1. s Scientific Innovations in Fundamental Sciences Published by: Faculty of Science Universiti Putra Malaysia 2014 Editors: Jayanthi Arasan Chen Chuei Yee Mohd Shafie Mustafa Risman Mat Hasim Siti Mahani Marjugi Loh Yue Fang

2. (Copyright) 2014 Faculty of Science, Universiti Putra Malaysia All rights reserved. No part of this book may be reproduced in any form without the permission in writing from the publisher, except by a reviewer who wishes to quote brief passage in a review written for inclusion in a magazine or newspaper. Fundamental Science Congress 2014: Scientific Innovations in Fundamental Sciences Faculty of Science, Universiti Putra Malaysia Editors: Jayanthi Arasan, Chen Chuei Yee, Mohd Shafie Mustafa, Risman Mat Hasim, Siti Mahani Marjugi, Loh Yue Fang. 3. CONTENTS FOREWORD by Vice Chancellor of UPM 1 Dean of Faculty of Science, UPM 2 Chairman of Organizing Committee of FSC 2014 3 BIOLOGY SYMPOSIUM 4 CHEMISTRY SYMPOSIUM 66 MATHEMATICS SYMPOSIUM 186 PHYSIC SYMPOSIUM 375 4. Let me begin by extending a very warm welcome to all participants of Fundamental Science Congress 2014 (FSC 2014). Congratulations to the Faculty of Science for being able once again, to organize this meaningful gathering of students and lecturers of the Faculty, as well as scientists from all over the country, for the sixth year in a row. To be one of the leading players in science and technology, a nation must have a very strong research in Fundamental Science as it provides the platform for significant invention and innovation in all scientific fields. The United States and Russia for example, would not be able to dominate outer space exploration, satellite building and defence technology without utilizing the skills of their fundamental scientists. Theories and results in Biology, Chemistry, Mathematics, and Physics have also been responsible for the improvement of technology in all types of equipment and instrument. A nation without the strength of fundamental science research will always be a mere follower of science and technology rather than a leader. The theme, `Scientific Innovations in Fundamental Sciences is befitting our country Vision 2020, since advancement of innovations in fundamental science will provide a leadership in science and technology, which is a characteristic of a developed nation. This congress is a welcoming effort to encourage collaboration of research among scientists across the four disciplines as well as other natural sciences fields. I would like to stress the importance of participation of scientists from areas such as Agriculture, Biotechnology, Engineering, Medicine or even Economy, in this congress since the significant and leading advancement of research in their areas depend largely on their understanding of how to utilize results in Fundamental Science research. Thank you to the Department of Mathematics, Faculty of Science for being the organizing committee for this year congress. I hope this event will be a huge success and will spark the collaboration research effort among all participants for times to come. Please have an enjoyable and fruitful congress. WITH KNOWLEDGE WE SERVE PROF. DATO DR. MOHD FAUZI HAJI RAMLAN D.M.S.M., J.S.M Vice Chancellor Universiti Putra Malaysia Foreword from the Vice Chancellor of Universiti Putra Malaysia 1 5. On behalf of the faculty and the organizers, it gives me great pleasure to welcome you to Fundamental Science Congress 2014. Organized by the Faculty of Science and first held in 2009, FSC has been in the academic calendar since. This year, the honor is given to the Department of Mathematics to host this special event of the faculty. Over the past 5 years, FSC has been the meeting ground for academics and fellow researchers to share their knowledge and deliver the outcomes of their research. It combines various fields of fundamental science mathematics, biology, physics and chemistry. Scientific Innovations in Fundamental Science is the theme for this years FSC. Through innovations we explore and introduce new ideas that cover the fundamentals of Science. This congress demonstrates the facultys continual commitment to contribute to the universitys success as a research university and its mission to be a leading centre of learning and research. Moreover, the Faculty of Science is a big contributor to the universitys scientific publications. Keynote and invited speakers for each parallel session include distinguished professors from our own faculty. In addition to this, the congress accommodates oral as well as poster presentations. My heartiest gratitude goes to everyone who participated and involved in sustaining the FSC throughout these years. I wish all of you a fruitful discussion and hopefully a future collaboration. Thank you. PROF. DR. ZAINAL ABIDIN TALIB Dean Faculty of Science Universiti Putra Malaysia Foreword from the Dean, Faculty of Science, Universiti Putra Malaysia 2 6. Welcome to the Fundamental Science Congress 2014. On behalf of the organizing committee, I would like to express my sincere gratitude to the management of the Faculty of Science for their support towards successful organization of this congress. I would also like to thank all members of the organizing committee for their hard work and cooperation that have ensured efficient and smooth run of this conference. My highest appreciation goes to the keynote and invited speakers who willingly shared their research ideas and outputs, as well as the participants whose presence have contributed towards success of this meeting. It is my sincere hope that the this years congress will be able to offer opportunities for those present to share their research findings, discuss and exchange ideas, and strengthen networking and research collaborations, hence leading to ideas of innovation in scientific research procedures and outputs. All these will in turn lead to development of new scientific researches and inventions . Finally, I wish all the participants the best and hope that this congress has been a productive and worthwhile event for all present. Thank you. PROF. DR. ADEM KILICMAN Chairman Organizing Committee of FSC2014 Foreword from the Chairman of the Organizing Committee FSC 2014 3 7. Biology Symposium Fundamental Science Congress 2014 4 8. 1. Abdul Latiff Mohamed Is Fundamental Research in Malaysia in Crisis? 2. Ibrahim Jaafar, Shahrul Anuar Md. Sah and Shahriza Shahrudin Amphibian Biodiversity in Peninsula Malaysia : A Precious Natural Heritage 3. Abdulmumin Y, Matazu K.I, Wudil A.M, Alhassan A.J and Imam A.A Bioactive Compound Determination and Curative Effects of Aqueous Stem Bark Extract of Boswellia Papyrifera (Del) in Carbon Tetrachloride Induced Liver Damage in Rats 4. Siti Khairani binti Mohd Kamarudin and Zaiton Hassan Anthocyanin Extraction, Antioxidant Activity, Total Phenolcontent and Color Stability of Purple-Fleshed Sweet Potato by Alpha Amylase Hydrolysis 5. Ramalingam Radhakrishnan, Sang-Mo Kang, Jae-Man Park, Soek-Min Lee and In- Jung Lee Isolation and Identification of Plant Growth Promoting Penicillium Species and Their Effect on Amino Acids to Promote Sesame Plant Growth 6. Nallusamy Sivakumar, Hamood Yahya Al Araimi and Saif Al Bahry Screening and Production of Cellulase by Thermophilic Bacteria Isolated from Hot Springs in Oman 7. Nallusamy Sivakumar, Saif Al Bahry and Abeer Al Busaidi Screening of Carboxymethyl Cellulase Producing Marine Bacteria and Optimization of NaCl Concentration 8. S.Lina, N.H. Hashida and H. Eliza Ameliorating Effect of Habbatus Sauda on androgen Receptor of Nicotine Treated Rats Seminal Vesicle 9. Laila Naher and Umi Kalsom Yusuf Mismatch Amplification Caused Non Specific Gene Isolation of Sadenosylmethionine Synthetase Cdna from Oil Palm (Elaeis Guineensis Jacq.) 10. Rosnida Tajuddin, David Johnson and Andy Taylor Imaging Phosphorus Transportation in Simple and Complex Ectomycorrhizal System 11. Luqman Abu Bakar and Hazlina Ahamad Zakeri Responses of Three Gracilaria (Rhodophyceae) Treated with Mercury(Ii) Nitrate 12. Najla Abu Shaala, Syaizwan Zahmir Zulkifli, Ahmad Ismail and Mohamed Noor Amal Azmai Effect of Tributyltin Chloride (TBTCL) on the Morphology of Nauplii Artemia Salina (Branchiopoda: Anostraca) Biology Symposium Fundamental Science Congress 2014 5 9. 13. Aqilah Mukhtar and Syaizwan Zahmir Zulkifli Determination of Benthos Food Web in Merambong Seagrass Area Using Stable Isotope Marker 14. Jafaru Malam Ahmed, Ahmad Ismail and Syaizwan Zahmir Zulkifli Potential Effects of Vanadium in the West Coast of Peninsular Malaysia 15. Hafidh Ali Almahrouqi, Lee Yan Yan, Hishamudin Omar and Ahmad Ismail Bioeconomic Study of Arthrospira Platensis (Spirulina) Cultured in Shaded Area 16. Geetha Annavi, Chris Newman, Hannah L. Dugdale, Christina D. Buesching, Yung Wa Sin, Terry Burke and David W. Macdonald Choosing Mates for offspring Genetic Quality: Does Extra-Group Paternity Enhance Fitness of the European Badger (Meles Meles)? 17. Bala S. Aliyu and Hamza Y. Kankia An Assessment of the Toxicity of Some Plants Used Locally as Aphrodisiacs in Northern Nigeria and Their Efficacy in Enhancing Erectile Function in Rats 18. Yahaya Mustapha Inheritance of Seed Coat Texture in Cowpea (Vigna Unguiculata (L.) Walp) 19. Amiruddin H.M., Ismail, A., Amal, M.N.A. and Rahman, F. Gonadosomatic Index and Fecundity of Tiny Scale Barb, Thynnichthys Thynnoides Bleeker 1852, During Their Mass Migration Season in Perak River, Perak 20. Safiya Yakubu, Nor Azwady Abd. Aziz, Mohamad Pauzi Zakaria, Noormashela Ulu Azmi, Nurul Shaziera Abdul Ghafar, Cheong Jee Yin, Azleen Ahmad Adli and Muskhazli Mustafa Evaluation of pH and Temperature Effects on the Mycoremediation of Phenanthrene by Trichoderma Sp. 21. Rohana Tair, Kamsia Budin and Kelly Iche Effect of Physical Characteristic of Crassostrea Iredalei on Heavy Metals Concentration in Its Tissue: An Approach Using Multiple Linear Regression Biology Symposium Fundamental Science Congress 2014 6 10. IS FUNDAMENTAL RESEARCH IN MALAYSIA IN CRISIS? A.Latiff University Kebangsaan Malaysia email: latiff@ukm.my ABSTRACT Malaysia has initiated, produced and launched a National Policy on Biological Diversity which contains policy statement, principles, objectives, rationales, strategies and above all action plans of programmes. The policy places great importance of taxonomy in realising the true dimension of biodiversity and that taxonomy is a cornerstone of biodiversity has long been understood. Yet, many taxonomic institutions such as National Museum and Herbarium are not within sight, taxonomy research is not prioritised within the R & D mechanism, capacity building is not undertaken with an accepted vigour, systematic research centres are not established, school and university curricula have not addressed taxonomy, biodiversity, proper training of taxonomists and parataxonomists are not planned, and data management is not in place. Ironically the rate of ecosystem and habitat degradation, species loss and genetic erosion are occurring at a rate unsurpassed in the past