 [pdf]**
*submitted on 2016-07-29 01:17:42*

**Authors:** B.Basavanagoud, V.R.Kulli, Vijay V. Teli

**Comments:** 8 Pages.

All graphs considered here are finite, undirected with no loops and multiple edges. We denote
by p the order(i.e number of vertices) and by q the size (i.e number of edges) of such a graph
G. Any undefined term and notation in this paper may be found in Harary [5].

**Category:** General Mathematics

[1709] **viXra:1607.0542 [pdf]**
*submitted on 2016-07-29 01:19:10*

**Authors:** R.Ponraj, M.Maria Adaickalam, R.Kala

**Comments:** 11 Pages.

A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate k-difference cordial labeling behavior of star, m copies of star and we prove that every graph is a subgraph of a connected k-difference cordial graph. Also we investigate 3-difference cordial labeling behavior of some graphs.

**Category:** General Mathematics

[1708] **viXra:1607.0541 [pdf]**
*submitted on 2016-07-29 01:20:14*

**Authors:** Tejaswini K M, Venkanagouda M Goudar

**Comments:** 12 Pages.

We investigate the relationship between nonsplit geodetic number and geodetic number. We also obtain the nonsplit geodetic number in the cartesian product of graphs.

**Category:** General Mathematics

[1707] **viXra:1607.0540 [pdf]**
*submitted on 2016-07-29 01:21:59*

**Authors:** Suha Yılmaz, Umit Ziya Savcı

**Comments:** 15 Pages.

In this paper, we investigate Smarandache curves according to type-2 Bishop frame in Euclidean 3- space and we give some differential geometric properties of Smarandache curves. Also, some characterizations of Smarandache breadth curves in Euclidean 3-space are presented. Besides, we illustrate examples of our results.

**Category:** General Mathematics

[1706] **viXra:1607.0517 [pdf]**
*submitted on 2016-07-27 01:42:32*

**Authors:** Nassim Abbas, Youcef Chibani, Bilal Hadjadji, Zayen Azzouz Omar, Florentin Smarandache

**Comments:** 8 Pages.

We propose in this paper a new writer-independent off-line handwritten signature verification (HSV) system using only genuine signatures. This system is based on a combination of two off-line individual HSV systems through the plausible and paradoxical reasoning theory of Dezert-Smarandache (DSmT).

**Category:** General Mathematics

[1705] **viXra:1607.0516 [pdf]**
*submitted on 2016-07-27 01:44:24*

**Authors:** Linfan MAO, Guohua Song

**Comments:** 168 Pages.

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** General Mathematics

[1704] **viXra:1607.0513 [pdf]**
*submitted on 2016-07-27 01:49:30*

**Authors:** A. Elhassouny, Florentin Smarandache

**Comments:** 32 Pages.

The purpose of this paper is to present an alternative of a hybrid method based on
Saaty’s Analytical Hierarchy Process and on the Technique for Order Preference by using the Similarity to Ideal Solution method (AHP-TOPSIS) and, based on the AHP and its use of pairwise comparisons, to extend it to a new method called α-D MCDMTOPSIS (α-Discounting Method for multi-criteria decision making-TOPSIS).

**Category:** General Mathematics

[1703] **viXra:1607.0512 [pdf]**
*submitted on 2016-07-27 01:50:59*

**Authors:** Florentin Smarandache

**Comments:** 7 Pages.

This paper is a theoretical approach for a potential neutrosophic quantum computer to be built in the future, which is an extension of the classical theoretical quantum computer, into which the indeterminacy is inserted.

**Category:** General Mathematics

[1702] **viXra:1607.0506 [pdf]**
*submitted on 2016-07-27 02:00:16*

**Authors:** F.Smarandache, N.Abbas, Y.Chibani, B.Hadjadji, Z.A.Omar

**Comments:** 6 Pages.

In this paper we use PCR5 in order to fusion the information of two sourcesproviding subjective probabilities of an event A to occur in the following form: chance that A occurs, indeterminate chance of occurrence of A, chance that A does not occur.

