Open Access Original Research Article

Coping with Mobile Backhaul Bandwidth Limitation: User Clustering and Content Sharing

Zhiwei Zhang, Qiang Ling, Lixiang Xu, Feng Li, Song Wang

Journal of Advances in Mathematics and Computer Science, Page 1-11
DOI: 10.9734/BJMCS/2016/23355

With the development of 4G mobile communication technologies, mobile networks are evolving into a new generation, which is expected to support not only the traditional services such as phone calls and text messages, but also the high-speed applications such as real-time videos, streaming music. Mobile networks are composed of Mobile Core Networks and Mobile Access Networks. Advances in wireless communication technologies have shifted the data traffic bottleneck to the mobile core network, which is either too expensive or too complex to upgrade. Caching technology is considered as a promising and economic technology to cope with this problem. In this paper, we propose a new caching architecture in which the cached contents in user mobile devices are shared among users with similar media content preference over the eNode in the Radio Access Network, thus decreasing the data traffic going through the Mobile Core Network. In order to make the best use of mobile devices' cache space, a centralized cache management algorithm, which is expected to run at the base station and control the cache objects in user mobile devices, is also designed Simulation shows that via cache content sharing among users with similar content preference, great cache performance gain can be achieved in terms of saving backhaul bandwidth and reducing the average user request delay.

Open Access Original Research Article

Preservation of Stability for Reduced Order Model of Large Scale Systems Using Differentiation Method

D. K. Sambariya, Hem Manohar

Journal of Advances in Mathematics and Computer Science, Page 1-17
DOI: 10.9734/BJMCS/2016/23082

In this paper, higher order single-input and single-output models are considered for reduction using differentiation method. The analysis of higher order models is complex, and time consuming. The application of model order reduction is necessitated to reduce the system with its core properties. The analysis of a low order system is easy, and fast process. The higher order systems are reduced using differentiation method and the results are compared with original and the reduced systems in literature. The performance comparison of the proposed reduced order model and original as well as systems in literature have been considered in terms of settling time, rise time, peak and peak time. The application of proposed differentiation method is applied to discrete system and found equally good to retain the stability in reduced order model.

Open Access Original Research Article

Scale Laws of Prime Number Frequencies by the Modified Chi-square Function

D. Lattanzi

Journal of Advances in Mathematics and Computer Science, Page 1-21
DOI: 10.9734/BJMCS/2016/23200

The methodology of experimental mathematics has not yet been applied to the frequencies of prime numbers thus the present report treats them as raw experimental data and as elements of larger and larger finite sequences {fm}≡{mp/Pm}. The modified chi-square function Xk2(A,x/μ) with its three parameters A, k and μ=μ(k) is the best-fit function of the differential distribution functions of the finite sequences {fm} thus showing that the property of scale invariance does not hold for the statistical distributions of prime frequencies. Moreover the function Xk2(A,x/μ) with the ad-hoc values of its parameters is the best-fit function of the finite sequences of prime frequencies {fm} from the analytical viewpoint too, what leads to induction algorithms and to relationships of the kind fm~f(mp), though within the precisions of the calculations and holding locally, showing that the property of scale invariance does not hold for prime frequencies even in the analytical case. An injective map can be set between these {fm} sequences and the {nα} truncated progressions through the parameter k of their common fit function Xk2(A,x/μ) in both the statistical and the analytical case. Moreover a general experimental elementary account of Riemann’s hypothesis is given in this frame.