few decades. Is there any crisis in Malaysian fundamental research? The taxonomic and fundamental research community is small and aged. The reference collections in research institutes and local universities are still small, the scientific productivity in term of publication of fundamental science papers to report new species, new records, phylogenetic relationships, variations, species loss and conservation efforts are still inadequate. An attempt is made here to relate the richness of biodiversity to taxonomy so that the latters impediments could be properly addressed. There must be a coordinated efforts to overcome the real taxonomic impediments and also the fate of fundamental research in parallel with applied research. Biology Symposium Keynote Speaker Fundamental Science Congress 2014 7 11. AMPHIBIAN BIODIVERSITY IN PENINSULA MALAYSIA : A PRECIOUS NATURAL HERITAGE Ibrahim Jaafar, Shahrul Anuar Md. Sah and Shahriza Shahrudin Sch. Of Distance Education Universiti Sains Malaysia 11800, Penang. email: jibrahim@usm.my Abstract About 107 species of amphibians are found in Pen. Malaysia. Despite this huge diversity of amphibian species here, not much research and studies have been carried on the biological and ecological aspects of these animals. They form an integral part of our Malaysian tropical habitats and play an important role in the ecological processes of the ecosystem. Only recently have amphibians been protected by Malaysian law and only 30 species are protected at that. With the current rate of rain forest destruction, 70 % of the Malaysian amphibian fauna will be endangered in the near future. The devastation of our lowland rain forests will place most of our amphibians under the threat of extinction, while the destruction of our montane forests will threaten some 27 montane species. Keywords: Amphibians, Diversity, Conservation Introduction Malaysia has a warm and wet equatorial climate and this augurs well for the survival and development of various species of amphibians. Berry (1975) recorded 86 species of frogs and toads in Peninsular Malaysia, Inger and Stuebing (1989) recorded 150 species in Sabah while Kiew (1984) listed about 160 species for the whole of Malaysia and recently Chan et al (2010) listed 107 species of amphibians in Pen. Malaysia. Despite this huge diversity of amphibian species here, not much research and studies have been carried on the biological and ecological aspects of these animals. This is mainly due to the lack of interest and support from the various authorities, minimal understanding of their ecological importance and last but not least, the feeling that frogs and toads are not attractive enough for high profile research. Malaysian Amphibian Diversity Except for the marine ecosystem, frogs and toads are found in all types of habitats and environment in Malaysia. Even so, one species (Rana cancrivora, Mangrove Frog) can tolerate brackish waters and can be found near the shore and in mangrove swamps. The most common species like the Grass Frog (Rana limnocharis), Pond Frog (R. erythrae), Common Biology Symposium Invited Speaker Fundamental Science Congress 2014 8 12. Puddle frog (Occidozyga laevis), Answering Froglet (Microhyla heymonsi), Noisy Froglet (Mi. butleri ) and Banded Bullfrog (Kaloula pulchra) are normally found in disturbed and cultivated areas, whereas the Common Toad (Bufo melanostictus) and Common Treefrog (Polypedates leucomystax ) are known as comensal of man. Less common species also found in disturbed habitats include Red-eyed Ground Toad (Leptobrachium hendricksonii), Granulated Puddle frog (Occidozyga lima), Siamese Frog (Rana doriae) and Torrent Frog (Staurois larutensis). Amphibians found in less disturbed areas include the Black-eyed Ground Toad (Leptobrachium nigrops), Malayan Giant Toad (Bufo asper), Straight-ridged Toad (B. parvus), Common Tree Toad (Pedostibes hosei), Malayan Giant Frog (Rana blythi), Copper- cheeked Frog (R. chalconota), Coarse Frog (R. paramacrodon) and Glandular Frog (R. glandulosa) while Poisonous Green Frog (Rana hosei), Mahagony Frog (R. luctuosa), Yellow-spotted Frog (R signata), Rhinoceres Frog (R. plicatella) and some others are found in least disturbed habitats and clean water rivers. Most others are only found in forested and undisturbed areas, and treefrogs from the genus Philautus, Ploypedates and Rhacophorus tend to inhabit forest trees and canopies, while montane species include the Montane Ground Toad (Leptobrachium montanum), Long-legged Horned Toad (Megophyrs longipes), Golden Tree Froglet (Philautis aurifasciatus), Hill Treefrog (Ph. larutensis), Twin-spotted Treefrog (Rhacophorus bipunctatus), Blue-webbed Treefrog (Rh. reinwardti) and White-lipped Froglet (Microhyla annectens), among others. Inventory studies in a few habitats have yielded some basic data on the species list and diversity of amphibians in the study areas. These include 24 species of amphibians in Ulu Endau (Kiew, 1987), 19 species in Tasek Bera (Norsham et al., 2000), 24 species in Temenggor (Kiew et al., 1995), 20 species in Pantai Acheh Forest Reserve, Penang (Ibrahim et al., 2003), 11 species in Sedilu Peat Swamp Forest, Sarawak (Ramlah, 2002), 15 species in Langkawi (Ibrahim et al., (2006) and 17 species in Pondok Tanjung Forest (Assalam, 2000) among others. Again this goes to show that some habitats and areas in Malaysia do harbour a wide variety and diversity of amphibian populations. Conservation Currently there are 160 species of amphibians from two orders and seven families found within Malaysian shores (Kiew, 1984). Almost all our amphibians are classified as indeterminate by Kiew (1984) because information is lacking for proper conservation status. He proposed that most of the frogs and toads be classified as vulnerable and we tend to agree with his point considering that we have no basic information and knowledge on these species. Only recently have amphibians been protected by Malaysian law (Conservation of Wildlife Act, 2010) and only 30 species are protected at that. Kiew (1984) mentioned that with the current rate of rain forest destruction, 70 % of the Malaysian amphibian fauna will be endangered in the near future and the devastation of all lowland rain forests will place most of our amphibians under the threat of extinction, while the destruction of our montane forests will threaten some 27 montane species. We hope the authorities will do something to remedy the situation and do it fast, before its too late. Biology Symposium Invited Speaker Fundamental Science Congress 2014 9 13. Acknowledgements The authors wish to extent their gratitude to Universiti Sains Malaysia for facilities provided. This paper was made possible through a Universiti Sains Malaysia research university grant No: 1001/PJJAUH/815030 to the first author. References 1. Assalam, M.A.A. (2000). Comparative Study on the Distribution of frog Populations in Four Disturbed and undisturbed Habitats. Unpublished Masters Thesis, USM. 2. Berry, P.Y. (1975). The Amphibian Fauna of Peninsula Malaysia. Kuala Lumpur :Tropical Press. 3. Chan, K.O., Belabut, D. and Norhayati, A. (2010). A revised checklist of the Amphibians of Pen. Malaysia. Russian Journal of Herpetology 17, 202-206. 4. Conservation of Wildlife Act (2010). Act 716. Malaysian Government. 5. Ibrahim, J., Shahrul, A.M.S. and Roswadi, Y. (2003). The Herpetofauna of Pantai Acheh Forest Reserve. In Chan, L.K. ed. Pantai Acheh Forest Reserve: The Case for a State Park. Penerbit USM : 137-144. 6. Ibrahim, J., Shahrul, A.M.S., Norhayati, A., Shukor, M.N., Shahriza, S., Nurul, E., Norzalipah, M. and MarkRayan, D. (2006). An annotated checklist of the herpetofauna of Langkawi island, Kedah. Malayan Nature Journal 57, 369-381. 7. Inger, R.F. and Stuebing, R.B. (1989). Frogs of Sabah. Kota Kinabalu : Sabah Parks Trustees. 8. Kiew, B.H. (1984). Conservation Status of Malaysian Amphibians. Malaysian Naturalist 37, 6-10. 9. Kiew, B.H. (1987). An annotated checklist of the herpetofauna of Ulu endau, Johore, Malaysia. Malayan Nature Journal 41, 413-423. 10. Kiew, B.H., Diong, C.H. and Lim, B.L. (1995). An annotated checklist of the amphibian fauna in the Temmenggor Forest Reserve, Malaysia. Malayan Nature Journal 48, 347- 351. 11. Norsham, Y., Lopez, A., Prentice, R.C. and Lim, B.L. (2000). A survey of the herpetofauna in the Tasek Bera Ramsar Site. Malayan Nature Journal 54(1), 43-56. 12. Ramlah, Z. (2002). Frog Diversity at Sedilu Peat Swamp Forest Reserve, Sarawak. Malayan Nature Journal 2002 56, 217 223. Biology Symposium Invited Speaker Fundamental Science Congress 2014 10 14. BIOACTIVE COMPOUND DETERMINATION AND CURATIVE EFFECTS OF AQUEOUS STEM BARK EXTRACT OF BOSWELLIA PAPYRIFERA (DEL) IN CARBON TETRACHLORIDE INDUCED LIVER DAMAGE IN RATS Abdulmumin Y1 , Matazu K.I2 , Wudil A.M3 , Alhassan A.J4 and Imam A.A5 1,2 Department of Science Laboratory Technology College of Science and Technology Hussaini Adamu Federal Polytechnic, Kazaure Jigawa State, Nigeria 3,4,5 Department of Biochemistry, Faculty of Science Bayero University, Kano P.M.B 3011, Kano, Nigeria email: yabdulmumin@yahoo.com ABSTRACT Phytochemical analysis of Boswelliapapryfera confirms the presence of alkaloids, flavonoids, tannins, saponins and cardiac glycosides in its aqueous stem bark extracts at different concentration with tannins being the highest (0.6110.002g%). The effects of Boswellia papyrifera aqueous stem bark extract was assessed in the cure of carbon tetrachloride (CCl4) induced liver damage in rats. Three different doses of the extract were administered (50mg/kg, 100mg/kg and 150mg/kg) daily to different groups of rats for two and four weeks periods after inducing with liver damage using CCl4 at a doses of 120mg/kg. The Serum Aspatateaminotransperase (AST) and serum alanine aminotransperase (ALT), serum Alkaline phosphatase (ALP), serum Albumin, Serum total and Direct Bilurubin serum and Total Protein were found to be no different from that of the normal control group and histopathology examination of the liver after four weeks showed normal liver architecture with distinct hepatocytes radiating from the venules. This shows the possible hepatocurative effect of aqueous stem bark extract of Boswellia papyrifera oral administration, which was found to be dose and time dependant. Keywords: Boswelliapapyrifera, aqueous extracts, Phytochemicals, carbon tetrachloride and liver enzymes Introduction Boswellia papyrifera (Del.) (Ararrabi) Hochst belongs to a tropical family of plant called Bruceraceae (Ergasa, 2007) which is distinguished by the presence of resin ducts in the bark (Grant, 1987). It is a deciduous tree which grow as tall as 12 m, with a rounded crown and a straight regular bole. (Ergasa, 2007). Liver disease is any condition that causes liver Biology Symposium Fundamental Science Congress 2014 11 15. inflammation or tissue damage. Loss of liver functions can cause significant effects to body systems. (Benjamin, 2011).And high dose of CCl4 generates an ideal hepatotoxicity model organism that