In this note, we propose a new class of orthogonal polynomials (named Bachok-Hasham polynomials of the first and second kind for order k, denote it as Z((i,n))(k)(x), i = {1, 2}, which is extension of the Chebyshev polynomials of the first and second kind respectively. It is found that Bachok-Hasham polynomials of first and second kind Z((i,n))(k)(x) are orthogonal with respect to weights w ((1,k)) (x)=-x(k-1)/root 1-x(2k), w((2,k)) (x) = x(k-1) root 1-x(2k) on the interval [-1,1], where k is positive odd integers. Spectral properties Bachok--Hasham polynomials of the first and second kind Z((i,n))(k)(x), i = {1,2} are proved. These properties are used to solve a special class of singular integral equations. Finally, numerical examples and comparison results with other methods are provided to illustrate the effectiveness and accuracy of the proposed method.

Transition from steady to chaos for the onset of Benard convection in porous medium was analyzed. The governing equation is reduced to ordinary differential equation and solved using built in MATLAB ODE45. The transition from steady to chaos take over from a limit cycle followed by homoclinic explosion.

A question of interest in the demographic and actuarial fields is the estimation of the complete sets of q(x) values when the data are given in age groups. When the complete life tables are not available, estimating it from abridged life tables is necessary. Three methods such as King's Osculatory Interpolation, Six-point Lagrangian Interpolation and Heligman-Pollard Model are compared using data on abridged life tables for Malaysian population. Each of these methods considered was applied on the abridged data sets to estimate the complete sets of q(x) values. Then, the estimated complete sets of q(x) values were used to produce the estimated abridged ones by each of the three methods. The results were then compared with the actual values published in the abridged life tables. Among the three methods, the Six-point Lagrangian Interpolation method produces the best estimates of complete life tables from five-year abridged life tables.

Actuaries, demographers and other professionals have always been aware of the critical importance of mortality forecasting due to declining trend of mortality and continuous increases in life expectancy. Heligman-Pollard model was introduced in 1980 and has been widely used by researchers in modelling and forecasting future mortality. This paper aims to estimate an eight-parameter model based on Heligman and Pollard's law of mortality. Since the model involves nonlinear equations that are explicitly difficult to solve, the Matrix Laboratory Version 7.0 (MATLAB 7.0) software will be used in order to estimate the parameters. Statistical Package for the Social Sciences (SPSS) will be applied to forecast all the parameters according to Autoregressive Integrated Moving Average (ARIMA). The empirical data sets of Malaysian population for period of 1981 to 2015 for both genders will be considered, which the period of 1981 to 2010 will be used as "training set" and the period of 2011 to 2015 as "testing set". In order to investigate the accuracy of the estimation, the forecast results will be compared against actual data of mortality rates. The result shows that Heligman-Pollard model fit well for male population at all ages while the model seems to underestimate the mortality rates for female population at the older ages.

The population of many countries might undergo dramatic changes in the coming decades due to continuous increases in life expectancy. The sustained reduction in mortality rates and its systematic underestimation has been attracting the significant interest of researchers in recent years because of its potential impact on population size and structure, social security systems, and (from an actuarial perspective) the life insurance and pensions industry worldwide. Among all projection methods, the Lee-Carter method has been widely accepted by the actuarial community. This paper explores the use of the Lee-Carter method to forecast the mortality rates for Malaysian population. The index of the level of mortality for each gender, and the shape and sensitivity coefficients for 18 age groups were obtained through the LeeCarter method. The Singular Values Decomposition (SVD) is used to forecast the general index for the time period that goes from 2011 to 2030. Since the model involves nonlinear equations that are explicitly difficult to solve, the Matrix Laboratory Version 7.0 (MATLAB 7.0) software will be used in the study. The empirical data sets of Malaysia population for the period of 1981-2010 and for both genders will be considered.

Improvement in life expectancies has driven further declines in mortality. The sustained reduction in mortality rates and its systematic underestimation has been attracting the significant interest of researchers in recent years because of its potential impact on population size and structure, social security systems, and (from an actuarial perspective) the life insurance and pensions industry worldwide. Among all forecasting methods, the Lee-Carter model has been widely accepted by the actuarial community and Heligman-Pollard model has been widely used by researchers in modelling and forecasting future mortality. Therefore, this paper only focuses on Lee-Carter model and Heligman-Pollard model. The main objective of this paper is to investigate how accurately these two models will perform using Malaysian data. Since these models involves nonlinear equations that are explicitly difficult to solve, the Matrix Laboratory Version 8.0 (MATLAB 8.0) software will be used to estimate the parameters of the models. Autoregressive Integrated Moving Average (ARIMA) procedure is applied to acquire the forecasted parameters for both models as the forecasted mortality rates are obtained by using all the values of forecasted parameters. To investigate the accuracy of the estimation, the forecasted results will be compared against actual data of mortality rates. The results indicate that both models provide better results for male population. However, for the elderly female population, Heligman-Pollard model seems to underestimate to the mortality rates while Lee-Carter model seems to overestimate to the mortality rates.