**Category:** General Mathematics

[1701] **viXra:1607.0505 [pdf]**
*submitted on 2016-07-27 02:01:36*

**Authors:** Jun Ye, Florentin Smarandache

**Comments:** 4 Pages.

This paper introduces a refined single-valued
neutrosophic set (RSVNS) and presents a similarity measure of RSVNSs. Then a multicriteria decisionmaking method with RSVNS information is developed based on the similarity measure of RSVNSs. By the similarity
measure between each alternative and the ideal solution (ideal alternative), all the alternatives can be ranked and the best one can be selected as well. Finally, an actual example on the selecting problems of construction
projects demonstrates the application and effectiveness of the proposed method.

**Category:** General Mathematics

[1700] **viXra:1607.0504 [pdf]**
*submitted on 2016-07-27 02:03:14*

**Authors:** Bui Cong Cuong, Pham Hong Phong, Florentin Smarandache

**Comments:** 12 Pages.

The traditional soft set is a mapping from a
parameter set to family of all crisp subsets of a universe.Molodtsov introduced the soft set as a generalized tool for modelling complex systems involving uncertain or not clearly defined objects. In this paper, the notion of
neutrosophic soft set is reanalysed. The novel theory is a combination of neutrosophic set theory and soft set theory. The complement, “and”, “or”, intersection and union operations are defined on the neutrosophic soft sets.
The neutrosophic soft relations accompanied with their compositions are also defined. The basic properties of the neutrosophic soft sets, neutrosophic soft relations and neutrosophic soft compositions are also discussed.

**Category:** General Mathematics

[1699] **viXra:1607.0503 [pdf]**
*submitted on 2016-07-27 02:04:43*

**Authors:** Florentin Smarandache, Huda E. Khalid, Ahmed K. Essa, Mumtaz Ali

**Comments:** 9 Pages.

In this paper, we introduce the concept of neutrosophic less than or equal to. The neutrosophy considers every idea < A > together with its opposite or negation < antiA > and with their spectrum of neutralities < neutA > in between them (i.e. notions or ideas supporting neither < A > nor < antiA >).

**Category:** General Mathematics

[1698] **viXra:1607.0502 [pdf]**
*submitted on 2016-07-27 02:05:55*

**Authors:** S. K. Patro, F. Smarandache

**Comments:** 7 Pages.

In this paper , the authors explore
neutrosophic statistics, that was initiated by Florentin Smarandache in 1998 and developed in 2014, by presenting various examples of several statistical distributions, from the work [1]. The paper is presented with more case studies, by means of which this neutrosophic version of statistical distribution becomes more pronounced.

**Category:** General Mathematics

[1697] **viXra:1607.0476 [pdf]**
*submitted on 2016-07-25 09:00:09*

**Authors:** Edgar Valdebenito

**Comments:** 6 Pages.

In this note show series for the Catalan's constant: G=0.915965594177...

**Category:** General Mathematics

[1696] **viXra:1607.0367 [pdf]**
*submitted on 2016-07-19 09:13:05*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

In this note we show series for the numbers: pi,ln 2,G,zeta(k),k=1,2,3,..., In this series appears the number phi.

**Category:** General Mathematics

[1695] **viXra:1607.0366 [pdf]**
*submitted on 2016-07-19 09:16:00*

**Authors:** Edgar Valdebenito

**Comments:** 21 Pages.

In this note we show some formulas for the number pi

**Category:** General Mathematics

[1694] **viXra:1607.0352 [pdf]**
*submitted on 2016-07-18 07:14:14*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

Imaginary. God. Transcendentality.
What's the probability for God to be in Number Theory?

**Category:** General Mathematics

[1693] **viXra:1607.0347 [pdf]**
*submitted on 2016-07-18 07:19:49*

**Authors:** Florentin Smarandache

**Comments:** 15 Pages.

This project is a part of a National Science Foundation interdisciplinary project proposal

**Category:** General Mathematics

[1692] **viXra:1607.0344 [pdf]**
*submitted on 2016-07-18 07:23:49*

**Authors:** Florentin Smarandache

**Comments:** 12 Pages.

This article is on Numeralogy or Properties of Numbers.

**Category:** General Mathematics

[1691] **viXra:1607.0343 [pdf]**
*submitted on 2016-07-18 07:24:47*

**Authors:** Florentin Smarandache

**Comments:** 13 Pages.

This article is on Numeralogy or Properties of Numbers.

**Category:** General Mathematics

[1690] **viXra:1607.0342 [pdf]**
*submitted on 2016-07-18 07:25:39*

**Authors:** Florentin Smarandache

**Comments:** 5 Pages.

This article is on Numeralogy or Properties of Numbers.