allows for evaluating the curative effects of medicinal plant rather than reporting natural healing (Alhassan et al, 2009). This paper is aimed at Determining the effect of aqueous stem bark extract of Boswellia papyrifera on liver function indices (serum AST,ALT, ALP, Albumin, Direct and total bilurubin, and total protein) on carbon tetrachloride(CCl4) induced liver damaged rats. Methods Serum Aspatate aminotransferase (AST), alanine aminotransferase (ALT) and serum Alkaline phosphatase (ALP) activities were determined by the method of (Reitman and Frankel, 1957), serum Albumin level by Grant (Grant, 1987)), Serum total and Direct Bilurubin levels by the method of Jendrassik and Grof (Jendrassik and Grof, 1938), serum Total Protein level by buiret method (Tietz, 1995). Histological Examination (Auwiro, 2010). The data was statistically analysed at P-value (p . The observations that have nhi 3> are categorized as suspected group while the others are categorized into a remaining group. Proposed influential detection methods as follow: i i i h s = 1 0)( , (3) where 0 = from the remaining group, and = group.suspectedfromif,groupremainingintoofaddition groupremainingfromif,groupremainingfromofdeletion )( ix ix i i i The large value of is means high influence of i-subject of p-covariate on estimation of p . Results and Discussion FIGURE 1. Influential diagnostics by the proposed method FIGURE 2. Influential diagnostics by score residual Mathematics Symposium Fundamental Science Congress 2014 274 278. FIGURE 3. Influential diagnostics by scaled score residual FIGURE 4. Influential diagnostics by lmax statistics From Fig. 1, we can see that our proposed method was detecting 3 influential observations from the simulation study. We compared the detected influential observations with score residual (Fig. 2), scaled score residual (Fig. 3), and lmax statistics (Fig. 4). From these 4 graphs, all the methods can identified the same influential observations and the 3 points detected by our proposed method were plotted with different symbol in the graphs of the other existing methods. Acknowledgement Participation in FSC2014 was funded by Vot no. 9424600. References 1. Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society. Series B (Methodological), 34(2), 187 - 220. 2. Cox, D. R. (1975). Partial Likelihood. Biometrika, 62(2), 269 - 276. 3. Cox, D. R. (1984). Analysis of Survival Data: London: Chapman and Hall. 4. Grambsch, P. M. and Therneau, T. M. (1994). Proportional Hazards Tests and Diagnostics Based on Weighted Residuals. Biometrika, 81, 515 - 526. 5. Pettitt, A. N. and Bin Daud, I. (1989). Case-weighted Measures of Influence for Proportional Hazards Regression. Appl. Statist, 38, 51 - 67. 6. Therneau, T. M., Grambsch, P. M., and Fleming, T. R. (1990). Martingale-Based Residuals for Survival Models. Biometrika, 77(1), 147 - 160. Mathematics Symposium Fundamental Science Congress 2014 275 279. ON FINDING THE SMALLEST REPLICATION NUMBER OF BOOTSTRAP CONFIDENCE INTERVAL USING COMPUTATIONAL SIMULATION Zaturrawiah Ali Omar1 , Noraini Abdullah2 and Zainodin Jubok3 1 Mathematics with Computer Graphics Programme 2,3 Mathematics with Economics Programme Mathematics and Statistical Applications Research Group Faculty of Science and Natural Resources Universiti Malaysia Sabah email: zatur@ums.edu.my Abstract A preliminary experiments were conducted to investigate if a smaller number of bootstrap replication B for confidence interval was able to be identified through computational simulation. Interest was particularly on small data sets and in this simulation, a data on Cadmium concentration were used with 28 observations. A stopping criteria were identified and it was still an ongoing work to see whether this experiment is conclusive. The standard errors when B = 1000 and B = 2000 were used as the benchmark. The results show that the stopping criteria were satisfied on specified cases of comparison up to d decimal point. As d increased, so did the number of B. Keywords: Small data set, Bootstrap replications, Confidence interval, Cadmium Introduction In this study, we were interested to look into the question of how many bootstrap replication B for the statistic of interest * was required for conducting 90 95 percent confidence intervals. As suggested by Carpenter et al. (2000) that B should be between 1000 and 2000. This is to ensure inadequate bootstrap sampling as explained in (Efron, 1987). Yet, getting a B as small as possible is still preferable. We, therefore dive to this question and try to investigate if B can be less than the range suggested through computational simulation. The data used were based on a case study of Cadmium (Cd) concentration. Other researches in answering the same question, though for different cases, were also done by Fisher & Hall (1991), Blair (1992), Pattengale et al. (2010) and Wang et al. (2013). Mathematics Symposium Fundamental Science Congress 2014 276 280. The investigation required a stopping criteria for identifying the smallest B and here we used the formula looking at bootstrap standard deviation = ! ! where S is the sample standard deviation and n is the sample size (Efron & Tibshirani, 1986). Since the true is not known, then S was used as an estimate. Yet S might not be a good estimator because the of small sample size (< 30 based on most textbooks). We therefore conducted the normality and randomness tests using shapiro.test() (Shapiro-Wilk normality test) and runs.test()functions in R version 3.0.1. The comparison on was on its rounded value in d number of digits. Here d = {2, 3, 4, 5, 6}. The range of B was from 30 to 5000. The boot() function then would iterated with the interval of 1, starting from B = 30 until the stop criteria was satisfied or, until B = 5000. When the stop criteria were satisfied, the B value of that iteration was saved. This was done repeatedly until 10 000 times for different cases of d. From the distribution of the saved B, the mean and median will be considered as the values of interest. Once the value of B was identified, another bootstrap was conducted based on the values and the * , and ! were saved. Since replication would only reduce the standard error (Wang et al. 2013), the ratio difference of the ! was compared with the ! when B = 1000 and B = 2000. Results and Discussion 28 observations were gathered with the following statistical information as shown in Table 1. TABLE 1. Descriptive Statistic of Cd n Mean Standard Deviation () Standard Error 28 0.25 0.12 0.02 Shapiro-Wilk normality test was conducted giving p-value = 0.573 (> = 0.05 ), thus accepting the null hypothesis that the sample follows a normal distribution. A run test was also conducted using function runs.test()from the randtests package in R. The results was p-value = 0.4411 (> = 0.05 ), therefore accepting the null hypothesis that the sample data was random. Based on both tests, we can conclude that the sample can be a good representative of the population. Results from conducting bootstrap when B = 1000 and B = 2000 then calculated as a benchmark for comparison with the simulation results. The values for its statistics of interest is shown in Table 2. TABLE 2. Results when B = 1000 and B = 2000 B = 1000 B = 2000 Mean * 0.2541025 0.2529234 Standard Deviation * 0.0222813 0.0220851 Standard Error ! 0.0007046 0.0004938 Mathematics Symposium Fundamental Science Congress 2014 277 Method 281. Results from the simulation is summarized in Table 3. Based on this preliminary experiment, as the number of the decimal point increased, so did the number of B. Interestingly, when d = 6, the mean and median of B had exceeded 1000, yet still below 2000. TABLE 3. Results of Simulation Decimal Point B Mean Standard Deviation Standard Error ! ! !!!""" / ! !!!""" ! ! !!!"""/ ! !!!""" d = 2 Mean 30 0.2554048 0.0246433 0.0044992 5.39 8.11 Median 30 0.2554048 0.0246433 0.0044992 5.39 8.11 d = 3 Mean 36 0.2522024 0.0192158 0.0032026 3.55 5.49 Median 34 0.2488550 0.0201217 0.0034508 3.90 5.99 d = 4 Mean 81 0.2481966 0.0229305 0.0025478 2.62 4.16 Median 71 0.2522284 0.0218194 0.0025895 2.68 4.24 d = 5 Mean 327 0.2531684 0.0229904 0.0012714 0.80 1.57 Median 276 0.2512164 0.0225958 0.0013601 0.93 1.75 d = 6 Mean 1581 0.2535610 0.0226246 0.0005690 -0.19 0.15 Median 1313 0.2528784 0.0216928 0.0005987 -0.15 0.21 References 1. Blair Hedges, S. (1992). The Number of Replications Needed for Accurate Estimation of The Bootstrap p Value in Phylogenetic Studies. Menopause (New York, N.Y.), 9(2), 366 369. 2. Efron, B. (1987). Better Bootstrap Confidence Intervals. Journal of the American Statistical Association, 82(397), 171185. 3. Efron, B. and Tibshirani., R. (1986). Bootstrap Methods for Standard Errors, Confidence Intervals an Other Measures of Statistical Accuracy. Statistical Science. 1(1), 54 77. 4. Carpenter, J. and Bithell, J. (2000). Bootstrap confidence intervals: when , which , what? A practical guide for medical statisticians. Statistics in Medicine, (August 1999), 1141 1164. 5. Fisher, N. I. and Hall, P. (1991). Bootstrap algorithms for small samples. Journal of Statistical Planning and Inference, 27(2), 157169. 6. Pattengale, N. D., Alipour, M., Bininda-Emonds, O. R. P., Moret, B. M. E. and Stamatakis, A. (2010). How many bootstrap replicates are necessary? Journal of Computational Biology: A Journal of Computational Molecular Cell Biology, 17(3), 33754. 7. Wang, Y., Sohn, M. D., Gadgil, A. J., Wang, Y., Lask, K. M. and Kirchstetter, T. W. (2013). How many replicate tests do I need? Variability of cookstove performance and emissions has implications for obtaining useful results. Lawrence Berkeley National Laboratory. Mathematics Symposium Fundamental Science Congress 2014 278 282. APPROXIMATE SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS BY REPRODUCING KERNEL HLBERT SPACE METHOD Ali Akgl Department of Mathematics, Education Faculty Dicle University TURKEY email: aliakgul00727@gmail.com Abstract In this paper, we use the reproducing kernel Hilbert space method (RKHSM) for finding approximate solutions of delay differential equations. Interpolation for delay differential equations have not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency. Keywords: Reproducing kernel method, Series solutions, Delay differential equations, Reproducing kernel space Introduction In this paper we consider delay differential equations in the reproducing kernel Hilbert space ].1,0[3 2W ,)1(,)0( ,10),())(( )( 1 ))((' )( 1 ))(('' )( 1 BuAu xxgxmu xq xhu xp xtu xs == 0.05 and the VIF were 0 is the steplength mostly obtained by an inexact line search, ! is the search direction define by ! = ! = 0 !!! = !!! + ! !, 1 (3) where ! is the gradient of at a point ! and ! is a scalar known as the conjugate gradient coefficient. Different selection of ! would lead to a different conjugate gradient methods (Hager and Zhang, 2006). Methodology We proposed a three-term conjugate gradient method by considering Davidon-Fletcher- Powell (DFP) of quasi Newton update (Fletcher, 2013) where at every iteration the inverse Hessian approximation is restarted with a multiple of identity matrix. The search direction is given by Mathematics Symposium Fundamental Science Congress 2014 302 306. !!! = ! ! !! ! !!!! !! !!! ! + !! ! !!!! !! !!! ! (4) where ! = !!! !, ! = !!! ! and ! = !! ! !! !! !!! !! !!! !! !!! ! !! !!! !! !!! , (5) is a positive scalar. In order to compare the performance of our proposed method we considered the problems used in Yuan and Wei (2009) in the Tables below. TABLE 1. Data for demand and price Price !