Open Access Original Research Article

Proclus Hypothesis

Alexey Stakhov

Journal of Advances in Mathematics and Computer Science, Page 1-22
DOI: 10.9734/BJMCS/2016/23100

This article substantiates a new look on Euclid's Elements and mathematics history, based on Proclus hypothesis. Proclus hypothesis answers the question about Euclid’s goal for writing his Elements. Two Greek mathematical achievements underlie Proclus hypothesis: the golden ratio, described in the Books II, VI and XIII, and Platonic solids, described in the final Book XIII of the Elements. As the golden ratio and Platonic solids expressed the Universe harmony in Greek science, it follows from Proclus hypothesis that the main Euclid’s goal in his Elements is to embody Pythagoras & Plato’s “Ideas of the Universe Harmony.” Euclid’s Elements are historically the first version of the Mathematics of Harmony as one of the main directions in mathematics development. This approach overturns our understanding of Euclid's Elements and mathematics history starting from Euclid. The article presents a general interest for all mathematicians, math teachers, mathematics students, and for all science representatives, who are interested for new ideas in the history of mathematics.

Open Access Original Research Article

A New Selection Index to Address within Course Competition and between Course Competition for Ranking Examination Scores

S. G. J. Senarathne, P. Wijekoon

Journal of Advances in Mathematics and Computer Science, Page 1-17
DOI: 10.9734/BJMCS/2016/23213

Different scenarios of examinations have to be handled separately when measuring the true performance of students. If examiners are required to compare the performance of different groups of students who follow different combinations of subjects in an examination, their combined raw scores have to be used. In this case the raw marks can be combined by using a linear equation called a selection index. A proper selection index should correctly address two types of competitions; namely within course competition and between course competition. Although different selection indices were introduced to address these issues in literature, these methods fail to fulfill the requirements expected from a proper selection method. The main objective of this study is to introduce a new selection index called Skewness based Common Currency Index (SCCI) which addresses both within course competition and between course competitions. The proposed method considers the relative subject effects, and these effects are identified by introducing a shape parameter to the selection index. The favorability of the proposed SCCI method is compared with three alternative selection indices. According to the statistical analysis it is found that there is a significant difference of ranks between the selected indices at 5% significance level. Further, the rank differences between the ranks of the SCCI method with the ranks of true student effects show smaller deviations with compared to the rank differences of the ranks of the other three selection methods. Based on the results of Wilcoxon rank sum test, it is revealed that the ranks of SCCI method are much closer to the ranks of the student effects than other three selection methods. Also the ranks of SCCI method have the highest correlation with the ranks assigned to the true student effects. According to the overall results it was confirmed that the new SCCI method can be used as a selection index to compare performance of examinees who follow different courses in an examination.

Open Access Original Research Article

Distributional SAdS BH Spacetime-Induced Vacuum Dominance

Jaykov Foukzon, A. Potapov, E. Meńkova

Journal of Advances in Mathematics and Computer Science, Page 1-54
DOI: 10.9734/BJMCS/2016/19235

This paper dealing with extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1], [2], [3], [4], [5], [6], [7] and [32]. The regularizations of singularities presented in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1], [2] and not classically. In this paper essentially new class Colombeau solutions to Einstein field equations is obtained. We leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. Using nonlinear distributional geometry and Colombeau generalized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates. However the Tolman formula for the total energy ET of a static and asymptotically at spacetime, gives ET = m, as it should be. The vacuum energy density of free scalar quantum field Φ with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum elds is forced, by the very same background distributional spacetime such distributional BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved spacetimes. In particular we obtain that the vacuum fluctuations ⟨ Φ2 ⟩ have a singular behavior on BHs horizon r+ : ⟨Φ2 (r) ⟩ ~ | r - r+|-2 .

Open Access Original Research Article

Equidistant Set of Two Congruent Spheres and Its Orthogonal Projection on Rk

M. Cristian M. Carvajal, P. Ronald A. Manríquez, S. Héctor R. Lorca, C. José A. González

Journal of Advances in Mathematics and Computer Science, Page 1-9
DOI: 10.9734/BJMCS/2016/23497

In this paper some properties of equidistant sets are presented,a relatively new concept. The equidistant concept is characterized and among two congruent spheres of ℝn. Afterwards the behavior of the orthogonal projection onto ℝk is studied, concluding that the projection of equidistant set of two congruent spheres, is a translation of the equidistant set of the spheres projections.