**Category:** General Mathematics

[1689] **viXra:1607.0329 [pdf]**
*submitted on 2016-07-18 07:46:54*

**Authors:** Florentin Smarandache

**Comments:** 6 Pages.

The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones.

**Category:** General Mathematics

[1688] **viXra:1607.0324 [pdf]**
*submitted on 2016-07-18 07:54:25*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

Acest articol va prezinta un aritmograf.

**Category:** General Mathematics

[1687] **viXra:1607.0307 [pdf]**
*submitted on 2016-07-18 08:20:25*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

This quatation is from a Letter of September 24, 1993, by Constantin M. Popa., an essayist
of the Paradoxist Literary Movement [a movement stating that <the sense has a non-sense and,
reciprocally, the non-eense has a sense too>], referring to the

**Category:** General Mathematics

[1686] **viXra:1607.0305 [pdf]**
*submitted on 2016-07-18 08:22:08*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

Analyzing the deterioration’s degree of the keys of a typing machine which functioned for more than 40 years at the clerk's office of a court of a Rumanian district (Vâlcea), one partitions them in the following groups.

**Category:** General Mathematics

[1685] **viXra:1607.0301 [pdf]**
*submitted on 2016-07-18 08:25:34*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

In this short paper one promotes the hypothesis that: There is no speed barrier in the universe, and one asks if it's possible to have an infinite speed?

**Category:** General Mathematics

[1684] **viXra:1607.0300 [pdf]**
*submitted on 2016-07-18 08:27:24*

**Authors:** Bencze MihÁly, Florin Popovici, Florentin Smarandache

**Comments:** 2 Pages.

In the "Octogon" Vo1.5, No.2, Zoltan Blazik, in the open problem OQ.102, asked if there exists a polynomial P(x,y) of at most second degree.

**Category:** General Mathematics

[1683] **viXra:1607.0268 [pdf]**
*submitted on 2016-07-18 05:23:46*

**Authors:** Florentin Smarandache

**Comments:** 15 Pages.

Ten birds are on a fence. A hunter shoots three of them.How many birds remain?
(Answer: none, because the three dead birds fell down from the fence and the other seven flew way!)

**Category:** General Mathematics

[1682] **viXra:1607.0256 [pdf]**
*submitted on 2016-07-18 05:36:52*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

This program takes an input number, performs some rule on it, and shows the output number that results.

**Category:** General Mathematics

[1681] **viXra:1607.0254 [pdf]**
*submitted on 2016-07-18 05:39:39*

**Authors:** Marian Dinca, Bencze MihÁly, SzilÁrd AndrÁs, Florentin Smarandache

**Comments:** 2 Pages.

In Octogon, April, 1998, Mihaly Bencze proposed the following open question.

**Category:** General Mathematics

[1680] **viXra:1607.0250 [pdf]**
*submitted on 2016-07-18 05:45:15*

**Authors:** Florentin Smarandache

**Comments:** 55 Pages.

The subsets T, I, F are not necessarily intervals, but may be any real subsets: discrete or continuous; single-element, finite, or (either countably or uncountably) infinite; union or intersection of various subsets; etc.

**Category:** General Mathematics

[1679] **viXra:1607.0248 [pdf]**
*submitted on 2016-07-18 05:56:49*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Si iata cii timpul trece ...
Mi-amintesc zilele cand luam intaia data contact cu revista Liceului "N. Grigorescu" din Campina, pc cand eram in tara, student la Facultatea
din Craiova, in 1979.

**Category:** General Mathematics

[1678] **viXra:1607.0230 [pdf]**
*submitted on 2016-07-18 06:29:29*

**Authors:** Florentin Smarandache

**Comments:** 7 Pages.

In this paper we present original or collected recreational mathematical problems.

**Category:** General Mathematics

[1677] **viXra:1607.0223 [pdf]**
*submitted on 2016-07-18 06:45:04*

**Authors:** Florentin Smarandache

**Comments:** 4 Pages.

Here it is a program created on T.I.-92 Graphing Calculator to simplify a given fraction.

**Category:** General Mathematics

[1676] **viXra:1607.0221 [pdf]**
*submitted on 2016-07-18 06:47:17*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

Les problèmes de mots croisés sont composés, on le sait, de grilles et de d3finitions.