($) 1 2 2 2.3 2.5 2.6 2.8 3 3.3 3.5 Demand !(500) 5 3.5 3 2.7 2.4 2.5 2 1.5 1.2 1.2 TABLE 2. Data for age and average height ! 2 3 4 5 6 7 8 9 10 11 ! 5.6 8 10.4 12.8 15.3 17.8 19.9 21.4 22.4 23.2 Observe that there exist a linear relationship between year demand of commodity and price. The relationship between the average height of the pine tree and it age is parabolic in nature. Both problems 1&2 can be solve using the least square method of regression analysis and its corresponding unconstrained optimization problem are given by min = ! 1, ! ! ! !!! andmin () = ! 1, ! ! ! !! !!! In order to solve the given problems with our proposed method we used the following algorithm. Algorithm Step 1. Select an initial starting point ! and determine !. Set ! = ! and k = 0 Step 2. Test the stopping criterion | ! | 10!!, if satisfied stop. Else go to step 3 Step 3. Determine the steplength ! by the following procedure: Given 0,1 and , !! with 0 < ! < ! < 1 i) Set = 1. ii) Test the following relation ! + ! ! ! ! (6) iii) If (6) is satisfied ! = and go to Step 4 else choose a new ! , ! and go to (ii) Step 4. Compute = ! + ! !, ! = and ! = ! ! Step 5. Compute! = ! ! ! !,! = ! ! ! ! ! = ! !!!! . Step 6. Update !!! = ! + ! ! !, ! 0 ! + ! !, Mathematics Symposium Fundamental Science Congress 2014 303 307. Step 7. Determine the search direction d!!! by (4) where! is computed by (5) Step 8. Set : = + 1 and go to Step 2. Different initial points were used so as to observe the robustness and efficiency of each method in estimating the regression parameters. The results are presented in the tables below. Results and Discussion TABLE 3. Numerical result for problem 1 = 6.5, 1.6 NA1 STCG Initial point ! ! ! (6.438301, -1.575289) (6.438285, -1.575314) 0.009929 0.009929 ! (6.438280, -1.575313) (6.438285, -1.575314) 0.009930 0.009929 ! (6.438285, -1.575314) (6.438285, -1.575314) 0.009929 0.009929 ! (6.438287, -1.575316) (6.438285, -1.575314) 0.009929 0.009929 TABLE 4. Numerical result for problem 2 = 1.33, 3.46, 0.11 NA1 STCG Initial point! ! ! (-1.296574, 3.450247, -0.107896) (-1.331363, 3.461742, -0.108712) 0.009407 0.000690 ! (-1.328742, 3.460876, -0.108650) (-1.331308, 3.461724, -0.108710) 0.000551 0.000682 ! (-1.328504, 3.460798, -0.108646) (-1.331363, 3.461742, -0.108712) 0.000585 0.000690 ! (-1.321726, 3.458558, -0.108483) (-1.331363, 3.461742, -0.108712) 0.002301 0.000690 Where ! = 1, 0.01 , ! = 1, 0.01 , ! = 1, 0.01 ,! = 1, 0.01 , ! = 1.1,3.0, 0.5 , ! = 1.2,3.2, 0.3 , ! = 0.003,7.0, 0.5 , ! = 0.001,7.0, 0.5 , are the initial points. is the approximate solution , ! is the approximated solution as the program terminate and is the relative error between and !. Table 3-4 gives the numerical results in comparison of the proposed method (STCG) with New line search method (NA1). From the presented results its worth saying that the error generated by each method,shows that the performance of our proposed method is encouraging. For future research, comparisons are going to be make through simulation study. References 1. Fletcher, R. (2013). Practical methods of optimization. John Wiley & Sons. 2. Hager, W. W. and Zhang, H. (2006). A survey of nonlinear conjugate gradient methods. Pacific journal of Optimization, 2(1), 35-58. 3. Yuan, G. and Wei, Z. (2009). New line search methods for unconstrained optimization. Journal of the Korean Statistical Society, 38(1), 29-39. Mathematics Symposium Fundamental Science Congress 2014 304 308. IMPROVED HOMOTOPY PERTURBATION METHOD FOR SOLVING FREDHOLM-VOLTERRA INTEGRO-DIFFERENTIAL EQUATION Fatimah Samihah Zulkarnain1 , Zainindin Eshkuvatov2 , Nik Mohd Asri Nik Long3 and Zahridin Muminov4 1,2,3,4 Department of Mathematics, Faculty of Science Universiti Putra Malaysia 2,3,4 Institute for Mathematical Research Universiti Putra Malaysia email: fsamihah88@gmail.com Abstract Improved homotopy perturbation method (HPM) has applied to solve the Fredholm- Volterro integro-differential equation of the third kindwith the derivative of order m. A few selective functions must be chosen and parameters has been added into the improvement of HPM. Only one iteration needed to obtain the solutions. Some examples are given to present the efficiency of the method. Keywords: Integral equation, Homotopy perturbation method, Numerical method Introduction Homotopy perturbation method (HPM) is the combination of two methods: the homotopy and perturbation method. In recent years, the application of HPM in mathematical problems has been applied by many researchers. This method deforms complicated problem to a simple problem which is easy to solve (He, 1999, 2000).Many mathematicians have made a few modifications on HPM by adding a few parameters that leads to approximate or exact solutions (Javidi, 2009). Our problem is to implement improved HPM for solving Fredholm- Volterra integro-differential equation of the third kindwith initial conditions ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ,,, 2211 dttutxKdttutxKxfxuxs x a b a m ++= (1) ( ) ( ) ( ) ( ) ( ) 1 1 1 1 0 ,,, === m m daudaudau , where ( ) ( )xfxs , are continuous function on [ ]ba, ,m is the order of differential, 1 and 2 are parameters, 1K and 2K are square integrable kernels and ( )xu is the function to be determined. We have the corresponding operators as follows ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )dttutxKxuKdttutxKxuKxuxsxLu b a b a m === ,,,, 222111 Mathematics Symposium Fundamental Science Congress 2014 305 309. Method Improved HPM has been presented by Ghorbani (2006). In this method the rateof convergence is accelerated, and at the same time it is possible find exact and approximate solutions. We construct the improved HPM as ( ) ( ) ( ) ( ),1,, 21 0 vKvKfLvpxgLvppvH N k kkm + = = (2) where k are called accelerating parameters and ( )xgk are the selective functions. The solution of the ( ) 0,, =pvHm is searched in the form of a power series ( ) ( ) = = 0 . n k k xvpxv (3) We applied series (3) into Eq. (2) with ( ) 0,, =pvHm , we obtain ( ) ( ) ( ) ( ) ( ) ... 00 2 0 1 00 +++= = = == = N k kk n k k n k k N k kk n k k xgxvpKxvpKfpxgxvpL Comparing the expressions with the same power of parameter p, we obtain ( ) ( ) .,3,2, , , 1211 0 02011 0 0 =+= ++= = = = nvKvKLv xgvKvKfLv xgLv nnn N k kk N k kk (4) Results and Discussion Example 1. Let us consider Fredholm-Volterra integro-differential equation as ( ) ( ) ( ) ( ) ( )dttutxtxdttutxxxx x x xu x x +++ + = + 0 22 1 0 46 1 6 1 20 3 4 5 2 2 24 2 4 with initial condition, ( ) 10 =u . It is easy to check that the exact solution is ( ) 2 31 xxu += . By using improved HPM (?) and the choose of selective functions ( ) 2,1,0, == kxxg k k , we obtain Mathematics Symposium Fundamental Science Congress 2014 306 310. v0 =0 +1 x +2 x2 , v1 = 1+ 2+ 1 2 0 + 1 4 1 + 1 6 2 " # $ % & 'x + 49 16 1 48 1 1 48 2 " # $ % & 'x2 + 5 48 1 24 0 1 48 1 1 36 2 " # $ % & 'x3 + 1 60 + 1 60 0 " # $ % & 'x5 + 1 144 + 1 144 0 + 1 144 1 " # $ % & 'x6 + 3 280 + 1 336 1 + 1 280 2 " # $ % & 'x7 + 3 640 + 1 640 2 " # $ % & 'x8 To find ( )210 ,, = we force that 01 =v it leads 032 ==== nvvv .The approximate solution is obtained as ( ) ( )xvxv 0= .The system of algebraic equations with respect to is .0 280 1 336 1 280 3 ,0 36 1 48 1 24 1 48 5 ,0 144 1 144 1 144 1 ,0 48 1 48 1 16 49 ,0 640 1 640 3 ,0 60 1 60 1 ,0 6 1 4 1 2 1 2 21210 1021 20210 =++= =++= =+=+=+++ (5) The solution of the Eq. (5) is 3,0,1 210 === Thus, we have solution ( ) 22 210 31 xxxxv +=++= which is identical with exact solution. From the example, we can see the accelerating parameter leads to the exact solution with only one iteration. Acknowledgement Participation in FSC2014 was funded by Kementerian Pelajaran Malaysia MyPHD. References 1. He, J.H. (1999). Homotopy perturbation technique, Comp. Meth. Appl. Mech. Engrg.,178, 257262. 2. He J.H. (2000). A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Internat. J. Non-linear Mech.,35, 3743. 3. Ghorbani, A., Saberi-Nadjafi, J. (2006). Exact solutions for nonlinear integral equations by a modified homotopyperturbation method, Comp. and Math. with Appl.,28, 1032-1039. 4. Javidi,M., Golbabai, A. (2009). Modified homotopy perturbation method for solving non- linear Fredholm integral equations, Chaos, Solitions and Fractals,40, 1408-1412. Mathematics Symposium Fundamental Science Congress 2014 307 311. TWO-STEP IMPROVED RUNGE-KUTTA NYSTROM METHOD FOR SOLVING SECOND-ORDER ODES Nafsiah Md Lazim1 , Faranak Rabiei2 , Fudziah Ismail3 and Norazak Senu4 1,2,3,4 Department of Mathematics, Faculty of Science Universiti Putra Malaysia 2,3,4 Institute for Mathematical Research Universiti Putra Malaysia email: faranak_rabiei@upm.edu.my Abstract Improved Runge-Kutta Nystrom (IRKN) methods are derived for solving a special class of second-order ordinary differential equation of the form " = , . The derivation of Improved Runge-Kutta Nystrom methods of the third, fourth and fifth order with 2, 3 and 4 stages, respectively is given. Numerical examples are given to show the efficiency of the methods compared with the existing Runge-Kutta Nystrom methods. Keywords: Improved Runge-Kutta Nysrom method, Second-order ordinary differential equations, Algebraic order conditions Introduction The special second-order ordinary differential equations are given by " = (, ), (!) = !, (!) = ! , : ! ! . (1) Rabiei et al. (2012) constructed Improved Runge-Kutta methods for solving ordinary differential equations. The general form of explicit Improved Runge-Kutta Nystrom method with s- stages are given by: !!! = ! + 3 2 ! ! 2 ! !!! + ! ! ! !!! ! !! , !!! = ! + (! ! !! !! + !(! !!)) ! !!! , ! = !, ! , !! = !!!, !!! , (2) ! = (! + !, ! + ! ! + ! !" !) !!! !!! , for = 2, , . !! = (!!! + !, !!! + ! !!! + ! !" !!), for = 2, , . !!! !!! Mathematics Symposium Fundamental Science Congress 2014 308 312. The order condition for IRKN method is derived by Rabiei et al. (2012) and additional Nystrom row sum condition is defined by ! ! ! ! = !", = 1, , .!!! !!! Method Consider the third order IRKN method with two-stages from formulas (2), the method is called IRKN3. To derive IRKN method, we need to satisfy the order conditions up to order three for ! and ! ! . Let ! = ! ! be free parameter, then we can obtain the value of other coefficients. Fourth order method with three stages (IRKN4) is derived by substituting ! = ! ! , ! = ! ! as free parameters. Fifth order method with four stages (IRKN5) is derived by substituting ! = ! ! , ! = ! ! , ! = ! ! as free parameters also we choose !" = ! ! , !" = 0. The obtained coefficients of IRKN3, IRKN4 and IRKN5 are given in Tables 1-3, respectively. TABLE 1. Table of coefficients IRKN method for order three with two stages ! = 2 3 !" = 2 9 !! = 1 8 ! = 7 8 ! = 5 8 ! = 5 12 TABLE 2. Table of coefficients IRKN method for order four with three stages ! = 1 3 !" = 1 18 ! = 2 3 !" = 0 !" = 2 9 !! = 1 8 ! = 9 8 ! = 1 2 ! = 7 8 ! = 1 3 ! = 1 12 TABLE 3.Table of coefficients IRKN method for order five with four stages ! = 1 4 !" = 1 18 ! = 2 4 !" = 0 !" = 2 9 ! = 3 4 !" = 1 9 !" = 0 !" = 49 288 !! = 1 45 ! = 46 45 ! = 1 15 ! = 1 10 ! = 29 45 ! = 49 180 ! = 7 180 ! = 19 180 Mathematics Symposium Fundamental Science Congress 2014 309 313. Results and Discussion Consider the tested Problem (see Lambert and Watson, 1976) !" = ! + 0.001 cos , ! 