##### Initial Value Problems for Second Order Neutral Impulsive Integro-differential Equations with Advanced Argument

Guobing Ye, Xuebin Wang, Xuan Ye, Chen Liu

Asian Research Journal of Mathematics, Page 1-9
DOI: 10.9734/arjom/2019/v13i130096

This paper discusses initial value problems for second order neutral impulsive integro-differential equations with advanced argument. By using the fixed point theorem of either Leray-Schaude or Banach, two existence results are obtained. By comparison, each of them has his own strong and weak points. If appropriate changes are made to some conditions for two results, the same results can be got. Two examples to illustrate our main results are given, which are compared with the existence results for impulsive differential equations from existing literature.

##### Estimating Stock Returns Volatility and the Risk-Return Nexus in the Nigerian Stock Market in the Presence of Shift Dummies

David Adugh Kuhe, Moses Abanyam Chiawa, Sylvester Chigozie Nwaosu, Jonathan Atsua Ikughur

Asian Research Journal of Mathematics, Page 1-15
DOI: 10.9734/arjom/2019/v13i130097

Volatility and the trade-off between risk and return in stock markets is an important subject in financial theory which play significant role in investment decision making, portfolio selection, options pricing, financial stability, hedging and pair trading strategy among others. This study estimates stock return volatility and analyses the risk-return trade-off in the Nigerian stock market using symmetric GARCH (1,1)-in-mean, asymmetric CGARCH (1,1)-in-mean and EGARCH (1,1)-in-mean models with Generalized Error Distribution and Student-t innovation. Data on daily closing all share prices of the Nigerian stock exchange for the period 2nd January, 1998 to 9th January, 2018 are utilised. The data is further divided into three sub-periods of pre-crisis, global financial crisis and post crisis periods to allow volatility behaviour and the risk-return trade-off to be investigated across the sub-periods. Results showed evidence of volatility clustering, leptokurtosis, high persistence of shocks to volatility and asymmetry without leverage effects across the study periods. The persistence of shocks to volatility increased during the global financial crisis period with delayed reactions of volatility to market changes. However, by incorporating the exogenous breaks into the volatility models for the full study period, the shock persistence drastically reduced with faster reactions of volatility to market changes. The results of this study also found supportive evidence for a significant positive risk-return relationship in Nigerian stock market across various study sub-periods and model specifications meaning that investors in Nigerian stock market should be compensated for holding risky assets. The empirical findings of this study further suggest that the recent global financial crisis have not altered the market dynamics to distort the risk-return trade-off in Nigerian stock market indicating that expected returns are not driven by changes in the stock market volatility. The study provides some policy recommendations for investors and policy makers in the Nigerian stock market.

##### Supply Chain Optimization for Farmer-Bepari System of Agricultural Products in Bangladesh

Asian Research Journal of Mathematics, Page 1-13
DOI: 10.9734/arjom/2019/v13i130098

This study, for the Farmer-Bepari system of agricultural products in Bangladesh, can be formulated as a mixed integer linear programming (MILP) model. Further, it will be investigated that the significant impact of profit the attributes such as labour cost, fertilizer cost, the raw material cost of different firms and also to estimate the product distribution in different locations. To solve this MILP model, with the help of a branch and bound algorithm by using A Mathematical Programming Language (AMPL). To investigate the model we have to collect data from seven locations of three districts in Bangladesh. Also, a numerical example presented this study, which objectives illustrate the models. From the sensitivity of the production, if the raw material cost, labour cost and fertilizer cost increase is about 5%, then decrease the profit by MILP model have 0.004%, 1.6% and 1.2% respectively. Labour cost is a significant factor in profit, which changes the profit more than the raw material cost and fertilizer cost of the product. The results are helping decision-makers to identify the desired agricultural production and distribution structure optimization strategy.

##### Exact Arithmetic of Zero and Infinity Based on the Conventional Division by Zero 0/0 = 1

Okoh Ufuoma

Asian Research Journal of Mathematics, Page 1-40
DOI: 10.9734/arjom/2019/v13i130099

The chief object of this work is to create an exact and consistent arithmetic of zero, denoted 0, and infinity (zero divisor), written as 1/0 and denoted ∞  , based on the conventional division by zero
$$\dfrac{{0}}{{0}}=1.$$
Manifold and undeniable applications of this arithmetic are given in this work in order to show its usefulness.

##### Adaptive Control of a Four-Dimensional Hyperchaotic System

Xuxia Li, Xinghua Fan, Jiuli Yin, Ying Zhang, Xiangxiang Lv

Asian Research Journal of Mathematics, Page 1-17
DOI: 10.9734/arjom/2019/v13i130100

This paper concerns the application of adaptive control method in a four-dimensional hyperchaotic
system. Firstly, we carry out a systematic dynamic analysis including the properties of equilibrium
point, stability, dissipation, Lyapunov exponent spectrum, and bifurcation. Both the existence
of two positive Lyapunov exponents and the Lyapunov dimension value show the hyperchaotic property of the system. Based on Lyapunov stability theorem, we then construct an adaptive controller and the adaptive law to suppress hyperchaos to the origin, which is an unstable equilibrium point under a certain parameter set. The effectiveness of the adaptive control is veried by theoretical analysis and numerical simulation. We nally brie y demonstrate the control eciency of self-linear feedback control and misaligned feedback control. For the fourdimensional hyperchaotic system, the adaptive control outperforms them from the view of control speed.