**Category:** General Mathematics

[1675] **viXra:1607.0210 [pdf]**
*submitted on 2016-07-18 07:03:49*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Analysant le degré de deterioration des touches d'une machine à écrire qui fonctionné,plus de 40 ans au greffe d'un tribunal d'un district roumain (Valcea) on les a réparties dans les groupes suivants.

**Category:** General Mathematics

[1674] **viXra:1607.0205 [pdf]**
*submitted on 2016-07-18 07:09:23*

**Authors:** Florentin Smarandache

**Comments:** 2 Pages.

Cate propozitii sunt adevarate si care anume dintre urmatoarele?

**Category:** General Mathematics

[1673] **viXra:1607.0182 [pdf]**
*submitted on 2016-07-15 09:06:41*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

In this note we give formulas, curves and surfaces,related with the constant pi.

**Category:** General Mathematics

[1672] **viXra:1607.0181 [pdf]**
*submitted on 2016-07-15 09:13:19*

**Authors:** Edgar Valdebenito

**Comments:** 18 Pages.

In this paper we give some formulas related with the numbers: pi,G(catalán),zeta(n),gamma(Euler-Mascheroni)

**Category:** General Mathematics

[1671] **viXra:1607.0135 [pdf]**
*submitted on 2016-07-11 09:14:07*

**Authors:** Edgar Valdebenito

**Comments:** 3 Pages.

In this note we exhibit an integral with one parameter. the integrand is of the form: A(t)cos(f(x,t))senh(g(x,t))+B(t)sen(f(x,t))cosh(g(x,t)),0<x<1,0<t<pi/2.

**Category:** General Mathematics

[1670] **viXra:1607.0134 [pdf]**
*submitted on 2016-07-11 09:18:10*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

we give some formulas related with the numbers pi,h(n,m),g(n,m),f(n,m).

**Category:** General Mathematics

[1669] **viXra:1607.0097 [pdf]**
*submitted on 2016-07-08 06:00:40*

**Authors:** Dmitri Martila

**Comments:** 15 Pages.

Solved all Millennium Problems. Each Problem deserves 1 000 000 USD from Clay University. Please help to get it. All rights reserved.

**Category:** General Mathematics

[1668] **viXra:1607.0043 [pdf]**
*submitted on 2016-07-04 09:05:12*

**Authors:** Edgar Valdebenito

**Comments:** 23 Pages.

In this paper we give some formulas for the number pi

**Category:** General Mathematics

[1667] **viXra:1607.0042 [pdf]**
*submitted on 2016-07-04 09:11:12*

**Authors:** Edgar Valdebenito

**Comments:** 5 Pages.

In this paper we give some formulas associated with an integral hidden in Gradshteyn and Ryzhik

**Category:** General Mathematics

[1666] **viXra:1606.0315 [pdf]**
*submitted on 2016-06-29 04:26:56*

**Authors:** Isaac Mor

**Comments:** 11 Pages.

The specific proof for e is not by me!!!
Only the proof for π and π-e are both mine!

**Category:** General Mathematics

[1665] **viXra:1606.0309 [pdf]**
*submitted on 2016-06-28 09:06:44*

**Authors:** Edgar Valdebenito

**Comments:** 25 Pages.

Some formulas related with the constant pi

**Category:** General Mathematics

[1664] **viXra:1606.0308 [pdf]**
*submitted on 2016-06-28 09:10:35*

**Authors:** Edgar Valdebenito

**Comments:** 8 Pages.

Some results related with the Generalized-Euler-Constant Function

**Category:** General Mathematics

[1663] **viXra:1606.0224 [pdf]**
*submitted on 2016-06-21 08:47:15*

**Authors:** Edgar Valdebenito

**Comments:** 11 Pages.

We give some formulas related with the continued radicals

**Category:** General Mathematics

[1662] **viXra:1606.0223 [pdf]**
*submitted on 2016-06-21 08:50:32*

**Authors:** Edgar Valdebenito

**Comments:** 16 Pages.

In this paper we give some formulas related with the constant pi

**Category:** General Mathematics

[1661] **viXra:1606.0183 [pdf]**
*submitted on 2016-06-17 22:39:41*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 18 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Relative Metric That Generates A Field Super-Set To The Fields Generated By
Various Distinct Relative Metrics’.