0 = 0.001, ! (0) = 0 !" = ! + 0.001 sin , ! 0 = 0, ! 0 = 0.001, Exact solution: y! = cos + 0.0005 sin , ! = sin 0.0005 cos . The numerical results of derived IRKN methods up to order 5 compared with existing Runge- Kutta Nystrom method of order four with four stages given Van derHouwen and Sommeijer (1987) are shown in Table 4. TABLE 4. Maximum global error for tested problem h IRKN3 IRKN4 IRKN5 RKN4 0.1 4.0810!! 3.1210!! 2.5710!! 6.4810!! 0.05 2.5310!! 1.9510!! 1.6010!! 2.8210!! 0.001 2.0010!! 3.1210!! 2.5710!!" 5.1510!! From Table 4, it is obvious that IRKN3 with only two stages requires lees number of function evaluation while existing RKN method starts with three stages. IRKN4 with three stages and IRKN5 with four-stages give more accurate results than RKN4 with four stages. Therefore, it can be concluded that IRKN method using less number of stages are more efficient than RKN method. Acknowledgement Participation in FSC2014 was funded by Vot no. FRGS 5524421. References 1. Rabiei, F., F. Ismail, S. Norazak, N. Abasi. (2012). Construction of Improved Runge-Kutta methods for solving ordinary differential equations, World Applied Sciences Journal 20(12), 1685-1695. 2. Lambert, J. D. and Watson, I. A. (1976). Symmetric multistep methods for periodic initial value problems. J. lnst. Mathematics Application 18, 189-202. 3. Van der Houwen, P.J. and B.P. Sommeijer, (1987). Explicit Runge-Kutta Nystrom methods with reduced phase errors for computing oscillating solutions. SIAM Journal Number Analysis, 24, 595-61. Mathematics Symposium Fundamental Science Congress 2014 310 314. SUBSETS OF PAIRWISE PARALINDELF BITOPOLOGICAL SPACES Hend Bouseliana1 and Adem Kiliman2 1,2 Department of Mathematics, Faculty of Science Universiti Putra Malaysia 2 Institute for Mathematical Research Universiti Putra Malaysia email: bouseliana2007@yahoo.com Abstract The goal of this work is to introduce some types of subsets of paralindelf bitopological spaces. Namely, (i,j)-paralindelf relative to X and iparalindelf-j relative to X where X is a bitopological space. Furthermore, their characterizations and properties have been studied. Also, we present the notion of (i,j)-regular relative to X and investigate some of its properties and relationships. Keywords: Bitopological space, (i,j)-paralindelf relative to X, i-paralindelf-j relative to X, (i,j)-regular relative to X Introduction In 1992, P. Daniel has introduced the idea of paralindelf set. A subset S of a topological space (X,) is said to be paralindelf set in X if every cover of S by -open subsets of X has -locally countable family of -open sets in X such that S {V: V }. A subset S of (X,) if regular relative to X if for each point s S and any -open subset U containing s, there is a -open subset V in X such that s V -cl(V) U (Kovaevi, 1984). During this paper, we define the ideas of paralindelf set and regular set in bitopological setting, namely, (i,j)- paralindelf relative to X and (i,j)-regular relative to X. The characterizations and the properties of these concepts have been studied and investigated. Method The ideas of paralindelf relative to bitopological space X defined here came from the modification and generalizations of paracompact sets and extension from paralindelf relative to single topology to bitopological settings. Mathematics Symposium Fundamental Science Congress 2014 311 315. Results and Discussion Theorem1: Let X be pairwise Hausdorff (j,i)-P-space and a subset A be (i,j)-paralindelf relative to X. Then, for any point x in X-A, there are disjoint i-open and j-open neighborhoods of x and A respectively. Consequently, A is i-closed set in X. Theorem 2: Let A be (i,j)-paralindelf and (i,j)-regular relative to j-P-space X. If U is i- open neighborhood of A, then there exists i-open set V containing A such that A V j- cl(V) U. Theorem 3: Let X be (j,i)-weakly P-space and A be a subset of X. If A is i-regular relative to X and i- paralindelf -j relative to X, Then, i-cl(A) is i- paralindelf -j relative to X. Theorem 4: In j-P-space X, if A is any (i,j)-regular relative to X and (i,j)- paralindelf relative to X, then every i-open cover of A has j-closed j-locally countable refinement. References 1. Kelly, J. C. (1963). Bitopological Spaces, Pro. London Math. Soc., 13(3), 71-89. 2. Kovaevi, I. (1984). Subsets and Paracompactenss, Review of research, faculty of science, University of Novi Sad, Mathematics Series, 14, 2. 3. Daniel Thanapalan, P. T. (1992). On Mappings of (Locally) Almost and (Locally) nearly paralindelof Spaces, Indian J. Pure Appl. Math, 23(12), 867-872. Mathematics Symposium Fundamental Science Congress 2014 312 316. FUZZY SOFT SEPARATION AXIOMS Azadeh Zahedi Khameneh1 and Adem Kilicman1,2 1 Institute for Mathematical Research Universiti Putra Malaysia 2 Department of Mathematics Faculty of Science Universiti Putra Malaysia email: azi_503@yahoo.com Abstract In this study, the notion of fuzzy soft point is introduced. We shall also investigate and compare the concept of fuzzy soft neighborhood and soft quasi neighborhood of a fuzzy soft point in a fuzzy soft topological space. Finally, separation axioms in fuzzy soft topological spaces are introduced. Some analogy theorems in general topology are considered for fuzzy soft spaces. Keywords: Fuzzy soft topology, Fuzzy soft point, Soft quasi-coincidence, Fuzzy soft ! -space Introduction Soft set theory, proposed by Molodtsov (1999), can be seen as a parameterization tool for modeling uncertainties by using a set-valued map: ! , where X is a set of objects and E is a parameter set. In Maji et al. (2001) combined the concept of soft set and fuzzy set to initiate the novel hybrid concept called fuzzy soft set. This new notion has more powerful potential to describe objects about which we have inexact information. The topological studies on fuzzy soft sets were begun by Tanay and Kandemir (2011). But Roy and Samanta (2012) showed the difficulty of Tanay and Kandemir's definition and improved definition of fuzzy soft topology. The main purpose of the present paper is to consider the separation axioms in fuzzy soft topological spaces. To obtain this aim, we first introduce the concept of fuzzy soft point and then consider different neighborhoods, including fuzzy soft neighborhood and soft quasi neighborhood, of a fuzzy soft point in a fuzzy soft topological space. Then we define the notion of separations in fuzzy soft spaces as an extension of the concept of separations in general topology. We also consider if our definition can be seen as a good extension of the crisp case. Mathematics Symposium Fundamental Science Congress 2014 313 317. Results and Discussion Definition1. The F.S-topological space (, , ) is called a i. fuzzy soft !-space if for every two distinct fuzzy soft points in X, there exists a fuzzy soft open set containing one of them and not intersection with the other. ii. fuzzy soft !-space if for every two distinct fuzzy soft points in X there exist two fuzzy soft neighborhood of them not containing the other. iii. fuzzy soft !-space if for every two distinct fuzzy soft points in X there exist two fuzzy soft neighborhood of them not intersection with each other. iv. fuzzy soft !-space if for any fuzzy soft closed set and a fuzzy soft point which is not belong to that set, there exist two fuzzy soft neighborhood of them not intersection with each other. Theorem 1. In the fuzzy soft topological space (, , )the following diagram is valid. Theorem 2. i. If (!, !, !) and (!, !, !)are fuzzy soft !-space, then (, , ) is fuzzy soft !-space. ii. If (!, !, !) and (!, !, !)are fuzzy soft !-space, then (, , ) is fuzzy soft !-space. Acknowledgement This work is partially supported by the Institute for Mathematical Research, Universiti Putra Malaysia, Grant No.5527179. References 1. D. Molodtsov, (1999) Soft set theory-first results, Computers and Mathematics with applications 37:19-31. 2. P. K. Maji, R. Biswas, A. R. Roy, (2001). Fuzzy Soft Set, Journal of Fuzzy Mathematics 9 (3): 589-602. 3. B. Tanay, M. B. Kandemir, (2011). Topological structure of fuzzy soft sets, Computers and Mathematics with Applications 61: 2952-2957. 4. S. Roy, T. K. Samanta, (2012) A note on fuzzy soft topological spaces, Annals of Fuzzy Mathematics and Information, 3(2), 305--311 5. A. Zahedi Khameneh , A. Kilicman and A. R. Salleh, (2014). Fuzzy soft product topology, Annals of Fuzzy Mathematics and Information, 7(6), 935-947. Mathematics Symposium Fundamental Science Congress 2014 314 318. ON SPECTRAL HOMOTOPY ANALYSIS METHOD FOR SOLVING SYSTEM OF FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS Zohreh Pashazadeh Atabakan1 , Adem Kilicman2 and Aliasghar Kazemi Nasab3 1,2,3 Department of Mathematics Faculty of Science Universiti Putra Malaysia email: pashazadeh2002@yahoo.com Abstract Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of second order system of Fredholm integro differential problems. Some examples are given to approve the efficiency and the accuracy of the proposed method. Keywords: Homotopy analysis method, Fredholm integro differential equation, Chebyshev differentiation matrix, Chebyshev pseudo-spectral method Introduction Motsa et al. (2010) suggested the so-called spectral homotopy analysis method (SHAM) using the Chebyshev pseudo-spectral method to solve the ordinary differential equations. The spectral homotopy analysis method is more efficient than the homotopy analysis method as it does not depend on the rule of solution expression and the rule of ergodicity. Pashazadeh et al. (2012-2013) solved different type of integro-differential equations using spectral homotopy analysis method. In this paper, the proposed method is applied to solve the second order system of Fredholm integro-differential equations. Method Consider the second order system of Fredholm integrodifferential equation 1 , , , , , , = ! + ! , , , , ! !! 2 , , , , , , = ! + ! , , , , ! !! (1) Subject to the boundary conditions 1 = 1 = 0, 1 = 1 = 0. In particular, consider that the given problem is wellposed and has unique solution. The algorithm is started with choosing initial approximations v0(x), w0(x) which are obtained by solving the following boundary value problems: Mathematics Symposium Fundamental Science Congress 2014 315 319. , , (2) Where !, ! are linear part of H1 and H2. The Chebyshev pseudo-spectral method is used to solve the system (2). The m-th order deformation equations are obtained as following: ! (!, ! !