In 2011 Alkhazaleh and Salleh defined the concept of soft expert sets where the user can know the opinion of all the experts in one model and give an application of this concept in decision-making problems. Also, they introduced the concept of the fuzzy soft expert set as a combination between the soft expert set and the fuzzy set. In 2010 Majumdar and Samanta presented the concept of a generalized fuzzy soft sets. The purpose of this paper is to combine the work of Alkhazaleh and Salleh (2011) and Majumdar and Samanta (2010), from which we can obtain a new concept: generalized fuzzy soft expert sets (GFSESs). We also introduce its operations, namely, complement, union intersection, AND and OR, and study their properties. The generalized fuzzy soft expert sets are used to analyze a decision-making problem. Also in our model the user can know the opinion of all experts in one model. In this work we also introduce the concept of a generalized fuzzy soft expert sets with multiopinions (four opinions), which will be more effective and useful. Finally, we give an application of this concept in decision-making problem.

This study investigates the sensitivity of pension costs to changes in the underlying assumptions of a hypothetical pension plan in order to gain a perspective on the relative importance of the various actuarial assumptions via a simulation analysis. Simulation analyses are used to examine the impact of actuarial assumptions on pension costs. There are two actuarial assumptions will be considered in this study which are mortality rates and interest rates. To calculate pension costs, Accrued Benefit Cost Method, constant amount ( CA) modification, constant percentage of salary ( CS) modification are used in the study. The mortality assumptions and the implied mortality experience of the plan can potentially have a significant impact on pension costs. While for interest rate assumptions, it is inversely related to the pension costs. Results of the study have important implications for analyst of pension costs.

A question of interest in the demografic and actuarial fields is the estimation of the complete sets of q(x)-values when the data are given in age groups. This study presents two techniques of expanding an abridged life table to a complete ones. The two expansion techniques used in the study are King's Osculatory Interpolation and Heligman-Pollard model. This work evaluated and compared the performance of King's Osculatory Interpolation and Heligman-Pollard model. For that purpose, empirical data sets on abridged life tables for Malaysian populations for the period of 1991, 1995 and 1999 for both gender were collected. Then each of the expanding techniques considered was applied to these abridged data sets. The complete sets of q(x)-values obtained from these two techniques were then used to produce the estimated abridged ones. These results were then compared with the actual values published in the abridged tables.

Background: Within a setting without organised breast cancer screening, the characteristics and survival of very early breast cancer were determined. Methods: All 4930 women diagnosed with breast cancer in University Malaya Medical Center, Malaysia from 1993 to 2011 were included. Factors associated with very early presentation (stage I) at diagnosis were identified. Tumour characteristics, management patterns, and survival of very early breast cancer were described, and where appropriate, compared with other settings. Results: Proportion of women presenting with stage I breast cancer significantly increased from 15.2% to 25.2% over two decades. Factors associated with very early presentation were Chinese ethnicity, positive family history of breast cancer, and recent period of diagnosis. Within stage I breast cancers, median tumour size at presentation was 1.5 cm. A majority of stage I breast cancer patients received mastectomy, which was associated with older age, Chinese ethnicity, postmenopausal status, and larger tumours. Chemotherapy was administered in 36% of patients. Five-year age-adjusted relative survival for women with stage I breast cancer was 99.1% (95% CI: 97.6-99.6%). Conclusions: The proportion of women presenting with very early breast cancer in this setting without organised screening is increasing. These women seem to survive just as well as their counterparts from affluent settings.

There are many researches showing that the life expectancy for most countries is increasing. Since the life expectancy at a particular age tends to increase over time for male and female, the mortality risk tends to be smaller over time. Therefore it is expected that pensioners tend to live longer and thereby cause increase in pension liabilities to the government. Countries are looking for solutions to decrease the effect of increased longevity on pension costs. The most common changes are to equalize the retirement age for male and female and to rise the retirement age. Therefore, we studied the longevity factor for the government pensioner in Malaysia at age 55, 56, 57, 58, 59 and 60 years respectively. Since the pensions are paid for the rest of the pensioner's life in the event of his death and also to his spouse and child if any in the form of a derivative pension, the longevity factor for pensioner, spouse and child will be formulated and estimated separately. To formulate and estimate these factors, the theory of annuities and the Pension Law of Malaysia need to be studied.

In the event of death of any government employee, their monthly pension will be given to their widow and their child. The government will stop paying that pension when the widow died and when the widow chooses to remarry. However, in 1st January 2002, the remarriage clause has been removed from the regulations. This would allow all widows who remarried to receive pension as usual. In view of this, there are possibilities that those widows who are still young might remarried. If many of the widows choose to remarry, it will be a burden to the government as it would increase the pension liabilities. However, we do not know how many of the widow will remarry. In view of this, the purpose of the study is to assess the impact to pension liabilities of government pension schemes on individual life due to removal clause of the remarriage by determining the pension factor and to assess to what extent the pension liabilities of government pension schemes would be affected.