**Category:** General Mathematics

[1660] **viXra:1606.0175 [pdf]**
*submitted on 2016-06-17 08:48:45*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 16 Pages.

In this research investigation, the author has presented a theory of ‘Universal
Holistic Beauty Primality Tree Of Any Set, Universal Growth Of Any Given Set’.

**Category:** General Mathematics

[1659] **viXra:1606.0145 [pdf]**
*submitted on 2016-06-15 00:18:49*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 14 Pages.

In this research investigation, the author has presented a theory of ‘Universal Natural Memory Embedding’.

**Category:** General Mathematics

[1658] **viXra:1606.0125 [pdf]**
*submitted on 2016-06-13 08:45:34*

**Authors:** Edgar Valdebenito

**Comments:** 8 Pages.

We remember a geometric method to estimate the constant pi

**Category:** General Mathematics

[1657] **viXra:1606.0124 [pdf]**
*submitted on 2016-06-13 08:50:09*

**Authors:** Edgar Valdebenito

**Comments:** 26 Pages.

In this paper we give some formulas related with the constant pi

**Category:** General Mathematics

[1656] **viXra:1606.0107 [pdf]**
*submitted on 2016-06-11 08:28:27*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 12 Pages.

In this research investigation, the author has presented the theory of ‘Primality
Tree Of Any Set’.

**Category:** General Mathematics

[1655] **viXra:1606.0106 [pdf]**
*submitted on 2016-06-11 08:32:04*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 12 Pages.

In this research investigation, the author has presented the theory of ‘Primality
Tree Of Any Set’.

**Category:** General Mathematics

[1654] **viXra:1606.0092 [pdf]**
*submitted on 2016-06-09 20:19:58*

**Authors:** Shi-YuangDong

**Comments:** 134 Pages.

The Power of DaYan—The Application of DaYan Matrix《威哉，大衍！》
Author:Shi-YuangDong/
The article on the 《Outline of Algorithm in Mathematics数术大略》./
我于2015 年 5月就已经完成了对秦九韶《数术大略》的解读，写了一些文章，集合成论文集《威哉，大衍！》，在此基础上又进一步完成了对秦九韶《数术大略》的全面的校勘工作。
这本论文集也联系过几个出版社，无果。现决定发表于此，免得埋没。

**Category:** General Mathematics

[1653] **viXra:1606.0091 [pdf]**
*submitted on 2016-06-09 20:43:28*

**Authors:** Shi-YuangDong

**Comments:** 61 Pages.

On The Method of Regulating the Day-ratio“调日法”分析-石远东
Comments: 60 Pages.
Category: General Mathematics
Abstract:
The method for regulating the day-ratio（调日法）is a fraction approximation one applied in astronomy in ancient China.
We have study the original article by He-ChengTian, Li-Rui, Qin-JiuShao, and Chen-JiuJin, and got the essence of the method for regulating the day-ratio.
摘要：调日法是中国古代长期用于天文历算的一种分数近似法，本文分析了何承天，李锐，秦九韶，陈久金关于调日法的论述，以揭示调日法的本质。

**Category:** General Mathematics

[1652] **viXra:1606.0087 [pdf]**
*submitted on 2016-06-09 06:33:15*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 10 Pages.

In this research investigation, the author has presented a ‘Scheme to Generate The Entire Elements Of A Field Given Three Elements Of It’.