(!!!, !!!) = !(!!!, !!!),. ! (!, ! !(!!!, !!!) = !(!!!, !!!),. (3) where h is the nonzero converging controlling auxiliary parameter and !(!!!, !!!) = 1 , !!!, ! !!!, "!!!, !!!, ! !!!, "!!! 1 ! ! ! , , !!!, !!! , ! !! !(!!!, !!!) = 2 , !!!, ! !!!, "!!!, !!!, ! !!!, "!!! 1 ! ! ! , , !!!, !!! , ! !! (4) Using the Chebyshev spectral differentiation matrix and Gauss-Lobatto collocation points, we rewritten the Eq.(3) to matrix form and get the following system (5) Finally, by using the initial approximation the recursive system (5) can be solved for m 1. Numerical Example 1. The following nonlinear second order Fredholm integro differential equation is solved by using spectral homotopy analysis method. 4" + + ! + 2 = 4! 2 + 2 +(2)! + 2(!!!! 1) ! !! " + + ! + ! + !!!! cos = 2 ! !! !!!! + 4! !!!! 1 + ! sin (2) + sin ()(!!!! 1) Mathematics Symposium Fundamental Science Congress 2014 316 320. FIGURE 1. The absolute errors of Example 1 for different orders of SHAM approximate solutions for h=-0.9. References 1. S. S. Motsa, P. Sibanda, and S. Shateyi. (2010). A new spectral-homotopy analysis method for solving a nonlinear second order BVP, Communications in Nonlinear Science and Numerical Simulation, 15(9), 2293 2302. 2. Z.P. Atabakan , A .Kilicman, and A. K. Nasab (2012). On spectral homotopy analysis method for solving linear Volterra and Fredholm integro-differential equations. Abstract and Applied Analysis, vol .2 012 ,Art icl eI D96 02 8 9 ,1 6pages. 3. Z. P. Atabakan, A. K. Nasab, A. Kilicman, and K. Z.Eshkuvatov. (2013). Numerical solution of nonlinear Fredholm integro-differentialequations using Spectral Homotopy Analysis method. Mathematical Problems in Engineering,v ol .2 01 3 ,A r tic leID67 43 6 4, 9 pages. 4. Z. P. Atabakan, A. Kazemi Nasab, A. Kilicman, Spectral Homotopy Analysis Method for Solving Nonlinear Volterra Integro Differential Equations Malaysian Journal of Mathematical Sciences, Accepted. Mathematics Symposium Fundamental Science Congress 2014 317 321. SKEW GENERALIZED CAUCHY NORMAL DISTRIBUTION Zahra Nazemi Ashani1 , Mohd Rizam Abu Bakar2 , Noor Akma Ibrahim3 and Mohd Bakri Adam4 1,2,3,4 Department of Mathematics, Faculty of Science Universiti Putra Malaysia 2,3,4 Laboratory of Computational Statistics and Operations Research Institute for Mathematical Research Universiti Putra Malaysia email: mrizam@upm.edu.my Abstract Skew symmetric distribution has been center of attention for statistical researchers during the last two decades because, in different fields, we need skew distributions for analyzing skew data. In this work, we evaluated a new generalization of univariate skew Cauchy distribution. We established some properties of this distribution and finally show !, ! can fit the skew data better than Cauchy and skew Cauchy normal distribution. Keywords: Skew-symmetric distribution, Cauchy distribution Introduction The journey of skew distribution started with papers of Azzalini (1985 & 1986). He introduced skew symmetric distribution with probability density function (pdf) 2 () where is a symmetric density function around 0, is an absolutely continuous distribution function such that ! is symmetric about 0 and is real constant. Azzalini (1985), with considering the pdf of skew distribution, introduced skew normal distribution as 2 () where and are respectively density function and distribution function of standard normal distribution. There is one limitation for model. For moderate value of , mass accumulates for positive and negative numbers specified by the sign of . Arelano-Valle et al. (2003) introduced a family of distribution that called skew-generalized normal ()so that it exhibited a better behaviour. In this paper we consider skew-Cauchy normal distribution and introduce skew Generalized Cauchy normal distribution and show that this is better than Cauchy distribution. Mathematics Symposium Fundamental Science Congress 2014 318 322. Method In this paper, at first, we introduced the new function, skew generalized Cauchy normal distribution, with the following structure = 2 1 (1 + !) ! 1 + ! ! < < , ! , ! 0 For evaluating , we simulated data from Cauchy distribution and then we estimated parameters by maximizing the likelihood function with respect to the components of = , , !, ! . Finally with using likelihood ratio test we concluded the skew generalize Cauchy normal distribution is better than Cauchy distribution, skew Cauchy normal distribution and skew curved Cauchy normal distribution. Results and Discussion We consider data of the height of 202 Australian athletes that part of them analyzed in Arellano-Valle et al. (2003). Below table presents the summary of these data. n ! ! ! 202 180.104 94.76038 -0.1978 3.4933 We compare skew Cauchy, skew Cauchy normal, skew curved Cauchy normal and Generalized Cauchy normal with using likelihood ratio test. We can see that the tests are not significant because skewness of data is not too much. Table1. Maximum likelihood parameter estimates for the height of Australian athletes data under the SGCN model and three special sub-models. Parameter estimates C SC SCCN SGCN 179.7238 177.8688 177.1067 177.2349 5.7045 5.7392 5.8355 5.8162 ! - 0.1671 0.3124 0.306 ! - - - 0.1203 Log-likelihood -777.3229 -776.3237 -775.7354 -775.7155 But when we use a set of skew data we can see that the likelihood ratio tests are significant and skew generalized Cauchy normal distribution is better than skew Cauchy and skew Cauchy normal distribution. Mathematics Symposium Fundamental Science Congress 2014 319 323. References 1. Azzalini, A. (1985). A class of distribution which include the normal ones. Scandinavian Journal Statistics, 12, 171-178. 2. Azzalini, A. (1986). Further results on a class of distribution which include the normal ones. Scandinavian Journal Statistics, 46, 199-208. 3. Arellano-valle, R.B., Gomez, H.W. and Quantana, F.A. (2003). A new class of skew- generalized normal distribution and its derivation. Proyecciones Journal of Mathematics, 33, 1465-1480. Mathematics Symposium Fundamental Science Congress 2014 320 324. REVIEW ON COMPUTABLE TOPOLOGICAL STRUCTURES Tayebeh Sepehrom1 and Adem Kilicman2 1,2 Department of Mathematics Faculty of Science Universiti Putra Malaysia email: tayebeh.sepehrom@gmail.com Abstract In this paper we review the type 2 theory of effectivity (TTE) as a computer-oriented theory for studying effectivity in mathematics. We go over the foundation of computable topology in the framework of Type-2 theory of effectivity, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations (Klaus, 2000). We survey computability of some topological structures such as metric spaces, subsets of metric space (Tanja, Matthias and Klaus, 2007), Euclidean spaces, regular topological spaces (Klaus, 2013), etc. from the viewpoint of computable analysis. Keywords: Computable analysis, Computable topology, TTE Introduction Computability and complexity theory are two central areas of research in mathematical logic and theoretical computer science. Computability theory is the study of the limitations and abilities of computers in principle. Computational complexity theory provides a framework for understanding the cost of solving computational problems, as measured by the requirement for resources such as time and space. The classical approach in these areas is to consider algorithms as operating on finite strings of symbols from a finite alphabet. Such strings may represent various discrete objects such as integers or algebraic expressions, but cannot represent general real or complex numbers, unless they are rounded (Klaus, 2000). Most mathematical models in physics and engineering, however, are based on the real number concept. Thus, a computability theory and a complexity theory over the real numbers and over more general continuous data structures is needed. During the last 70 years various mutually non-equivalent models of real number computation have been proposed. Among these models the representation approach (type-2 theory of effectivity, TTE) introduced for real functions by Grzegorczyk and Lacombe seems to be particularly realistic, flexible and expressive (Klaus and Tanja, 2009). Mathematics Symposium Fundamental Science Congress 2014 321 325. In this paper we make a review on computability of topological structures. We go over the literature about computable topology, as this is a very recent hot topic in the area of computable analysis. Method In order to make a review on computable topology structures the method of library research should be followed. Reading, analyzing and synthesizing mathematics research literature in the related area for the purpose of obtaining data or information that is needed to pose extensions of the project and fulfill the objectives of the research. Acknowledgement The authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the GP-IBT Grant Scheme having project number GP- IBT/2013/9420100. References: 1. Klaus Weihrauch. (2000). Computable Analysis, Springer. 2. Klaus Weihrauch and Tanja Grubba. (2009). Elementary computable topology. Journal of Universal Computer Science, 15(6), 1381-1422. 3. Klaus Weihrauch. (2013). Computably regular topological spaces. Logical Methods in Computer Science, 9(3.5), 1-24. 4. Tanja Grubba, Matthias Schroder, and Klaus Weihrauch. (2007). Computable metrization. Mathematical Logic Quarterly, 53(4-5), 381-395. 5. Rettinger, R., Weihrauch, K. (2013). Products of effective topological spaces and a uniformly computable Tychonoff Theorem, Logical Methods in Computer Science 9(4). Mathematics Symposium Fundamental Science Congress 2014 322 326. THE IMPORTANCE OF TIME MANAGEMENT: UNDERGRADUATE UNIVERSITY STUDENTS PERCEPTION USING FACTOR ANALYSIS Suriani Hassan1 , Rajeswary Ramachandran2 , Norlita Ismail3 , Khadizah Ghazali4 and Kamsia Budin5 1,2,3,4,5 Faculty of Science and Natural Resources Universiti Malaysia Sabah email: surianihassan@gmail.com.my Abstract This research is mainly about the perception on the importance of time management in academic achievement among undergraduate university students. The main objective of this study is to identify the factors of systematic time management. The data was collected by distributing the questionnaires to undergraduate university students. There were 350 students who had participated in this survey. There were few tests used in this research to analyze the data. Those are pretest and factor analysis. The factor analysis result showed that four factors were extracted, which are Factor 1, long and short range goal planning, Factor 2, goal-oriented time management, Factor 3, the level of awareness in managing and controlling the time and Factor 4, perception on the importance of time management in academic achievement. Keywords: Factor analysis, Time management Introduction Time is something that cannot be repeated and it will not stop for everyone. According to Claessen et al. (2007), time management is defined as behavior that seeks to achieve an efficient use of time and at the same time, engage in focused and beneficial activities. According to a study by Kearns and Gardiner (2007), the failure to manage time properly is one of the causes that give rise to feelings of stress and anxiety. This showed that time management is very important in a student's academic achievement. Noraini (1994) stated that students need to learn good time management method. Mercanlioglu (2010) concluded that time management includes identifying goals and lists them in order of importance to perform certain tasks within the stipulated time. This study was conducted in Universiti Malaysia Sabah, in Kota Kinabalu campus to study the factors affecting UMS undergraduate students perception on the importance of time management in academic achievement among undergraduate university students. Method The data that had been used throughout this study was primary data, which were collected from the questionnaire. SPSS was used to perform statistical analysis of the data collected Mathematics Symposium Fundamental Science Congress 2014 323 327. from the survey forms. The analysis performed in this study were the pretest and factor analysis. Chua (2009) suggested that factor analysis is the procedure which always been used by the researchers to organize, identify and minimize big items from the questionnaire to certain constructs under one dependent variable in a research. Chatfield and Collins (1994) stated the model for factor analysis was: pjeff jmjmjj ,1,=...X 11 +++= where jX is the original variable, p is the number of variable(s), m is the general number of factors, jk is the number of loading factors, kf is the k-general factor (k = 1,2,m) and je is the j-specific factor. Results and Discussion There were 350 undergraduate university students participated in this study. The pretest analysis result showed that the Cronbachs Alpha was 0.904. Abu & Tasir (2001) stated that the reliability coefficient of more than 0.6 is always used. Since the reliability analysis result, which was 0.904 is more than 0.6, therefore, there were internal consistency of the scales. Hence, this instrument used in this study had a high reliability value. The results value of Bartletts test of sphericity were large (X2 = 3031.613) and the value is significant (p 0 such that ! ! ! ! ! ! , then ! is often called a sufficient descent direction. The descent property is very important for the iterative method to be global convergent, especially for the conjugate gradient method by An et.al (2011). We say that ! satisfies the Armijo condition if ! + ! ! ! + ! ! ! ! (5) holds, where 0 < < 1. One such reset method was proposed by Powell (1977), based on an earlier version proposed by Beale. This technique restarts if there is very little orthogonality left between the current gradient and the previous gradient. This is tested with the following inequality: !!! ! ! ! ! !. (6) If this condition is satisfied, the search direction is reset to the negative of the gradient. Modified Conjugate Gradient Method To ensure the convergence of the conjugate gradient methods, we intend to modify a standard conjugate gradient algorithm by forcing it to satisfy sufficient descent condition. Our strategy is to choose the search direction as ! and subsequent algorithm. ! = ! + ! !" !!!, ! ! ! ! ! ! !!! ! ! < ! ! ! ! , (7) where we propose = !!!! ! !!!! !!!! ! !!!! (8) Algorithm Modified Conjugate Gradient (MCG) Step 0: Choose an initial point ! !and set ! = !. Given constants (0,1), let = 0. Step 1: = 10!! , check ! , if yes, stop. Else, go to step 2. Step 2: Compute ! by (15) where ! = 0.01 and ! = 0.2 are used. Step 3: Find the step size ! satisfying (6). (We choose = ! ! ). Step 4: Let the next iteration be !!! = ! + ! !. Step 5: Let = + 1 and go to step 1. Mathematics Symposium Fundamental Science Congress 2014 364 368. Numerical Results and Discussion Our experiments are performed on the unconstrained optimization test function collection by Andrei (2008). Since we are interested in large-scale of unconstrained optimization problems, we set the number of variables at 1000, 5000 and 10000. The numerical results of the total number of iteration and function evaluations of 15 functions are listed in table 1. TABLE 1. Test result for standard and modified Conjugate Gradient Method Number of iteration Number of function evaluations Standard Conjugate Gradient Method 792 5027 Modified Conjugate Gradient Method-1 768 1527 Observing the tables, we see that the modified conjugate gradient works better than standard conjugate gradient method with less total number of iteration and function evaluations. The numerical experiments show that modified conjugate gradient method can potentially be used to solve unconstrained optimization problems with higher dimensions. Conclusion In this paper, motivated by sufficient descent condition, we have proposed modified conjugate gradient method for solving large-scale unconstrained optimization problems. The numerical results indicate that it works well on some selected test functions and more efficient than the standard conjugate gradient method. Acknowledgement The first author is supported by University Putra Malaysia Graduate Research Fellowship 2014. References 1. An, X. M., Li, D. H. and Xiao, Y. (2011). Sufficient descent directions in unconstrained optimization, Computational Optimization and Applications, 48(3), 515-532. 2. Andrei, N. (2008). An unconstrained optimization test functions collection, Adv. Model. Optim, 10(1), 147-161. 3. Gilbert, J. C. and Nocedal, J. (1992). Global convergence properties of conjugate gradient methods for optimization, SIAM Journal on optimization, 2(1), 21-42. 4. Leong, W. J., Hassan, M. H. and Yusuf, M. W. (2011). A matrix-free quasi-Newton method for solving large-scale nonlinear systems, Computer& Mathematics with Applications, 62(5), 2354-2363. 5. Leong, W. J. and Hassan, M. A. (2011). Scaled memoryless symmetric rank one method for large-scale optimization, Applied Mathematics and Computation, 218(2), 413-418. 6. Powell, M. J. D. (1977). Restart procedures for the conjugate gradient method, Mathematical programming, 12(1), 241-25. Mathematics Symposium Fundamental Science Congress 2014 365 369. NUMERICAL SOLUTION OF SYSTEMS OF SECOND-ORDER BOUNDARY VALUE PROBLEMS BY USING CHEBYSHEV WAVELET FINITE DIFFERENCE METHOD Aliasghar Kazemi Nasab1 , Zohreh Pashazadeh Atabakan2 and Adem Kilicman3 1,2,3 Department of Mathematics, Faculty of Science Universiti Putra Malaysia email: a.kazeminasab@gmail.com Abstract Chebyshev wavelet finite difference method (CWFD) is proposed to solve systems of second-order boundary value problems. A comparison is made among the current technique, other well-known methods and the exact solution to clarify the accuracy and the efficiency of the introduced method. Keywords: Wavelet analysis method, Chebyshev wavelet finite difference method, System of ODEs, Boundary value problems, Chebyshev polynomials Introduction Chebyshev wavelet finite difference method is based on Chebyshev wavelets and Chebyshev finite difference methods. Chebyshev wavelet finite difference method and Chebyshev wavelets analysis method were proposed for solving different types of problems varying ordinary differential equations of fractional order, Lane-Emden equations and singular boundary value problems by Kazemi Nasab et al. (2013), Kazemi Nasab et al. (2014), Kazemi Nasab et al. (2013), and Kazemi Nasab et al. (2013). In this paper, we consider the second- order systems of boundary value problems on interval [0, 1) with the following general form: !"()=g1(x,u1(x), , ! , !(), , !()) !"()=g2(x,u1(x), , ! , !(), , !()) !"()=gn(x,u1(x), , ! , !(), , !()) (1) subject to the boundary conditions ! 0 =1,! 1 = ! ! 0 =2,! 1 = ! ! 0 =n,! 1 = ! (2) where gi are linear or nonlinear functions of ui and u'i and i !, i=1,2, , n are constants. Mathematics Symposium Fundamental Science Congress 2014 366 370. Methodology Suppose the interval [0, 1) is divided into 2!!! subintervals ! = !!! !!!! , ! !!!! , n=1, 2, , 2!!! . We define the Chebyshev-Gauss-Lobatto collocation points !", r=0, 1, , M as follows, !" = 1 2!!! cos + 2 1 . In order to solve system (1) - (2), we first collocate Eq. (1) at the !", r =1, , M 1, while approximate ! , ! ! , ! !! , = 1, 2, , in terms of chebyshev wavelet finite difference basis functions and their derivatives as follows, 1 2 01 ( )u (x) (x), k M M i nm nm mn P c == = where the summation symbol with double primes denotes a sum with both the first and last terms halved. Moreover, cnm ,n 1,2,...,2k 1 ,m 0,1,...,M, are the expansion coefficients of the function u(x) at the subinterval !!! !!!! , ! !!!! and , (x)n m are defined as following: /2 2 /2 0 0 1 2 1 2 . u ( ) ( ) . u ( )cos( ) (2 ) 2 M M nm i np nm np i npk k p pm m mp c t t t p M p M M = = = = and 2 1 1 1 2 2 , 2 (2 2 1), , (x) 0, otherwise, k k k k n n m m n m p T x n x + < = where 1 2 , 0 , 0,1,..., . , 1 m m p m M m = = = In addition, we can approximate ! ! and ! !! as below, ( ) ( ) , , 0 ( ) u ( ), 1,2 M r r i nm n m j i nj j u t d t r = = = where 1 (1) , , 1 0,( ) 4 ( ) ( ), M k j k n m j nk nj nl nm k l k l odd l k l k d t t M c p p = = + = 2 22 (2) , , 2 0,( ) 2 2 ( ) ( ) ( ), kM k j k n m j nk nj nl nm k l k l even l k l k k l d t t M c p p = = + = Mathematics Symposium Fundamental Science Congress 2014 367 371. with ! = ! = ! ! , ! = 1 for j=1, 2, , M-1. Besides, it is necessary that ! and its rst derivative be continuous at the interface of subintervals. We also get other 2n equations from boundary condition (2) which we will totally have 2!!! ( + 1) equations and by solving them, we get solutions ! , = 1, 2, , to system (1) (2). For example, consider the nonlinear singular system of boundary value problems ! !! + ! + ! ! = () ! !! + ! ! + ! = () with the following boundary conditions, ! 0 = ! 1 = 0; ! 0 = ! 1 = 0, where f(x) = -! sin + ! + ! 3! + 2! g(x) = cos + ! 3! + 8 6. The exact solutions are ! = sin and ! = ! 3! + 2. We solve this example with M=10, k=5. A comparison between approximate solution ! and exact solution ! is made in Table 1. TABLE 1. Comparison between the approximate solution ! and the exact solution! x Approximate solution Exact solution Absolute error 0.2 0.58778525229247307342 0.58778525229247312917 5.610!!" 0.4 0.95105651629515346829 0.95105651629515357212 1.310!!" 0.6 0.95105651629515346950 0.95105651629515357211 1.010!!" 0.8 0.58778525229247307626 0.58778525229247312914 5.310!!" 0.9 0.30901699437494739445 0.30901699437494742412 3.010!!" Conclusion As can be seen from example, by using of the proposed method, we can get very accurate solutions to system of boundary value problems. References 1. Kazemi Nasab, A., Klman, A., Pashazadeh Atabakan, Z. and Abbasbandy, S. (2013, December). Chebyshev Wavelet Finite Difference Method: A New Approach for Solving Initial and Boundary Value Problems of Fractional Order. In Abstract and Applied Analysis (Vol. 2013). Hindawi Publishing Corporation. 