**Category:** General Mathematics

[1651] **viXra:1606.0064 [pdf]**
*submitted on 2016-06-07 00:48:21*

**Authors:** Shi-YuanDong

**Comments:** 284 Pages.

Outline of Algorithm in Mathematics《数术大略》
Author：[Song]Qin-JiuShao(1247) 秦九韶[Emendator：Shi-YuanDong石远东]
Comments: 284 Pages.
Category: General Mathematics
目录Contents
石远东序Preface by Shi-YuanDong
秦九韶序Preface by Qin-JiuShao
Chapter 1. 大衍类DaYanShu(Indeterminate Analysis)
Chapter 2. 天时类Astronomic, Calendar, Weather
Chapter 3. 田域类Area of Cropland
Chapter 4. 测望类Survey
Chapter 5. 赋役类Taxation and Levies of Labor
Chapter 6. 钱谷类Money and Rice
Chapter 7. 营建类Buildings and Fortifications
Chapter 8. 军旅类Military Affairs
Chapter 9. 市易类Price and Accrual
Qin-JiuShao
Qin-JiuShao (秦九韶, 1208–1268), courtesy name: DaoGu (道古).
Qin-JiuShao was born in AnYue, SiChuan province(普州，四川省安岳), his family came from ShanDong province(鲁郡).
His work is the ShùShùDàLüè《数术大略》(Outline of Algorithm in Mathematics,1247); It’s called ShùXuéJiǔZhānɡ《数学九章》in SìKùQuánShū《四库全书》; and now, everyone know It’s called ShùShūJiǔZhānɡ《数书九章》.
This treatise covered a variety of topics including indeterminate equations and the numerical solution of certain polynomial equations up to 10th order, as well as discussions on military matters and surveying.
In the treatise Qin-JiuShao included a general form of the Chinese remainder theorem that used DaYanShu (大衍术) or algorithms to solve it.
In geometry, he discovered “Qin-JiuShao's formula” for finding the area of a triangle from the given lengths of three sides. This formula is the same as Heron’s formula, proved by Heron of Alexandria about 60 BC, though knowledge of the formula may go back to Archimedes.

**Category:** General Mathematics

[1650] **viXra:1606.0059 [pdf]**
*submitted on 2016-06-06 08:42:44*

**Authors:** Edgar Valdebenito

**Comments:** 13 Pages.

Some formulas related with the constant pi

**Category:** General Mathematics

[1649] **viXra:1606.0054 [pdf]**
*submitted on 2016-06-05 18:58:56*

**Authors:** Clive Jones

**Comments:** 2 Pages.

An attempt to recreate the missing margin proof of Pierre de Fermat

**Category:** General Mathematics

[1648] **viXra:1606.0053 [pdf]**
*submitted on 2016-06-05 19:02:44*

**Authors:** Clive Jones

**Comments:** 1 Page.

An attempt to square the circle & circle the square

**Category:** General Mathematics

[1647] **viXra:1606.0040 [pdf]**
*submitted on 2016-06-04 07:00:01*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 8 Pages.

In this research investigation, the author has presented the notion of ‘Relative Metric And Field Generation Based On The Same’.

**Category:** General Mathematics

[1646] **viXra:1606.0034 [pdf]**
*submitted on 2016-06-03 12:29:14*

**Authors:** Damodar Rajbhandari

**Comments:** 5 Pages. Attribution 4.0 International (CC BY 4.0)

This paper's aim is to tackle any oscillatory sequence in a new way. I introduced a new method to find any term of the specific sequence given on a title. This method not only focus for that problem, but also can apply in such a sequence whose terms are repeated after some period.

**Category:** General Mathematics

[1645] **viXra:1606.0033 [pdf]**
*submitted on 2016-06-03 12:55:59*

**Authors:** Damodar Rajbhandari

**Comments:** 7 Pages. Attribution 4.0 International (CC BY 4.0)

In this article, I presented the proved of Taylor expansion in one variable by the concept of binomial theorem, Taylor series concepts in curves and an expository piece on the asymptote of an algebraic curves as an example of this expansion.

**Category:** General Mathematics

[1644] **viXra:1606.0017 [pdf]**
*submitted on 2016-06-02 06:14:07*

**Authors:** Ramesh Chandra Bagadi

**Comments:** 6 Pages.

In this research investigation,
the author has presented a
‘‘Universal Sequence Of Primes
Of Rth Order Space’ Finding Algorithm’.

**Category:** General Mathematics

[76] **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

[75] **viXra:1609.0066 [pdf]**
*replaced on 2016-09-13 19:42:46*

**Authors:** Marvin Ray Burns

**Comments:** 11 Pages. This pdf from a structured and clear Mathematica notebook is complemented plots and with some more traditional forms of the mathematics language. For further research or copying formulae, download marvinrayburns.com/ClosedformsGelfond.nb .

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 shall struggle with numeric evaluation to compare the limit of the ratios of
integral(Sin[Pi*I*x]x^(1/x),{x,1,a})/integral(Sin[Pi*I*x]x^(1/x),{x,1,a-1}) and integral(Cos[Pi*I*x]x^(1/x),{x,1,a})/integral(Cos[Pi*I*x]x^(1/x),{x,1,a-1}).
Through some simple plotting and taking into consideration oscillatory nature of the sine and cosine functions, we make judgments on whether we should invest in finding their limit.