2. Kazemi Nasab, A., Klman, A., Pashazadeh Atabakan, Z. and Leong, W. J. (2014). A numerical approach for solving singular nonlinear Lane-Emden type equations arising in astrophysics. New Astronomy. In Press. 3. Kazemi Nasab, A., Klman, A., Babolian, E. and Pashazadeh Atabakan, Z. (2013). Wavelet analysis method for solving linear and nonlinear singular boundary value problems. Applied Mathematical Modelling, 37(8), 5876-5886. 4. Kazemi Nasab, A., Pashazadeh Atabakan, Z. and Klman, A. (2013). An Efficient Approach for Solving Nonlinear Troeschs and Bratus Problems by Wavelet Analysis Method. Mathematical Problems in Engineering, 2013. Mathematics Symposium Fundamental Science Congress 2014 368 372. ON SOLUTIONS OF MATRIX INITIAL VALUE PROBLEMS BY USING DIFFERENTIAL TRANSFORMATION METHOD AND CONVOLUTIONS Omer Altun1 and Adem Kiliman2 1,2 Department of Mathematics, Faculty of Science Universiti Putra Malaysia email: omeraltun@msn.com Abstract In this study, we consider some nonlinear matrix initial value problems by using differential transformation method and convolutions. Differential transformation method is a transformation technique which is based on the Taylor series expansion. With this method exact solution may be obtained without any cumbersome work. All implementation are calculated by Maple. Keywords: Differential Transformation Method, Initial Value Problems, Convolutions Introduction The differential transform method (DTM) concept was first proposed and applied to solve linear and nonlinear initial value problems in electric circuit analysis by Zhou (1986). It is an approximation to the exact solutions which are differentiable and it has high accuracy with minor error. This method is different than the traditional high order Taylor series because Taylor series requires the computation of the necessary related derivatives (Ozgumus and Kaya, 2010). This method finally gives series solution but truncated series solution in the practical applications. Method Consider a function with two variables y(x,t) and Y(k,j) is transformed function. Definition 1: Two dimensional differential transformation for partial differential equation is defined as : , = ! !!!! !!!! !"!!"! (, ) (1) Definition 2 : Differential inverse transform of Y(k,j) is defined as the following: , = (, )! !! !!! ! !!! (2) Mathematics Symposium Fundamental Science Congress 2014 369 373. By substituting equation (2) into equation (1), we have the following : , = !!!! !"!!"! (, )! !!! ! !!! (3) We can easily see that, equation (3) implies the two dimensional of Taylor series expansion. Theorem 1: !, !, ! = !, !, ! !, !, ! , !, !, ! = (!, !, !) (!, !, !) Theorem 2 : !, !, ! = !, !, ! !, !, ! = (!, !, !) Results and Discussion Definition 3: Let = !"() !,! ! = !"() !,! ! The convolution product is a matrix functions defined for t 0 as follows (whenever integral exists): = !" !" = !" !" ! ! ! !!! = !"() !"()! !!! The Convolution product can be used in the matrix differential equations as in the following example: Example 1: Consider the following non-linear second order matrix initial value problem: !! + ! = () 0 = ! ! ! ! ! ! ! ! ! , 0 = ! ! ! ! ! ! ! ! ! = ! ! ! ! ! ! ! ! ! By applying matrix differential transform operator on above non=linear system for k=0,1,2,,n, we get Mathematics Symposium Fundamental Science Congress 2014 370 374. !!! ! !! U(k + 2) + + 1 + 1 = , =0 0 = ! ! ! ! ! ! ! ! ! 1 = ! ! ! ! ! ! ! ! ! And so on where U(k) and C(k) are the differential transform of u(t) and c(t) respectively. Thus we transform the system into the algebraic equations then by using the MAPLE. We can solve it, see for example (Kiliman and Altun, 2012, 2014). References 1. Zhou, J. K. (1986). Differential Transformation and its Application for Electrical Circuits, Huazhong University Press, Wuhan China. 2. Ozgumus, O. O. and Kaya, M. O. (2010). Vibration analysis of a rotating tapered Timoshenko beam using DTM. Meccanica, 45(1), 33-42. 3. Kiliman, A. and Altun, O. (2012). On the solution of some boundary value problems by using differential transformation method with convolution terms, Filomat, 26(5), 917-928. 4. Kiliman, A. and Altun, O. (2014). On higher-order boundary value problems by using differential transformation method with convolution terms. Journal of the Franklin Institute, 351(2), 631-642. Mathematics Symposium Fundamental Science Congress 2014 371 375. DERIVATIONS OF LOW DIMENSIONAL ASSOCIATIVE ALGEBRAS A. O. Abdulkareem1 , M. Abubakar2 and I. S. Rakhimov3 2,3 Department of Mathematics Faculty of Science Universiti Putra Malaysia 1,3 Laboratory of Cryptography, Analysis & Structure Institute for Mathematical Research Universiti Putra Malaysia email: risamiddin@gmail.com Abstract It has been proven that any Lie algebra over a field of characteristic zero which has non degenerate derivations is nilpotent. The converse of this assertion has been given a negative answer by constructing an example of a nilpotent Lie algebra all of whose derivations are nilpotent (hence degenerate). Lie algebras whose derivations are nilpotent have been called characteristically nilpotent. The concern of this study is on the derivations of four dimensional associative algebras. As a consequence, we describe the derivation algebras of all four-dimensional associative algebras. Keywords: Derivation, Associative algebras, Dimension of derivation Introduction The interest in the study of derivations of algebras goes back to a paper by (Jacobson, 1955). There, Jacobson proved that any Lie algebra over a field of characteristic zero which has non degenerate derivations is nilpotent. In the same paper, he asked for the converse. In the paper by (Dixmier and Lister, 1957), the authors have given a negative answer to the converse of Jacobson's hypothesis by constructing an example of a nilpotent Lie algebra all of whose derivations are nilpotent (hence degenerate). Lie algebras whose derivations are nilpotent have been called characteristically nilpotent. The result of (Dixmier and Lister, 1957) is assumed to be the origin of the theory of characteristically nilpotent Lie algebras. A few years later in 1959, (Leger and Togo, 1959) published a paper showing the importance of the characteristically nilpotent Lie algebras. The results of Leger and Togo have been extended for the class of non-associative algebras. The theory of characteristically nilpotent Lie algebras constitutes an independent research object since 1955. Until then, most studies about Lie algebras were oriented to the classical aspects of the theory, such as semi-simple and reductive Lie algebras (see (Tits, 1966)). Mathematics Symposium Fundamental Science Congress 2014 372 376. In this study, we concentrate on the derivations of associative algebras in dimension four. An algorithm is found to describe the derivations and this algorithm has been found to be efficient in computing the derivations of other classes of algebras as well. Thereafter, the algorithm is applied to find the derivations of four-dimensional associative algebras. The algorithm The derivation of an algebra A is a linear operator AA satisfying D(xy)=D(x)y+xD(y) for any x,y from A. Being a linear transformation, D can be represented in matrix form (shown below) with respect to a basis {!, !, !, , !} of the algebra A: = !" = !! !! !! !" . The set of all derivations of is denoted by (). The set () is a Lie algebra with respect to commutator operation. This algebra plays a crucial role in structural theory of algebras and its dimension is an important invariant. Therefore the study of derivations is of great interest. In this project, we give an algorithm to find derivations of algebras and apply the algorithm to the class of associative algebras. Let be an -dimensional associative algebra and be its derivation. Let {!, !, !, , !} be a basis of . Then representing the derivation in matrix form = (!"), we construct the system of equations with respect to !"on the basis {!, !, !, , !} as follows. !! !! = !" ! !!! !! , where !() = is the left multiplication operator on the algebra A. Solving the system of equations with respect to variables !", we find the derivation in matrix form as shown below. All the algebras considered are over the field of complex numbers . Results and Discussion As a consequence of the algorithm given above, we give the derivations of four dimensional associative algebras. There are 54 isomorphism classes of four dimensional associative algebras. Due to space constraints, we give the derivations of a few isomorphism classes and the dimensions of their derivation algebras in the following table form. Mathematics Symposium Fundamental Science Congress 2014 373 377. TABLE 1. Derivations of four dimensional associative algebras S/N Algebra Derivation Dimension 1 ! ! : ! ! = !, ! ! = !. !! 0 0 0 0 !! 0 0 !" !" 2!! 0 !" !" 0 2!! 6 2 ! ! : ! ! = !, ! ! = !. !! 0 0 0 0 !! 0 0 !" !" !! + !! 0 !" !" 0 !! + !! 6 3 ! ! : ! ! = !, ! ! = !. !! 0 0 0 !" !! 0 0 !" 0 !! 0 !" !" !" !! + !! 6 4 ! ! : ! ! = !, ! ! = !, ! ! = ! . !! 0 0 0 0 0 0 0 0 !" !! 0 !" 0 0 2!! 4 5 ! ! : ! ! = !, ! ! = !, ! ! = !. !! 0 0 0 0 !! 0 0 !" !" 2!! 0 !" !" 0 2!! 5 Acknowledgement The authors would like to thank Assoc. Prof. Dr. Leong Wah June for his financial support. Participation in FSC2014 was funded by Putra Grant with vote number 9419300. References 1. Jacobson, N. (1955). A note on automorphisms and derivations of Lie algebras, Proc. Amer. Math. Soc., 6(2), 281-283. 2. Tits, J. (1966). Sur les constantes de structure et le theorem dexistence des algebres de Lie semisimples, Publ. Math. I.H.E.S., 31(1), 21-58. 3. Dixmier J., Lister W.G. (1957). Derivations of nilpotent Lie algebras. Proc. Amer. Math. Soc., 8, 155-158. 4. Leger, G. and Togo, S. (1959). Characteristically nilpotent Lie algebras. Duke Math. J., 26(4), 623-628. Mathematics Symposium Fundamental Science Congress 2014 374 378. Physics Symposium Fundamental Science Congress 2014 375 379. 1. Abdul Kariem Arof Evaluating Transport Properties of Polymer Electrolytes Using Impedance Spectroscopy 2. M. Deraman, M.A.R. Othman, M.M. Ishak, N.S.M. Nor, R. Omar, R. Daik, S.Soltaninejad, A.A. Aziz and A.R.M. Ali Supercapacitor Based on Carbon Electrode from Precarbonized Dragon Fruit Peels and Graphene 3. K. Hariharan Materials Development for Solid State Ionic Devices 4. S.A. Halim, A.Arlina, S.K. Chen ,M.M. Awang Kechik ,K.P.Lim, and M.Navasery AC Susceptibility Study of Y-123 Addition on Bi(Pb)-2223 Superconductors 5. W. Mahmood Mat Yunus Photothermal and Surface Plasmon Resonance Techniques for Characterizing Thermal and Optical Properties of Liquid and Solid Samples 6. Lim Kean Pah, Abdul Halim Shaari, Chen Soo Kien, Chin Hui Wei, Albert Gan, Ng Siau Wei and Chong Kuen Hou Grain Size Effect on the Structural, Electrical, Magnetic and Magnetoresistance Properties of Polycrystalline La0.67Sr0.33MnO3 7. Abubakar Yakubu, Zulkifly