**Category:** General Mathematics

[74] **viXra:1609.0066 [pdf]**
*replaced on 2016-09-11 20:28:10*

**Authors:** Marvin Ray Burns

**Comments:** 14 Pages. This pdf from a structured and clear Mathematica notebook is complemented plots and with some more traditional forms of the mathematics language. For further research or copying formulae, download marvinrayburns.com/ClosedformsGelfond.nb .

First we will consider
integral(cos[pi*I*x]x^(1/x),{x,1,a}) and
integral(sin[pi*I*x]x^(1/x),{x,1,a}),
where our numerical analysis makes it appear that the limit of the ratio of a to a-1,as a goes to infinity,in integral(Sin[Pi*I*x]x^(1/x),{x,1,a})/integral(Sin[Pi*I*x]x^(1/x),{x,1,a-1}) and integral(Cos[Pi*I*x]x^(1/x),{x,1,a})/integral(Cos[Pi*I*x]x^(1/x),{x,1,a-1}) 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 shall struggle with numeric evaluation to compare the ratios of
integral(Sin[Pi*I*x]x^(1/x),{x,1,a})/integral(Sin[Pi*I*x]x^(1/x),{x,1,a-1}) and integral(Cos[Pi*I*x]x^(1/x),{x,1,a})/integral(Cos[Pi*I*x]x^(1/x),{x,1,a-1}).
We shall compare the ratios of
integral(sin[pi*I*x]x^(1/x),{x,1,a}) and integral(sin[pi*I*x]x^(1/x),{x,1,a-1}) with
integral(sin[pi*x]x^(1/x),{x,1,a}) and integral(sin[pi*x]x^(1/x),{x,1,a-1}).
In later parts of this paper we will look at the ratio of three deceptively similar looking integrals which have the terms of MKB for integrands. (For reference we call these the "later integrals."
Finally, after finding no closed form for the ratios of the later integrals, we will settle for closed forms of their differences.

**Category:** General Mathematics

[73] **viXra:1609.0066 [pdf]**
*replaced on 2016-09-07 15:48:30*

**Authors:** Marvin Ray Burns

**Comments:** 12 Pages. This pdf from a structured and clear Mathematica notebook is complemented with the traditional form of the mathematics language. For further research or copying formulae, download marvinrayburns.comClosedformsGelfond.nb .

First we will consider
integral(cos[pi*I*x]x^(1/x),{x,1,a}) and
integral(sin[pi*I*x]x^(1/x),{x,1,a}),
where it appears that the limit of the ratio of a to a-1,as a goes to infinity, in
integral(Sin[Pi*I*x]x^(1/x),{x,1,a})/integral(Sin[Pi*I*x]x^(1/x),{x,1,a-1}) and integral(Cos[Pi*I*x]x^(1/x),{x,1,a})/integral(Cos[Pi*I*x]x^(1/x),{x,1,a-1})
is Gelfond's Constant,(e^pi). We will consider that the hypothesis and provide hints for a proof using L'Hospital's Rule since, with the hyperbolic integrals, we have indeterminate forms as a goes to infinity.
We shall compare the ratios of
integral(sin[pi*I*x]x^(1/x),{x,1,a}) and integral(sin[pi*I*x]x^(1/x),{x,1,a-1}) with
integral(sin[pi*x]x^(1/x),{x,1,a}) and integral(sin[pi*x]x^(1/x),{x,1,a-1}).
In later parts of this paper we will look at the ratio of three deceptively similar looking integrals which have the terms of MKB for integrands. (For reference we call these the "later integrals."
Finally, after finding no closed form for the ratios of the later integrals, we will settle for closed forms of their differences.

**Category:** General Mathematics

[72] **viXra:1607.0097 [pdf]**
*replaced on 2016-07-11 11:23:09*

**Authors:** Dmitri Martila

**Comments:** 15 Pages.

Solved all Millennium Problems. Each Problem deserves 1 000 000 USD from Clay University. Please help to get it. All rights reserved.

**Category:** General Mathematics

[71] **viXra:1607.0097 [pdf]**
*replaced on 2016-07-10 14:01:17*

**Authors:** Dmitri Martila

**Comments:** 15 Pages.

Solved all Millennium Problems. Each Problem deserves 1 000 000 USD from Clay University. Please help to get it. All rights reserved.

**Category:** General Mathematics